A Brief Introduction to Stochastic Calculus

1 IEOR E4706: Foundations of Financial Engineeringc 2016 by Martin HaughA Brief Introduction to Stochastic CalculusThese notes provide a very Brief Introduction to Stochastic Calculus , the branch of mathematics that is mostidentified with financial engineering and mathematical finance. We will ignore most of the technical details andtake an engineering approach to the subject. We will only introduce the concepts that are necessary forderiving the Black-Scholes formula later in the course. These concepts include quadratic variation, stochasticintegrals and Stochastic differential equations . We will of couse also introduce It o s Lemma, probably the mostimportant result in Stochastic Martingales, Brownian Motion and Quadratic VariationWe make the following assumptions throughout.

2 There is a probability triple( ,F,P)where Pis the true orphysicalprobability measure is the universe of possible outcomes. We use to represent a generic outcome, typically asample path of a Stochastic process. the setFrepresents the set of possibleeventswhere an event is a subset of . There is also afiltration,{Ft}t 0, that models the evolution of information through time. So for example,if it is known by timetwhether or not an event,E, has occurred, then we haveE Ft. If we are workingwith a finite horizon,[0,T], then we can takeF=FT. We also say that a Stochastic process,Xt, isFt-adapted if the value ofXtis known at timetwhen theinformation represented byFtis known.

3 All the processes we consider will beFt-adapted so we will notbother to state this in the sequel. In the continuous-time models that we will study, it will be understood that the filtration{Ft}t 0will bethe filtrationgeneratedby the Stochastic processes (usually a Brownian motion,Wt) that are specified inthe model Martingales and Brownian MotionDefinition1A Stochastic process,{Wt: 0 t }, is astandard Brownian 02. It has continuous sample paths3. It has independent, stationary N(0,t).Definition2 Ann-dimensional process,Wt= (W(1)t,..,W(n)t), is astandardn-dimensional Brownianmotionif eachW(i)tis a standard Brownian motion and theW(i)t s are independent of each Stochastic process,{Xt: 0 t }, is amartingalewith respect to the filtration,Ft, andprobability measure,P, ifA Brief Introduction to Stochastic [|Xt|]< for allt [Xt+s|Ft] =Xtfor allt,s 1(Brownian martingales)LetWtbe a Brownian motion.

4 ThenWt,W2t tandexp( Wt 2t/2)are all latter martingale is an example of anexponential martingale. Exponential martingales are of particularsignificance since they are positive and may be used to define new probability (Conditional expectations as martingales)LetZbe a random variable and setXt:= E[Z|Ft]. Show thatXtis a Quadratic VariationConsider a partition of the time interval,[0,T]given by0 =t0< t1< t2< .. < tn= a Brownian motion and consider the sum of squared changesQn(T) :=n i=1[ Xti]2(1)where Xti:=Xti Xti (Quadratic Variation) The quadratic variation of a Stochastic process,Xt, is equal to thelimit ofQn(T)as t:= maxi(ti ti 1) quadratic variation of a Brownian motion is equal toTwith functions with which you are normally familiar, continuous differentiable functions, have quadraticvariation equal to zero.

5 Note that any continuous Stochastic process or function1that has non-zero quadraticvariation must have infinitetotal variationwhere the total variation of a process,Xt, on[0,T]is defined asTotal Variation:= lim t 0n i=1|Xtk Xtk 1|.This follows by observing thatn i=1(Xtk Xtk 1)2 (n i=1|Xtk Xtk 1|)max1 k n|Xtk Xtk 1|.(2)If we now letn in (2) then the continuity ofXtimplies the impossibility of the process having finite totalvariation and non-zero quadratic variation. Theorem 1 therefore implies that the total variation of a Brownianmotion is infinite. We have the following important result which proves very useful if we need to price optionswhen there are multiple underlying Brownian motions, as is the case with quanto options for (Levy s Theorem) A continuous martingale is a Brownian motion if and only if its quadraticvariation over each interval[0,t]is equal sample path of a Stochastic process can be viewed as a Brief Introduction to Stochastic Calculus32 Stochastic IntegralsWe now discuss the concept of a Stochastic integral, ignoring the various technical conditions that are requiredto make our definitions rigorous.

6 In this section, we writeXt( )instead of the usualXtto emphasize that thequantities in question are timeof the filtrationFtis a random time, , such that the event{ t} Ftfor allt > non-mathematical terms, we see that a stopping time is a random time whose value is part of the informationaccumulated by that say a process,ht( ), iselementaryif it is piece-wise constant so that there exists a sequenceof stopping times0 =t0< t1< .. < tn=Tand a set ofFti-measurable2functions,ei( ), such thatht( ) = iei( )I[ti,ti+1)(t)whereI[ti,ti+1)(t) = 1ift [ti,ti+1) Stochastic integral of an elementary function,ht( ), with respect to a Brownian motion,Wt, is defined as T0ht( )dWt( ) :=n 1 i=0ei( )(Wti+1( ) Wti( )).]]]

7 (3)Note that our definition of an elementary function assumes that the function,ht( ), is evaluated at theleft-hand pointof the interval in whichtfalls. This is a key component in the definition of the stochasticintegral: without it the results below would no longer hold. Moreover, defining the Stochastic integral in thisway makes the resulting theory suitable for financial applications. In particular, if we interpretht( )as a tradingstrategy and the Stochastic integral as the gains or losses from this trading strategy, then evaluatinght( )atthe left-hand point is equivalent to imposing thenon-anticipativityof the trading strategy, a property that wealways wish to a more general process,Xt( ), we have T0Xt( )dWt( ) := limn T0X(n)t( )dWt( )whereX(n)tis a sequence of elementary processes that converges (in an appropriate manner) 2We want to compute T0 WtdWt.

8 Towards this end, let0 =tn0< tn1< tn2< .. < tnn=Tbe apartition of[0,T]and defineXnt:=n 1 i=0 WtniI[tni,tni+1)(t)whereI[tni,tni+1)(t) = 1ift [tni,tni+1)and is0otherwise. ThenXntis an adapted elementary process and, bycontinuity of Brownian motion, satisfieslimn Xnt=Wtalmost surely asmaxi|tni+1 tni| 0. The2 Loosely speaking, a functionf( ) isFtmeasurable if its value is known by Brief Introduction to Stochastic Calculus4stochastic integral ofXntis given by T0 XntdWt=n 1 i=0 Wtni(Wtni+1 Wtni)=12n 1 i=0(W2tni+1 W2tni (Wtni+1 Wtni)2)=12W2T 12W20 12n 1 i=0(Wtni+1 Wtni)2.(4)By Theorem 1 the sum on the right-hand-side of (4) converges in probability toTasn.]]]

9 And sinceW0= 0we obtain T0 WtdWt= limn T0 XntdWt=12W2T that we will generally evaluate Stochastic integrals using It o s Lemma (to be discussed later) withouthaving to take limits of elementary processes as we did in Example define the spaceL2[0,T]to be the space of processes,Xt( ), such thatE[ T0Xt( )2dt]< .Theorem3(It o s Isometry)For anyXt( ) L2[0,T]we haveE ( T0Xt( )dWt( ))2 = E[ T0Xt( )2dt].Proof: (For the case whereXtis an elementary process)LetXt= iei( )I[ti,ti+1)(t)be an elementary process where theei( ) s andti s are as defined in Definition therefore have T0Xt( )dWt( ) := n 1i=0ei( )(Wti+1( ) Wti( )).]

10 We then haveE ( T0Xt( )dWt( ))2 = E (n 1 i=0ei( )(Wti+1( ) Wti( )))2 =n 1 i=0E[e2i( )(Wti+1( ) Wti( ))2]+ 2n 1 0 i<j n 1E[eiej( )(Wti+1( ) Wti( ))(Wtj+1( ) Wtj( ))]=n 1 i=0E e2i( ) Eti[(Wti+1( ) Wti( ))2] =ti+1 ti + 2n 1 0 i<j n 1E eiej( )(Wti+1( ) Wti( ))Etj[(Wtj+1( ) Wtj( ))] =0 A Brief Introduction to Stochastic Calculus5= E[n 1 i=0e2i( )(ti+1 ti)]= E[ T0Xt( )2dt]which is what we had to (Martingale Property of Stochastic Integrals)The Stochastic integral,Yt:= t0Xs( )dWs( ), is a martingale for anyXt( ) L2[0,T].Exercise2 Check that t0Xs( )dWt( )is indeed a martingale whenXtis an elementary process.