Lecture 10 Stochastic Process & BS Model

Lecture 10 Stochastic Process & BS Model - Lecture...

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Lecture 10: (Ch12 & 13) Discrete - time stochastic process is one where the value of the variable can change only at certain fixed points in time, whereas a continuous - time stochastic process is one where changes can take place at any time. In a continuous variable process, the underlying variable can take any value within a certain range, whereas in a discrete variable process, only certain discrete values are possible. Markov Process Stochastic process for which only the present value of a variable is relevant for predicting the future. The past history of the variable is irrelevant. Markov property of stock prices is consistent with the weak form of market efficiency. o The Markov property implies that the probability distribution of the price at any particular future time is not dependent on the particular path followed by the price in the past. Wiener Process A Wiener process is a particular type of Markov stochastic process with mean change of zero and a variance rate of 1.0 per year. A variable z follows a Wiener process if it has the following two properties: Property 1 : The change in z during a short period of time Δ t is *where E has a standardized normal distribution φ (0,1) Property 2 : The value change of z for any two different short intervals of time are independent. This property implies that z follows a Markov process * φ (0,1) a Normal probability distribution with mean ‘m’ and variance ‘v 2 ’. Generalized Wiener process A generalized Wiener process for a variable x is
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where a and b are constants. Δ x is normally distributed with mean a Δ t and standard deviation Wiener Process: Continuous Time In the limit Δ t 0, we can write the generalized Wiener process as dx = a dt + b dz The ‘a dt’ term implies that x has an expected drift rate of a per unit of time. The ‘b dz’ term on the right-hand side of the equation can be regarded as adding noise or variability to the path followed by x . Ito Process An Ito process is a generalised Wiener process where a and b are functions of the values of the underlying variable x and time t : dx = a(x,t)dt + b(x,t)dz Both the drift rate and variance rate of an Ito process are liable to change over time. Process of Stock Prices One type of Ito process takes the form: dS is the change in S over a small time interval dt μ is the expected rate of return per unit time for S σ is the volatility of S. This is called geometric Brownian motion This is the most widely used model of stock price behaviour. S is lognormally distributed If we use the following geometric Brownian motion process to model stock price S:
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continuously compounded returns are normally distributed. Lower bound of S is 0
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Lecture 10 Stochastic Process & BS Model - Lecture...

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