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1、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 Brow
2、nian 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 mathematica
3、l 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 sequence n Random WalknConsider a time period 0,T, which can be divided into N equal inter
4、vals. 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
5、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
6、(0)=0,W(t) is a continuous function of t.n2) Normal increments: For any t0,W(t) N(0,t), and for 0 s 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 ti
7、me only, then the investment strategy can only be adjusted on trading days, and the gain (loss) at the time interval is nTherefore the total profit in 0,T isDefinition of It IntegralnIf f(t) is a non-anticipating stochastic process, such that the limit exists, and is independent of the partition, th
8、en 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 a
9、t 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
10、of It Integral 2nIn the above proof process : since the quadratic 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
11、 function.It FormulanLet , where is a stochastic process. We want to know nThis 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
12、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 Differe
13、ntial 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
14、 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 SDE then Proof of Theorem 4.2nBy the Taylor expansion nBut
15、Proof 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 SDE nthen Proof of Theorem 4.3n n By the Ito formula, Proof of Theorem 4.3 cont.nSubstituting them into above formulanThus the Th
16、eorem 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 4.4 cont.nThus by Theorem 4.3, we haven nTheorem is proved.RemarknTheorems 4.3-4.4 tell us: nDue to the change in the Chain Rule for differentiating comp
17、osite function of the Wiener process, the product rule and quotient rule for differentiating functions of the Wiener process are also changed.nAll these results remind us that stochastic calculus operations are different from the normal calculus operations!Multidimensional It formulanLet be independ
18、ent standard Brownian motions,n where Cov denotes the covariance:n Multidimensional EquationsnLet be stochastic processes satisfying the following SDEs where are known functions.Theorem 4.5nLet be a differentiable function of n+1 variables, are stochastic processes , thenwhere Summary 1nThe definiti
19、on of the Brownian motion is the central concept of this chapter. Based on the quadratic variation theorem of the Brownian motion, we have established the basic rules of stochastic differential calculus operations, in particular the Chain Rule for differentiating composite function-the Ito formula,
20、which is the basis for modeling and pricing various types of options.Summary 2nBy the picture of the Brownian motion, we have established the relation between the discrete model (BTM) and the continuous model (stochastic differential equation) of the risky assetnprice movement. This sets the ground for further study of the BTM for option pricing (such as convergence proof).
