Continuous Processes. Brownian motion Stochastic calculus Ito calculus

Transcription

1 Continuous Processes Brownian motion Stochastic calculus Ito calculus

2 Continuous Processes The binomial models are the building block for our realistic models. Three small-scale principles in continuous processes The value can change at any time and from moment to moment Any real number can be taken as a value. The process changes continuously the value can not make instantaneous jumps. Remark: Jump-diffusion processes (we don t worry about it in this class).

3 Louis Bachelier (1900) analyzed the motion of the Paris stock exchange in his Ph.D. dissertation. The term Brownian motion is in honor of the English botanist Robert Brown, and it refers to the physical phenomenon that minute particles move about randomly in a fluid. The Brownian motion is one of the simplest stochastic processes, --- it is a limit of simple random walks. The mathematical model of Brownian motion can be used to describe stock market fluctuations.

4 /3/ /3/ /3/ /3/ /3 / /3/ /3 / /3 / /3 / /3/ /3/ /3/ /3/ /3/2006 6/3 / Google stock price Brownian motion 1 /3/ /3 / /3/2005

5 A special family of discrete binomial processes --- the random walk W n (t). Definition Another description

6 Random walks of 25 and 100 steps

7 R code The distribution of W n (t) tends to N(0,t). --- Why? Question: (1) What happens if we change binomial to N(0,1)? (2) What happens if we only assume E(X)=0, Var(X) is finite?

8 Definition: The process W = (W t : t 0) is a P-Brownian motion if and only if (i) W t is continuous and W 0 = 0; (ii) The value of W t is distributed, under P, as N(0,t); (iii) The increment W s+t W s is distributed as N(0,t), under P, and is independent of F s, the history of what the process did up to time s. Question: Why is P-measure? What is the difference between W t and W n (t)?

9 Covariance function Using the independent increments assumption, we could find the covariance of the Brownian motion. For instance, if 0<s < t,

10

11

12 Odd properties of Brownian motion (BM) BM is continuous but nowhere differentiable. BM could hit any real value, but, with probability one, it will be back down again to zero. BM is self-similar. BM is also called a Wiener process.

13 Brownian motion with drift Questions What is the covariance function? What is P(S t > a)?

14 Geometric Brownian motion (GBM) GBM with drift is a model often used for stock prices. Assume that µ is a drift factor and σ is a noise factor. Question: What is the covariance function? What is P(S t > a)?

15 Continuous Processes Brownian motion Stochastic calculus Ito calculus

16 Stochastic Calculus Consider functions of Brownian motion, can we establish certain calculus rules? Recall that, in Newtonian differentials Differentiable functions can be approximated by piecewise linear functions. The old differential tools can NOT be used for Brownian motion. Why?

17 Stochastic Calculus In stochastic differentials, Brownian motions have self-similarity property, in another word, we can NOT approximate Brownian motions (or functions of them) by piecewise linear functions. A stochastic process X will have a Newtonian term based on dt and a Brownian term, based on the diffusion term, dw t. drift volatility

18 Stochastic Calculus A stochastic process X is a continuous process (X t : t 0) such that X t can be written as where σ and µ are random F-previsible processes such that The differential form of this equation is:

19 Stochastic Calculus Uniqueness of volatility σ and drift µ

20 Stochastic Calculus Uniqueness of volatility σ and drift µ A stochastic differential equation (SDE) for X is: In a simple case, where σ and µ are constants, the SDE for X becomes What is the solution of