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,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,*,*,Chapter 4,Brownian Motion,& It,Formula,Stochastic Process,The price movement of an underlying asset is a,stochastic process,.,The French mathematician,Louis Bachelier,was the first one to describe the stock share price movement as a Brownian motion in his 1900 doctoral thesis.,introduction to the Brownian motion,derive the continuous model of option pricing,giving the definition and relevant properties Brownian motion,derive stochastic calculus based on the Brownian motion including the Ito integral & Ito formula.,All of the description and discussion emphasize clarity rather than mathematical rigor.,Coin-tossing Problem,Define a random variable,It is easy to show that it has the following properties:,& are independent,Random Variable,With the random variable, define a random variable and a random sequence,Random Walk,Consider 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,A,random walk,is defined in ,0,T,:,is called the,path,of the random walk.,Distribution of the Path,Let,T,=1,N,=4,=1/4,Form of Path,the path formed by linear interpolation between the above random points. For,=1/4,case, there are 24=16 paths.,t,S,1,Properties of the Path,Central Limit Theorem,For 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.,Consider limit as,0.,Definition of Winner Process(Brownian Motion),1),Continuity of path,: W(0)=0,W(t) is a continuous function of t.,2),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.,Partition 0,T by:,If 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 is,Therefore the total profit in 0,T is,Definition of It,Integral,If f(t) is a non-anticipating stochastic process, such that the limit,exists, and is independent of the partition, then the limit is called the,It,Integral,of f(t), denoted as,Remark of It,Integral,Def. of the,Ito Integral,one of the,Riemann integral,.,- the Riemann sum under a particular partition.,However, f(t) -,non-anticipating,Hence in the value of f must be taken at the left endpoint of the interval,not,at an arbitrary,point,in,.,Based 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,.,So, for a,Wiener process, if the Riemann sum is calculated over,arbitrarily,point in, the Riemann sum has,no limit,.,Remark of It,Integral 2,In 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 Formula,This indicates a corresponding,change,in the,differentiation,rule,for the,composite function,.,It,Formula,Let , where is a stochastic process. We want to know,This 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,The 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.,Expansion,By the Taylor expansion ,Then neglecting the higher order terms,Example,1,Differential of Risky Asset,In a risk-neutral world, the price movement of a risky asset can be expressed by,We want to find,dS(t)=?,Differential of Risky Asset cont.,Stochastic Differential Equation,In 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),V,is differentiable,both variables. If satisfies SDE,then,Proof of Theorem 4.2,By the Taylor expansion,But,Proof of Theorem 4.2 cont.,Substituting it into ori. Equ., we get,Thus Ito formula is true.,Theorem 4.3,If are stochastic processes satisfying respectively the following SDE,then,Proof of Theorem 4.3,By the Ito formula,Proof of Theorem 4.3 cont.,Substituting them into above formula,Thus the Theorem 4.3 is proved.,Theorem 4.4,If are stochastic processes satisfying the above SDE, then,Proof of Theorem 4.4,By Ito formula,Proof of Theorem 4.4 cont.,Thus by Theorem 4.3, we have,Theorem is proved.,Remark,Theorems 4.3-4.4 tell us:,Due to the change in the Chain Rule for differentiating composite function of the Wiener process, the product rule and quotient rule for differentiating functions of the Wiener process are also changed.,All these results remind us that stochastic calculus operations are different from the normal calculus operations!,Multidimensional It,formula,Let be independent standard Brownian motions,where,Cov,denotes the covariance:,Multidimensional Equations,Let be stochastic processes satisfying the following SDEs,where are known functions.,Theorem 4.5,Let be a differentiable function of n+1 variables, are stochastic processes , then,where,Summary 1,The definition 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, which is the basis for modeling and pricing various types of options.,Summary 2,By 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 asset,price movement. This sets the ground for further study of the BTM for option pricing (such as convergence proof).,
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