Wiener chaos and the Cox Ingersoll Ross model

Transcription

1 Wiener chaos and the Cox Ingersoll Ross model M. R. Grasselli and T. R. Hurd Dept. of Mathematics and Statistics McMaster University Hamilton ON L8S 4K1 Canada March, 4 Abstract In this paper we recast the Cox Ingersoll Ross model of interest rates into the chaotic representation recently introduced by Hughston and Rafailidis. Beginning with the squared Gaussian representation of the CIR model, we find a simple expression for the fundamental random variable X. By use of techniques from the theory of infinite dimensional Gaussian integration, we derive an explicit formula for the nth term of the Wiener chaos expansion of the CIR model, for n =, 1,,.... We then derive a new expression for the price of a zero coupon bond which reveals a connection between Gaussian measures and Ricatti differential equations. Key words: Interest rate models, Wiener chaos, functional integrals, squared Gaussian models. Research supported by the Natural Sciences and Engineering Research Council of Canada and Mathematics of Information Technology and Complex Systems, Canada 1

2 1 Introduction In the present paper we shall study the best known example of a term structure with positive interest rates, namely the CIR model [3, in the context of the chaotic approach to interest rate dynamics introduced recently by Hughston and Rafailidis [6 (see also [1). By an interest rate model we mean the specification of a spot rate process r t and of a market price of risk process λ t both under the natural or physical measure in the economy P. In the chaotic approach, the random nature of the model is assumed to be given by a probability space (Ω, F, P ) equipped with a Brownian filtration (F t ) t. The essence of the approach by Hughston and Rafailidis is the specification of the most general term structure with positive interest rates in terms of a single unconstrained random variable they denote X. They then apply a Wiener chaos expansion to X, and interpret the resulting terms as building blocks for models of increasing complexity. In one version of the CIR model, r t is governed by the equation dr t = a(b r t )dt + c r t d W t, (1) for some positive constants a, b, c with 4ab > c, where W t is a standard one dimensional P Brownian motion, and the market price of risk is taken to be proportional to r. By embedding this model inside a general class of squared Gaussian models, we will be led to a natural choice for the basic square integrable random variable X associated to it. As we shall then demonstrate, the resulting stochastic process X t = E t [X admits an explicit chaos expansion, one which in general includes terms of every chaotic order. A central idea in the present paper is the link between the Wiener chaos expansion and the theory of Gaussian functional integration, an essential tool invented to study the mathematical structure of quantum field theory. In fact, the mathematics underlying our example is a consequence of certain basic results in that theory, and can be found in for example [5. The organization of the paper is as follows. In section we review the essential ingredients for the construction of positive interest rates models in both Flesaker Hughston and the state price density approaches, and then compare these approaches to the recently introduced chaotic representation of Hughston and Rafailidis [6. We end the section by describing the structure of the Wiener Itô chaos expansion and show how it can be expressed in terms of a certain generating functional acting on the space L (R N ).

3 In section 3, we describe the squared Gaussian formulation of the CIR model and show the spot rate process can be explicitly computed. Based on this representation, we state the form of the random variable X, and give a proof that it lies in L. In section 4, we state the exponential quadratic formula which is the main technical tool in this paper. It is a formula for the generating functional of random variables of the form X = e Y for Y lying in a general class of elements in the second chaos space H. In section 5, we compute the generating function for the random variables X t in the CIR model and, as the main result of the paper, derive their chaos expansion. In section 6, we show that the usual CIR bond pricing formula has a natural derivation within the chaotic framework. Three appendices focus on the theory of Gaussian functional integration and its relation to the Wiener chaos expansion. Appendix A explores the white noise calculus. Appendix B states and provides a proof of the generating functional theorem. Appendix C provides a proof of the exponential quadratic formula. Positive Interest Rates.1 State price density and the potential approach Rather than focus on the spot rate process, one can model the system of bond prices directly. Let P tt, t T denote the price at time t for a zero coupon bond which pays one unit of currency at its maturity T. Clearly P tt = 1 for all t < and furthermore, positivity of the interest rate is equivalent to having P ts P tu, () for all t u s. A general way to model bond prices [14, 13 is to write P tt = E t[v T V t, (3) for a positive adapted continuous process V t, called the state price density. Positivity of the interest rates is then equivalent to V t being a supermartingale. In order to match the initial term structure, this supermartingale needs to be chosen so that E[V T = P T. If we further impose that P T as T, then V t satisfies all the properties of what is known in probability 3

4 theory as a potential (namely, a positive supermartingale with expected value going to zero at infinity). It follows from the Doob Meyer decomposition that any continuous potential satisfying ( ) E sup Vt < (4) t can be written as V t = E t [A A t, (5) for a unique (up to indistinguishability) adapted continuous increasing process A t with E(A ) <. Therefore, the model is completely specified by the process A t, which can be freely chosen apart from the constraint that E [ AT T = P T T, for a.a. T. (6). Related quantities and absence of arbitrage An earlier framework for positive interest rates was introduced by Flesaker and Hughston [4, eq. (8), who observed that any arbitrage free system of zero coupon bond prices has the form P tt = T h sm ts ds, for t T <. (7) h t s M ts ds Here h T = P T is a positive deterministic function obtained from the initial T term structure and M ts is a family of strictly positive continuous martingales satisfying M s = 1. Any such system of prices can be put into a potential form by setting V t = t h s M ts ds. (8) The converse result is less direct and was first established by Jin and Glasserman [9, lemma 1. These equivalent ways of modelling positive interest rates can now be related to other standard financial objects. A particularly straightforward path is to follow [14, proposition 1: given a strictly positive supermartingale V t, there exists a unique strictly positive (local) martingale Λ t such that the 4

5 process B t = Λ t /V t is strictly increasing and V = Λ. We identify B t with a riskless money market account initialized at B = 1 and write it as B t = exp( t r s ds), (9) for an adapted process r s >, the short rate process. A sufficient condition for an arbitrage free bond price structure in the potential approach is to require that the local martingale Λ t be in fact a martingale, since it can then be used as the density for an equivalent martingale measure. It is an interesting open question in the theory to isolate what general conditions on the potential V t would suffice for that. The formulation up to this point is quite general, in the sense that is does not make use of any particular structure of the underlying filtration (other than the usual conditions). Let us now assume that (F t ) t is actually generated by an N-dimensional Brownian motion W t. The market price of risk then arises as the adapted vector valued process λ t such that dλ t = Λ t λ tdw t, Λ = 1, (1) where we suppress vector indices by adopting a matrix multiplication convention, including for transpose. It is also immediate to see that the state price density process is the solution to dv t = r t V t dt V t λ tdw t, V = 1. (11) so that the specification of the process V t is enough to produce both the short rate r t and the market price of risk λ t. It has already been remarked by Flesaker and Hughston [4 that in the Brownian filtration with finite time horizon any positive interest rate model in their formulation corresponds to a model in the HJM family with positive instantaneous forward rates f tt. The converse result that any interest rate model in HJM form with positive instantaneous forward rates can be written in the Flesaker Hughston form was also obtained by Jin and Glasserman [9, theorem 5. In order to prove this result they found a rather technical necessary and sufficient condition for positivity in terms of the volatility structure of the HJM form, confirming that the HJM formulation is not the most natural one to investigate positive interest rates. 5

6 .3 The Chaotic Approach We have seen in the potential approach that the fundamental ingredient to model the random behaviour of the interest rates is the increasing process A t in the decomposition V t = E t [A A t, whereas in the Flesaker Hughston construction the corresponding role is played by the martingales M ts. In [6, Hughston and Rafailidis introduced an elegant construction of general positive interest rate models based on a Brownian filtration using simpler fundamentals. If we assume that V t is potential for which the process Λ t is a true martingale, then integrating (11) on the interval (t, T ), taking conditional expectations at time t and the limit T, one finds that [ V t = E t r s V s ds. (1) In particular [ E Now let σ t be a vector valued process such that Due to (13), we can define the random variable X = t r s V s ds <. (13) σ t σ t = r t V t. (14) and it follows from the Itô isometry that σ s dw s. (15) V t = E t [X E t [X, (16) which is called the conditional variance representation of the state price density V t. To obtain the connection between this representation and the Flesaker Hughston framework, observe that a direct comparison between (1) and (8) gives that h s M ts = E t [ σ s σ s. (17) Similarly, by comparing the conditional variance representation (16) with the decomposition (5), we see that E t [X X t = E t [A A t, 6

7 where X t = E t [X. It follows from the uniqueness of the Doob-Meyer decomposition that A t = [X, X t, that is, the quadratic variation of the process X t. Conversely, given a zero-mean random variable X L (Ω, F, P ), the representation (16) defines a potential V t, which can then be used as a state price density to obtain a system of bond prices. The issue of absence of arbitrage can then be addressed in terms of necessary and sufficient conditions on X, and is by and large an open question at this point. The construction in [6 flows in the opposite direction, in the sense that the authors first enumerate a series of axioms to be satisfied by an arbitrage-free interest rate model and then obtain a square integrable random variable X corresponding to it..4 Wiener chaos As Hughston and Rafailidis also observed, the L condition on X is necessary and sufficient for X to have the type of orthogonal decomposition known as a Wiener chaos expansion [7, 11, 15. They interpret the different orders of this decomposition as basic building blocks for models of increasing complexity. Let W t be an N dimensional Brownian motion on the filtered probability space (Ω, F, {F t } t R+, P ). We introduce a compact notation τ = (s, µ). = R + {1,..., N} and express integrals as. f(τ)dτ = µ. f(τ)dw τ = For each n, let µ f(s, µ)ds f(s, µ)dw µ s (18) H n (x) = ( 1) n e x / dn dx n e x / (19) 7

8 be the nth Hermite polynomial. For h L ( ), let h = h(τ) dτ and define the Gaussian random variable W (h) := h(τ)dw τ. The spaces H n. = span{hn (W (h)) h L ( )}, n 1, H. = C form an orthogonal decomposition of the space L (Ω, F, P ) of square integrable random variables: L (Ω, F, P ) = n=h n Each H n can be identified with L ( n ) via the isometries J n : L ( n ) H n given by f n J n (f n ) = f n (τ 1,..., τ n )dw τ1... dw τn () n. where n = {(τ1,..., τ n ) τ i = (s i, µ i ), s 1 s s n < }. With these ingredients, one is then led to the result that any X L (Ω, F, P ) can be represented as a Wiener chaos expansion X = J n (f n ) (1) n= where the deterministic functions f n L ( n ) are uniquely determined by the random variable X. A special example arises by noting that for h L ( ) n!j n (h n ) = h n H n ( W (h) h ) () and furthermore [ exp W (h) 1 h(τ) dτ = 8 ( ) h n W (h) H n n! h n= (3)

9 In the notation of quantum field theory (see Appendix A), this example defines the Wick ordered exponential and Wick powers [. : exp[w (h) : = exp W (h) 1 h(τ) dτ : W (h) n. : = n!j n (h n ) (4) Generating functionals provide one systematic approach to developing explicit formulas for the terms of the chaos expansion in specific examples. Theorem.1 For any random variable X L (Ω, F, P ), the generating functional Z X (h) : L ( ) C defined by Z X (h) =. [ [ E X exp W (h) 1 h(τ) dτ (5) is an entire analytic functional of h L ( ) and hence has an absolutely convergent expansion Z X (h) = F (n) X (h) (6) n where F (n) X (h) = f (n) X (τ 1,..., τ n )h(τ 1 )... h(τ n )dτ 1... dτ n. (7) n The n-th Fréchet derivative of Z X at h =, f (n) X (τ 1,..., τ n ), lies in L ( ). Finally, the Wiener Itô chaos expansion of X is X = n Proof: See appendix B. n f (n) X (τ 1,..., τ n )dw τ1... dw τn (8) 3 Squared Gaussian models A number of authors [1,[8, [13 have observed that the CIR model [3 with an integer constraint N =. 4ab N c + \ {, 1} lies in the class of so called squared Gaussian models. By introducing an R N valued Ornstein Uhlenbeck process R t, governed by the stochastic differential equation dr t = a R t dt + c dw t (9) 9

10 where W t is N dimensional Brownian motion, the Itô formula together with Lévy s criterion for Brownian motion shows that the square r t = R tr t satisfies (1) where W t = t (R sr s ) 1/ R s dw s is itself a one dimensional Brownian motion. We focus on a general family of interest rate models which includes this example, the so-called extended CIR model and more. Note that we work always in the physical measure and thus to specify the term structure model one needs to determine the market price of risk vector λ t as well as the spot rate process r t. Definition 3.1 A pair (r t, λ t ) of (Ω, F, F t, P ) processes is called an N dimensional squared Gaussian model of interest rates (N ) if there is an R N valued Ornstein Uhlenbeck process such that r t = R tr t and λ t = λ(t)r t. R t satisfies dr t = α(t)( R(t) R t )dt + γ(t)dw t, R t= = R (3) where α, γ, λ are symmetric matrix valued and R vector valued deterministic Lipschitz functions on R + and W is standard N dimensional Brownian motion. In addition we impose boundedness conditions that there is some constant M > such that λ (t) MI, α(t) M 1 I, α(t) + γ(t) λ(t) M 1 I, γ (t) M 1 I, for all t. where The exact solution of (3) is easily seen to be R t = R(t) + R(t) = K(t, )R + t t K(t, t 1 )(γdw ) t1 (31) K(t, t 1 )α(t 1 ) R(t 1 )dt 1 (3) and K(t, s), t s is the matrix valued solution of { dk(t, s)/dt = α(t)k(t, s) s t K(t, t) = I t which generates the Ornstein Uhlenbeck semigroup. 1 (33)

11 In accordance with (11), we define the state price density process to be [ t ( V t = exp R s (I + λ(s) ) ) R s ds + R s λ(s)dw s (34) We thus have a natural candidate for the random variable X : where the R N valued process t (. σ t = exp [ R s X = ( I + λ(s) 4 σ t dw t (35) ) R s ds + 1 R s λ(s)dw s ) R t (36) is the natural solution of σ t σ t = r t V t. Before proceeding to analyse X, we show that squared Gaussian models give rise to a state price density V t satisfying the conditions of the previous section, and consequently E[X = 1. Proposition 3. For squared Gaussian models, the process ( t Λ t = exp λ s dw s 1 t ) λ sλ s ds (37) is a martingale for t T and the state price density V t defined in (34) is a potential. Proof: To show that the positive local martingale Λ t is a martingale, it suffices to show that E[Λ T = Λ. The proof, which we sketch, follows from a similar proof in [. We introduce a new process d R t = α(t) R(t) [α(t) + γ(t) λ(t) R t + γ(t)dw t and two sequences of stopping times for n N τ n = inf{t : τ n = inf{t : t t (R s λ (s)r s ds n} T, ( R s λ (s) R s ds n} T. 11

12 Under the conditions of definition 3.1, we have that both lim P {τ n = T } = 1 and lim P { τ n = T } = 1. n n Moreover, the processes λ (n) t = λ t 1l {t<τn} satisfy the Novikov condition, so that from the Girsanov theorem we obtain their stochastic exponentials Λ (n) t are densities of equivalent measures Q (n) P. Therefore E[Λ T = lim n E[Λ T 1l {τn=t } = lim n E[Λ (n) T 1l {τ n=t } = lim E Q(n) [1l {τn=t n } = lim E[1l { τn=t n } = 1, where we have used the fact that the distribution of R t τn under Q (n) is the same as the distribution of R t τn under P. To prove that the the supermartingale V t defined in (34) is a potential, let < ɛ < M and write V T = e Y 1 Y where T Y 1 = R t (I ɛ λ(t) ) R t dt and Y = 1 T [ ɛ R t(i + ɛ) λ(t) R t dt + (I + ɛ)r λ(t)dw t t. By the Hölder inequality E[V T ( E[e (1+1/ɛ)Y 1 ) ɛ/(1+ɛ) ( E[e (1+ɛ)Y ) 1/(1+ɛ) with the second factor less than or equal to 1, since e (1+ɛ)Y is a positive local martingale. Now Y 1 is a positive random variable for which a direct computation using the lower bound on γ shows mean(y 1 ) = C 1 NT (1 + O(T )) (38) var(y 1 ) = C NT (1 + O(T )) (39) for positive constants C 1, C. An easy application of Chebyshev s inequality ( Prob Y 1 C ) 1NT O( 1 NT ) (4) then implies that lim T E[V T =. 1

13 4 Exponentiated second chaos The chaos expansion we seek for the CIR model will be derived from a closed formula for expectations of e Y for elements Y = A + B(τ 1 )dw τ1 + C(τ 1, τ )dw τ1 dw τ (41) in a certain subset C +. H = H H 1 H. In the integrals above, recall that compact notation using τ s carries a summation over vector indices as well as integration over time. The formula we present is well known in the theory of Gaussian functional integration [5, Chapter 9. In probability theory, this result gives the Laplace transform of a general class of quadratic functionals of Brownian motion. Many special cases of this result have been studied in probability theory, see for example [16, Chapter and the references contained therein. If in (41) we define C(τ 1, τ ) = C(τ, τ 1 ) when τ 1 > τ, then C is the kernel of a symmetric integral operator on L ( ): [Cf(τ) = C(τ, τ 1 )f(τ 1 )dτ 1 (4) Recall that Hilbert-Schmidt operators on L ( ) are finite norm operators under the norm: C HS = C(τ 1, τ ) dτ 1 dτ We say that Y H is in C + if C is the kernel of a symmetric Hilbert Schmidt operator on L ( ) such that (1 + C) has positive spectrum. Proposition 4.1 Let Y C +. Then E[e Y = [det (1 + C) 1/ [ exp A + 1 B(τ 1 )(1 + C) 1 (τ 1, τ )B(τ )dτ 1 dτ (43) Remark 4. The Carleman Fredholm determinant is defined as the extension of the formula det (1 + C) = det(1 + C) exp[ Tr(C) (44) from finite rank operators to bounded Hilbert Schmidt operators; the operator kernel (1+C) 1 (τ 1, τ ) is also the natural extension from the finite rank case. 13

14 Proof: See Appendix C. Using this proposition, it is possible to deduce the chaos expansion of the random variables e Y, Y C +, a result known in quantum field theory as Wick s Theorem: Corollary 4.3 If Y = C(τ 1, τ )dw τ1 dw τ C +, then the random variable X = e Y has Wiener chaos coefficient functions { K G G f n (τ 1,..., τ n ) = n g G [C(1 + C) 1 (τ g1, τ g ) n even n odd where K = [det (1 + C) 1/ and for n even, G n is the set of Feynman graphs on the n marked points {τ 1,..., τ n }. Each Feynman graph G is a disjoint union of unordered pairs g = (τ g1, τ g ) with g G g = {τ 1,..., τ n }. Proof: The generating functional for X = e Y is [ ( Z X (h) = E X exp h(τ)dw τ 1 ) h(τ) dτ [ ( = E exp h(τ)dw τ 1 ) h(τ) dτ C(τ 1, τ )dw τ1 dw τ, so we can use Proposition 4.1 with A = 1 h(τ) dτ and B(τ) = h(τ), which yields Z X (h) = det (1 + C) 1/ [ exp 1 h (τ 1 )[δ(τ 1, τ ) (1 + C) 1 (τ 1, τ )h(τ )dτ 1 dτ.(46) Using the last part of Theorem.1, the result comes by evaluating the nth Fréchet derivative at h =, or equivalently by expanding the exponential and symmetrizing over the points τ 1,..., τ n in the n/th term. 14

15 5 The chaotic expansion for squared Gaussian models We now derive the chaos expansion for the squared Gaussian model defined by (3). In view of (35) it will be enough to find the chaos expansion for σ µ T, T <. We start by finding its generating functional Z σ µ. For h, k L ( ), T define the auxiliary functional Z(h, k) = E [ e Y T with Y T = T R t ( I + λ T ) R t dt + 1 T h (t)h(t)dt T R t λ(t)dw t T h (t)dw t k (t)r t dt. (47) Proposition 5.1 Z(h, k) is an entire analytic functional on L ( ) L ( ). Moreover δz(h, k) lim = Z t T δk µ (t) σ µ (h) (48) k= T where Z σ µ (h) is defined by (5) with X = σµ T T, µ = 1,..., N. Proof: Analyticity in (h, k) follows by repeating the argument given in Appendix B. By the definition of Fréchet differentiation and continuity of the t T limit, (48) follows. We want to use Proposition 4.1 in order to compute Z(h, k). Substitution of (31), into the first term of (47) leads to T R t ( I + λ(t) ) R t dt = 4 + T ( I R (t) + λ(t) ) R(t)dt 4 T [ T + R (s) (I + λ(t) [ T γ(t 1 ) K T (t 1, s) (I + λ(t) T { [ T ( I + tr γ(t) K T (t, s) + λ(t) 4 ) K T (s, t)ds γ(t)dw t ) K T (s, t )ds γ(t )dw t1 dw t ) } K T (s, t)ds γ(t) dt, 15

16 where we define K T (t 1, t ) = 1l(t 1 T )K(t 1, t ). For the second and the last terms of (47) we have 1 T R λ(t)dw t t = 1 T R (t) λ(t)dw t + 1 T [ T tr γ(s)k T (s, t) λ(t)ds dt + 1 ( γ(t 1 )K T (t 1, t ) λ(t) + λ(t)k ) T (t 1, t )γ(t ) dw t1 dw t, T T T ( T ) k (t)r t dt = k (t) R(t)dt + k (s)k T (s, t)ds γ(t)dw t. Thus the exponent Y T appearing in (47) has the form of (41) with T [ ( I A T = R (t) + λ(t) ) R(t) h (t)h(t) k (t) R(t) dt T { [ T ( I + tr γ(t) K T (t, s) + λ(t) ) } K T (s, t)ds γ(t) dt 4 1 T [ T tr γ(s)k T (s, t) λ(t)ds dt, B T (t) = h(t) γ(t) T +γ(t) T K T (t, s)k(s)ds + 1 λ(t) R(t) ) K T (t, s) ( I + λ(t) [ T C T (t 1, t ) = γ(t 1 ) K T (t 1, s) (I + λ(t) + 1 R(s)ds ) K T (s, t )ds γ(t ) [ γ(t 1 )K T (t 1, t ) λ(t) + λ(t)k T (t 1, t )γ(t ) It is clear that the operator C T has ( Hilbert-Schmidt ) norm C T HS = O(T ). Moreover, if we denote by γk T 1 + λ K T γ, γk λ T and λk T γ the operators whose kernels appear in the expression above, then C T can be written as C T = γk T K T γ + 1 (γk T λ + 1)( λk T γ + 1) 1 (49) from which we see that (1+C T ) is positive. Therefore, we can use Proposition 4.1 for E[e Y T, leading to a general formula for the generating functional Z(h, k): 16

17 ( Z(h, k) = det (1 + C T ) 1/ exp 1 ) trc T [ ) { T exp { 1 exp T ( I R (t) [ h (t 1 ) + + λ(t) 4 T R (s) (I + λ(t) [ h(t ) + γ(t ) T T γ(t ) K T (t, s) R(t) + 1 } h (t)h(t) k (t) R(t) dt k (s)k T (s, t 1 )γ(t 1 )ds 1 R (t 1 ) λ(t) ) K T (s, t 1 )γ(t 1 )ds (1 + C T ) 1 (t 1, t ) K T (t, s)k s ds 1 λ(t) R(t ) (I + λ(t) ) } R(s)ds dt 1 dt Differentiation once with respect to k then yields { T [ ( I Z σt (h) = M T exp R (t) + λ(t) ) R(t) + 1 } 4 h (t)h(t) dt [ { R λ(t) R + K T γ(1 + C T ) 1 h γk T (I + λ(t) ) } R (T ) { [ ( 1 R λ(t) exp h R I + λ(t) ) K T γ (t 1 )(1 + C T ) 1 (t 1, t ) [ λ(t) R h γk T (I + λ(t) ) } R (t )dt 1 dt (51) where (5) M T = e 1 trc T (det (1 + C T )) 1/ = (det(1 + C T )) 1/. (5) By comparing (5) and (51), the reader can observe our use of an operator notation which suppresses some time integrals, for example in the very last term [ γk T (I + λ ) and similarly for other terms. R (t) =. T ( γ(t) K T (t, s) I + λ ) 17 R(s)ds,

18 These formulas simplify considerably if the function R vanishes, which is true in the simple CIR model of (9) when r =. In this case we have α(t) = a I and γ(t) = c I, so that K T (s, t) = e a(s t)/ 1l(t s T ) and C T (t 1, t ) = c 4a (I + λ ) [e a t 1 t e a (t 1+t T ) + c λe a t 1 t (53) Moreover, the previous expression for Z σt (h) reduces to [ Z σt (h) = M T KT γ(1 + C T ) 1 h (T ) [ exp 1 T h (t)h(t)dt + 1 h (t 1 )(1 + C T ) 1 (t 1, t )h(t )dt 1 dt. We can then easily evaluate the nth Fréchet derivative of Z σt at h = as in the proof of Corollary 4.3 and determine the following partly explicit form for the nth term of the chaos expansion. Theorem 5. The nth term of the chaos expansion of σ T for the CIR model with initial condition r = is zero for n even. For n odd, the kernel of the expansion is the function f (n) T ( ) : n R f T (t 1,..., t n ) = M T L(g) (54) where G Gn g G [C T (1 + C T ) 1 (t g1, t g ) T / g L(g) = (K T γ(1 + C T ) 1 )(T, t g ) T g (55) Here, G n is the set of Feynman graphs, each Feynman graph G being a partition of {t 1,..., t n, T } into pairs g = (t g1, t g ). The chaos expansion for X itself is exactly the same, except that the variable T is treated as an additional Itô integration variable. The explicit 18

19 expansion up to fourth order is: X = M T [K T γ(1 + C T ) 1 (T, t 1 )dw t1 dw T + M T [K T γ(1 + C T ) 1 (T, t 3 )[C T (1 + C T ) 1 (t 1, t )dw t1 dw t dw t3 dw T 4 + M T [K T γ(1 + C T ) 1 (T, t )[C T (1 + C T ) 1 (t 1, t 3 )dw t1 dw t dw t3 dw T 4 + M T [K T γ(1 + C T ) 1 (T, t 1 )[C T (1 + C T ) 1 (t, t 3 )dw t1 dw t dw t3 dw T 4 (56) 6 Bond pricing formula In this section we give a derivation of the price of a zero coupon bond in the CIR model. Recall from section that these are given by P tt = E t [V 1 t V T. (57) To keep things as simple as possible, but not any simpler, we take λ = so V t = exp[ t r sds, or in terms of the squared Gaussian formulation, [ t V t = exp R sr s ds. (58) As we have seen in the previous section, for t s T R µ s = K T (s, t)r µ t + c s hence log[vt 1 V T = T µ t (Rµ s ) ds can be written as [ 4 T T c (Rµ t ) C T (t, t) + 4Rµ t C T (t, s)dw s µ + c µ t t +N where C T (s 1, s ) is given by (53) with λ =. 19 K T (s, s 1 )dw µ s 1, (59) T t t s t C T (s 1, s )dw µ s 1 dw µ s C T (s, s)ds (6)

20 Taking the conditional expectation of V 1 t V T by use of Proposition 4.1 leads to the desired formula P tt = [det(1 + C T ) N/ (61) [ exp 4 c (Rµ t ) ( C T (1 + C T ) 1) (t, t) (6) µ Thus P tt has the exponential affine form exp[ β(t, T )r t α(t, T ) with β(t, T ) = 4 c [C T (1 + C T ) 1 (t, t) α(t, T ) = N log[ det(1 + C T ) (63) The known formula has the same form, with (e ρ(t t) 1) β(t, T ) = (ρ + a)(e ρ(t t) 1) + ρ, ρ = a + c α(t, T ) = ab [ (a+ρ)(t t)/ ρ log ρe (ρ + a)(e ρ(t t) 1) + ρ (64) which can be derived as solutions of the pair of Ricatti ordinary differential equations β t α t = c β + aβ 1 = abβ (65) One can demonstrate using power series expansions that (63) do in fact solve the Ricatti equations and hence agree with the usual formula. This example points to the rather subtle general relationship between kernels such as (1 + C T ) 1 and solutions of Ricatti equations deserving of further study. 7 Discussion We have shown how the CIR model, at least in integer dimensions, can be viewed within the chaos framework of Hughston and Rafailidis as arising from a somewhat special random variable X. This random variable can

21 be understood as derived from exponentiated second chaos random variables e Y, Y C + ). Such exponentiated C + variables form a rich and natural family which is likely to include many more candidates for applicable interest rate models. Although their analytic properties are complicated, there do exist approximation schemes which can in principle be the basis for numerical methods. On the theoretical side, this family is distinguished by its natural invariance properties. Most notably, as will be investigated elsewhere, it is invariant under conditional F t expectations: log E t [e Y C + whenever Y C +. Note in particular that this implies that these can be used as the Radon Nikodym derivatives of measure changes which generalize the Girsanov transform, and which can greatly enrich the tools applicable in finance. The results of this paper rest on the special properties of squared Gaussian processes which in turn work within the filtration F W of N dimensional Brownian motion W. However, the CIR process r t itself is adapted to the much smaller filtration F W of one dimensional Brownian motion W. It is an important unsolved problem to construct an analytical chaos expansion for the CIR model adapted to the natural filtration F W. However, since the expansion we present is certainly not adapted to F W, we do not expect a simple relation between the two constructions. Our application of the chaos expansion to squared Gaussian models also illustrates a deep connection between methods developed for quantum field theory and the methods of Malliavin calculus. Many of the very rich analytic properties of this example reflect well known techniques widely used in mathematical physics. To conclude, even if the representation of the CIR model and its generalizations we present is not simple, it does show the way to the use of a powerful set of analytical techniques which may well prove to be very useful in financial modelling. A White noise calculus Here we describe the white noise calculus, which can be regarded as a reformulation of the calculus of Wiener measure [1 into concepts familiar to practitioners in quantum field theory such as Gaussian functional integration [5 Let S be the Schwartz space of smooth functions on R + of rapid decrease, 1

22 and S its topological dual, the space of tempered distributions on R +. If φ S and f S, we write φ(f). = φ, f for their canonical pairing. We also use the formal notation φ(f) = f(s)φ s ds R + to represent the smearing of the distribution φ over the test function f. It acquires a rigorous meaning, however, in the cases where φ is itself a function on R + for which the pointwise product φ(s)f(s) is integrable for all f S. Define now the functional S{f}. = e 1 f = e 1 f,f L, (66) where, L denotes the real-valued inner product in L (R + ). From its properties, it follows from the Bochner-Minlos theorem that there exists a unique Borel probability measure µ on S such that S{f} corresponds to a moment generating functional, that is, for all f S S e iφ(f) dµ(φ) = S{f} = e f /. (67) The measure space (S, B, µ), where B is the Borel σ-algebra of S, is called the white noise probability space. In Euclidean quantum field theory, a given Borel measure µ on S characterizes the family of fields φ S through the properties of the random variables φ(f) : S R obtained for each f S. One tries to construct measures µ so that the generating functional S{f} satisfy the so-called Osterwalder- Schrader axioms, in order to guarantee that the fields φ have certain required physical properties. The family of Euclidean free fields is obtained when S C {f}. = e f,cf = S e iφ(f) dµ C (φ), where C is the integral kernel of a positive, continuous, nondegenerate Euclidean covariant bilinear form C on S S. We see that the special case

23 S{f} = e f / is obtained when C(s, t) = δ(t s), called the ultralocal covariance. For each φ S, the random variables {φ(f) : f S} L (S, B, µ)} form a Gaussian family with mean zero and covariances E µ [φ(f)φ(g) = f(s)g(t)c(s, t)dsdt = f, g. (68) R + R + The theory of martingales makes its appearance in white noise calculus through the concept of Wick ordered random variables. We begin by defining the Wick ordered exponential for any f S to be Then we have Proposition A.1 For any f S, : e φ(f) := e φ(f) f /. (69) : e φ(f) := n f n n! ( ) φ(f) H n, (7) f where H n is the nth Hermite polynomial. Moreover, for any f, g S, [ ( ) ( ) ( ) n φ(f) φ(g) f, g E µ H n H m = δ nm (n!) (71) f g f g Proof: For any a, b R we have the absolutely convergent expansions e a b / = e (b a/b) /+a /(b ) = b n ( ) n! ( 1)n e a /(b ) dn e x / dx x=a/b n n = b n n! H n(a/b) (7) n where the last line makes use of the defining property of the Hermite polynomials. Using this with a = φ(f), b = f gives (7). The orthogonality relation (71) follows by expanding the identity E µ [: e φ(f) :: e φ(g) : = e f,g (73) in powers of f, g and comparing to the expansion derived from (7). 3

24 From this we define the Wick ordered monomials as the random variables ( ) φ(f) : φ(f) n := f n H n, (74) f so that linearity and convergent power series imply that : e φ(f) := n : φ(f) n :. n! Formally, we express the Wick ordered monomials as : φ(f) n := f(s 1 )... f(s n ) : φ s1... φ sn : ds 1... ds n R n + and from the orthogonalization (71) we can define the Wick products : φ(f 1 )... φ(f n ) : = f 1 (s 1 )... f n (s n ) : φ s1... φ sn : ds 1... ds n,(75) R n + H n is defined to be the span of {: φ(f) n : f S} and consists of precisely the random variables (n!) 1 f(s R n 1,..., s n ) : φ s1... φ sn : ds 1... ds n + where f lies in T n the L completion of the space of symmetric functions in S n. These subspaces form an orthogonal decomposition of L (S, B, µ). In quantum field theory this is known as the Fock space decomposition of the Hilbert space of quantum states into n particle sectors for n. The relationship between Wiener measure and the white noise measure is to identify φ(f) =. f(s)φ s ds with W (f) =. f(s)dw s for all f S L (R + ). One is then lead to the formal relation φ s = dw s /ds, that is white noise is the derivative of Brownian motion. This identification extends to all orders in the chaos expansion via: (n!) 1 f(s 1,..., s n ) : φ s1... φ sn : ds 1... ds n = f(s 1,..., s n )dw s1... dw sn R n + n (76) where the bijection f f between T n and L ( n ) is the restriction map and its inverse. Finally, this leads to the identification L (S, B, µ) L (Ω, F, P ). One useful consequence of Wick products and the chaos expansion is that conditional expectations can be handled systematically. 4

25 Proposition A. Let f L (R + ) and t [, ). Then E[: e φ(f) : F t = : e φ(f (t)) : E[: (φ(f)) n : F t = : (φ(f (t) )) n : (77) where [f (t) (s) = 1l(s t)f(s). In other words, the processes : e φ(f (t)) :, : (φ(f (t) )) n : are martingales. B Generating functional for chaos coefficients We derive Theorem.1 in the notation of the white noise calculus. To begin, we recall the definition of analyticity for a function between complex Banach spaces. Definition B.1 Let f : B C where B, C are complex Banach spaces. Then f is analytic at a point x B if 1. for any finite set {x 1,..., x n } B of points the function F : C n C F (ζ 1,..., ζ n ) = f(x + i ζ i x i ) (78) is analytic at C n ;. f is continuous at x. Proof of Theorem.1 One can check directly that the map h : e φ(h) : is entire analytic between L C (R +) and L C (S, B, µ). Similarly, Z X (h) is analytic for any h L C (R +). Finally, to verify (8), it is enough to take the expectation of the equation multiplied by : φ(g) n : for arbitrary g L ( ), n : E[X : φ(g) n : = dn dλ Z X(λg) n λ= whereas by (71), E[ (m!) 1 m R m + = n! n f (n) X (s 1,..., s n )g(s 1 )... g(s n )ds 1... ds n (79) f (m) X (s 1,..., s m ) : φ(s 1 )... φ(s m ) : ds 1... ds m : φ(g) n : = n! g(s 1 )... g(s n )f (n) X (s 1,..., s n )ds 1... ds n (8) R m + 5

26 C The exponential quadratic formula Here we prove the formula (43) in the context of one dimensional white noise calculus. The proof extends easily to the multidimensional case. Proof of Proposition 4.1: 1. (Step 1) First we note that Y C + can be approximated in the Hilbert space H by random variables of the form Y = A + M [b i φ(g i ) + c i : φ(g i ) : / (81) i=1 with {g i } a finite orthonormal set in L ( ) and numbers c i > 1, b i. Then X i = φ(g i ) form a collection of independent N(, 1) random variables. For Y of this type, there is a factorization E[e Y = e A (π) M/ exp[ b i x (1 + c i )x / + c i /)dx (8) i R into one dimensional Gaussian integrals. Each integral gives the factor (π) 1/ (1 + c i ) 1/ exp 1 [c i + b i (1 + c i ) 1 (83) leading to a formula for E[e Y which agrees with (4.1) for Y of this form.. (Step ) Since the formula is true for Y in a dense subset of C +, it is now enough to prove that the map Y E[e Y is continuous in C provided the kernel C satisfies the stated conditions. By the definition of the Carleman Fredholm determinant det (1 + C) = exp (Tr[log(1 + C) C). (84) Since log(1 + x) x = O(x ) for x we see that det (1 + C) is welldefined and continuous for C > 1 and Hilbert Schmidt. Therefore, the entire right hand side of (4.1) is continuous in C for Y C +. 6

27 Acknowledgements: We thank L.P. Hughston, D. Brody and L.C.G. Rogers for discussions and suggestions. We are grateful for the comments by the participants of the Financial and Insurance Mathematics talks at ETH Zurich, May 3, the Probability Seminars at University of Cambridge, June 3, and the Financial Mathematics and Applied Probability seminars at King s College London, July 3, where this work was presented. References [1 D. Brody and L. P. Hughston. Chaos and coherence: a new framework for interest rate modelling. to appear in Proc. Roy. Soc., 4. [ P. Cheridito, D. Filipovic, and R. Kimmel. Market Price of Risk Specifications for Affine Models: Theory and Evidence. Working paper available at 3. [3 J. C. Cox, J. E. Ingersoll, Jr., and S. A. Ross. A theory of the term structure of interest rates. Econometrica, 53():385 47, [4 B. Flesaker and L. P. Hughston. Positive interest. Risk Magazine, 9(1):46 49, [5 J. Glimm and A. Jaffe. Quantum physics. Springer-Verlag, New York, A functional integral point of view. [6 L. P. Hughston and A. Rafailidis. A chaotic approach to interest rate modelling. to appear in Finance and Stochastics 4. [7 K. Ito. Multiple Wiener integral. J. Math. Soc. Japan, 3: , [8 F. Jamshidian. A simple class of square-root interest rate models. Appl. Math. Fin, (1):61 7, [9 Y. Jin and P. Glasserman. Equilibrium positive interest rates: a unified view. Rev. Fin. Studies, 14(1): , 1. 7

28 [1 Y. Maghsoodi. Solution of the extended CIR Term Structure and Bond Option Valuation. Math. Finance, 6:89-19, [11 D. Nualart. The Malliavin calculus and related topics. Probability and its Applications. Springer-Verlag, New York, [1 B. Øksendal. An introduction to Malliavin calculus with applications to economics. Unpublished lecture notes available at [13 L. C. G. Rogers. The potential approach to the term structure of interest rates and foreign exchange rates. Math. Finance, 7(): , [14 M. Rutkowski. A note on the Flesaker Hughston model of the term structure of interest rates. App. Math. Finance, 4: , [15 N. Wiener. The homogeneous chaos. Amer. J. Math., 6: , [16 M. Yor. Some aspects of Brownian motion. Lectures in Mathematics ETH Zurich. Birkhäuser Verlag, Basel,