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1、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 Bro
2、wnian 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
3、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
4、. 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 t
5、he 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
6、(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 t, W(t)-W(s) is normally distributed with mean 0 and variance t-s, i.e., 3) Independence of increments: for any choice of in 0,T with the increments are independent.,Continuous Models of Asset Price M
7、ovement,Introduce the discounted value of an underlying asset as follows: in time interval t,t+t, the BTM can be written as,Lemma,If ud=1, is the volatility, letting then under the martingale measure Q,Proof of the Lemma,According to the definition of martingale measure Q, on t,t+t, thus by straight
8、forward computation,Proof of the Lemma,Moreover, since,Proof of the Lemma cont.,by the assumption of the lemma, input these values to the ori. equation. This completes the proof of the lemma.,Geometric Brownian Motion,By Taylor expansion neglecting the higher order terms of t, we have,Geometric Brow
9、nian Motion cont.,By definition therefore after partitioning 0,T, at each instant , i.e.,Geometric Brownian Motion cont.-,Geometric Brownian Motion cont.-,This means the underlying asset price movement as a continuous stochastic process, its logarithmic function is described by the Brownian motion.
10、The underlying asset price S(t) is said to fit geometric Brownian motion. This means: Corresponding to the discrete BTM of the underlying asset price in a risk-neutral world (i.e. under the martingale measure), its continuous model obeys the geometric Brownian motion .,Definition of Quadratic Variat
11、ion,Let function f(t) be given in 0,T, and be a partition of the interval 0,T: the quadratic variation of f(t) is defined by,Quadratic Variation for classical function,Theorem 4.1,Let be any partition of the interval 0,T, then the quadratic variation of a Brownian motion has a limit as follows:,Path
12、 of a Brownian motion,For any let be an arbitrary partition of the interval and be the quadratic variation of the Brownian motion corresponding to the partition , then by Theorem 4.1, Referring to the conclusion regarding the differentiable function, we have: The path of a Brownian motion W_t as a r
13、andom walk of a particle is continuous everywhere but differentiable nowhere.,Remark,If dt 0 (i.e. 0), let denote the limit of then by Theorem 4.1, Hence neglecting the higher order terms of dt, i.e. neglecting higher order terms, the square of the random variable is a definitive infinitesimal of th
14、e order of dt.,An Example,A company invests in a risky asset, whose price movement is given by Let f(t) be the investment strategy, with f(t)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 b
15、y: 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
16、, 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 q
