K. Ito's stochastic calculus isacollection of tools which permitusto perform operations

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1 BROWNIAN MOTION AND ITO CALCULUS K. Ito's stochastic calculus isacollection of tools which permitusto perform operations such as composition, integration and dierentiation, on functions of Brownian paths and more general random functions known as Ito processes. As we shall see, Ito calculus and Ito processes are extremely useful in the formulation of nancial risk-management techniques. These notes are intended to introduce the reader to stochastic calculus in a straightforward, intuitive way. For rigorous treatments ofthis rich subject the reader can consult, for instance, Ikeda andwatanabe (North Holland-Kodansha, 989), Varadhan (98) Karatzas and Shreve (Springer 988).. Brownian Motion Intuitively, Brownian motion corresponds to the concept of a homogeneous, continuoustime, continuous random walk. Oneway to visualize Brownian paths is to consider a simple random walk on the realline, in which the walker starts atposition X = andmoves up or down by an amount p dt after each time-interval of duration dt. If X n denotes the position of the walker after the nth jump, we have X n = X n; p dt n = ::: () where the + and ; signs occur with probability =. This process is called a simple random walk. Note that the magnitude of the jump and the lag between successive jumps are chosen so that the variance of the displacement of the walker after time T (where T is an integer multiple of dt) is exactly T. N. Ikeda ands.watanabe, Stochastic Dierential Equations and Diusion Processes, nd Ed., North Holland-Kodansha, Amsterdam and Tokyo, 989 S.R.S. Varadhan Lectures on Brownian Motion and Stochastic Dierential Equations, Tata Institute offundamental Research, Bombay and I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 988.

2 A continuous path can be built from the variables X n byinterpolating linearly between the dierent points: X(t) = X n + (t ; ndt) ( X n+ ; X n ) for ndt t (n +)dt : () These paths have the following properties:. If t = ndt and a >, the increment X(t + a) ; X(t) isindependent ofthe \past" X(s) s t. E ; X(t) = 3. E ; X(t) = t. For small time-increments (dt ), these paths have nearly independent increments (neglecting the small persistence eect due to the linear interpolation). Moreover, the mean and variance of the walker's displacement areindependent ofdt. For dt, each increment ofx(t) isasumofmany independent binomial random variables with mean zero and nitevariance. Therefore, by the Central Limit Theorem, the limiting probability distribution of the increments ofx(t) asdt! isgaussian (or normal). More precisely, wehave lim P X(t + a) ; X(t) x = dt! Z x + p e ; y a dy a for all x. This property, together with the independence of the increments, characterizes the statistics of the paths X(t) in the limit. This discussion motivates Denition : Brownian Motion is a probability distribution on the set of real-valued functions Z(t) t < with the following properties. Z() = with probability. For all t > and a >, the increments Z(t + a) ; Z(t) are Gaussian with mean zero and variance a and 3. Z(t + a) ; Z(t) is independent of f Z(s) s t g. Items { 3 completely specify the probability distribution of any n-tuple ( Z(t ) Z(t ) Z(t 3 ) ::: Z(t n ))

3 where t < t < ::: < t n are arbitrary times. It is easy to see that this distribution is a multivariate Gaussian with covariance E f Z(t i ) Z(t j ) g = Min (t i t j ) :. Elementary properties of Brownian paths The rst important property is the continuity of Brownian paths, namely P fz() is a continuous function g = : The proof of this fact is mathematically non-trivial (see Ikeda and Watanabe or Varadhan). Brownian paths are continuous but also very irregular. Any student of random walks (or, for the matter, of nancial time series) has noticed that the sample paths of random walks are non-smooth and appear to have an innite slope ( making trends dicult to predict.) One way to see that Brownian paths are not dierentiable is to consider their quadratic variation. Proposition. Let =t < t < ::: < t n = T represent a partition of the time interval [ T] and let dt = max j (t j ; t j; ).Set With probability one, Brownian paths satisfy Z j = Z(t j ) ; Z(t j; ) : lim dt! 8 < : 9 = ( Z j ) = T : (The right-hand sideofthis last equation is known as the quadratic variation of the path Z() onthe interval [ T].) This result is is a direct consequence of the Law of Large Numbers of probabilitytheory. In fact, the random variables ( Z j ) are independent and Notice that the statistical distribution of the paths for nite dt is more complicated, because increments have multinomial distributions with dierent parameters, according to the number of elementary jumps between the times t and t + a. In this respect, Brownian motion is a simpler object than a random walk with a nite jump size. 3

4 have means E f( P Z j ) g = t j ; t j;.therefore, as dt!, the sumconverges to its expected value, j (t j ; t j; ) = T. This result implies that Brownian paths are not dierentiable. To see this, we note that if the function f(t) isdierentiable, then we have necessarily 3 lim dt! 8 < : But then, in view of the inequality j f j j 9 = Z T jf (s) j ds : ( f j ) max j j f j j j f j j and the factthat max j j f j j = O ( dt ) we conclude that dierentiable functions f satisfy ( f j constant Z T jf (s) j ds A dt : In other words, dierentiability implies that the quadratic variation must vanish. The niteness of the quadratic variation of Brownian Motion implies therefore that itspaths are not dierentiable. Another remarkable property of Brownian motion is statistical self-similarity. For any parameter >, the transformation Z(t) 7! ; Z(t) maps Brownian paths into Brownian paths. This means that if one considers, for instance, the ensemble of Brownian paths on the interval [ ] and \stretches" them according tothe above transformation with = =, the result isbrownian motion on the interval [ ]. This self-similarity is consistent with the fact that Brownian paths are \fractal" objects. 3 Using the Intermediate Value Theorem. 4

5 3. Stochastic Integrals Denition. A function f(t) is said to be non-anticipative with respect to the Brownian motion Z(t) if, for all t >, f(t) = ~ f ( fz(s) s t g t) i.e. if the value of the function at time t is determined by the values taken by the history of the path Z() up to time t. We also include in this denition deterministic functions{ these have a \trivial" dependence on the random paths. The point of introducing the concept of a non-anticipative function is to distinguish between general functions of Brownian paths and those which are determined by the natural \ow of information" associated with the path Z() as time progresses. Examples. The function f (t) = 8 >< >: if max s t Z(s) < 5 if max Z(s) 5 s t is non-anticipative. (This function is equal to zero at time t if the walk has not reached the value 5 by time t and is equal to otherwise.) On the other hand, f (t) = 8 >< >: if max s Z(s) < 5 if max s Z(s) 5 is not. The reason is that the value of the latter function at any time t< is determined by the realization of the path Z() over the entire interval [ ]{the information gained by knowing Z(s) for s t is insucient todetermine f (t). Non-anticipative functions are the \natural" objects to perform integration with repect Brownian increments. Proposition. Let f() be a continuous, non-anticipative function such that E 8 < Z T : jf(t)j dt 5 9 = < :

6 Then, given any sequence ofpartitions of the interval [ T] with mesh size t!, we have lim t! f(t j; ) (Z(t j ) ; Z(t j; ) = lim t! f(t j; ) Z(t j ) exists and is independent of the sequence used to take the limit. This limit is, by denition, the Ito integral of f. It is denoted by Z T f(t) dz(t) : Stochastic integration is a natural operation associated with Brownian paths: a path is \sliced" into consecutive Gaussian increments, each increment is multiplied by a random variable and these numbers are then added together again to reconstruct the stochastic integral. Thus, the stochastic integral can be viewed as a random walk with increments which have dierent amplitudes, or conditional variances { a sort of \inhomogeneous random walk". In this respect, it is important to emphasize the role of the non-anticipative assumption. Consider the j th incrementafter multiplication bytherandom variable f(t j; ): f(t j; ) dz j = f(t j; ) (Z(t j ) ; Z(t j; )) : (3) Once the history of the path uptotime t j; is revealed, the value of f(t j; ) is also known. Therefore, the increment of the stochastic integral over the next period conditional on the past up to time t is Gaussian with mean zero and variance f(t j; ) (t j ; t j; ).Ifthe function f(t) was anticipative (for lack of a better word), the two factors in (3) need not be conditionally independent. In the latter case, the mean of the increment may not be zero necessarily and the variance cannot be calculated in explicit form. Thus, non-anticipative functions are the correct functions to dene a continuous, inhomogeneous random walk through the stochastic integral. Some basic properties of the stochastic integral are given in Proposition 3: Under the above assumptions, we have E 8 < Z T : f(t) dz(t) 6 9 = = (4)

7 E 8 >< T >: f(t) dz(t) 9 8 >= < Z A T > = E : jf(t) j dt 9 = : (5) R t Moreover, t 7! probability. f(s) dz(s) is a non-naticipative process which is continuous with Proofs of Propositions and 3 are sketched in the Appendix. The main idea behind the denition of the stochastic integral is the fact that Brownian motion increments dz point \to the future" of f at each discretization time. Example. Consider a function f(t) = (t) which is deterministic (i.e. independent of the Brownian motion). 4 In this case, the stochastic integral X(t) = (s) dz(s) t (6) is a Gaussian random process, in the sense that(x(t ) X(t ) ::: X(t N ))isamultivariate normal vector for any set of times t < t < ::: < t N.Thereasonisthat the stochastic integral is a limit (in the sense of the mean-square norm) of sums of independent Gaussian random variables. Thus, if the integrand (t) is deterministic, the stochastic integral is a Gaussian random walk with time-dependent local variance. The variance of any increment is given by the formula E n (X(t + a) ; X(t)) o = Zt+a t (s) ds : In Options Theory, stochastic processes of type (6) are often used to model the term structure of volatility. Mathematicians have a dierent way of thinking about X(t): in fact, suppose that we dene a new time scale (t) by the equation (t) = (s) ds : The reader may think of as the time that a special clock would give you whenever the \real' time was t. Dening anew process ~ X() X(t) (the displacement with repect to the \rubber clock") we have, 4 Weusethe notation (t) to suggest the notion of a \local variance". 7

8 E ~X() = E n ( X(t)) o = (s) ds = : Thus the process induced by the stochastic integral can be regarded as a Brownian motion with respect to the new clock. This is a nice result, because it characterizes the probability distribution of Ito integrals in the caseofdeterministic integrands. 5 Example. Suppose that f(t) = f(z(t)), i.e. that f depends on the current value of the Brownian path. This is, of course, a non-anticipative function so the Ito integral can be dened. Intuitively, the stochastic integral corresponds to a walk in which the local variance of increments is a function of the auxiliary Brownian path. The stochastic integral f(z(s)) dz(s) will be non-gaussian in general. We should make atthis point an important remark, which is related to the results of the previous section and to what lies ahead. If Brownian paths were smooth, or rather, if one ignored the fact that they are non-smooth, then one might be tempted to consider the anti-derivative (primitive) of f and to write whereby df (Z(t)) = f(z(t)) dz(t) f(z(s)) dz(s) = F (Z(t)) ; F () : (7) 5 This trick ofchanging time has also made itsway into nancial modeling. Some exchanges close while others remain open and interest accrual on deposits may take into account time at which thereis no trading. Because of this, the localvolatility of certain assets over holidays and quiet periods can be lowered to reect the lack of strong trading activity. Similarly, the short-term volatility parameter might be increased on the date of an important economic or political announcementwhich will have a large impact on prices or rates. To take into account these eects in valuation models it is useful to introduce the notion of \nonlinear" time. 8

9 This is wrong!! (Unless, f is constant.) For instance, if the primitive F (Z) is positive and vanishes at Z = we would conclude from (7) that the stochastic integral is positive. This cannot be, since we know from Proposition 3 that stochastic integrals have mean zero. Consider, for instance, the elementary case f(z) = Z (and hence F (Z) = Z =). Using the denition the stochastic integral and standard notation, we have Z T Z(s) dz(s) X j Z j; dz j = X j Z j; (Z j ; Z j; ) = ; X j (Z j ; Z j; ) + X j Z j (Z j ; Z j; ) = ; X j (Z j ; Z j; ) + X j ; Z j ; Zj; ; X j Z j; (Z j ; Z j; ) ;T + Z (T ) ; Z T Z(s)dZ(s) (8) where we used the result on the quadratic variation of Brownian motion (Proposition ). Thus, we conclude that Z T Z(s) dz(s) = Z(T ) ; T : (9) The right-handsidenowhas expectation zero (at itshould). The additional term T, which is missing inthe \naive" formula (7) comes from the quadratic variation P (dz j ). Example 3. Suppose that one wants to implement the idea of a random walk with conditionally Gaussian increments (i.e. a stochastic integral) in which the local variance depends on the position of the walk. This means that the \elementary increment" of the stochastic integral should have the form 9

10 X(t) = (X(t)) Z(t) : More formally, we would like to dene a process that satises X(t) = ( X(s)) dz(s) : () As opposed to Example 3, the stochastic integral on the right-hand side depends on the left-hand side of the equation. This is therefore an integral equation for X(t), which is often written in dierential form dx(t) = ( X(t)) dz(t) () in which case it is termed a stochastic dierential equation. If Z(t) were smooth, then () has a clear meaning and can be solved by standard methods of Ordinary Dierential Equations. In the case of Brownian dierentials, () should be interpreted as the integral equation (). The existence and uniqueness of solutions of stochastic dierential or integral equations such as () and () is treated in most books on stochastic calculus (Ikeda and Watanabe, Varadhan and Karatzas and Shreve, among many others.) The Ito integral, or stochastic integral, is a powerful analytic tool for constructing stochastic processes which are similar to Brownian motion but have local characteristics which dependontime, the value of the process itself or more general non-anticipative factors. The implications for nancial modeling are very interesting: stochastic integrals and stochastic dierential equations can be used to model heterogeneity of the local price volatility, a key theme in lectures to follow.

11 4. Ito's Lemma We now discuss a systematic approach for evaluating Ito integrals and functions of Brownian motion. This result can be viewed as the analogue of of the Fundamental Theorem of Calculus for functions of Brownian motion. Proposition 4. Let F (Z t) be asmooth function of two real variables Z and t with bounded derivatives of all orders. Then F (Z(T ) T) = F ( ) + (Z(s) s) + Z (Z(s) s) (Z(s) s) ds : () This proposition is known as Ito's Lemma. It provides the correction to formula (7) that would result from a naive application of standard Calculus. The additional term term is Z F (Z(s) s) ds : (Compare with (9).) The Fundamental Theorem of Calculus (cf. (7)) does not involve second derivatives, the reason for this being that the contribution to the integral due to quadratic and higher-order terms is negligible. (The quadratic variation of a smooth function is zero). In contrast, the (dz) -terms contribute tothe dierential in the case of Brownian motion because the quadratic variation of the paths is non-trivial. The beauty of Ito's Lemma isthat itprovides a \closed-form" expression for (F (Z(t) t) for any reasonable smooth function F,elucidating the eect of the quadratic variation. This avoids having togothrough manipulations of sums like the ones done inthe Example of the previous section. Sketch ofthe proof of Ito's Lemma. 6 We consider the Taylor expansion of of F about some point (Z t). Formally, wehave F = F Z Z + F t t + F ZZ (Z ) + F Zt + F tt (t ) + ::: (4) 6 Later on, in the study of hedging in imperfect markets with transaction costs, we willneed to revisit the mathematical derivation of Ito's Lemma.

12 Now, assume that wehave a partition of the interval [ T], ft j g,andthat Z = Z(t j; ) t= t j; t = t j ; t j; and Z j = j ; j; : Considering the Taylor expansion (4) and adding up the successive increments, we nd that the rst term in the right-handside of (4) gives the contribution R F Z (Z(s) s) ds to (). Similarly, the second term in the Taylor expansion (4) contributes to the integral R Ft dt in (). These are the twoterms that weexpectfromstandard dierential Calculus. Let us turn to the higher-order terms in the Taylor expansion. Since we have E fjz j p (t ) q g _ (t ) p=+q P the contribution to df j which arises from adding the (N = T=t) terms proportional to (Z j ) p (t) q has order ( t ) p=+q ;. The conclusion is that the only terms with p + q which contribute are those with p =andq =, i.e.the (Z) terms. All other terms vanish asymptotically as t!. This means that we should study the asymptotic behavior of the sums F ZZ (Z(t j; t j; )(Z j ) (5) recalling that each increment DZ j Riemann sum points tothe future of t j;. This sum is equal to the F ZZ (Z(t j; t j; )t approximating the integral in (3). To establish Ito's Lemma, it suces to prove that the sums F ZZ (Z(t j; t j; ) Z j ) ; t (6)

13 converge to zero as t! in a suitable sense. To see this, we analyze the mean and the variance of the sum. This analysis is similar to the oneofthe proof of Propositions and 3 given in the Appendix. The crucial point is that the increments in (6) have mean zero and point towards the future. Because of the latter property, we have E F ZZ (Z(t j; t j; ) Z j ) ; t = : The expected value of the sums (6) is therefore zero. Consider now the variance of (6), 4 E 8 >< N X F ZZ (Z(t j; t j; ) Z j ) 9 >= ; t A > : We observe two things: rst, the variables F ZZ (Z(t j; t j; ) Z j ) ; t and F ZZ (Z(t k; t k; ) Z k ) ; t are uncorrelated for j 6= k. The reasonisthat if, say, t j < t k, F ZZ (Z(t j; t j; ) Z j ) ; t F ZZ (Z(t k; t k; ) and (Z k ) ; t are independent conditionally on the past until time t k. Second, the latter random variable has expectation zero. This guarantees that the expectation of their product is zero. Consequently, the variance is given, to leading order, by 4 E n (F ZZ (Z(t j; t j; )) o E n ; (Zj ) t t E n (F ZZ (Z(t) t)) o dt ; t o A a negligible quantity as t!. Here, we used the fact that E (Z ; t ) = (t) E (N ; ) = (t) 3

14 where N isastandard normal. 7 Wehave thus shown that the sums in (5) converge to the integral (3) (the \Ito correction term") in the sense that the dierence has mean zero and variance converging to zero as t!. Example. Consider the following function of Brownian motion: S(t) = S e Z(t) +t t where and are constants. This random process is sometimes called geometric Brownian motion. Let us apply Ito's Lemma to S(t), with the object of nding the integral equation satised by it. Accordingly, applying () to the function F (Z t) = S exp f Z + tg, wendthat S(t) ; S() = S() dz() + S() + d : (8) If we use dierential notation, we obtain ds(t) = S(t) dz(t) + S(t) + dt : (9) This stochastic dierential equation states that geometric Brownian motion has the property that the innitesimal relative increments ds(t) S(t) = dz(t) + + are normal with mean + and variance.thisisthe Black&Scholes \world" for option pricing theory, which we will discuss shortly using stochastic calculus ideas. dt 7 From the explicit form of the moment-generating function for the standard normal distribution, E e N = e,itfollows that E N k = (k)!= k k! for all integers k >. 4

15 4. Ito Processes and Ito Calculus What isthe mostgeneral class of random processes that can be described in terms of sums of stochastic integrals and standard integrals? The answer is the classofito processes. Denition: We say that a random process X(t) t is an Ito Processif there exists abrownian motion measure and two non-anticipative functions (t) and b(t) t such that X(t) = X() + (s) dz(s) + b(s) ds t > () or, in terms of dierentials, dx(t) = (t) dz(t) + b(t) dt : () Intuitively speaking, Ito processes are continuous random functions with innitesimal increments which, conditionally on the past until time t, are Gaussian with mean b(t) and variance (t). 8 An important class of Ito processes which we already encountered in the lecture are functions of Brownian motion, i.e. X(t) = F (Z(t) t) : () In fact, Ito's Lemma shows that the local parameters associated to this process are (t) and b(t) F (Z(t) : The next result is very important for applications. It states that any smooth function of an Ito process is also an Ito process and gives a formula for the local parameters. 8 is importantto observe that, in general, the increments will not be Gaussian over a nite time interval. 5

16 Proposition 5 (Generalized Ito Lemma) Suppose that X(t) is an Ito process with local parameters (t) and b(t), i.e that () or () hold. Let F (X t) be smooth function of (X t) with bounded derivatives of all orders. Then Y (t) = F (X(t) t) is an Ito process with local parameters (t) and B(t) given by (t) (t) (3) and B(t) + F (X(t) t) : (4) In other terms, the Generalized Ito Lemma states that, for all t >, the integral equation F (X(t) t) = F (X() ) (s) dz(s) + F (X(s) s) ds (5) holds for all suciently smooth functions F (X t). It is convenient to express equation (5) in dierential form, like we did in for the case of Brownian motion. Accordingly, df (X(t) t) = (t) b(t) dt dt + F (X(t) dt 6

17 ( (t) dz(t) + b(t) dt dt + F (X(t) dt dx(t) dt + F (X(t) dx(t) + F (X(t) (dx(t)) (6) where we used equation () and the formal relation (dx(t)) (t) dt : (7) The latter equation is a convention: it expresses in synthetic form the contributions from quadratic terms ( (dx(t)) or (dz(t)) -terms). Thus, the Generalized Ito Lemmacanbe written concisely in the form df (X(t) t) = dx(t) + F (X(t) (dx(t)) : The result is simple to remember: the innitesimal increment of a smooth function of an Ito process is obtained by making a Taylor expansion of order in dt, oforder in dx and using the convention (7 ). This convention often referred to as the Ito multiplication rule. Note that if (t) and b(t), we recover the \little" Ito Lemma (Proposition 4) of the previous section. In summary, stochastic dierential calculus is an extension of standard calculus in which we use the Ito multiplication rule (7) to account for the eect of non-trivial quadratic variations. The proof of the Generalized Ito Lemma is very similar to the proof of Proposition 4. We omit it. The interested reader should consult for instance Ikeda and Watanabe, Varadhan, or Karatzas and Shreve. 7

18 APPENDIX: PROPERTIES OF THE ITO INTEGRAL The aim of this Appendix is to provide additional technical elements to understand the basic properties of the stochastic integral stated in Propositions and. 9 Let us consider Proposition. Notice that any \approximating sum" s N = f(t j; )(Z(t j ) ; Z(t j; )) f j; dz j (8) has mean zero and uniformly bounded variance. In fact, since f(t) is non-anticipative, we have E ff j; dz j g = E f E ff j; dz j jz(s) s t j; gg = E f f j; E fdz j jz(s) s t j; gg = where E fjz(s) s t j; g represents the conditional expectation operator given the history of the path uptotime t j;. The conditional expectation of dz j vanishes because the increments ofz(t) areindependent oftheir past. We conclude that the sum in (8) has mean zero. To compute itsvariance, we must evaluate the sumofterms of the form E ff j; dz j f k; dz k g where j k N. Thekey observation is that this expectation vanishes if j 6= k. In fact, if j < k, we have E ff j; dz j f k; dz k g = E f E ff j; dz j f k; dz k jz(s) s t k; gg = E f f j; dz j f k; E fdz k jz(s) s t k; gg = 9 These are in lieu of formal proofs, which would take ustoofarfromthe subject of these lectures into Measure Theory. I encourage the interested reader to consult the aforementioned references to obtain mathematically precise hypotheses on f(t), on the types of convergence of random variables used in the proofs, etc. So much to say, so little time... 8

19 The variance on (8) is therefore equal to the sum of the \diagonal" terms E n ( f j; dz j ) o = n o E ( f(t j; )) (t j ; t j; ) Z T E n ( f(t)) o dt : (This assumes implicitly that the sum can be approximated by the integral, i.e. E n ( f(t)) o is continuous.) that We have established that the random variables s N have bounded mean andvariance. Next, we show that the sums converge to a limit as the partitions become ner and ner. n o Consider a sequence of partitions t () N j,where max t () j j N ; t () j;! as! and the the corresponding sequence of sums s N.Weclaimthat lim! E n ; sn ; s N o = : Consider two such sums, s N and s N, correspondingto dierent partitions. The idea is to \merge" the two partitions to form a ner one which combines the nodes of each ofthem. A moment of reection shows that, with respect this rened partition (which we call ft j g to xideas) both sums can be written in the form and f(t j;) (Z(t j ) ; Z(t j; )) f(t j; )(Z(t j) ; Z(t j; )) where the t j sandthe t j are times (of the original partitions) such that 9

20 t j t j t j t j (9) and the dierence t j ; t j converges to zero as!.the dierence of the two sums can therefore be expressed in the form ; f(t j;) ; f(t j;) ( Z(t j ) ; Z(t j; )) Taking into account (9), it is easy to show that this sum has mean zero and variance E n ; f(t j;) ; f(t j;) o ( t j ; t j; ) : (3) If f(t) satises suitable boundedness and continuity assumptions, e.g, if f(t) is bounded and n o lim E ( f(t + h ) ; f(t)) = h! the sum in(3)tends to zero with the mesh size. This argument shows the existence of the Ito stochastic integral. The properties stated in Proposition 3 also follow from these calculations.