Chapter 4Brownian Motion & It FormulaStochastic ProcessnThe price movement of an underlying asset is a stochastic process. nThe French mathematician Louis Bachelier was the first one to describe the stock share price movement as a Brownian motion in his 1900 doctoral thesis. nintroduction to the Brownian motion nderive the continuous model of option pricingngiving the definition and relevant properties Brownian motionnderive stochastic calculus based on the Brownian motion including the Ito integral & Ito formula. n All of the description and discussion emphasize clarity rather than mathematical rigor.Coin-tossing ProblemnDefine a random variablenIt is easy to show that it has the following properties:n & are independentRandom VariablenWith the random variable, define a random variable and a random sequencen Random WalknConsider a time period 0,T, which can be divided into N equal intervals. Let =T N, t_n=n ,(n=0,1,cdots,N), then nA random walk is defined in 0,T:n is called the path of the random walk.Distribution of the PathnLet T=1,N=4,=1/4,Form of Pathnthe path formed by linear interpolation between the above random points. For=1/4 case, there are 24=16 paths.tS1Properties of the PathCentral Limit TheoremnFor any random sequence where the random variable X N(0,1), i.e. the random variable X obeys the standard normal distribution:E(X)=0,Var(X)=1. Application of Central Limit Them.n Consider limit as 0.Definition of Winner Process (Brownian Motion)n1) Continuity of path: W(0)=0,W(t) is a continuous function of t. n2) Normal increments: For any t0,W(t) N(0,t), and for 0 0(0) denoting the number of shares bought (sold) at time t. For a chosen investment strategy, what is the total profit at t=T?An Example cont.nPartition 0,T by: nIf the transactions are executed at time only, then the investment strategy can only be adjusted on trading days, and the gain (loss) at the time interval isnTherefore the total profit in 0,T isDefinition of It IntegralnIf f(t) is a non-anticipating stochastic process, such that the limitexists, and is independent of the partition, then the limit is called the It Integral of f(t), denoted asRemark of It IntegralnDef. of the Ito Integral one of the Riemann integral. n - the Riemann sum under a particular partition. nHowever, f(t) - non-anticipating, nHence in the value of f must be taken at the left endpoint of the interval, not at an arbitrary point in. nBased on the quadratic variance Them. 4.1 that the value of the limit of the Riemann sum of a Wiener process depends on the choice of the interpoints. nSo, for a Wiener process, if the Riemann sum is calculated over arbitrarily point in , the Riemann sum has no limit.Remark of It Integral 2nIn the above proof process : since thequadratic variation of a Brownian motion is nonzero, the result of an Ito integral is not the same as the result of an ormal integral.Ito Differential Formulan nThis indicates a corresponding change in the differentiation rule for the composite function.It FormulanLet , where is a stochastic process. We want to knownThis is the Ito formula to be discussed in this section. The Ito formula is the Chain Rule in stochastic calculus.Composite Function of a Stochastic Process nThe differential of a function is the linear principal part of its increment. Due to the quadratic variation theorem of the Brownian motion, a composite function of a stochastic process will have new components in its linear principal part. Let us begin with a few examples.ExpansionnBy the Taylor expansion ,nThen neglecting the higher order terms, Examplen1 Differential of Risky AssetnIn a risk-neutral world, the price movement of a risky asset can be expressed by,nWe want to find dS(t)=?Differential of Risky Asset cont.n Stochastic Differential EquationnIn a risk-neutral world, the underlying asset satisfies the stochastic differential equation where is the return of over a time interval dt, rdt is the expected growth of the return of , and is the stochastic component of the return, with variance . is called volatility.Theorem 4.2 (Ito Formula)n V is differentiable both variables. If satisfies SDEthenProof of Theorem 4.2nBy the Taylor expansionnButProof of Theorem 4.2 cont.nSubstituting it into ori. Equ., we getn nThus Ito formula is true.Theorem 4.3nIf are stochastic processes satisfying respectively the following SDEnthenProof of Theorem 4.3n n By the Ito formula,Proof of Theorem 4.3 cont.nSubstituting them into above formulanThus the Theorem 4.3 is proved.Theorem 4.4nIf are stochastic processes satisfying the above SDE, thenn Proof of Theorem 4.4nBy Ito formula n Proof of Theorem