DERIVATIVE PRICING IN LÉVY DRIVEN MODELS. Thesis submitted for the degree of Doctor of Philosophy at the University of Leicester

Transcription

1 DERIVATIVE RICING IN LÉVY DRIVEN MODELS Thesis submitted for the degree of Doctor of hilosophy at the University of Leicester by Alexander Kushpel Dr Department of Mathematics University of Leicester 05

2 Contents List of tables 6 List of figures 7 Remarks on notation 9 Introduction General definitions 4. Market and derivative instruments The time value of money Arbitrage theorem Lévy processes and characteristic exponents 9. Introduction Basic results Equivalent martingale measures Characteristic exponents and density functions KoBoL family Recovery of density functions in jump-diffusion models Introduction A class of stochastic systems Representation of density functions by Fourier modes Recovery of density functions by sk splines and radial basis functions Shannon s Information Theory and recovery of density functions... 73

3 3.6 The problem of optimal recovery of density functions Option pricing Introduction The equivalent martingale measure condition for basket options ricing of high-dimensional basket options Kolmogorov s n-widths in the Theory of ricing Introduction Kolmogorov s n-widths and optimal approximation Spherical KoBoL process Introduction Estimates of expectations Lower bounds Upper bounds Conclusion 9 Appendix I: Measure and integral 37 L p spaces Fubini and Tonelli theorems Radon-Nikodym s theorem Appendix II: Harmonic Analysis 4 Wiener s spaces Riesz-Thorin and Riesz theorems Appendix III: Martingales 45 Martingale methods and pricing Decomposition and conversion of sub-martingales into martingales Appendix IV: Topology and Functional Analysis 50 Topological spaces Topological vector spaces Weak and weak star topologies

4 Bibliography 53 3

5 Abstract We consider an important class of derivative contracts written on multiple assets which are traded on a wide range of financial markets. More specifically, we are interested in developing novel methods for pricing financial derivatives using approximation theoretic methods which are not well-known to the financial engineering community. The problem of pricing of such contracts splits into two parts. First, we need to approximate the respective density function which depends on the adapted jump-diffusion model. Second, we need to construct a sequence of approximation formulas for the price. These two parts are connected with the problem of optimal approximation of infinitely differentiable, analytic or entire functions on noncompact domains. We develop new methods of recovery of density functions using sk-splines (in particular, radial basis functions), Wiener spaces and complex exponents with frequencies from special domains. The respective lower bounds obtained show that the methods developed have almost optimal rate of convergence in the sense of n-widths. On the basis of results obtained we develop a new theory of pricing of basket options under Lévy processess. In particular, we introduce and study a class of stochastic systems to model multidimensional return process, construct a sequence of approximation formulas for the price and establish the respective rates of convergence. 4

6 Acknowledgements It is a distinct pleasure for me to extend my gratitude to my supervisor rofessor Dr. Jeremy Levesley for his support, patience, understanding, guidance and advice throughout the preparation and writing of this dissertation. I feel significantly enriched by ennumerous discussions of a wide range of cardinal mathematical problems we had since 994. I feel honored for having been a part of the Department of Mathematics at the University of Leicester. I am very thankful for providing me with an academic home and proper financial support during the past four years. I also thank all the participants of the Applied and Financial Math seminars for providing me with opportunities to talk on my research and to learn from their talks. I am especially grateful to my colleagues, Dr. Stephen Garrett, Dr. Steve Hales and Mrs. Tara Chakraborti for support and sharing their academic experience. 5

7 List of tables. Table. p. 54. Error of approximation at x = ; ; 5 of Gaussian density p (x) = () = exp ( x =) using oisson method with truncation parameter = 4 and M = 4, 6 and 6 terms.. Table. p. 03. Simulation result for a European call option. European call opton parameters: r = 0:0, T = 0:5, S 0 = 00, K = S 0 exp ( 0:5). Model (KoBoL) parameters: = 0:5, c + = c = 0:6506, + = :9458, = :087, = 0: Contour parameter = 9. Truncation parameter = 6. 6

8 List of figures. Figure. p. 36. Real part of characteristic exponent (z), z C of KoBoL process with parameters T = 0:5, c + = c = 0:6506, + = :9458, = :087, = 0:5, = 0: Figure. p. 36. Imaginary part of characteristic exponent (z), z C of KoBoL process with parameters T = 0:5, c + = c = 0:6506, + = :9458, = :087, = 0:5, = 0: Figure 3. p. 37. Absolute value of characteristic function (z), z C of KoBoL process with parameters T = 0:5, c + = c = 0:6506, + = :9458, = :087, = 0:5, = 0: Figure 4. p. 39. Absolute value of characteristic function (u ; u ) from Example Figure 5. p. 4. Absolute value of characteristic function 3 (u ; u ) from Example Figure 6. p. 4. Absolute value of characteristic function 4 (u ; u ) from Example Figure 7. p. 46. Absolute value of characteristic function F (v ; v ;), from Example. Correlation parameter = Figure 8. p. 47. Absolute value of characteristic function F (v ; v ;) from Example 5. Correlation parameter = 0:. 9. Figure 9. p. 47. Absolute value of characteristic function F (v ; v ;) from Example 5. Correlation parameter =. 0. Figure 0. p. 53. Approximant g (p; M; ; x) of Gaussian density function p (x). Truncation = 4. The number of terms M = 0. 7

9 . Figure. p. 54. Error of approximation p (x) g (p; M; ; x), = 4, M = 0.. Figure. p. 55. Two-dimensional Gaussian density function p (x; y). 3. Figure 3. p. 55. Approximant g (p; M; ; x; y) of Gaussian density function p (x; y). Truncation = 4. The number of terms M = Figure 4. p. 56. Error of approximation p (x; y) g (p; M; ; x; y), = 4, M = Figure 5. p. 57. Approximant g (p; M; ; x) of density function p (x) from Example 6. Truncation = 4. The number of terms M = Figure 6. p. 58. Two-dimensional approximant g (p; M; ; v ; v ) of density function p (x; y) from Example. Truncation =. The number of terms M = 8. Correlation parameter = Figure 7. p. 58. Two-dimensional approximant g (p; M; ; v ; v ) of density function p (x; y) from Example. Truncation =. The number of terms M = 8. Correlation parameter = 0:. 8. Figure 8. p. 59. Two-dimensional approximant g (p; M; ; v ; v ) of density function p (x; y) from Example. Truncation =. The number of terms M = 8. Correlation parameter =. 9. Figure 9. p. 0. Absolute value of restrictions of characteristic function of KoBoL process with parameters T = 0:5, c + = c = 0:6506, + = :9458, = :087, = 0:5, = 0:39563 onto shifted contours of integration. 0. Figure 0. p. 6. Optimal distribution of data points on S.. Figure. p. 7. Optimal coefficients a b l, b =.. Figure. p. 8. Optimal radial basis function for approximation on S. 8

10 Remarks on notation N,, R and C are, respectively, the sets of all positive integers, all integers, all real numbers, and all complex numbers. + and R + are the collections of nonnegative elements of and R; respectively. R n is the n-dimensional Euclidean space with the canonical basis e ; ; e n. Its elements x = (x ; ; x n ) and y = (y ; ; y n ) are vectors with n real components. The inner product in R n is hx; yi = n j= x n =. jy j ; the norm is jxj = j= j x C n is the n-dimensional complex space. Its elements z = (z ; ; z n ) are vectors with n complex components. Similarly we define N n and n. For a matrix A = (a j;k ), A T = (a k;j ) means its transpose. For sets A and B, A B means that all elements of A belong to B. Let X be a vector space over reals (see Appendix IV). For A; B X, z X, and c R, A+z = fx + z jx Ag, A z = fx z jx Ag, ca = fcx jx Ag, A = f x jx Ag, A n B = fx jx A&x = B g. Minkovski s sum and difference of A X and B X are defined as A + B = fx + y jx A; y B g () and A B = fx y jx A; y B g respectively. Vol n (B) is the Lebegue measure of a set B R n. B is the indicator function of a set B, that is, B (x) = for x B and 0 for x = B. The abbreviation a.s. denotes almost surely, that is, with probability. The abbreviation a.e. denotes almost everywhere, or almost surely, with respect to the Lebesgue measure. Similarly, -a.e. denotes almost everywhere, or almost every, with respect to a measure. The symbol a represents the probability measure concentrated at a R n. The expression represents the convolution of finite measures and ; m is the m-fold convolution of. When m = 0, m is understood to be 0. 9

11 Let f : R n! R be an integrable function, f L (R n ). Define the Fourier transform Ff (y) = exp ( R n i hx; yi) f (x) dx and its formal inverse as F f (y) = () R n exp (i hx; yi) f (x) dx: n (A) is the probability of an event A. E [X] is the expectation of a random variable X. Sometimes E [X] is written as EX. var [X] is the variance of a real random variable X. I is the identity matrix. A T and A are, respectively, the transpose and the conjugate of a matrix A. Let X be a Banach space and f be a function, f X. The notation kf (; )k X means that we are taking the norm of f (; ) with respect to the argument denoted by (). Let X and Y be Banach spaces. The norm kak := sup fkaxk Y jkxk X g of a linear operator A : X! Y is denoted by ka jx! Y k and the space of bounded linear operators A is denoted by L (X; Y ). Let X, Y and be Banach spaces A L (X; Y ) and B L (Y; ) then the composition of A and B is denoted by B A : X!. The expression f (x) g (x) means that lim x! f (x) =g (x) =. We shall write f (x). g (x) if lim x! f (x) =g (x) and A B n, n N if B n is a sequence of formal approximants to A without regard of any type of convergence. Different positive universal constants are mostly denoted by the letter C. We did not carefully distinguish between the different constants, neither did we try to get good estimates for them. The same letter will be used to denote different universal constants. For the easy of notation we put a m b m for two sequences, if a m Cb m for m N and a m b m if C b m a m C b m for all m N and some constants C, C, and C. Through the text [a] means integer part of a R. Sometimes a subscript is written larger in parentheses, such as X t (!) = X (t;!), X t = X (t), x m = x (m). 0

12 Introduction There are six chapters, conclusion and four appendices in this manuscript. They can be divided into three parts. Chapters, and Appendices I-IV constitute the basic part. Essential results, examples and major tools for the analysis are given in Chapter and Appendixes. Chapter deals with more specific results which are important in our applications. It gives description of equivalent martingale measures, representations of characteristic and density functions, introduces and studies the so-called KoBoL model. Chapters 3 and 6 are the second part. They are dealing with the problem of recovery of high-dimensional density and characteristic functions in jump-diffusion models. In Chapter 3 we introduce and study a class of stochastic systems to model multidimensional return process and develop new methods of recovery of highdimensional density functions in jump-diffusion models. We consider two approaches here - approximation of density functions by complex exponents and by sk-splines, in particular, by Wiener spaces. Also, the problem of optimal recovery of density functions is considered. More specifically, we derive lower bounds for Alexandrov s cowidths of sets of density functions which are important in applications. Chapters 4 and 5 constitute the third part. Chapter 4 is devoted to the problem of pricing of a basket options. This is a deep problem of Financial Mathematics. We develop an original and simple method of pricing. Recall that the price V of the common spread option at time 0 is given by V = exp( rt )E Q ['] ; where ' is the reward function and the expectation E Q ['] is taken with respect to the equivalent martingale measure Q. Observe that in many important cases the reward function ' grows exponentially. Hence the characteristic function of our model process must admit an analytic extension to guarantee convergence of the pricing integral V.

13 We specify the equivalent martingale measure condition for our model and present a detailed study of a general system of stochastic equations introduced in Chapter 3. In particular, we give an explicit analytic expression of the characteristic exponent which is important in our applications and construct a sequence of approximants which is based on the appropriately scaled exponential hyperbolic cross. Finally, we derive explicit approximation formulas for the price V in multidimensional settings [83, 84, 85]. Chapter 5 is devoted to the problem of optimal approximation of the pricing integral using a wide range of approximation algorithms. This is the first attempt in Financial Mathematics to employ far advanced methods of Functional Analysis. This approach is in spirit of Kolmogorov s n-widths. The main problem here is that the domain of integration is not compact, hence we need to use the notion of average dimension introduced by Tikhomirov [3]. This notion has been inspired by Whittaker-Kotel nikov-shannon s theorem which has its roots in the Information Theory. We introduce a set of regular Lévy processes of exponential type ( ; + ) ; < 0 < + which are adapted to a given payoff function ' (x), ' (x) = 0; x 0, i.e. for some!; <! < 0, the function ' (x) exp (!x) is square integrable on [0; ). Denote by G ( ) the set of such functions. For a given KoBoL model process we calculate upper bounds for the respective Kolmogorov s n-widths with respect to the average dimension [8]. Our result shows exponential rate of decay as!. Chapter 6 deals with spherical KoBoL processes. The key problem here is to approximate spherical integrals by their discrete analogs. A fundamental problem in this important direction is to construct in an explicit form the best possible method of recovery of functions from classical Sobolev s classes in L. Observe that in the case of S it is not possible to construct, in general, equidistributed partition since there are finitely many polyhedral groups. Different attempts to find sets of points which imitate the role of the roots of unity on the unit circle usually led to the deep problems from the Geometry of Numbers, Theory of otential, etc., and usually these approaches give just a measure of uniform distribution of points, e.g. cup discrepancy or minimum energy configurations. Extensive computations for optimal configurations have been reported in a number of articles (see, e.g. [49]). The next, and probably much more complicated step in this direction is to construct in an explicit form a method of approximation of smooth functions using their values on such irregular sets of points. It was remarked in [6] that "the problem of finding a set of interpolation points that yields a good uniform approximation, or understanding how good such approximants can be, remains elusive". We give the solution of this well-known problem. Namely, we construct asymptoti-

14 cally (as m! ) best possible set of basis functions ; ; m (which are radial basis functions) and points x ; ; x m on S to reconstruct smooth functions f () from Sobolev s classes in L using linear methods m k= f (x k) k (). The solution splits onto two parts. We establish lower bounds on a wide range of manifolds including S n, the n-dimensional sphere using a probabilistic approach [80]. To get upper bounds we construct a quasi interpolant [74]. Remark that the problem (the Laplace-Legendre problem) of optimal reconstruction of functions and functionals on manifolds is of independent interest and remained open since the end of the XVIII th century [80]. Finally, in the Appendices I-IV we collected all necessary results which we use in the text. In particular, observe that in the majority of text books on Financial Engineering, Fubini s theorem has been used instead of Tonelli s theorem which is valid under much more general conditions (see Appendix I for more information). We present Radon-Nikodym s theorem which is important in the robability Theory. In Appendix II we collect fundamental facts from Harmonic Analysis which are useful in ricing Theory, such as Wiener spaces of entire function on R n, aley-wiener s, lancherel s, Riesz and Riesz-Thorin theorems. Appendix III presents two results on martingale conversion, the Doob-Meyer decomposition and Girsanov s theorem which are important ingredients of the Theory of ricing. In the Appendix IV we present all important results which constitute the basis of the proof of the "Fundamental theorem of asset pricing". They are: Tychonov s theorem, axiom of Choice, Hahn-Banach and Banach-Alaoglu theorems. The results obtained have been presented and discussed on Applied and Financial Math seminars of the Department of Mathematics, University of Leicester 00-04, International Workshop-Radial Basis Functions, 04, Birkbeck College, University of London, seminar of the Department of Economics, Mathematics and Statistics, Birkbeck College, University of London, 04, Actuarial Teachers and Reseachers Conference 0, European Numerical Mathematics and Applications- 0, Leicester, Festivals of hd students, University of Leicester, 0 and 0, British Mathematical Colloquium-0, Leicester, and many other National and International meetings. 3

15 Chapter General definitions. Market and derivative instruments A market is a system of institutions, procedures, social relations and infrastructures where parties engage in exchange. Market participants consist of all the buyers and sellers of a good who influence its price. A market allows any tradable item to be evaluated and priced. In general, the structure of a well-functioning market can be approximated as following:. Many small buyers and sellers.. Buyers and sellers have equal access to information. 3. roducts are comparable. An investor is someone who puts money into something with the expectation of a financial return. Assets are economic resources, i.e. value of ownership which has a positive economic value and that can be converted into cash. Finance is the study of how investors allocate their assets over time under conditions of certainty and uncertainty. A derivative instrument is a contract between two parties that specifies conditions under which payments are to be made between the parties. We say that a financial contract is a derivative security (or a contingent claim) if its value at expiration date T is determined exactly by the market price of the underlying cash instrument at time 0. An option (in finance) is a derivative instrument that specifies a contract between two parties for a future transaction on an asset (commonly a stock, a bond, a currency or a futures contract) at a reference price (the strike). A stock represents the original capital invested in the business by its founders. A bond 4

16 is a negotiable certificate that acknowledges the indebtedness of the bond issuer to the holder. A forward contract is an obligation to buy (or sell) an underlying asset at a fixed price (forward price) on a known date T. A European call option on a security S t is the right to buy the security at a fixed strike price K at the expiration date T. The call option can be purchased for a price C t (called the premium) at time t < T. A European put option gives the owner the right to sell an asset at a specified price at expiration T. Instead, American options can be exercised at any time t, 0 < t T: Before the option is first written at time t, its value C t is unknown. That is why it is important to get some estimates of what this price will be if the option is written. Hence, the problem is to get a good approximation for C t as a function of the underlying assets price and the relevant market parameters. Observe, that at the time of the expiration T, the price of the derivative asset F (T ) is completely determined by S T, the value of the underlying asset, i.e. F (T ) = S T. It simply means that at the expiration date T the value of the future contract F (T ) should be equal to its cash equivalent S T. The ask (bid) price is the price at which a market maker offers to sell (buy) a derivative security. The bid-ask spread is the difference between the bid and ask price. To simplify our model we assume that our market is such that:. There are no commissions and fees (the price of an asset in trade is much bigger then commissions and fees).. The bid-ask spreads on S t and C t are zero (the market is in equilibrium). With these two assumptions we have the following two possibilities. If S T K (the option is out-of-money) then the option will have no value. Hence, C T = 0. Otherwise, if S T > K (the option is in-the money) then (since by our assumption, there are no commissions and bid-ask spreads) the net profit will be C T = S T K > 0. Joining these possibilities we get C T = max fs T K; 0g = (S T K) + ; where (a) + = a; a > 0; 0; a 0: A state security (or an Arrow-Debreu security, see [, ] for more information) is a security which pays off $ iff a given state of nature occurs. A state of nature is said to be insurable state when there exists a portfolio which has a non-zero return in that state. For a market where every state is insurable, a price vector can be uniquely determined. Hence, a complete market can be defined as a market in which all the contingent claims are attainable. 5

17 A complete market can be defined with respect to the concept of a viable financial market. If any strategy which is implemented at the initial time with a zero cost has a zero terminal payoff then we have the absence of riskless arbitrage opportunities. A viable financial market is defined as a market where there is no profitable riskless arbitrage opportunities. Note that there is an important relationship between arbitrage and the martingale property of securities prices. It means that the best estimation of the future price is derived from the latest information, i.e only the most recent information matters. A financial market is viable iff there is a probability Q which is equivalent to a historical probability, under which the discounted asset prices have the martingale property. We say that a viable market is complete iff there is such a probability Q.. The time value of money The time value of money is one of the central concepts in finance theory which states that a unit of currency today is worth more than the same unit of currency tomorrow due to its potential earning capacity. In other words, $ paid now is worth more than $ paid in a year because by depositing $ in the bank today, one gets more then a pound in a year. resent value (or present discounted value) is a future value of an asset that has been discounted to reflect its value today. Similarly, future value is the value of an asset in the future which is equivalent to a specified sum at present. For a fixed time period [T ; T ], interest is the additional gain between the beginning T and the end T of the time period. resent value of a future sum F can be obtained using continuous compound interest rate r as = exp ( rt) F, where t = T T. In general, if r is a function of t, then = F exp T T r(t)dt :.3 Arbitrage theorem All known methods of pricing derivatives employ the notion of arbitrage. An arbitrage can be defined as a way to make guaranteed profit from nothing by selling an asset at time T and then settling accounts at T. An existence of arbitrage provides an investment opportunity with infinite rate of return. Hence, investors would try to use arbitrage to make money without putting up anything at time T. Consequently, to eliminate this possibility we need to introduce so-called Efficient Market 6

18 Hypothesis which are essentially are:. All known information is reflected on prices of all securities.. The current prices are the best estimates of the values of securities. 3. The prices will instantaneously adjust according to any new information. 4. An investor cannot outperform the market price using all known information. To give an analytic definition of an arbitrage consider a simple model with two time points T and T, T < T and zero interest. Fix a measure space (; F), where := ft ; T g, F =. Assume that the value of a stock S(T ) at T is S(T ). Let a be the value of S at T with probability p and b be the value of S at T with probability p, a < b. By this way we specify on (; F). Hence, we get a probability space (; F; ). Consider a portfolio (N; MS(T )) consisting of N units of money and M units of stocks. The value V T of this portfolio at T is V T = N + MS and at T is V T = N + MS(T ). We say that there exists an arbitrage opportunity if there exists a portfolio (N; MS(T )) such that V T = 0, V T 0 and (V T > 0) > 0. It is possible to show that there exist no arbitrage opportunities iff a < S(T ) < b [37]. Theorem (Fundamental theorem of asset pricing) There exist no arbitrage opportunities iff there exist a probability measure Q equivalent to the original probability measure such that the stock price process (S(T ); S(T )) satisfies E Q [S(T ) js(t )] = S(T ): A probability measure Q is called an equivalent martingale measure. Observe that Theorem explicitly relates a fundamental notion of arbitrage to a far advanced theory of martingales. In the case of multi-period model we have a similar result [35]. Theorem There are no arbitrage opportunities in the multi-period model iff for every t, the one-period model (S t ; S t+ ), with respect to the filtration (F t ; F t+ ), admits no arbitrage opportunities. See Appendix III for more information. settings (see, e.g. [35] ). Consider the case of continuous-time Definition 3 A probability measure Q on a measure space (; F) is called equivalent martingale measure if it is equivalent to and S is a martingale with respect to Q. The collection of all equivalent martingale measures on the measure space (; F) is denoted by M S (; F). 7

19 The change of measure spaces (; F; )! (; F; Q) is based on the Radon- Nikodim s theorem (see Appendix I, Theorem 70). Theorem 4 There are no arbitrage opportunities iff there exists an equivalent martingale measure. The proof of this statement is based on the separating hyperplane theorem (in the finite dimensional case) and on the Hahn-Banach theorem (in the infinite dimensional settings) for locally convex topological vector spaces (see Appendix IV, Theorem 88). Also, Banach-Alaoglu theorem is an important ingredient of the proof (see Appendix IV, Theorem 87). 8

20 Chapter Lévy processes and characteristic exponents. Introduction We start with the one-dimensional case. We shall study a common frictionless market consisting of a riskless bond and stock which is modeled by an exponential Lévy process S t = S 0 exp (X t ) under a fixed equivalent martingale measure Q with a given constant riskless rate r > 0. Since in our model the stock does not pay dividends then the discounted stock price exp ( rt) S t must be a martingale under Q. Consider a contract (European call option) which gives to its owner the right but not the obligation to buy the underlying asset for the fixed strike price K at the specified expiry date T. We need to evaluate its price V. In this case the payoff has the form '(x) = (S 0 exp (x) K) + ; (.) where for any a R, (a) + = maxfa; 0g, K is the strike price. In the classical Merton-Black-Scholes model the price of a stock follows the Geometric Brownian motion defined as S t = S 0 exp(x t ), where X t, t 0 is the Brownian motion with the probability density function p t (x) = t = (x t) exp t for the increments X t+t X t and parameters and are known as drift and volatility respectively [8], p.. 9

21 It is well-known that Merton-Black-Scholes theory becomes much more efficient if additional stochastic factors are introduced. Consequently, it is important to consider a wider family of Lévy processes. Stable Lévy processes have been used first in this context by Mandelbrot [08] and Fama [38]. From the 90 s Lévy processes became more popular (see, e.g. [09, 0, 7, 8, 63] and references therein).. Basic results We start with basic definitions and results. A probability space (; F; ) is a triplet of a set, an admissible family F of subsets, F and a mapping : F! [0; ] such that. F and ; F:. If A n F for any n N; then [ \ A n F; A n F: n= n= 3. If A F, then A c F: 4. 0 (A) ; () = ; and (;) = 0: 5. If A n F for any n N and A n \ A m = ;, 8n; m N, n 6= m; then! [ A n = n= X (A n ) : n= A family F satisfying, and 3 is called a -algebra and a mapping with the properties 4 and 5 is called a probability measure. Let (; F; ) be a probability space. Let B (R n ) be the collection of all Borel sets on R n which is the algebra generated by all open sets in R n ; i.e. the smallest algebra that contains all open sets in R n : A real valued function is called measurable (Borel measurable) if it is B (R n ) measurable. A mapping X :! R n is an R n valued random variable if it is F-measurable, i.e. for any B B (R n ) we have f!jx(!) Bg F: A stochastic process X = fx t g tr+ is a one-parametric family of random variables on a common probability space (; F; ). The trajectory of the process X is a map 7 R +! R n t! X t (!) ; 0

22 where! and X t = (X ;t ; ; X n;t ). For a fixed 0 t 0 < t < < t m ; m N and Borel measurable sets B k R n ; 0 k m consider the map B (R mn )! R + Q km B k 7! [X t B ; ; X tm B m ] ; which defines a probability measure on B (R mn ) : The system of finite-dimensional distributions of X is the family of all such measures over all choices 0 t 0 < t < < t m ; m N. Two stochastic processes X and Y are identical in law, written as X d = Y (or X = Ymod (law)) if the system of their finite-dimensional distributions are identical. Consider the -algebra F generated by the cylinder sets, known as Kolmogorov s -algebra. Theorem 5 (Kolmogorov s extension theorem) Suppose that for any 0 t < t m and m N a distribution t ;;t m is given. If for any B ; ; B m B(R n ) we have t ;;t m m Y s= B s! = t ;;t k ;t k+ ;;t m Y sm;s6=k B s! ; B k = R n then there exists a unique probability measure on F that has t ;;t m of finite-dimensional distributions. as its system Different proofs of this statement can be found in [6] and [0]. X = fx t g tr+ is called a Lévy process (process with stationary independent increments) if. The random variables X t0 ; X t X t0 ; ; X tm X tm, for any 0 t 0 < t < < t m and m N are independent (independent increment property).. X 0 = 0 a.s. 3. The distribution of X t+ X t is independent of (temporal homogeneity or stationary increments property). 4. It is stochastically continuous, i.e. lim [jx X t j > ] = 0!t for any > 0 and t 0.

23 5. There is 0 F with ( 0 ) = such that, for any! 0 ; X t (!) is rightcontinuous on [0; ) and has left limits on (0; ): A process satisfying (-4) is called a Lévy process in law. An additive process is a stochastic process which satisfies (,,4,5) and an additive process in law satisfies (,,4). The convolution = of two distributions and on R n is defined as (B) = B (x + y) (dx) (dy) < ; R n R n where B (x) := ; x B; 0; x =B is the characteristic function of a Borel (Lebesgue) measurable set B R n. A probability measure is called infinitely divisible if for any m N there is a probability measure (m) such that = (m) (m) : {z } m Consider the set L of Lévy processes X = fx t g tr+ on a probability space (; F; ). Let us denote the set of infinitely divisible measures on R by M (R) : It is possible to show that Lmod (law) = M (R) [4]. Let x; y R n, x = (x ; :::; x n ), y = (y ; :::; y n ), hx; yi be the usual scalar product in R n, i.e. nx hx; yi = x k y k R k= and jxj := hx; yi = : Let C(R n ) be the space of continuous functions on R n and L p (R n ) be the usual space of p-integrable functions equipped with the norm R kfk p = kfk Lp(R n ) := jf(x)j p dx =p R ; p < ; n ess sup xr n jf(x)j; p = : The holomorphic Fourier transform on classes of square-integrable functions f supported on R is defined as Ff() = exp ( ix) f(x)dx; R

24 where for = + i, ; R we have > 0. The inverse Fourier transform is F u(x) = exp (ix) u()d: as R For a finite measure on R n (i.e. if (R n ) < ) we define its Fourier transform F (y) = exp ( R n and its formal inverse i hx; yi) (dx) (dx) = F F (dx) = () n R n exp (i hx; yi) (F) (y) dy: It is known that if is infinitely divisible then there exists a unique continuous function : R n! C such that (0) = 0 and exp ( (y)) = F (y) (see, e.g. [], p. 37). The characteristic function (x; t) of the distribution of X t of any Lévy process can be formally defined as (x; t) := E [exp (i hx; X t i)] = exp ( t (x)) = () n F p t (x) ; where p t (x) is the density function of X t, x R n, t R + and the function (x) is uniquely determined. This function is called the characteristic exponent. Vice versa, a Lévy process X = fx t g tr+ is determined uniquely by its characteristic exponent (x). In particular, p t can be expressed as p t () = () n R n exp ( i h; xi t (x)) dx = () n F (exp ( t (x))) () : The characteristic function (z) of a probability measure is formally defined as (z) := exp (i hz; xi) (dx) = () n F (z) ; z R n : R n We say that a matrix A is nonnegative-definite (or positive-semidefinite) if x Ax 0 for all x C n (or for all x R n for the real matrix), where x is the conjugate transpose. A matrix A is nonnegative-definite iff it arises as the Gram matrix of some set of vectors v ; ; v n, i.e. A = (a ij ) = hv j ; v i i. The following classical result, plays a key role in our analysis. 3

25 Theorem 6 (Lévy-Khintchine formula) Let X = fx t g tr+ be a Lévy process on R n. Then its characteristic exponent admits the representation (y) = hay; yi ihb; yi R n ( exp (ihy; xi) ihy; xi D (x)) (dx); (.) where D (x) is the characteristic function of D := fx R n ; jxj g, A is a symmetric nonnegative-definite n n matrix, b R n and (dx) is a measure on R n such that minf; hx; xig(dx) < ; (f0g) = 0: (.3) R The density of is known as the Lévy density and A is the covariance matrix. In particular, if A = 0 (or A = (a j;k ) j;kn, a j;k = 0) then the Lévy process is a pure non-gaussian process and if = 0 the process is Gaussian. The following important result relates distributions on R n and nonnegative definite functions (see e.g. [], p. 8). Theorem 7 (Bochner) Let be a distribution on R n : Then (0) = ; j (z)j and (z) is uniformly continuous and nonnegative-definite in the sense that, for each n N nx nx (z j z k ) j k 0 j= k= for any z k R n ; k n and k C; k n: Conversely, if a function ' (z) : R n! C with ' (0) = is continuous at z = 0 and nonnegative-definite, then ' (z) is the characteristic function of a distribution on R n : We say that the Lévy process has bounded variation if its sample paths have bounded variation on every compact time interval. A Lévy process has bounded variation iff A = 0 and R n min fjxj ; g (dx) < ; (f0g) = 0 (see, e.g. [5], p. 5). The systematic exposition of the theory of Lévy processes can be found in [4, 43, 44,, 4, 3]. 4

26 .3 Equivalent martingale measures If a probability measure is estimated using historical return data for the underlying stock, the measure is referred to as the market measure (or the physical measure, or historical measure). Asset prices are modeled by stochastic processes (S t ) t>0 whose evolutions are determined by a fixed probability measure. In the theory of arbitrage pricing there exists a risk neutral probability measure under which asset prices are arbitrage free. The absence of arbitrage is equivalent to the existence of a risk neutral equivalent martingale measure Q for (S t ) t>0 making the underlying process become a martingale. Under the equivalent martingale measure all assets have the same expected rate of return which is the risk free rate. It means that under no-arbitrage conditions the risk preferences of investors acting on the market do not enter into valuation decisions [7]. Consider a frictionless market consisting of a riskless bond B and stock S. In this market S is modeled by an exponential Lévy process S = S t = S 0 exp (X t ) under a chosen equivalent martingale measure Q. Assume that the riskless rate r is constant. Theorem 8 Let D be the domain of Q () and R [ f ( i) = r. ig D, then in our notations roof. The discounted price process which is given by ~S t = exp( rt)s t = exp( rt)s 0 exp(x t ) must be a martingale under a chosen equivalent martingale measure Q, i.e. for any 0 l < t T the martingale condition must hold, ~S l = E Q h ~St jf l i (see Appendix III for more information). t (0; T ] we have In particular, let l = 0 then for any ~S 0 = S 0 exp( r0) = S 0 = E Q [S 0 exp( rt) exp(x t )jf 0 ] = E Q [S 0 exp( rt) exp(x t )] = S 0 E Q [exp( rt) exp(x t )] : (.4) Since S 0 > 0 then E Q [exp( rt) exp(x t )] = : 5

27 In particular, let t = T then exp(rt ) = E Q [exp(x T )] : Since ( i) is defined then by the definition of the characteristic exponent exp( T ( i)) = E Q [exp(i( i)x T )] = E Q [exp(x T )] : Hence, since T > 0, then from (.4) it follows that r = ( i). A commonly used condition on Q () is that it admits the analytic continuation into the strip fzj =z 0g (see, e.g. [9] p. 83). In general, Q is not unique [8]. We assume that Q has been fixed and all expectations will be computed with respect to this measure..4 Characteristic exponents and density functions Definition 9 We say that the process is purely discontinuous if A = (a j;k ), a j;k = 0, j; k n and b = 0 in (.). Consider the one-dimensional case. It follows from (.) that for a purely discontinuous process () can be written as () = exp (ix) + ix [ ;] (x) (dx): (.5) R Applying (.5) we get the following representation for the density function p t (y); p t (y) = exp ( iy) E[exp (ix t )]d R = F (exp ( t ())) (y) = exp ( iy t ()) d R = exp iy t exp (ix) + ix [ ;] (x) (dx) d: (.6) R R To find an explicit expression of the integral (.6) we need to assume a very specific form of (dx). Consequently, an explicit representation in this case is a real rarity. For instance, Barndorff and Nielsen [7] obtained the Lebesgue density function for generalised hyperbolic distributions which depends on five parameters (x; ; ; ; ; ) = = + (x ) ( =)= () = = + K 6

28 K = + x = exp ( (x )) ; where K (x) is the modified Bessel function of the third kind, K (x) = x! exp d: 0 Motivated by the idea to consider a wider class of models we need to decide at which stage we should fall into numerical methods. If we still want to apply with an explicit form of the characteristic exponent but still agree to apply numerical methods to compute the density function p t (y) then we apply the following standard trick. The sense of this manipulation is based on Cauchy s theorem (see, e.g. [7]) and will be clarified in details in the following sections. Here we remark just the following. Let T be the expiry date, t < T and ' (X T ) > 0 be the terminal payoff at the expiry date T, where ' (x) is defined by (.). The price V of the European call option formally can be obtained from exp ( rt) V = E [exp ( rt ) '(X t )jx t = x] ; where exp (x) = S t (see, e.g. [8], p. 04). It means that V can be formally written as V = exp ( r) '(x + y)p (y)dy R = exp ( r) p (y x)'(y)dy; (.7) R where := T t is the time to expiry. The integral (.7) is well-defined if I (x) := jp (y x)'(y)jdy = = R ln K = R p (y x) (exp (y) K) + dy ln K p (y x) (exp (y) K) dy exp (y) p (y ln K x)dy exp (y) exp ( i(y x) ()) d dy < (.8) R 7

29 for any fixed x R. It means that to satisfy (.8) we need to assume an extra condition on the characteristic exponent (). This condition is given in an implicit form and can be satisfied if () has an analytic extension at least onto the strip = [!; 0];! < with some additional technical conditions if! = to guarantee (.8). In what follows we shall not discuss the case! =. Assume that! <, and lim R! lim R! R i!+r R i! R exp ( iy ()) d = 0 (.9) exp ( iy ()) d = 0 (.0) for any y R and 0. Applying Cauchy s theorem we get I exp ( iy ()) d = 0; where is the rectangular contour [ R; R] [ [R; i! + R] [ [i! + R; i! R] [ [i! R; R] and R > 0: Letting R! and applying conditions (.9) and (.0) we get p (y) = exp ( iy ()) d R = lim exp ( iy ()) d R! = lim R! = lim R! = [ R;R] i!+ i! [i! + [i!+r;r] [i! + exp ( iy ()) d R;i!+R] [ R;i! R] exp ( iy ()) d R;i!+R] exp ( iy ()) d: Changing variable in the last integral, z = i! we get p (y) = exp ( iy (i! + z) (i! + z)) dz R 8

30 = exp (!y) exp ( iyz (i! + z)) dz: (.) R Substituting (.) into (.8) we get I (x) := = exp ( < : exp (y) exp (! (y x)) exp ( i (y x) z (i! + z)) dz dy R!x) exp (( +!) y) exp ( i (y x) z (i! + z)) dz dy ln K R ln K Observe that ( +!) < 0: Summarising, we get the following three conditions on () :. () has the form (.5) where (dx) satisfies (.3).. The condition (.8) must be satisfied. In particular, () should be analytically extendible into the strip = [!; 0];! < : 3. The function x (y) := exp (( +!) y) R exp ( i (y x) z (i! + z)) dz should be integrable on R + for each fixed x R. In general, if the payoff function G has a more complicated structure, we proceed as following. Assume that the characteristic exponent () has an analytic extension into the strip < = < +, where < 0 < +. Choose! + and! such that <! < 0 <! + < + and consider two rectangular contours and := [ R; R] [ [R; i! + R] [ [i! + + R; i! + R] [ [i! + R; R] := [ R; R] [ [R; i! R] [ [i! + R; i! R] [ [i! R; R]: Applying the same as the above line of arguments we get p (y) = exp (y! +) exp ( iyz (z + i! + )) dz R 9

31 and p (y) = exp (y! ) exp ( iyz (z + i! )) dz: R Hence p (y) = exp ( y! + ) + exp ( y! ) exp ( iyz) (exp ( (z + i! + )) + exp ( (z + i! ))) dz: R In particular, if! =! + =! then p (y) = 4 (cosh(!y)) exp ( iyz) (exp ( (z + i!)) + exp ( (z i!))) dz: R Observe that more general representations of density functions using contour deformation technique has been obtained in [8]..5 KoBoL family In this section we study characteristic exponents of so-called KoBoL family. The idea is based on a simple observation. From the Lévy-Khintchine formula (.5) it follows that it is possible to find () explicitly if we can compute explicitly the inverse Fourier transform of (dx): Therefore, it was suggested by the authors of [8] to consider the following form of (dx); (dx) = jxj exp ( jxj) dx; where and are fixed parameters. The following definitions are based on this observation. Definition 0 We say that a Lévy process is a regular Lévy process of exponential type if its density has at the origin a power type singularity and decays exponentially at infinity. 30

32 The characteristic exponent () of a regular Lévy process of exponential type admits an analytic extension onto the strip = ( ; + ) continuous on the boundary and admits the representation where () = i + () ; () c jj,!, = ( ; + ) : It is possible to show that for any regular Lévy process of exponential type the density function p t (y) is infinitely differentiable on R and exponentially decays with all of its derivatives as y!. To formalise the properties of the KoBoL exponent which are useful for analysis the following definitions have been introduced. Definition Let 0 <, < 0 < + be some fixed parameters. Assume that. () admits the analytic extension into the strip < = < () := () + i is asymptotically positively homogeneous of order as! in any closed strip! =! + <!! + < +, i.e. for any 6= 0 lim jj!;! =! + 0 () jj 0 () = : In this case we say that the corresponding Lévy process X is a regular Lévy process of exponential type ( ; + ) and order. and Let < 0 < +, + (; + ; dx) = (max fx; 0g) exp ( + x) dx (; ; dx) = (max f x; 0g) exp ( x) dx; where < : Definition A Lévy process is called a KoBoL process of order < if it is purely non-gaussian with the Lévy measure of the form (dx) = c + + (; + ; dx) + c (; ; dx); where c + > 0; c > 0; < 0 < + : 3

33 We call the order of the process, + and the steepness parameters and c + and c the intensity parameters of the process. The parameter ( + respectively) determines the rate of the exponential decay of the right (left respectively) tail of the density function. It is easy to see that the condition (.3) is satisfied, i.e. min ; x c + + (; + ; dx) + c (; ; dx) < : R Moreover, if < then min f; jxjg c + + (; + ; dx) + c (; ; dx) < ; R i.e. a KoBoL process is of finite variation iff < : In what follows we shall adopt the standard notations, z = exp ( ln z), where ; z C such that z 6 ( ; 0] and ln z denotes the branch of ln z defined on C n ( ; 0] and such that that ln() = 0. Lemma 3 ([8], p. 70) If (0; ) [ (; ) then () = i + c ( ) (( ) ( i) ) +c + ( ) + ( + + i) : (.) If = 0; then () = i + c [ln ( i) ln ( )] +c + [ln ( + + i) ln + ] : If = ; then () = i + c [( ) ln ( ) ( i) ln ( i)] +c + [ + ln + ( + + i) ln ( + + i)] ; where R; c > 0; and < 0 < +. The proof of Lemma 3 presented in [8] is incomplete. The next statement gives a complete proof of the representation (.) which is important in our applications. Theorem 4 Let (0; ) then in our notation () = i + c ( ) (( ) ( i) ) +c + ( ) + ( + + i) ; where is a real parameter. 3

34 roof. It is sufficient to prove the statement just for the + (; ; dx), i.e. to find + () := exp (ix) ix [ ;] (x) + (dx) = = = R 0 i 0 0 R exp (ix) ix [ ;] (x) x + exp ( x) dx exp (ix) ix [ ;] (x) x exp ( x) dx (exp (ix) ) x exp ( x) dx x exp ( x) dx := I (; ; ) ib (; ) ; where B (; ) := R x exp ( x) dx and 0 I (; ; ) = 0 (exp ( ( i) x) exp ( x)) dx = (exp ( ( i) x) exp ( x)) x j 0 ( ( i) exp ( ( i) x) + exp ( x)) x dx = i 0 0 exp ( ( i) x) x dx ( ) := I ( ) : Making change of variable z = ( i) x in I we get ( i) I = exp ( z) z dz; where is the ray fz jz = ( i) x; > 0; Rg, and are fixed parameters and x 0. Assume that 0. The case 0 can be treated similarly. Consider the contour := [ [ 3 [ 4, where := fz = exp (i) j0 arg ( i) ; > 0; Rg ; := fz j z R; z Rg ; 33

35 3 := fz = R exp (i) j0 arg ( i) ; > 0; Rg ; 4 := fz jz = ( i) x; jzj Rg : The function exp ( z) z is analytic in the domain bounded by, hence from the Cauchy s theorem it follows that I exp ( z) z dz = 0 and since 0 then for some > 0 we get = + arg ( i) 0: Hence lim R! exp ( z) z dz 3 arg( i) = lim exp ( R exp (i)) R exp ( i) Ri exp (i) d R! 0 lim R! exp ( R cos ) exp R = 0: Observe that lim!0 exp ( 3 Hence lim!0 0 z) z dz lim!0 + = 0: exp ( exp (i)) exp ( i) i exp (i) d exp ( z) z dz = exp ( R + z) z dz = ( + ) = ( ) : Consequently, I = ( i) exp ( y) y dy = ( ) ( i) and + () = ( ) ( (( i) )) + ib (; ) : 34

36 Finally, the term ib (; ) can be considered as a part of i, where R is a free parameter. It is easy to see that the parameters (; c + ; c ; + ; ) determine the probability density. For larger and c we get a larger peak of the probability distribution. The parameters c + and c control asymmetry of the probability distribution while and + determine the rate of exponential decay as! : In [9, 93] it is shown that there exist < and d 0, <d 0 > 0 such that as!, 0 ( exp (i)) = d + exp (i) + O( ); [0; = 0]; 0 ( exp (i)) = d exp (i( + )) + O( ); [= + 0; ]; 0 ( exp (i)) = d 0 exp (i( + )) + O( ); [ ; = 0]; 0 ( exp (i)) = d 0 exp (i) + O( ); [ = + 0; 0]; where = = 0 means that = exp (i) is of the form = iz 0, where z > 0 and = = 0 means that = exp (i) is of the form = iz 0, where z < 0. Consider the asymptotic behavior of KoBoL exponent () in the strip = ( ; + ) as jj!. The following statement can be proved by a direct calculation. Lemma 5 =( lim where 8 < (x) := : ; + );jj! () (jj) = ; ix ( ) x i c + exp + c exp i ; (0; ) [ (; ) ; i (x + (c + c ) x ln x) + (c ++c )x ; = ; x + (c + + c ) ln x; = 0: In particular, if c + = c = c then 8 < ix c ( ) x cos ; (0; ) [ (; ) ; (x) := ix + cx; = ; : x + c ln x; = 0: Example 6 Let, in particular, T = 0:5, c + = c = 0:6506, + = :9458, = :087, = 0:5, = 0:39563 (see [9, 93]). (x) = i0:39563x + ( 0:5) 0:6506 (:087) 0:5 (:087 ix) 0:5 + ( 0:5) 0:6506 :9458 0:5 (: ix) 0:5 ; (x) = (exp ( 0:5 (x))) : 35

37 z 0 y x Figure. Re ( (x + iy)) y z x Figure. Im ( (x + iy)). 36

38 y z x 5 5 Figure 3. j (x + iy)j. Example 7 At this point we present two more important examples of characteristic exponents () which are of practical interest in empirical studies of financial markets. Remark that Madan and collaborators [99]-[0] were first who applied Variance Gamma processes in studies of financial markets. The respective characteristic exponent has the form () = i + c + [ln( i) ln( )] + c [ln( + + i) ln( + )]; where < 0 < +, c > 0 and R. A Variance Gamma process with these parameters is also a Lévy process of exponential type ( ; + ). So-called Normal Inverse Gaussian processes were introduced and studied by Barndorff- Nielsen [7]-[]. The respective characteristic exponent is h () = i + ( + i) i = ( ) = : 37

39 Chapter 3 Recovery of density functions in jump-diffusion models 3. Introduction Remind that the pricing formula has the form V = exp ( rt ) E Q ['], where Q is a fixed equivalent martingale measure. Since the reward function ' has usually a simple structure the main problem is to approximate the respective risk-neutral density function p Q T, where T > 0 is a maturity time. Observe that almost uniformly distributed lattice points in R n can be used to reproduce trigonometric polynomials just with the spectrum inside a properly scaled symmetric hyperbolic cross [65, 64, 48]. Unfortunately, characteristic exponents of density functions which correspond to a jump-diffusion processes, which can be used in pricing formulas, should admit an analytic extension into a proper domain to guarantee the existence of the pricing integral. This is a very restrictive condition. The shape of the level surfaces of such characteristic exponents is quite far from the hyperbolic crosses. Hence, a number theoretic lattice points can not be effective in such situations. Consider several examples of characteristic functions to illustrate this situation. Example 8 Let Wt and Wt be risk-neutral Brownian motions with correlation and ; > 0. Consider the vector S t = (S ;t ; S ;t ) with components S k;t = S k;0 exp r k= t + k W k t ; k = ; : 38

40 The joint characteristic function of X T = (ln S ;T ; ln S ;T ) has the form (u; T ) = exp i u;rt e T T u; u T ; where u = (u ; u ) ; e = (; ) ; = ( ; ), = and u T means u transposed. Direct calculation shows (u; T ) = (u ; u ) = exp T ir u + u u + u + ir u + u iru iru : The parameters are [8, 55]: r = 0:, T =, = 0:5, = 0:, = 0:. For such set of parameters the function simplifies as (u ; u ) = exp 0:004iu + 0:0u u + 0:u + 0:00iu + 0:u 0:iu 0:iu : z y x0 5 5 Figure 4. j (u ; u )j. 39

41 Example 9 Consider a three factor stochastic volatility model [8] which is defined as dx = r dt + = dw ; dx = r dt + = dw ; d = ( ) dt + = dw ; where dw ; dw and dw have correlations E dw ; dw = dt; E dw ; dw = dt; E dw ; dw = dt; X t = (log S ;t ; log S ;t ) and t is the squared volatility. The characteristic function has the form! ( exp ( T )) (u) = (u ; u ) = iu ln S ;0 + iu ln S ;0 + 0 ( ) ( exp ( T )) where +i hu; (re )i T ( ) ( exp ( T )) log + ( ) T ;! := u + u + u u + i u + u ; := i ( u + u ) :=! ; := ( ; ) ; e := (e ; e ) : Let us fix parameters as in [8], p. 6: r = 0:, T =, = 0:5, = 0:5, = 0:5, = 0:05, = 0:05, = 0:5, = :0, 0 = 0:04, =, = 0:04, = 0:05, S ;0 = 96, S ;0 =

42 y z x 0 40 Figure 5. j (u ; u )j. Example 0 Following [0] consider the V.G. process. The Lévy measure in this case is (x) = exp ( a +x) [0;) (x) + exp (a x) ( ;0] (x) ; > 0; a + > 0; a > 0; x where A (x) := ; x A; 0; x = A; A R and the characteristic function is t Yt (u) = + i u + u : a a + a a + Let Y k;t, k = ; ; 3 be three independent V.G. processes with common parameters a +, a, = = ( ), 3 =, [0; ] : The log return X k;t = log S k;t ; k = ; is given by X k;t = X k;0 + Y k;t + Y 3;t ; k = ; : The characteristic function has the form 3 (u; T ) = 3 (u ; u ; T ) 4

43 = + i (u + u ) + (u + u ) a a + a a + ( )T + i u + u a a + a a + ( )T + i u + u : a a + a a + Let us put T = ; a + = ; a = 3; = ; = 0:5.! T 0 y z x 0 Figure 6. j 3 (u ; u )j. In the multidimensional settings application of polynomial splines meets various technical difficulties such as partitioning problem and growing computational complexity as the dimension grows. Hence it is important to construct a simple, saturation free method well adapted to the multidimensionality of approximation of density functions which are important in the theory of spread options. We consider in detail two methods of approximation of density functions. Our first method is based on the oisson summation formula and approximation of density functions by harmonics in the respective exponential hyperbolic cross. The advantage of this approach is that the application of the oisson summation formula gives a periodic extension of the density function of the same smoothness as the original function. The second approach is based on approximation of density functions by sk-splines, in particular by Wiener spaces. 4

44 3. A class of stochastic systems In this section we introduce a class of stochastic systems to model multidimensional return processes. Let X ;t ; ; X n;t and ;t ; ; n;t be independent random variables, with the density functions p () ;t (x ) ; ; p () n;t (x n ) and p () ;t (x ) ; ; p () n;t (x n ) and characteristic exponents () s and () m ; s; m n respectively. Let X t = (X ;t ; ; X n;t ) T, t = ( ;t ; ; n;t ) T and A = (a j;k ) be a real matrix of size n n. Consider random vector U t = (U ;t ; ; U n;t ) T ; U t = X t + A t : (3.) A matrix A reflects dependence between the processes U ;t ; ; U n;t in our model. Assume for simplicity that E [X s;t ] = 0 and E [ s;t ] = 0; s n, var (X s;t ) = var ( s;t ) = v t and a s;k = ; s; k n. It is easy to check that for any s and l, s 6= l n the correlation coefficient (U s;t ; U l;t ) := E [U s;t U l;t ] E Us;t E U = l;t between U s;t and U l;t, where nx U s;t = X s;t + a s;k k;t ; U l;t = X l;t + k= nx a l;k k;t is (U s;t ; U l;t ) = n(n + ). This reflects our empirical experience: if the market is in crisis then the prices of stocks are highly correlated (see for more information). The next statement gives us an explicit form of the characteristic function of the return process U t : Theorem Let U t = X t +A t ; A = (a m;k ) : Then in our notation the characteristic function (v;t) of U t has the form Y n! n! Y (v;t) = () n F p () (v) F A T v ; = exp t s= nx s= s;t () s (v s ) + where A T = (a k;m ) is the transpose of A. nx m= 43 k= () m m= p () m;t!!! nx a k;m v k ; k=

45 roof. Consider the transformation R n! R n defined as U t = X t + A t ; t = t : The inverse is given by X t = U t A t ; t = t ; or Xt t I A = 0 I Ut t and the Jacobian J of this transformation is I A J = det = ; 0 I where I = I nn is an identity. The density function is given by ep t (u; z) = ep t (u ; ; u n ; z ; z n ) ep t (u; z) = ny s= p () s;t u s! nx Y n a s;m z m p () l;t (z l) : m= l= This means that the density function p t (u) of U t is p t (u) = ep t (u; z) dz R n and the characteristic function has the form (v;t) := E [exp (i hu t ; vi)] := exp ( t (v)) = exp (i hu; vi) p t (u) du R n = exp (i hu; vi) R n ep t (u; z) dz R n du 44

46 = exp (i hu; vi) R n = R n ny s= R p () s;t u s ny R n s= p () s;t u s! nx Y n a s;m z m m= m=!! nx Y n a s;m z m exp (iu s v s ) du s m= p () m;t (z m ) dz m=! du p () m;t (z m ) dz: (3.) In the last line we applied Fubini s theorem (see Appendix I, Theorem 68). Let s = u n s m= a s;mz m ; s n: Then R = p () s;t R u s! nx a s;m z m exp (iu s v s ) du s m= p () s;t ( s ) exp i s + = exp iv s nx!! nx a s;m z m v s d s m= a s;m z m! m= R! nx = exp iv s a s;m z m F m= Comparing (3.) and (3.3) we get (v;t) = = = = = R n p () s;t ( s ) exp (i s v s ) d s p () s;t! ny nx exp iv s a s;m z m F s= ny F s= ny F s= p () s;t p () s;t ny F s= ny F s= m= (v s ) R n (v s ) exp R n p () s;t p () s;t (v s ) : (3.3) p () s;t! Y n (v s ) m=!! ny nx Y n exp iv s a s;m z m s= i nx s= v s m= z m;t (z m ) dz m=!! nx Y n a s;m z m m= (v s ) exp (i hv; Azi) R n (v s ) exp i A T v; z R n ny m= m= p () m;t (z m ) ny m=! p () m;t (z m ) p () m;t (z m ) dz p () m;t (z m ) dz dz! dz 45

47 = ny F s= p () s;t (v s ) F n Y m= p () m;t! A T v ; where A T = (a k;j ) is the transpose of A. Hence ny ny (v;t) = exp t () s (v s ) exp t () m s= = exp t nx s= m= () s (v s ) + nx m= () m!! nx a k;m v k k=!!! nx a k;m v k : k= Example Let X t, X t and t, t be independent identically distributed KoBoL processes with the same parameters as in Example 6, and characteristic function. Let a a A = ; a a where a = a = ; a = a = 0 and > 0 be a free parameter. In this case the respective characteristic function has the form F (v ; v ; ) = (v ) (v ) (v + v ) : Consider characteristic functions F (v ; v ; ) for different values of. 0 z y x 0 Figure 7. jf (v ; v ; )j, = 0. 46

48 z y x 0 0 Figure 8. jf (v ; v ; )j, = 0: z y 0 0 x 0 Figure 9. jf (v ; v ; )j, =. 47

49 3.3 Representation of density functions by Fourier modes Assume that all characteristic exponents ();Q s ; s n and ();Q m ; m n correspond to a KoBoL process and hence are analytically extendable into the strips s; < s; < 0 < s;+ < s;+, s n and 0 s; < 0 s; < 0 < 0 s;+ < 0 s;+, s n respectively. Let a k;m 0, k; m n. It is easy to check that nx s= ();Q s (v s ) + nx m= ();Q m! nx a k;m v k k= is analytically extendable into the domain! n[ T n := fim v s [ s; ; s;+ ]g s= 0 8 n[ < : Im v s s= or a ;s v s a +;s ; where 8 < a ;s := max : s; ; 0 s; 8 < a +;s := min : s;+; 0 s;+ 4 0 s; nx k= nx k=! nx a k;m ; 0 s;+ k=! 9 = a k;m ; ;! 9 = a k;m ; ;! 39 nx = a k;m 5 A ; ; k= s n and a k;m 0, k; m n. In this case Q (v;t) = Q (v ; ; v n ;t) admits an analytic extension into the same domain T n C n : Let a + := (a +; ; ; a +;n ) and a := (a ; ; ; a ;n ). Theorem 3 Let ();Q s ; s n and ();Q ; m n be defined by (.), i.e. ();Q s ( s ) = i s s + c ;s ( s ) (( ;s ) s ( ;s i s ) s ) + c +;s ( s ) s +;s ( +;s i s ) s m 48

50 and ();Q m ( m ) = i m m + c ;m ( m ) (( ;m ) m ( ;m i m ) m ) + c +;m ( m ) m +;m ( +;m i m ) m : Then the respective density function p Q T () can be represented as p Q T () = () n (exp (h; a + i) + exp (h; a i)) exp ( i h; vi) Q (v ia + ;T ) + Q (v ia ;T ) dv: R n In particular, if a = a + := a then Let p Q T () = () n (cosh (h; ai)) exp ( i h; vi) Q (v+ia;t ) + Q (v ia;t ) dv: (3.4) R n Q (v;t ) := Q (v+ie a ;T ) + Q (v ie a ;T ) ; Then Q k (v;t ) := Q k (v+ie ka k ;T ) + Q k (v ie ka k ;T ) ; k n: p Q T (x) = n n! ny cosh (a s x s ) exp ( i hx; vi) Q n (v;t ) dv: (3.5) R n s= roof. We shall prove just (3.4) since (3.5) follows in a similar manner. In our notation the density function can be represented as p Q T () = () n R n exp ( i h; vi) Q (v;t ) dv = () n F Q (v;t ) () : Recall that ();Q s ( s ), s n admits an analytic extension into the strip = s [ s; ; s;+ ], where s; < s; < 0 < s;+ < s;+, s n and ();Q m ( m ), 49

51 m n admits an analytic extension into the strip = m 0 m; ; 0 m;+, where 0 m; < 0 m; < 0 < 0 m;+ < 0 m;+, m n: By the Lemma 5 we have where lim = s [ s; ; s;+ ]; j s j! l (x) = i l x ( l ) ();Q s ( s ) s (j< s j) = l (0; ) [ (; ) ; l = s; m: c l;+ exp lim = m [ 0 m; ; 0 m;+]; j m j! il + c l; exp Observe that sgn ( ) cos > 0; (0; ) [ (; ) : Then, from (3.6) it follows that ();Q m ( m ) m (j< m j) = ; il x ; (3.6) lim jvj!; vt n Q (v;t ) = 0: (3.7) Hence, applying Cauchy s theorem n times in the domain T n, which is justified by (3.7), we get p Q T () = () n R n exp ( i h; vi) Q (v;t ) dv = () n R n +ia + exp ( i h; vi) Q (v;t ) dv or Similarly, = () n R n exp ( i h; x ia + i) Q (x ia + ;T ) dx = exp ( h; a + i) () n R n exp ( i h; xi) Q (x ia + ;T ) dx; p Q T () exp (h; a +i) = () n R n exp ( i h; xi) Q (x ia + ;T ) dx: (3.8) p Q T () exp (h; a i) = () n R n exp ( i h; xi) Q (x ia ;T ) dx: (3.9) Comparing (3.8) and (3.9) we get the proof. We will need the following result which is known as the oisson summation formula. 50

52 Theorem 4 ([7] p. 5) Suppose that for some A > 0 and > 0 we have max ff (x) ; Ff (x)g A ( + jxj) n : Then X f (x + m) = X n m n m n Ff for any > 0. The series converges absolutely. ut M T = M T M T := () n Q n ; a s= m k ;m k 6=0 exp ( R n i m exp hm; xi i h; vi) Q n (v;t ) dv fix T > 0; > 0 and select such N that Y n X cosh L(R n ) m k a s : (3.0) Theorem 5 Let Q n := fx jx = (x ; ; x n ) R n ; jx k j ; k ng be the unit cube in R n and a = a + := a. Then in our notation pq T (x) X i Q n m; T exp hm; xi n=p ; m n L p( Qn) where p : roof. Using Theorem 3 we get p Q T (x) n n ny cosh (a s x s )! M T : (3.) s= Applying (3.) we can check that the conditions of Theorem 4 are satisfied. Hence using condition (3.0) we get pq T (x) X p Q T (x + m) m n L ( Qn) 5 ;

53 = M T Observe that X m n f0g n Y p Q T X s= m k ;m k 6=0 (x + m) L( Qn) cosh m k a s ": Q ( x;t ) = () n F p Q T ( x) = () n () n exp (i hx; yi) p Q T ( R n = exp (i h x; yi) p Q T (y) dy R n Consequently, pq T (x) and = Fp Q T (x) : X m n p Q T = pq T (x) n = pq T (x) pq T (x) X n (x + m) L ( Qn) X m n Fp Q T X Q n m n m n Q y) dy i m exp hm; xi i m; T exp hm; xi L ( Qn) L ( Qn) i m; T exp hm; xi n : L ( Qn) Finally, applying Riesz-Thorin interpolation theorem (see Appendix I, Theorem 75) we obtain pq T (x) X i Q n m; T exp hm; xi n=p : m n L p( Qn) 5

54 Observe that the function Q Hence the series X n m n Q i m; T exp hm; xi m; T exponentially decays as jmj!. converges absolutely and represents an infinitely differentiable function. Consider several examples. Example 6 To demonstrate the power of our approach we start with a simple onedimensional example where the Fourier transform is explicitly known. Let x p (x) = () = exp be a Gaussian density then its Fourier transform is exp ( x ). For a fixed M and consider the approximant g (p; M; ; x) := X k Fp exp ikx : jkjm y Figure 0. g (p; 4; 0; x). x 53

55 y Figure. p (x) g (p; 4; 0; x). Table indicates that a small error is obtained with a small number of terms M in the expansion. Error at M = 4 M = 8 M = 6 x = : : : x = : : : x = 3 : : : 35 0 x = 4 : : : x = 5 : : : Table. Error of approximation at x = ; ; 5 of Gaussian density p (x) = () = exp ( x =) using oisson method using M = 4; 6; 6 terms. x Example 7 Let x p (x; y) = () + y exp : 54

56 0.5 z y x 4 Figure. p (x; y) = () exp ( (x + y )). The respective approximant has the form g (p; M; ; x; y) := X X k Fp ; jkjm jsjm s exp ikx + isy : 0.5 z y x Figure 3. g (p; 4; 0; x; y). 55

57 .0e 3 7.5e 4 5.0e z.5e 4 0.0e+0 0.5e 4 y 5.0e 4 0 x e 4 Figure 4. p (x; y) g (p; 4; 0; x; y). Example 8 Let the characteristic exponent (x) be the same as in Example 6 then the approximant g (p; M; ; x) for the density function p can be written as 0 g (p; M; ; x) = X k exp ikx A ; jkjm where is the respective characteristic function. 56

58 y Figure 5. g (p; 6; 4; x). x Example 9 Let the characteristic function F be the same as in Example then the approximant g (p; M; ; v ; v ) for the density function p (v ; v ) can be written as g (p; M; ; v ; v ) : 0 = X X F jkjm jsjm k ; s k ; exp i v ; s v A : Let us present several examples of density functions for different values of. 57

59 4 3 z x y.0.0 Figure 6. g (p; 8; ; v ; v ), = z x y Figure 7. g (p; 8; ; v ; v ), = 0:. 58

60 .5 z x y.0.0 Figure 8. g (p; 8; ; v ; v ), =. The next statements deal with approximation of functions using M-term exponential sums. Theorem 30 Let Q (v;t ) ; % := v R n ; Q (v;t ) % ; { n = n +! nx k k= s (0; ), s n, where Then d s := ( s ) cos s ny s= ( + s )! ny s= (c s;+ + c s; ) ; s n: (d s T ) s! ; Card Q (v;t ) ; % \ n { n ln % n k= s ; %! 0: roof. Observe that p Q T () = () n F Q (v;t ) () : 59

61 From (3.6) it follows that Q C (R n ) \ L (R n ). Hence by lancherel s theorem F p Q T (v) = () n Q (v;t ) ; or Q (v;t ) = exp (i hv; xi) p Q T (x) dx: R n Since p Q T (x) 0 and R p Q R n T (x) dx =, then max v R n Q (v;t ) = Q (0;T ) = : (3.) Let s (0; ) ; s n. From (3.6) and (3.) it follows that Q (v;t ) can be asymptotically majorised as jv s j! ; s n; v T n ; Q (v;t ) = exp T nx s= Q s (v s ) + nx m= Q m!!! nx a k;m v k k= ' ny exp s= ny exp s=. = ny s= T exp ny exp ( s= T Q s (v s ) i s x s ( s ) c s;+ exp T ( ( s )) cos T d s jx s j s ) = exp is + c s; exp is jx s j s s (c s;+ + c s; ) jx s j s! nx (d s T ) s s x s ; (3.3) s= where jx s j! ; x s = Re v s and d s > 0; s n: For a fixed 0 < % < consider the set Q (v;t) ; % := v R n ; Q (v;t) % : From (3.3) it follows that Q (v;t ) ; % ( 0 % := x R n ; exp! ) nx (d s T ) s s x s % s= 60

62 or 0 % := ( x R n ; nx ds T ln % s= s x s s ) : Now we have Card 0% \ n ' Vol n 0 % ; as %! 0 and where Vol n 0 % = ny ln % s= B ( ; ; n ) := ( d s T s Vol n (B ( ; ; n )) ; x = (x ; ; x n ) R n ; and > 0; ; n > 0: It is known [36] that ) nx jx s j s s= Hence Vol n B ( ; ; n ) = n ny s= ( + s ) ( + n s= s) : Card Q (v;t ) ; % \ n { n ln % n s= s ; %! 0: Theorem 3 Let p ; =p + =p 0 =, n M := { n (ln R) n s= s : Then in our notation X E (M) := Q n m n i m; T exp hm; xi 6

63 n n X m n \( ) 0 =R 0 M ( n s= s Q ) { n i m; T exp hm; xi Lp( Qn) n ( n s= s ) A M ( n s= s ) ( n s= s )=p 0 ; as M;!. roof. Observe that the system of functions i m (x) := n= exp hm; xi ; m n ; x Q n is uniformly bounded j m (x)j n= ; and orthonormal in L 8m n Q n. Let %!. Then in our notation Vol n 0 = ' { n (ln ) n s= s := V () : Applying Riesz theorem (see Appendix I, Theorem 76) we get X i E (M) = Q n m; T exp hm; xi m R n n( ) 0 \ =R n L p( Qn) X = Q n= m; T m (x) m R n n( ) 0 \ =R n L p( Qn) n= (n=) =p 0 n Observe that I = R p0 dv () R 6 0 =p p0 dv () := n=p I =p 0 : () n=p0

64 = { n nx s= s! Let > and > 0. Then R p0 (ln ) n s= s d: (3.4) since = R x (ln x) dx + x + (ln x) j R = R + (ln R) + R R + x + (ln x) x (ln x) dx x dx = R + (ln R) ; R! ; (3.5) lim R! R R x (ln x) dx R R x (ln x) dx = 0: Comparing (3.4) and (3.5) we get! I ' { nx n p 0 s R p0 (ln R) n s= s ; R! : Hence s= E (M) n=p () n=p0 { n p 0 nx s= s!! =p 0 R (ln R) ( n s= s )=p 0 ; R! : This means that using n M := { n (ln R) n s= s harmonics from 0 =R \ n we get the error of approximation E (M) n=p () n=p0 { n p 0 nx s= s!! =p 0 63

65 0 M ( n s= s ) { n n ( n s= s ) A M ( n s= s = n=p () 0 ) ( n { n n=p0 p 0 s= s )=p 0 n M ( n s= s nx s= { n s ) { n!! =p 0 n ( n s= s ) ( n s= s )=p 0 n ( n s= s ) A M ( n s= s ) ( n s= s )=p 0 = p 0 =p 0 { =p0 n 0 M ( n s= s () n=p0 nx s= ) { n s! =p 0 n n ( n s= s ) A M ( n s= s ) ( n s= s )=p 0 0 n M ( n s= s ) { n n ( n s= s ) A M ( n s= s ) ( n s= s )=p 0 ; as M;!. Comparing Theorem 5 and Theorem 3 we get the following statement. Corollary 3 Let p. Then in our notation p Q T (x) X i Q n m; T exp hm; xi m( ) 0 =R \n Lp( Qn) n=p p + n M 0 as M!. ( n s= s ) exp { n () n n M ( n s= s ) ; 64

66 3.4 Recovery of density functions by sk splines and radial basis functions Recall that p Q T () can be written as p Q T () = () n R n exp i h; xi T Q (x) dx = () n F Q (x;t ) () ; where Q (x;t ) is the characteristic function of X T. Let := fx k g be an additive group of lattice points in R n and K () be a fixed kernel function (the reader should t mix the strike price K with the kernel function K ()). Assume that the interpolant sk Q ; x := X c k K (x x k ) for Q (x;t ) exists and is unique. Then, formally, we get p Q T () () n X c k Q (x k ;T ) F (K (x x k )) () = () n F (K) () X c k Q (x k ;T ) exp (i h; x k i) : In what follows we give an explicit form of c k Q (x k ;T ) and F (K (x x k )) () which will give us an approximant for the density function p Q T. Remark that in many important cases the coefficients c k Q (x k ;T ) decay exponentially fast as jkj!. Let us consider in more details the problem of interpolation in R n. Let f be a continuous function on R n ; f C (R n ) and X c j K (x k x j ) = f (x k ) ; x k ; x j : x j Of course, it is very difficult (or in general impossible) to get an explicit solution of the interpolation problem. But, if we assume some regularity condition on the data points fx k g it is still possible to solve the interpolation system. We give here 65

67 an explicit solution of the interpolation problem in the case of a uniform mesh on R n. Let us describe first the one-dimensional situation on T. For a given m N let m = f0 = x 0 < < x m < x m = g be an arbitrary partition of [0; ) and K be a continuous function. Then sk(x) = X c k K(x x k ); c k R: x k m Denote by SK( m ) the space of sk-splines, i.e. SK( m ) = linfk(x x m ); x k m g: See [68] for more information. Let y R be a fixed parameter and y k = y + x k, k m be the points of interpolation. If the interpolation problem has a unique solution then the spline interpolant sk(x; y; m ; f) = sk(x; f) with knots x k, k m and points of interpolation y k can be written in the form sk(x) = X f(y k ) sk ~ k (x); k m where ~ sk k (y s ) = ; k = s; 0; k 6= s; are the fundamental sk-splines. It is important in various applications to have an explicit form of the Fourier series expansions for the fundamental sk-splines. As a motivating example consider sk-splines on the uniform grid m = fx k = =m; k mg, with the points of interpolation y ; ; y m, where y R is a fixed parameter, mx mx sk(x) = c 0 + c k K(x x k ); c k = 0; c k R; k m; k= In this case fundamental splines are just the shifts of sk(), ~ where sk(y ~ ; k 0(modn); k ) = 0; otherwise. Let, in particular, K(x) = D r (x) = X k= k= k r cos kx + r 66 ; r N

68 be the Bernoulli monospline. Then the space SK( m ) is the space of polynomial splines of order r, defect, with knots x k, k m and points of interpolation y k, k m. The first Fourier series expansions of fundamental splines were obtained by Golomb [47] in the case r = 4, and y = 0, i.e. in the case of cubic splines. It was shown that where sk(x; ~ 0) = sk(x) ~ = m + mx j (x) m j= j (0) ; j (x) = mx j cos D 4 x m = : m In the case of a general kernel function K C (T ) and an arbitrary y R the respective results were established in [68]. Namely, it was shown that where sk(x; ~ y) = sk(x) ~ = m + mx j (x) j (y) + j (x) j (y) m j= j (y) + j (y) ; j (x) = j (x) = mx j cos K m = mx j sin K m = x x ; m ; m Of course, to guarantee existence of fundamental splines for a given y we need to assume that maxf j(y); j(y); j mg > 0. A detailed study of such kind of conditions in terms of Fourier coefficients of the kernel function K can be found in [68, 66]. Different analogs of these results in multidimensional settings, on T d can be found in [48, 69, 94, 88]. We remark that the problem of convergence of skspline interpolants and quasi-interpolants was considered in [67, 76, 94, 78, 79] where it was shown that the rate of convergence of sk-splines has the same order as the respective n-widths (5.), (3.). We give representations of cardinal sk-splines on a uniform mesh in R n and apply these results to the problem of recovery of density functions which are important in the theory of pricing [86]. A similar problem has been considered in [3] in the case of Gaussian kernel K (). 67

69 Let a = (a ; ; a n ), a k > 0, k n be a fixed mesh parameter m = (m ; ; m n ) n and a := f(a m ; ; a n m n ) jm n g R n be a mesh in R n. Let A := diag (a ; ; a n ), then the mesh points are x m := Am T. For a fixed continuous kernel function K : R n! R, the space SK ( a ) of sk-splines on a is the space of functions representable in the form sk (x) = X m n c m K (x x m ) ; where c m R. Let f (x) be a continuous function, f : R n! R. Consider the problem of interpolation where sk (x m ) = f (x m ) ; sk (x m ) = X m n c mk (x x m ) ; c m R: Even in the one-dimensional case the problem of interpolation does not always have a solution. If the solution exists then the sk-spline interpolant can be written in the form sk (x) = X m n f (x m ) f sk (x x m ) ; where f sk (x x m ) are fundamental sk-splines, i.e. fsk (x m ) = ; m = 0; 0; m 6= 0: Theorem 33 Let K : R n! R be such that K L (R n ) \ C (R n ) ; X m n F (K) z+a m T 6= 0; 8z Q a ; where Q a := fxjx = (x ; ; x n ) R n ; 0 x k =a k ; k ng ; and the function (z) := m F (K) (z+a m T ) n 68

70 can be represented by its Fourier series, i.e. for any z R n, (z) = X s n s exp i As T ; z : Then fsk (x) = det (A) () n R n (z) F (K) (z) exp (i hz; xi) dz and this representation is unique. roof. First, we show that det (A) () n (z) F (K) (z) exp (i hz; xi) dz SK ( a ) : R n Since (z) = X s n s exp i As T ; z then fsk (x) = = = det (A) () n det (A) () n det (A) () n R n F (K) (z) X s n s R n (z) F (K) (z) exp (i hx; zi) dz X s n s exp R n F (K) (z) exp i x Since K L (R n ) then by lancherel s theorem fsk (x) = det (A) X s n s K x As T ; i As T ; z! exp (i hx; zi) dz As T ; z dz: so that f sk (x) SK ( a ). Second, we check the condition fsk Am T = ; m = 0; 0; m 6= 0: : 69

71 Indeed, fsk Am T det (A) = () n (z) F (K) (z) exp i z; Am T dz R n det (A) X = () n (z) F (K) (z) exp i z; Am T dz A l T +Q a l n det (A) = () n X F (K) z+a lt l m F (K) (z+a l T +A m T ) exp i z+a l T ; Am T dz n n = Q a det (A) () n = = = = = = = X l n det (A) () n det (A) () n det (A) () n det (A) () n det (A) () n Qa F (K) z+a l T m n F (K) (z+a m T ) exp i z+a l T ; Am T dz Q a l n F (K) z+a l T m n F (K) (z+a m T ) exp i z+a l T ; Am T dz exp i z+a l T ; Am T dz Q a exp i z; Am T + i hl; mi dz Q a exp i z; Am T dz Q a! nx i a k m k z k dz exp Q a =ak k= det (A) ny () n k= 0 ; mk = 0; k n; 0; otherwise. exp (ia k m k z k ) dz k Finally, we need to show that the representation of fundamental sk spline is unique. It is sufficient to show that the functions sk f (x x m ) ; x m a are linearly independent. Let a m R; m n be such that not all a m are zero. Let, in particular, a s 6= 0 for some s n : Consider a linear operator, L : C(R n )! R f () 7! f (x s ) : 70

72 Assume that m n a m f sk (x 0 = L X m n a m f sk (x xm ) xm ) 0 then! = a s L fsk (x xs ) = a s ; which is a contradiction. Theorem 34 In the assumptions of Theorem 33 p Q T () det (A) F (K) () () n m F (K) (+A m T ) n X m n Q (x m ;T ) exp (i h; x m i)! ; as max fa k ; k ng! 0: roof. Observe that Hence = since det (A) fsk (x) = () n = det (A) F R n p Q T () = () n F Q (x;t ) () F (K) (z) exp (i hz; xi) dz m F (K) (z+a m T ) n F (K) (z) (x) : m F (K) (z+a m T ) n () n F X m n Q (x m ;T ) f sk (x x m ) det (A) () n F X m n Q (x m ;T ) F = det (A) () n! ()! F (K) (z) (x x m F (K) (z+a m T m ) () ) n! F (K) () X Q (x m ;T ) exp (i h; x m i) ; m n m n F (K) (+A m T ) F (K) (z) m n F (K) (z+a m T ) L (R n ) : 7

73 Example 35 Let K (x) be a Gaussian of the form! nx K (x) = K (x ; ; x n ) = exp b k x k and B = diag (b ; ; b n ), then F (K) (z) = n= (det (B)) = exp Applying the oisson summation formula and lancherel s Theorem we get X m n F (K) z+a m T k= nx k= z k 4b k = X Y n F exp b k yk z k + m k a m n k k= ny X = F exp b k yk z k + m k a k k= m k ny ak X = exp (ia k m k z k ) F F exp = k= ny k= = det (A) m k ak () X exp (ia k m k z k ) exp m k ny Hence, in this case X k= m k exp (ia k m k z k ) exp det (A) F (K) (z) () n m F (K) (z+a m T ) n n= (det (B)) = n exp = Y n () n m k exp (ia km k z k ) exp k= k=! : ak m k b k ak m k b k ak m k b k : zk 4b k b k : a k m k 7

74 3.5 Shannon s Information Theory and recovery of density functions In this section we discuss a multidimensional generalisation of a well-known Nyquist- Whittaker-Kotel nikov-shannon theorem which explains why Wiener spaces W (R) are so important. Observe that the Nyquist-Whittaker-Kotel nikov-shannon theorem has its roots in the Information Theory first developed by Shannon [, 3, 4]. Definition 36 Let a be a fixed positive vector in R n, i.e. a = (a ; ; a n ) R n, a k > 0, k n and A = diag(a ; ; an ) be a diagonal matrix generated by a. Consider the set of points in R n, a := z m = Am T j m n ; m n : Observe that m z m = ; ; m n a a n for any fixed m n : Let Q a := fx j x = (x ; ; x n ) R n ; jx k j a k ; k ng : Definition 37 Denote by W a (R n ) the space of functions f L (R n ) such that supp Ff Q a : Let C (C n ) and C(R n ) be the spaces of continuous functions on C n and R n respectively. We construct a family of linear operators a, a : C (C n )! W a (R n ) + iw a (R n ) C(R n ) + ic(r n ) f (z) 7! ( a f) (z) such that k a jc (C n )! C(R n ) + ic(r n )k < and ( a f) (z) = f (z) for any f (z) W a (R n ): The sign "+" means the Minkowski sum () of two vector spaces C(R n ) and ic(r n ) endowed with the induced topology of C (C n ) C(R n ) + ic(r n ): 73

75 Theorem 38 Let f (z) W a (R n ) and a : R n! R be any continuous function such that a (y) = if y Q a and a (y) = 0 if y R n n Q a ; then where f (x) = X m n f Am T J m;a (x) = X m X m n f m ; ; m n J m ;;m a a n; (x a ;;an ; ; x n ) ; n J m;a (x) := n (deta) F a x iam T = n (deta) (F a ) x + Am T : roof. For any f W a (R n ) we have f(x) = (Ff) (y) exp (i hy; xi) dy () n R n = () n a (y) (Ff) (y) exp (i hy; xi) dy Q a because supp Ff Q a and a (y) = if y Q a and a (y) = 0 if y R n nq a. Since the set % m (y; a) := n= (det A) = exp i Am T ; y = n= n Y k= a = k! exp nx k=! i m k y k ; m n a k is an orthonormal basis in L (Q a ) then (Ff) (y) can be represented as (Ff) (y) = X m n m % m (y; a); where m are the Fourier coefficients of Ff. Recall that f W a (R n ) L (R n ). We understand the convergence in L (Q a ) in the sense that X lim (Ff) (y) m %! m (y; a) = 0: mq a 74 L (Q a)

76 Observe that instead of Q a we can take any neighborhood of 0 R n. lancherel s theorem and the fact that supp Ff Q a we find m = (Ff) (y)% m (y; a)dy = Q a (Ff) (y)% m ( Q a y; a)dy = (Ff) (y)% m ( R n y; a)dy = () n n= (det A) = F Ff Am T Using = () n n= (det A) = f Am T : Applying lancherel s theorem again we get f(x) = X () n n= (det A) = f () n R n m n Am T a (y) % m (y; a) exp (i hx; yi) dy = X m n n= (det A) = f Am T n= (det A) = Q a a (y) exp i hx; yi + i Am T ; y dy = n (det A) X mn f Am T Q a a (y) exp i hx; yi + i Am T ; y dy: Changing the index of summation and simplifying we get f(x) = X m n f Am T J m;a (x); where J m;a (x) = n (det A) = n (det A) = n (det A) = n (det A) Q a a (y) exp i hx; yi Q a a (y) exp i x () n F a x Am T : (F a ) x + Am T : 75 Am T ; y dy i Am T ; y dy

77 Consider a particular form of -deformation. Let ny a (x) = a ;;a n (x ; ; x n ) := ak (x k ) ; (3.6) k= where 8 >< a (x) := >: 0; x a; a x + ; a x a=; ; a= x < a=; a x + a= x < a; 0; x a; ; Direct calculation shows (F a ) (y) = a= = 4 cos ay ay a a=! a + + exp ( a= a= cos (ay) ; a > 0: ixy) a (x) dx Lemma 39 roof. k a jc (R)! C (R)k < : 834: k a jc (R)! C (R)k := sup = sup sup xr = sup xr ( sup xr X m f a m X jj m;a (x)j m X m a (Fa ) n o a f kfk C(R) ) J m;a (x) kfk C(R) x + m a 76

78 X = a sup 4 xr a (ax m) m X = sup xr (ax m) m Observe that the function ' (x) := X (ax m) m is =a-periodic. Consequently, Clearly, ax cos ax cos ax cos m m cos (ax m) cos (ax m) : m sup f' (x) jx Rg = sup f' (x) jx [0; =a)g sup X + sup X A x[0;=a) jmj (ax m) sup X x[0;=a) m 4 X m cos m = 4 x[0;=a) m=f ax m (ax m) 6 ;0;g ax cos : cos (ax m) : m cos (ax m) cos (ax m) Similarly, sup x[0;=a) m X (ax m) ax cos m cos (ax m) 4 X m m = 4 6 = 3 : and sup x[0;=a) m=f X ;0;g (ax m) ax cos m cos (ax m) 77

79 0:375 + : : Hence k a jc (R)! C (R)k : : 834: 3 Corollary 40 Let a is defined by (3.6). Then and ny k a jc (R n )! C (R n )k ak jc (R)! C (R) : 834 n k= k a jc (C n )! C(C n ))k : 834 n : Consider the set A ; UM of analytic functions f (z) = u (x; y) + iv (x; y), z = x + iy in the strip jim zj such that ju (x; y)j M for any z fzj jim zj g and f (x) R for any x R. It is known (see, [33], p. 50) that such functions can be represented in the form f (z) = (z x) cosh g (x) dx; R where jg (x)j M: Conversely, the function (cosh ( (x the strip fz jjim zj < g and! (z x) Re cosh > 0; (z x) cosh dx = : R Then the function f (x + iy) = (x + iy s) cosh g (s) ds R iy) =)) is analytic in is analytic in the strip jyj < ; x R and jre f (x + iy)j M if ess sup jg (x)j M. Similarly, in the multidimensional settings, the set A ; UM; = ( ; ; n ) 78

80 of functions f (z) ; z = (z ; ; z n ) C n which are analytically extendable with respect to z k, k n into the strip jz k j k admits representation f (z) = (K g) (z) := K (z R n x) g (x) dx; where ny K (z) := K (k) (z k ) (3.7) k= and Let K (k) (x k ) := zk cosh : k k E (f; W a (R n ) ; L p (R n )) := inf n o kf gk Lp(R n ) jg W a (R n ) be the best approximation of f(z) by the space W a (R n ) in L p (R n ) and E (K; W a (R n ) ; L p (R n )) := sup fe (f; W a (R n ) ; L p (R n )) jf Kg be the best approximation of a function class K L p (R n ) by W a (R n ) in L p (R n ) : In what follows we use entire functions of exponential type a = (a ; ; a n ) defined in Appendix II. Lemma 4 Let # a (x) := ny # ak (x k ) k= and # ak (x k ) be an integrable entire function of exponential type a k > 0: Let g (y) be a bounded function, i.e. jg (y)j M; 8y R n. Then the function % a (x) := # a (x y) g (y) dy R n is of exponential type a = (a ; ; a n ) : 79

81 roof. It follows from the Definition 73 (see Appendix II) that it is sufficient to show that! nx j% a (x)j = j% (x ; ; x n )j M exp a l jx l j ; l n for some absolute constant M and any x C n. Expanding # al (x l into the power series with respect to x l we get X (# al ( y l )) (s) # al (x l y l ) = x s l : s! s=0 Hence j% a (x)j = # a (x y) g (y) dy R n! ny ny = # ak (x k y k ) g (y ; ; y n ) dy k R n k= k=! ny X (# al ( y l )) (s) ny = x s l g (y ; ; y n ) dy k R s n l= s l =0 l! k=! ny X (# al (y l )) (s) Y n = x s l g ( y ; ; y n ) ( ) n dy k R s n l= s l =0 l! k= 0 ny X (# al (y l )) (s) ny jx l j s A dy k R s! n l= s=0 k= ny X (# al (y l )) (s) M jx s l j dy l : s! l= R s=0 k= Applying Bernstein s inequality [33] p. 3 (# a (y)) (s) dy a s j# a (y)j dy; s N; R R which is valid for any entire function # a (y) of exponential type a; we get ny! X a s l j% a (x)j M l s! jx lj s l= s=0 80 y l ) ; l n

82 = M where l :=! ny l exp (a l jx l j) ; l= R j# al (y)j dy: This means that % a (x) is of exponential type a. Lemma 4 Let K is defined by (3.7), i.e. ny! zk K (z) = cosh : k k Let k= k () := ( cosh ( k )) ; C k () := X ( ) s k ((s + ) a k + ) ; k n; s=0 ak # ak (x k ) := k ( k () C k (a k ) C k (a k + )) cos x k d; # a (x) = 0 ny # ak (x k ) ; k= S ak := k a k + k (a k ) + k (a k ) : k Let kgk L(R n ) M. Then E (K g; W a (R n ) ; L (R n )) MkK # a k L (R n ) ny Sak Mn exp ( min fa k k j k ng) : k k= Let kgk L (R n ) L. Then E (K g; W a (R n ) ; L (R n )) LkK # a k L (R n ) ny Sak Ln exp ( min fa k k j k ng) : k k= 8

83 roof. We prove Lemma just in the case L (R n ). The case L (R n ) follows in a similar way. It is easy to check that # ak (x k ) is a function of exponential type a k and # ak (x k ) L (R) : Hence # a (x) L (R n ) : Consider the set of functions K g, where jgj M and kgk L (R n ) L. In this case, using Young s inequality (see Appendix I), # a g k# a k L (R n ) kgk L (R n ) L k# ak L (R n ) : R n L (R n ) Hence, by Lemma 4, (# a g) W a (R n ) and n o E (K g; W a (R n ) ; L (R n )) sup kk g # a gk L(R n ) jgj M; kgk L (R n ) L M kk # a k L (R n ) : (3.8) Observe that for any complex numbers ;m and ;m ; m n we have (see, e.g. [94]) ny m= ;m n Y nx ;m = m= m= m Y ;m ;m n Y ;r r= r=m+ ;r : (3.9) It easy to check the following facts: sup z k R k cosh zk k 0 k () ; R; k ; (3.0) and jc k (a k + )j k (a k ) ; 0 a k jc k (a k )j k (a k ) ; 0 a k : Hence j# ak (x k )j S ak : (3.) Comparing (3.8) - (3.) we get E ((K g) ; W a (R n ) ; L (R n )) 8

84 Mn ny k= Sak k n (cosh max (k )) # ak () L(R n) j k n o : It is known (see, e.g. [33], p. 30) that for sufficiently large a k ; (cosh (k )) # ak () L (R n ) k X s=0 Observe that X ( ) k (k + ) cosh ((k + ) ) < cosh () k=0 = Hence, exp () + exp ( ) < exp ( ) : E (K g; W a (R n ) ; L (R n )) ny Sak Mn exp ( min fa k k j k ng) : k k= ( ) s (s + ) cosh ((s + ) a k k ) : Lemma 43 In our notation exp ( T ()) X m n exp ( + :834 n ) Mn roof. Since ny k= T Am T J m;a () Sak k L (R n ) exp ( min fa k k j k ng) : a () = X m n a Am T J m;a () ; 8 a W a (R n ) ; then applying Lemma 4 and Corollary 40 we get exp ( t ()) X t Am T J m;a () m n exp 83

85 = exp ( T ()) a () + a () X m n exp E (K g; W a (R n ) ; L (R n )) + X a () T Am T J m;a () m n exp T Am T J m;a () L (R n ) E (K g; W a (R n ) ; L (R n )) ( + k a jc (R)! C (R)k) ny ( + :834 n Sak ) Mn exp ( min fa k k j k ng) : k k= 3.6 The problem of optimal recovery of density functions In this section we discuss the problem of optimal (recovery) approximation of density functions using a wide range of approximation methods. In problems of optimal recovery arise quantities which are known as cowidths. Let (X; d) be a given metric (Banach) space, Y a certain set (coding set), A X, a family of mappings : A! Y, then the respective cowidth can be defined as where co (A; X) = inf sup y(a) (y) = fx jx X; (x) = (y)g : diam (y) \ A ; (3.) In particular, let Y be R m and : A! R m be a linear application, = L (A; R m ), then we get a linear cowidth m (A; X). It is easy to check that m = d m, where d m is the Gelfand s m-width defined by d m (A; X) = inf fl m X jsup fkxk jx A \ L m gg ; where inf is taken over all subspaces L m of codimension m. Letting Y be the set of all m-dimensional complexes in X and = C (A; Y ) be the set of all continuous mappings : A! Y, then we get Alexandrov s cowidths a m (A; X). 84

86 Let > 0, f% k (x) ; k Ng be a set of continuous orthonormal functions on the n-dimensional torus, T n := R n = n with a normalised measure dx = n dy, where dy is the restriction of Lebesgue measure on R n onto T n. Let L := sup kn k% k k and := f k ; k Ng be a fixed decreasing sequence of positive numbers. For any f L ( T n ) we can construct a formal Fourier series s [f] = X c k (f) % k ; c k (f) := f % k dx: T n k= Consider the set of functions := ff jjc k (f)j k ; k Ng : It is easy to check that is a convex and symmetric set. Also, is compact in C ( T n ) if X k < : k= Theorem 44 Let k= k < then a m (; L ( T n )) L m+ : roof. Let us remind some basic definitions. Let X be a Banach space with the norm kk and the unit ball B and A be a convex, compact, centrally symmetric subset of X. Let L m+ be an (m + )-dimensional subspace in X. Bernstein s m- width is defined as b m (A; X) = sup fl m+ X jsup f > 0 jb \ L m+ Agg : The Alexandrov s m-width is the value a m (A; X) = inf mx inf sup fkx (x)k jx A; g ; :A! m where the infimum is taken over all m-dimensional complexes m, lying in X and all continuous mappings : A! m. The Urysohn s width u m (A; X) is the infimum of those > 0 for which there exists a covering of A by open sets (in the sense of topology induced by the norm kk in X) of diameter < in X and multiplicity m + (i.e. such that each point is covered by m + sets and some point is covered by exactly m + sets). Observe 85

87 that the width u m (A; X) was introduced by Urysohn [34] and inspired by the Lebesgue-Brouwer definition of dimension. It is known [9] p. 90 that for any compact set A in a Banach space X b m (A; X) a m (A; X) [3] p., a m (A; X) u m (A; X) and [9] p. 90, Hence Let us fix u m (A; X) a m (A; X) : b m (A; X) a m (A; X) : (3.3) L m+ = lin f% k ; k m + g and consider the set ( Clearly QL m+ := QL m+ t m+ = m+ X k= c k % k ; jc k j m+ ) and, by the Riesz theorem (see Appendix II, Theorem 76), where m+ B ( T n ) \ L m+ L QL m+ ; B ( T n ) := ff jkfk g : It means that b m (; L ( T n )) L m+ : (3.4) 86 :

88 Finally, comparing (3.4) and (3.3) we get a m (; L ( T n )) L m+ : For simplicity we assume that A = 0. In this case! nx Q ();Q (v;t ) = exp T s (v s ) : Consider the function class ( := s= f (x) = X m n c m % m (x) ) ; where jc m j m ; i % m (x) := exp hm; xi ; m n and m = Q m;t : Observe that the system f% m (x) ; m n g is orthonormal and L =. Using Theorem 44, Theorem 30 and the standard embedding arguments we get the following statement. Theorem 45 In our notations min a M (; L q ( T n )) ; M (; L q ( T n )) exp { n M ( n q ; M! : s= s ) ; 87

89 Chapter 4 Option pricing 4. Introduction ricing of high-dimensional options is a deep problem of Financial Mathematics. The main aim of this chapter is to develop new simple and practical methods of pricing of basket options. As a motivating example consider a frictionless market with no arbitrage opportunities with a constant riskless interest rate r > 0. Let S j;t, j n; t 0, be n asset price processes. Consider a European call option on the price spread S n ;T j= S j;t. The common spread option with maturity T > 0 and strike K 0 is the contract that pays ' = S n ;T j= S j;t K at time T, where (a) + := max fa; 0g. There is a wide range of such options traded across different sectors of the financial markets. Assuming the existence of a riskneutral equivalent martingale measure Q we get the following pricing formula for the value V of the European call option at time 0, V = exp ( rt ) E Q ['] ; where ' is a reward function and the expectation is taken with respect to the equivalent martingale measure. There is an extensive literature on spread options and their applications. In particular, if K = 0 a spread option is the same as an option to exchange one asset for another. An explicit solution in this case has been obtained by Margrabe []. Margrabe s model assumes that S t; and S t; follow a geometric Brownian motion whose volatilities and do not need to be constant, but the volatility of S t; =S t; is a constant, = ( + ) ; where is the correlation 88 +

90 coefficient of the Brownian motions S ;t and S ;t. Margrabe s formula states that V = exp ( q T ) S 0; N (d ) exp ( q T ) S 0; N (d ) ; where N denotes the cumulative distribution for a standard Normal distribution, d = S0; ln + q T = q + T ; S 0; d = d T = and q ; q are the constant continuous divident yields. Unfortunately, in the case where K > 0 and S t;, S t; are geometric Brownian motions, no explicit pricing formula is known. In this case various approximation methods have been developed. There are three main approaches: Monte Carlo techniques which are most convenient for high-dimensional situation because the convergence is independent of the dimension, fast Fourier transform methods studied in [4] and DEs. Observe that DE based methods are suitable if the dimension of the DE is low (see, e.g. [6, 36, 8, 37] for more information). The usual DE s approach is based on numerical approximation resulting in a large system of ordinary differential equations which can then be solved numerically. Approximation formulas usually allow quick calculations. In particular, a popular among practitioners Kirk formula [60] gives a good approximation to the spread call (see also Carmona-Durrleman procedure [3, 96]). Various applications of the fast Fourier transform have been considered in [8, 98]. Different approaches of pricing basket options using geometric Brownian motion have been discussed in [4, 95, 59,, 5]. It is well-known that the Merton-Black-Scholes theory becomes much more efficient if additional stochastic factors are introduced. Consequently, it is important to consider a wider family of Lévy processes. Stable Lévy processes have been used first in this context by Mandelbrot [08] and Fama [38]. From the 90th Lévy processes became very popular (see, e.g. [09, 0, 7, 8, 9] and references therein). 4. The equivalent martingale measure condition for basket options In this section we specify the equivalent martingale measure condition for our model. Under the equivalent martingale measure all assets have the same expected rate of return which is a risk free rate. This means that under no-arbitrage 89

91 conditions the risk preferences of investors acting on the market do not enter into valuation decisions. Recall that in general, Q is not unique [8]. We assume that Q has been fixed and all expectations will be computed with respect to this measure. We specify now the equivalent martingale measure condition for the system (3.). Theorem 46 Let the stock prices be modeled by S s;t = S s;0 exp (U s;t ) ; s n; and the domain D R n +ir n of the characteristic exponent Q contains R n [ ([ n k= f ie kg) where fe k ; k ng is the standard basis in R n : Then Q ( ie s ) = r; s n: (4.) roof. Observe that for any s n the discount price process S s;t must be a martingale under a chosen equivalent martingale measure Q: Let Q s (x s ) be the characteristic exponent of U s;t. Then exp t Q s (x s ) = E Q [exp (ix s U s;t )] = E Q [exp (hix;u s;t e s i)] = exp t Q (x s e s ) : Thus by Theorem we get r = Q s ( i), which gives a system of n equations Q ( ie s ) = r; s n: Observe that in general riskless interest rate may depend on s. In this case we get the system Q ( ie s ) = r s ; s n: 4.3 ricing of high-dimensional basket options In applications it is important to construct a pricing theory which includes a wide range of reward functions '. For instance, the European call reward function which is given by! nx ' = ' (x) = ' (x ; ; x n ) = S 0; exp (x ) S 0;j exp (x j ) K 90 j= +

92 admits an exponential growth with respect to x as x!. Hence we need to introduce the following definition. Definition 47 We say that the model process S t = fs j;t ; j ng is adapted to the payoff ' if E Q ['] <. Clearly, if E Q ['] = then the option can not be priced. Recall that the operator of expectation is taken with respect to the density function p Q t which satisfies the equivalent martingale measure condition (4.). The next statement reduces the European call reward function to a canonical form. Lemma 48 In our notation V = K exp ( rt ) exp (y ) R n! nx exp (y j ) p Q T (y j= + b) dy; where b := (b ; ; b n ) ; b j = ln S0;j K ; j n: roof. Recall that V = exp ( ' = S ;T nx j= S j;t rt ) E Q [']. In our case! K ; + where S j;t = S 0;j exp (U j;t ) ; j n: This means that V = exp ( rt ) S 0; exp (x ) R n nx S 0;j exp (x j ) j=! K + p Q T (x) dx; = K exp ( rt ) exp x + ln R n S0; K nx j= 9 exp x j + ln S0;j K! + p Q T (x) dx;

93 where S 0;j ; j n are the respective spot prices. Making the change of variables S0;j y j = x j + ln ; K j n; we get where V = K exp ( rt ) exp (y ) R n b := (b ; ; b n ), b j = ln S0;j K! nx exp (y j ) j= ; j n: + p Q T (y b) dy; Let (z) be the gamma function, () := 0 x exp ( x) dx; Cn f N[ f0gg : We will need the following result [55, 54]. Theorem 49 Let n. For any real numbers = ( ; ; n ) with m > 0 for m n and < n m= m;! nx exp (x ) exp (x m ) = () n exp (i hu; xi) FS (u) du; m= R + n +i where x = (x ; ; x n ) and, for u = (u ; ; u n ) C n ; FS (u) = (i n m= u m ) ny m= (iu + ) ( iu m ) : Theorem 50 Let (x) := exp (x )! nx exp (x m ) : m= + 9

94 Then for any m n and = ( ; ; n ) with m > 0 for m n and < n m= m; we have i exp R m + ; x (x) dx n = i n s= m s roof. By the Theorem 49 (x) = () n R n +i n s= s n Y s= i m s + s i m + : exp (i hu; xi) FS (u) du = () n R n exp (i hz+i; xi) FS (z+i) dz = () n exp ( h; xi) exp (i hz; xi) FS (z+i) dz: R n Hence (x) exp (h; xi) = () n R n exp (i hz; xi) FS (z+i) dz: Since (x) exp (h; xi) L (R n ) [55], applying lancherel s theorem (see Appendix II) we get F ( (x) exp (h; xi)) (z) = exp ( i hu; xi) (x) exp (h; xi) dx = FS (u+i) R n = = (i ((u + i ) + i n m= (u m + i m )) ) (i (u + n m= u m) (i (u + i ) + ) n ny m= m ) (iu + ) 93 m= ny m= ( i (u m + i m )) ( iu m + m ) :

95 This means that i exp R n = i hm; xi (x) exp (h; xi) dx = FS n s= m s n s= s n Y s= i m s + s i m + : i m+i The next statement gives a general approximation formula for the European call options which is important in various applications. Observe that it does not show the rate of convergence. This problem will be discussed later. At this stage we just explain how to construct the approximation formula. Theorem 5 Let b := (b ; ; b n ), b j = ln S0;j K ; j n n m= m. Then the formal and = ( ; ; n ), i T n, m n, < approximation formula can be written as K exp ( rt hb; i) X V Re Q n m + i; T exp and m( ) 0 =R i n s= m n s s= s Y n i m s + s s= i m ; R! ;! ; + ( nx 0 s =R = x R n ds T s ) ; x s ; ln R d s = ( s ) cos s= roof. Applying Lemma 48 we get s (c s;+ + c s; ) ; 0 < s < : i hm; bi V = exp ( rt ) E Q ['] 94

96 where = exp ( rt ) S 0; exp (x ) R n = K exp ( rt ) exp (y ) R n b := (b ; ; b n ) ; b j = ln S0;j K nx S 0;j exp (x j ) j=! nx exp (y j ) j= ; j n: + K! p Q T (y + p Q T (x) dx; b) dy; Assume that i T n. Applying Cauchy s theorem n times in the tube Tn, 0 which is justified by (3.6), we get p Q T (y) = () n R n exp ( i hy; xi) Q (x; T ) dx = () n R n +i exp ( i hy; xi) Q (x; T ) dx = () n R n exp ( i hy; x + ii) Q (x + i; T ) dx = exp (hy; i) () n R n exp ( i hy; xi) Q (x + i; T ) dx: Let y Q n. Recall that Q n = fx jx = (x ; ; x n ) R n, jx k j g. Then from Corollary 3 we get p Q T (y) exp (hy; i) X! i n m + i; T exp hm; yi and p Q T n (y b) exp (hy b; i) X Q m( ) 0 =R m n Q m + i; T exp i i hm; bi exp hm; yi : Finally, assuming = ( ; ; n ), j n; < n j= j, using Theorem 50 and the fact that the domain 0 =R is centrally symmetric we obtain! nx V = K exp ( rt ) exp (y ) exp (y j ) p Q T (y b) dy R n j= 95 +

97 = = K exp ( rt ) n exp (y ) R n X m( ) 0 =R Q! nx exp (y j ) j= K exp ( rt hb; i) n exp (y ) R n K exp ( rt hb; i) n X m( ) 0 =R m + i; T exp + Q! nx exp (y j ) j= X m( ) 0 =R exp (hy + Q i hm; bi i b; i) exp hm; yi dy m + i; T exp exp (hy; i) A i exp m + i; T exp i hm; bi hm; yi dy i hm; bi i n s= m s n s= s n Y s= i m s + s i m + : Theorem 5 Let, in our notations, M (; R) := R () n exp (i h n b; xi) Q ( x + i; T ) dx n X m( ) 0 =R Q m + i; T exp i and V e be the approximant for V from Theorem 5, then := V V e i hm; bi exp hm; i ; L(R n ) K exp ( rt ) ( n ny s= s ) ( ) s= ( s ) 96

98 + n exp { n () n n M ( n s= s ) M ( n s= s ) +K exp ( rt ) M (; R)! nx exp (y ) exp (y j ) j= + exp (h; i) ; M! ;! : L(R n n( kbk )Q n) roof. Let V e be the approximant for V, then assuming that = V V e n 0! nx = K exp ( rt exp (y ) exp (y j ) R n j= () n exp (i hy b; xi) Q ( x + i; T ) dx R n X m( ) 0 =R Q := K exp ( rt ) exp (y ) R n m + i; T exp i! nx exp (y j ) j= + i T 0 we get exp (hy; i) A i C hm; bi exp hm; yi A dy + exp (hy; i) (y) dy: From the Corollary 3 it follows that for chosen > 0 and M > 0 we have j (y)j := R () n exp (i hy b; xi) Q ( x + i; T ) dx n n X m( ) 0 =R + n exp Q m + i; T exp i i hm; bi exp hm; yi { n () n n M ( n s= s ) M ( n s= s ) 97

99 for 8y Q n b. Observe that in our case p 0 =. Let us put m = 0 in the Theorem 50. Then it follows that! nx L := exp (y ) exp (y j ) exp (hy; i) = ( n s= s ) ( ) j= ny s= ( s ) + L (R n ) for a chosen. Observe that Q n b kbk Qn, where kbk := max fjb k j ; k ng and! nx exp (y ) exp (y j ) exp (hy; i) (y) dy ( kbk )Q n L + n exp j= Then we have 0 := exp (y ) R n n( kbk )Q n M (; R) exp (y ) { n () n n M ( n s= s ) M ( n +! nx exp (y j ) j=! nx exp (y j ) j= + + s= s ) : exp (hy; i) (y) dy exp (hy; i) : L(R n n( kbk )Q n) Example 53 Remind that in the one-dimensional case this pricing formula can be written as V = exp ( rt ) E Q ['] = exp ( rt ) R ' (y) p Q T (y) dy; where ' (y) is a reward function and p Q T (y) is a density function, p Q T (y) = exp ( ixy) Q (x; T ) dx: R 98

100 Assume that Q (x; T ) is analytic in the closed strip fx j Im x + g, where 0 +, then for a fixed, < 0 we get p Q T (y) = exp ( R+i ixy) Q (x; T ) dx = exp ( i ( + i) y) Q ( + i; T ) d R = exp (y) exp ( iy) Q ( + i; T ) d R Let, as before, S t = S 0 exp (X t ), where S 0 is a fixed spot price and X t is a model process. In the case of European call option ' (y) = max fs 0 exp (y) K, 0g and V = exp ( rt ) (S 0 exp (y) K) p Q T (y) dy: ln(k=s 0 ) Let us fix market parameters, r = 0:0, T = 0:5, S 0 = 00, K = S exp(0:5) and model (KoBoL) parameters, c + = c = 0:6506, + = :9458, = :087, = 0:5, = 0: In this case characteristic function (x) has the form where (x) = exp ( T (x)) = exp ( 0:5 (x)) ; (x) = ix + ( ) c + (( ) ( ix) ) + ( ) c + ( + + ix) = i0:39563x + ( 0:5) 0:6506 (:087) 0:5 (:087 ix) 0:5 + ( 0:5) 0:6506 :9458 0:5 (: ix) 0:5 : Let = 9 (i.e. we shift the contour of integration by 9i). Observe that 9 > = :087. It means that the contour of integration R 9i is in the domain of analyticity of (x). The optimal choice of depends on the model parameters. If > is getting closer to then we get a better damping factor exp (y) in the representation of the density function p Q T (y), but from the other hand, a higher oscillation of the respective approximant of the density function. 99

101 5 0 5 z y x Figure 9. Q (x; T ). The approximant for the pricing formula has the form = ln(k=s 0 ) ev = e V (M; ) = exp ( rt ) = exp ( rt ) S 0 exp ( rt ) S 0 exp ( 0 (S 0 exp (y) K) exp (y) X jkjm X jkjm rt ) K + exp ( rt ) K = exp ( rt ) S 0 Re Im X jkjm X jkjm X jkjm k k Re Im Re X jkjm k ik + i exp y A dy = k + i exp (( + ) y) cos ln(k=s 0 ) y dy = k + i exp (( + ) y) sin ln(k=s 0 ) y dy k k k = k + i exp (y) cos ln(k=s 0 ) y dy = k + i exp (y) sin ln(k=s 0 ) y dy 00 + i

102 exp ( + ) + K cos S 0 exp ( rt ) S 0 X jkjm ( ) k ( + ) k k ln K S 0 + k sin Im exp ( + ) + k K sin ln K S 0 exp ( S 0 rt ) K X jkjm k ln K S 0 + cos k k + i k ( ) k k Re exp ( ) k + 4 k k cos k ln K S 0 + sin k k K S 0 cos k ln K S 0 + k + exp ( rt ) K X jkjm + 4 k Im k exp ( ) k+ + 4 k K S 0 sin k ln K S 0 k k + 4 k + i K S 0 sin k + i K S 0 cos k ln K S 0 A ln K S 0 k k ln K S 0 A ln K S 0 Hence we need to specify > ln (K=S 0 ) and M 0. The choice of depends on the interval of approximation [ =; =]. A : A 0

103 Let = 6 then 0 exp ( rt ) (S 0 exp (y) K) exp (y) [=;) exp ( rt + ( + ) =) S 0 ( + ) X k k X jkjm k + i < 6:3 0 9 : The oisson s approximant for the pricing formula in this case has the form ev (M; 6) = exp ( 0:0 0:5) 3 (00 exp (y) 00 exp(0:5)) exp ( 9y) 0:5 MX Re 6 k= M k 6 9i exp iky 6! dy: + i exp iky A dy The benchmark price for the selected model is 7: (see [9] p., [93]). Inserting multipliers k = ( (k=m) n ), n = 0 we get oisson-ygmund s approximants. M (the number of terms) oisson rice oisson-ygmund rice 9: : : : : : : : : : : : : : Table. Simulation result for a European call option. European call option parameters: r = 0:0, T = 0:5, S 0 = 00, K = S 0 exp ( 0:5). Model (KoBoL) parameters: = 0:5, c + = c = 0:6506, + = :9458, = :087, = 0: Contour parameter = 9. Truncation parameter = 6. 0

104 Chapter 5 Kolmogorov s n-widths in the Theory of ricing 5. Introduction Our main aim is to approximate V. Remark that in [57, 9] and [56] to approximate the price the authors used a piecewise linear approximation on a uniform mesh with step h > 0. Observe that such kind of approximation has saturation of order O(h ) and can not reflect analytic properties of the characteristic exponent (). ossible use of polynomial splines of higher degree, r significantly increases complexity of computations and gives the order of saturation O(h r+ ) which is also too far from the optimal rate of convergence. The same argument is applicable to the possible wavelet approximations. Since any sequence of algebraic polynomials is not uniformly convergent on R (except of stationary sequences) then any continuous function which is not a polynomial can not be uniformly approximated. To compare and determine which apparatus of approximation is better over a wide range of methods of approximation we will need to set up the respective extremal problem. The key problem here is that the information regarding smoothness of characteristic exponents () (which are important in practical applications) is given implicitly. This fact creates a range of significant difficulties of a fundamental nature. Of course, in such settings an analytic structure of the best possible algorithm would be too complex to be useful in applications. In the next section we present optimal (in the sense of n-widths) rates of convergence of approximation formulas to the pricing integral. 03

105 5. Kolmogorov s n-widths and optimal approximation To compare different methods of approximation we need to introduce Kolmogorov s n-width of a symmetric with respect to the origin set A in a Banach space X, d n (A; X) := inf L nx sup xa inf kx yk X ; (5.) yl n where L n runs over all collection of n-dimensional spaces in X. To be able to consider n-width of function classes defined on locally compact Abelian groups we use the notion of average dimension introduced by Tikhomirov [3]. Let M be a homogeneous locally compact space on which a group G acts. Assume that M is equipped with the invariant measure d and metric d (; ) with respect to G. For any fixed x M define M = M (x) := fz M jd (x; z) g : Consider the collection of all subspaces L L p (M) such that the operator of restricting, 7 : L! M (x) \ L p (M) f! f is compact and the quantity K (; L; L p (M)) := min fn + j d n (L \ U p (M ); L p (M )) < g ; n o where U p (M) := g g L p (M)& kgk p : We put K (; L; L p (M)) = + if fn + j d n (L \ U p (M ); L p (M )) < g = ;: It is easy to check that the function! K (; L; L p (M)) is non-decreasing for any > 0 and! K (; L; L p (M)) is non-increasing for any > 0: Definition 54 Let # : R +! R + be a non-decreasing function. The quantity K (; L; L p (M)) dim (L; L p (M); ') := lim lim inf!0! # () is called the # average dimension of L in L p (M): In particular, if M = R and # () =, then the # average dimension is called the average dimension and is denoted by dim (L; L p (R)) : 04

106 The idea of this general definition is based on Theorem 55 (Whittaker-Kotel nikov-shannon) Let f W then f(x) = X k f k sin ( (x (x k=)) : k=) It means that any function f W can be recovered from its values on the set of points fk=g k : It is shown in [3], p. 367 that dim (W ; L (R)) = which is the inverse of the distance between the mesh points fk=g k : Similarly, to recover any trigonometric polynomial t n (x) T n := lin f; cos kx; sin kx; k ng we need to know its values at n + points fk= (n + ) ; k n + g : Remark that dim T n = n + : Observe that Shannon []-[4] was the first who introduced the notion of average dimension in order to compare massivity of sets. His idea is based on the notion of entropy of a random object. Later, Kolmogorov defined an average dimension in terms of -entropy of a set of non-random functions. This line of research has been developed by Tikhomirov [3, 30], Din ung [3], Le Chyong Tung [90], Magaril-Il yaev [3], [0]-[07]. Definition 56 Let A be centrally symmetric set, A L p (M): The average Kolmogorov #-width is d # (A; L p (M)) := inf sup inf kf gk gl L ; p(m) fa where inf is taken over all subspaces of dimension > 0: Let f be a measurable function on R. rearrangement of f, i.e. By f + we denote the non-increasing where f + (t) := inff > 0 j () t; 0 < t < g; () := measfx Rjf(x) g: Let [a k ; b k ), k N be a finite system of disjoint intervals. Consider the family S of simple sets S representable in the form S = [ N k= [a k; b k ), where N N: 05

107 Define the Jordan measure m of S as m (S) = N k= (b k a k ) : For a bounded set B R define its inner Jordan measure as m (B) = sup fm (S) js B g and its outer Jordan measure as m (B) = inf fm (S) js B g ; where the supremum and infimum are taken over all simple sets S: If m (B) = m (B) then we say that B is Jordan measurable. We say that f J (R) if for any a > 0 the set ft R j jf(t)j ag is Jordan measurable. It is known [3] that Theorem 57 Let K L (R), FK J (R), # > 0. Then d # (K U (R); L (R)) = (FK()) + : Definition 58 We say that a regular Lévy process of exponential type ( ; + ) ; < 0 < + is adapted to the payoff function ' (x) ; ' (x) = 0; x 0 if for some!; <! < 0; we have ' (x) exp (!x) CU (R + ) for some absolute constant C > 0: We denote by G ( ) such class of payoff functions. Remind that for any regular Lévy process of exponential type ( ; + ) we have () = i + () ; where R and () jj as! in = ( ; + ) : Theorem 59 Let X t be a regular Lévy process of exponential type ( ; + ) and order adapted to the class of payoff functions G ( ). Then d # (p t G ( ) ; L (R)) C exp ( t!) exp ( C t j#j ) ; #! : In particular, let X t be a KoBoL process of order 0 < < with the intensity parameter c = c + = c ; then 8 < exp (ct ( ) cos (=) # ) ; 0 < < ; 6= 0; ; d # (p t G ( ) ; L (R)) C 3 exp ( ct#) ; = ; : # ct ; = 0; where C ; C and C 3 are some positive absolute constants. roof. Consider a regular Lévy process of exponential type ( ; + ) which is adapted to the class of payoff functions G ( ). For a fixed R > 0 define the contour = [ R; R] [ [R; i! + R] [ [i! + R; i! R] [ [i! R; R] : 06

108 Since () is analytic in the strip = ( ; + ) then applying Cauchy s theorem we get I exp ( iy t ()) d = 0: Hence, since () is a characteristic exponent of a regular Lévy process of exponential type ( ; + ) then for some R we have () = i + () ; where () jj as! in = ( ; + ) : It means that for any t > 0; exp ( iy t ()) d! lim exp ( t R! jrj ) = 0 and lim R! + [R;i!+R] [i! R; R] p t (y) = exp ( iy t ()) d R = lim exp ( iy t ()) d R! [ R;R] = lim R! = lim R! = [i! [i! + [R;i!+R] [i! + exp ( iy t ()) d R;i!+R] [i! R; R] exp ( iy t ()) d R;i!+R] exp ( iy t ()) d: ;i!+] Changing variables in the last integral, z = i! we get p t (y) = exp ( iy (i! + z) t (i! + z)) dz R = exp (!y) exp ( iyz t (i! + z)) dz: R Since a regular Lévy process of exponential type ( ; + ) is adapted to the class G ( ) then for some absolute constant C > 0 we have exp (!y) G ( ) CU (R). Observe that K (y) := R exp ( iyz t (i! + z)) dz = F (exp ( t (i! + ))) ( y) 07

109 and by the aley-wiener theorem we get FK (#) = F F (exp ( t (i! + ))) ( #) = exp ( t (i! #)) : Since () = i + () ; where () jj as! in = ( ; + ) then for some absolute constant C > 0, jfk (#)j = jexp ( t (i! #))j = jexp ( t ( i (i! #) + (i! #)))j = exp ( t!) jexp ( t (i! #))j exp ( t!) exp ( Ct j#j ), #! : Consequently, (FK (#)) + exp ( t!) exp ( Ct j#j ), #! and by Theorem 57 d # (p t U (R), L (R)) exp ( t!) exp ( Ct j#j ), #! : In particular, d # (p t G ( ), L (R)) C exp ( t!) exp ( Ct j#j ), #! : Applying Theorem 57 and Lemma 5 we get d # (p t G ( ), L (R)) d # (p t U (R), L (R)) 8 < exp (ct ( ) cos (=) # ) ; 0 < < ; 6= 0; ; exp ( ct#) ; = ; : # ct ; = 0: Remark that ( ) cos(=) 0 for any (0; ). 08

110 Chapter 6 Spherical KoBoL process 6. Introduction In this chapter we study characteristic exponents and density functions of the spherical KoBoL family. Let d be the Haar measure on S n. A known class of high-dimensional models is based on the so-called KoBoL family which is given by (dx) = exp ( () ) d 0 (d ()) ; where 0 (d) is a finite measure on the unit sphere S n and : C (S n )! R + [8]. The respective characteristic exponent has the form () = i h; i + ( ) (( ()) ( () i ha; i) ) 0 (d ()) ; S n where (0; ) [ (; ) ; R n and A is a positive-definite matrix. Clearly () = i h; i + C C () ; where C := ( ()) 0 (d ()) S n and C () := ( () i ha; i) 0 (d ()) : S n 09

111 Let in particular 0 (d) = cd; where c > 0. Then the problem is to approximate the integral C () := c ( () i ha; i) d () : S n This problem is computationally difficult especially if n is big. Hence it is important to approximate "well" spherical integrals on sets of smooth functions on S n by their discrete analogs, NX ( () i ha; i) d () k ( ( k ) i ha; k i) ; S n k= where k S n and k R, k N must be selected. We start with general definitions. Let (; ; ) be a measure space, where is a compact domain in R n ; and K C () be a given set of real continuous functions, f :! R. Let fx ; ; x N g be a fixed set of points. It is natural to approximate the integral fd by a cubature formula NX k f (x k ) ; k= and to minimise the error of approximation N (K) := inf inf sup fx ;;x N g ( ;; N )R N fk fd NX k f (x k ) : The theory of cubature formulas has a long history and the first references are coming from ancient times. In the modern epoch simple cubature (quadrature) formulas have been constructed by Kepler and Torricelli (664), Simpson (743), Newton and Cotes (7). Different important methods of computing integrals have been developed by Lagrange, Chebyshev, Bernstein, Krylov, Nikol skij, Sobolev and many others. In the one-dimensional case, on S, the unit circle, it is known that the formula of rectangles f (x) dx NX k f S N N k= 0 k=

112 is optimal on Sobolev s classes W r (S ) = f f (r) U (S ), where U (S ) = f jkk g and r N. An analogous result for fractional values of r > 0 is unknown to the author. The problem of numerical integration over the surface of the unit sphere S n in R n, n ; is one of the most important in Numerical Analysis and Applications. The theory of functions on S was initiated in the eighteenth century in the works of Laplace and Legendre when the first cubature formulas appeared. Consequently, the problem of an optimal cubature formula on S (in general, S n, n ) has remain open since that time. Therefore, it is natural to call the problem on the best cubature formula on S or, in general, on compact Riemannian manifolds M n the Laplace-Legendre problem. A fundamental problem in this area is connected with an optimal distribution of data points x ; ; x N and computation of optimal coefficients ; ; N to approximate the integral "well". Even in the case of S, the two-dimensional sphere in R 3, it is not possible to construct, in general, an equidistributed set of data points since there are finitely many polyhedral groups. Different attempts to find sets of points on the sphere which imitate the role of the roots of unity on the unit circle usually led to deep problems of the Geometry of Numbers, Theory of otentials, etc., and usually these approaches give just a measure of a uniform distribution like cup discrepancy or minimum energy configurations. We consider cubature formulas for the Sobolev s classes W r (M n ) C (M n ) on a compact two-point homogeneous manifold M n which we define later. The respective extremal problem can be formulated as following. Let f W r (M n ) and fx ; ; x N g M n. Consider an information operator T N L C (M n ) ; R N ; Let T N : C (M n )! R N f () 7! (f (x ) ; ; f (x N )) : % (f (x ) ; ; f (x N )) = % T N f be a given function, % : R N problem N (W r (M n ) ; R) :=! R (a recovery operator). Consider the extremal inf inf fx ;;x N gm n sup %R fk fd M n % T N f ; (6.) where R is a given class of functions % : R N! R and d is the invariant normalised measure on M n. We shall write N (W r (M n )) instead of N (W r (M n ) ; R)

113 if R =R RN (the set of all functions % : R n! R). In particular if R is the set of all linear functionals % over R N then for some ; ; N, % T N f = NX k f (x k ) ; k= and (6.) reduces to the Laplace-Legendre problem 0 N (W r (M n )) := inf inf sup fd fx ;;x N gm n f ;; N gr M n fw r (M d ) NX k f (x k ) : (6.) k= The problem of reconstruction of optimal cubature formulas splits into two parts, finding of a lower bound in (6.) or (6.) and the obtaining of respective upper bounds. The lower bounds are of independent interest because they allow us to compare and classify a wide range of cubature formulas. We develop a new method for obtaining lower bounds for N (W r (M n )), r > 0 on general compact two-point homogeneous manifolds M n which are sharp in the power scale in the case of S [80]. As a first step we apply the result of Smolyak [6] to reduce the problem (6.) to the linear case (6.). Then, to find respective lower bounds for the rate of convergence of a cubature formula on Sobolev s classes W r (M n ) we consider the set W r (M n ) \ ker T N \ T M, where ker T N := ff jf C (M n ) ; T N f = 0g and T M is the set of polynomials of order M on M n and dim T M := m N. Then, applying Bernstein s inequality we reduce the problem to the consideration of the set m r=n U (M n )\T M, where U (M n ) is the unit ball in L (M n ). Finally, we need to find a polynomial ; deg CM, where C > 0 is an absolute constant, such that T N = 0, kk = and the value R d is sufficiently big. Even in the case of M n the circle, S, it is very difficult to construct a polynomial with such properties. We show the existence of such objects using methods of geometry of Banach spaces. Remark that in applications we have a little information concerning special convex bodies in R m which are connected with the structure of a fixed system of spherical harmonics on M n. This is a source of fundamental difficulties which occur if we try to apply the results of Geometry of Banach spaces to various open problems in different spaces of functions. A useful tool in this range of problems are Lévy means. We employ estimates of Lévy means in combination with the Bieberbach inequality and the Brunn-Minkowski theorem. Note that estimates of Lévy means connected with different orthonormal systems have been obtained in [70]-[77], [87]. The results we derive are apparently

114 new even in the one-dimensional case and the method s possibilities are not confined to the statements proved but can be applied in studying more general problems. 6. Estimates of expectations Let X be a Banach space with the norm kk, dim X = m; m N. In this section we estimate expectations E [kk] with respect to the Haar measure d concentrated on S m, the unit sphere on R m. Definition 60 Given a measure space (; ; ). Let = f k g k= be a set of orthonormal functions in L (; ; ). Suppose that there exists an increasing sequence fk j g j= N; k =, such that for any j N; x and some C > 0; k j+ X k=k j j k (x)j Cd j -a.e. on, where d j = k j+ k j. Then we say that ; ; ; ; fk j g k= K. Let L p = L p (; ; ) be the usual set of p-integrable functions. Suppose that ; ; ; ; fk j g k= K. Since all the functions k are -a.e. bounded, then for an arbitrary function L p ; p we can construct the sequence fc k ()g k= where c k () = k d and consider the formal Fourier series X l= k l+ X k l c k () k : Recall that a Riemannian manifold M n is called homogeneous if its group of isometries G acts transitively on it. Recall that a group action is transitive if for every pair of elements x; y M n there is a group element g G such that gx = y. Let be the Laplace-Beltrami operator defined on a compact homogeneous manifold M n. It is well-known that is an elliptic operator which is second order, self-adjoint and invariant under isometries. 3

115 The eigenvalues k, k 0 of are discrete, nonnegative and form an increasing sequence 0 0 with + the only accumulation point and k k, k!. Corresponding eigenspaces H k ; k 0 are finite dimensional, d k := dim H k, orthogonal and L (M n ) = k=0 H k. It is possible to show that d k k n, m = dim T M M n, where T M := M k=0 H k [45, 5]. Let us fix a real orthonormal d k of H l= k. For an arbitrary L p (M n ), p with the formal Fourier series basis Y k l X kn d k X l= c k;l () Yl k ; c k;l = Yl k d; M n the rth fractional integral r, r > 0 is defined as r c 0 + X kn r= k d k X l= c k;l () Y k l ; c 0 R: (6.3) Recall that d is the normalised invariant measure on M n.the last equation defines the operator of fractional integration r = I r. The function (r) L p (M n ), p, is called the rth fractional derivative of if (r) X kn r= k d k X l= c k;l () Y k l : Sobolev s classes Wp r (M n ) are defined as sets of functions with formal Fourier expansions (6.3) where kk p and R d = 0 (see [6] for details). M d Let H j, j 0 be any eigenspace of the Laplace-Beltrami operator, d j = dim H j and Y j ; ; Y j d j be any orthonormal basis of H j. Then d j s= jy j s (x)j = d j for any x M n (see, e.g. [45]). Hence, any compact, connected, orientable, n-dimensional C ; homogeneous Riemannian manifold M n ; with C metric has the property K (in particular, the unit sphere S n R n ). Let = ( ; ; m ) R m, = ( ; ; m ), h; i = m k= k k. Let jj = n o kk () = h; i = be the Euclidean norm on R m, S m = R m ; kk () = be the unit sphere in R m. Let us fix a norm on R m and denote by E the Banach space E = (R m ; kk) with the unit ball B E : Lévy mean M BE is the expectation of kk on S m, i.e. M = M BE = M (R m ; kk) = kk d () ; S m 4

116 where d () denotes the normalised rotation invariant measure on S m Lévy mean M e BE is defined by. Modified em = M e BE = M e (R m ; kk) = S m = kk d () : We shall obtain upper bounds for modified Lévy means M e since they have a wide range of applications and M M. e Let us specify now the norm kk on R m whose Lévy mean we want to estimate. Let be a compact space with a probability measure, m := f k g m k= be a set of orthonormal functions in L (; ; ) and X be a Banach space of functions on. Assume that ; ; ; ; fk j g k= K. Consider the coordinate isomorphism J : R m! lin ( m ) that assigns to = ( ; ; m ) R m the function mx J = = k k lin ( m ) : The definition k= kk (p) = k k Lp() induces a norm on R m. Theorem 6 Let ; ; ; ; fk j g k= K and dj = k j+ for any m N; M = M R m ; kk (p) M e R m ; kk (p) C where C > 0 is an absolute constant. roof. By Jensen s inequality (see Appendix I) M R m ; kk (p) M e R m ; kk (p) : k j, M j= d j = m. Then p = ; p < ; (log m) = ; p = ; Hence it is sufficient to obtain upper bounds for M. e Let dg () = exp kk () d be the Gaussian measure on R m. For an arbitrary continuous function f C (S m ) we define an extension f e to R m n f0g by! ef () := kk () f : kk () 5

117 From the uniqueness of a normalised invariant measure d on S n it follows that m f () d () = f e () dg () : (6.4) S m R m To find the constant m we take f (), so that m = dg () = m t exp t dt = m : (6.5) R m kk () Let fr k ()g kn denote the sequence of Rademacher functions, r k () := sgn sin k ; [0; ] : For l N and s m put l s () := l = r (s )l () + + r sl () : Let h : R m! R be a continuous function satisfying the condition! mx h () = h ( ; ; m ) exp j s j! 0 uniformly when mx j s j! : s= R s= Then (see [89], Lemma., p. 585)! l () h () dg () = lim h R m l! 0 () ; ; l m () d: (6.6) = () = Applying (6.4)-(6.6) for the functions and f () = kk (p), Sm h () = e f () = kk (p), R m 6

118 we obtain em := = m S m = m lim l! kk (p) d () kk (p) dg () = R m m 0 l () () = ; ; l m () () = We will need the Kahane inequality [97], p. 74, 0 bx r j () x j () j= r X da =r k( ; ; m )k (p) dg ( ; ; m ) R m! d (6.7) (p) C r 0 bx r j () x j () j= X d (6.8) which is valid for any Banach space X, < r < and every finite subset fx j ()g b j= X, b N. Comparing (6.7) and (6.8) we get em C lim m= l! = C lim m= l! 0 0 l () ; ; l m () d (p) mx l s () s () s= Lp() d: (6.9) Setting e (s get )l+k () = l = s () in (6.9) for s m; k l and m N we mx Xml l s () s () = r j () e j () ; s= j= so that we can rewrite (6.9) in the form em C Xml lim r m= j () l! e j () = C lim m= l! 0 0 j= L p() Xml r j () e j (x) j= p 7 d d (x)! =p d: (6.0)

119 Therefore, from the Khintchine inequality, Jensen s inequality (see Appendix I), and (6.0) we obtain for p <, em C lim m= l! C c (p) m = 0 l! 0 Xml j= Xml r j () e j (x) j= p d! e p= j (x) d (x) A! =p d (x)! =p Cp = m = = Cp = ; M X j= d j! = since ; ; ; ; fk j g k= K. Consider the case p =. Let (6.) K m (x; y) := mx s (x) s (y) : s= It is easy to check that K m (x; y) = K m (x; z) K m (z; y) d (z) = = and! mx s (x) s (z) s= l= mx s (x) s (y) s=! mx l (z) l (y) d (z) K m (y; x) = K m (y; x) : Therefore, by Hölder s inequality (see Appendix I), for any x; y we have jk m (x; y)j = K m (x; z) K m (z; y) d (z) 8

120 jk m (x; z) K m (z; y)j d (z) kk m (x; )k kk m (; y)k = kk m (x; )k K m (y; ) m = kk m (x; )k = X s (x) s () = mx j s (x)j s= Cm, s= since ; ; ; ; fk j g k= K. This implies kk m (; )k Cm. (6.) For any polynomial t m (y) = mx s s (y) ; t m (y) m := lin f s ; s mg s= we have = = K m (x; y) t m (y) d (y)! mx l (x) l (y) l= mx s s (x) = t m (x) : s=! mx s s (y) d (y) s= Hence, applying (6.) and Hölder s inequality we get kt m k kk m (; )k kt m k Cm kt m k : This means that ki jl () \ T N! L () \ T m k Cm; 9

121 where I is an embedding operator. It is easy to see that ki jl () \ T m! L () \ T m k = : Application of Riesz-Thorin interpolation theorem (see Appendix I, Theorem 75) gives kt m k (Cm) =p kt m k p, p : (6.3) Finally, using (6.) and (6.3) with p = ln N we get kk () d () (Cm) =p kk (p) d () S m S m C =p m =p C p = = C C = ln m m = ln m (ln m) = (log m) = ; m! : 6.3 Lower bounds We prove lower bounds just in the case of two-point homogeneous spaces M n to avoid unnecessary technical definitions. A complete classification of the two-point homogeneous spaces was given in [4]. For information on this classification see, e.g. [5, 5, 6]. They are: the spheres S n ; the real projective spaces n (R), n =, 3,; the complex projective spaces n (C), n = 4, 6,; the quaternionic projective spaces n (H), n = 8,, ; the Cayley elliptic plane 6 (Cay). Theorem 6 Let M n be a two-point homogeneous space. Then for any r > 0 and " > 0 we have N (W r (M n )) = 0 N (W r (M n )) C " N r=n (ln N) (+") ; where C " depends just on " > 0. roof. It was discovered by S. Smolyak and then published in [6] that for the recovery of linear functionals using linear information it is sufficient to use linear 0

122 methods. More precisely, let K be a convex centrally symmetric subset of Banach space X, T N : X! R N, D E D E T N x = x 0 ; x ; ; x 0 N; x ; x 0 k X 0 ; k N; and a linear functional, (x) = x 0 0; x ; x 0 0 X 0. Then there is an optimal linear method of recovery, that is N inf sup j (x) % T N xj = inf sup %R (a ;;a N )R N (x) X a k Dx 0 k; xe ; xk where R = R RN is the set of all functions % : R N! R. Recall that the Sobolev classes W r (M n ) are convex and centrally symmetric. Hence, from (6.) and (6.) we get N (W r (M n )) = 0 N (W r (M n )) inf sup T N fd ; fw r (Mn )\ker T N M n where T N f = (f (x ) ; ; f (x N )) : Clearly, codim ker T N N. In what follows we show that for any fx ; ; x N g M n there is a polynomial xk k= (x ;;x N ) W r (M n ) \ T M \ ker T N ; dim T M = m N; such that for any " > 0 M n (x ;;x N )d C " N r=d (log N) (+") ; (6.4) where C " > 0 depends just on ". Let us fix a norm kk on R m. Let V be a convex centrally symmetric body in R m which is a unit ball in E (R m, kk) and Vol m be the standard m-dimensional volume of subsets in R m. A direct calculation (see, e.g. [8]) shows that Vol m (V ) A = kk Vol m B m d () ; () m S m

123 where n o B(p) m := x x R m ; kxk (Lp) ; p : Applying convexity arguments we get S m =m kk m d () S m Hence, by Theorem 6 it follows that kk d () = M : Vol m B m () C m (log m) m= Vol m B m () : (6.5) Let L s R m be any s-dimensional subspace, (L s )? be the orthogonal complement of L s in R m and (Ls) B m? () be the orthogonal projection of B() m onto (L s)?. It is easy to check that B() m Bm () and therefore (Ls)? Bm () (Ls)? Bm () and Vol m s (Ls)? Bm () Vol m s B m s () : Hence, Vol m B() m = dx = B() m (Ls)? Vol s B() m B m () \ (y + L s ) dy: Thus, involving standard arguments connected with the Brunn-Minkowski theorem we get Vol s B m () \ (y + L s ) Vol s B m () \ L s for any y (Ls)? B N s (). Consequently, Vol m B() m Vols B() m \ L s Volm s (Ls)? Bm () Vol s B m () \ L s Volm s B m s () : (6.6)

124 Hence, by (6.5) and (6.6), Vol s B m () \ L s C m (log m) m= Vol m Vol m B() m s B m s () Comparing (6.7) with the Bieberbach inequality [], p. 93, 0 =s diam (V Vol s (V ) A Vol s B() s : (6.7) which is valid, in particular, for any convex centrally symmetric body V R s we get the lower bound for the diameter of the set B m () \ L s, diam B() m \ L s C m=s (log m) Vol m Recall that Vol m B() m = N= m + and Hence, 0 0 s (z) = z z = exp ( z) () = + O z ; z! Vol m s Vol m B() m A B m s () Vol s B() s m! ; s! ; s m: =s Vol m B() m A B m s Vol s B() s () m = m =(s) ; s This means that for any 0 < < and s = [m] we have diam B m () \ L s C (log m) =() ; or for any fixed fx ; ; x N g M n ; N = m s, there exists a polynomial t m T M ; dim T M = m, such that t m (x k ) = 0; k N; kt m k = and kt m k C (log m) =(). Consider the polynomial = (x ;;x N ) = (dim T M ) r=d t m: 3 =s :

125 Clearly, T M, 0, kt m k = (dim T M ) r=d, (x k ) = 0; k N and M n d = (dim T M ) r=n kt m k C (dim T M ) r=n (log m) = : Applying Bernstein s inequality [33] (which is valid for any two-point homogeneous space) we get or C r;n (dim T M ) r=n U (M n ) \ T M W r (M n ) CW r (M n ) \ T M ; from which (6.4) follows. 6.4 Upper bounds We consider here just the case of S = fx jx R 3 ; jxj = g. Let d be the normalised rotation invariant measure on S. The space L (S ) has the orthogonal decomposition L S = k=0h k ; where H k is the space of spherical harmonic polynomials of degree k, dim H k = k +. Let Yl k (x) k be an orthonormal basis for H l= k k. For each k = 0; ; ;, H k is an eigenspace of the Laplace-Beltrami operator for the sphere,, corresponding to the eigenvalue k = k (k + ). A function is zonal with respect to a pole S is it is invariant under the action of all rotations of S which fix, i.e. (x) = (x) for all x S, and SO (3) with =. Then (x) = e (hx; i) for some e defined on [ ; ]. The real zonal polynomial k (x) H k have a simple expression in terms on the Legendre polynomials k, which can be defined in terms of the generating function t + X = = k (t) ; k=0 4

126 where 0 jj < and jtj <. It is known that k (x) = (k + ) k (hx; i) : We parametrise the points x S by their spherical coordinates x = (; ) [0; ] [0; ). Let us consider the equiangular grid points x r;k = ( r ; k ), where r = r= (b), 0 r b, k = k=b, 0 k b. Figure 0 gives the distribution of equiangular grid points in the case b =. From the estimate (6.9) it follows that this is optimal in the power scale distribution of data points z x y.0.0 Figure 0. Optimal distribution of data points. Let and (b= ) l := ; 0 k [(b= ) =] ; k= (b= ) ; [(b= ) =] + l b= ; a b r := b= X (s + ) r b s + sin b s=0 K (r; k ) b= (x) := a(b) r b= X l=0 (b= ) l (r;k ) (x) : 5

127 Theorem 63 In our notation b sup f (x) X f ( r ; k ) K (r; k ) fw r (S ) r;k=0 b= (x) b r, b! : Clearly, b N =, where N is the number of data points ( r ; k ). a(l,b) Figure. Optimal Coefficients a b l, b =. Figure presents graph of the function b= (x) r; k) r = + K( K (r; k ) b= in the case b = 6, x = ( r ; k ), r = 5=4, k = 05=6 (see [76]). l 6

128 then [75, 74], b sup f (x) X where fw r (S ) N = mx k=0 Figure. Optimal radial basis function on S. r=0 dim H k b : b X k=0 f (x r;k ) K (r; k ) b= (x) From (6.8) and Theorem 6 it follows that N r=, N! ; (6.8) C " n r= (ln n) (+") 0 n W r S n r=, n! : (6.9) Define an information operator T N, T N L C (S ) ; R N ; T N : C S! R N f () 7! (f (x ) ; ; f (x N )) ; where fx ; ; x N g S. Consider the extremal problem A N W r S ; L := inf T N inf sup LL(R N ;L (S )) fw r (S ) 7 kf L T N fk L(S ):