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Ito_lemma_ver3

Ito_lemma_ver3 - Its Formula o Background From standard...

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Itˆo’s Formula Background. From standard Brownian motion W ( t ) one constructs arithmetic Brownian motion by dX ( t ) = μdt + σdW ( t ) (1) where the drift μ and the diffusion coefficient σ are constants. There is an easy solution to this SDE, viz. X ( t ) = X (0) + μt + σW ( t ) (2) As convenient as this might be, this version of Brownian dynamics is not really suitable for modeling stock prices. Two important objections are (i) the returns on a stock in this model do not depend on the price of the stock but rather are the same for every price level, and (ii) stock prices in this model can be negative. Introduce geometric Brownian motion with two parameters to obtain a stock pricing model which responds to the two objections above. The two parameters are: μ – the average annual continuously compounded rate of return of the stock. σ – the annual standard deviation of the rate of return. The Itˆo process for this model is: dS ( t ) = μS ( t ) dt + σS ( t ) dW ( t ) (3) where μ and σ are constants. Note that the mean and variance of the changes in the stock price do depend on the magnitude of the price. We will see shortly that the stock prices in this model are always positive. 1
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The big question is how to find an explicit expression for S ( t ). Itˆo’s formula is the main computational tool for these stochastic processes. For motivation, consider a basic example. Suppose that X ( t ) is an asset (like a stock) price process and that f ( t, X ( t )) is the price process of a derivative contract (like an option) written on the value of the asset. Itˆo’s formula will give the relation between the two dynamics in the SDE for the new stochastic process Y ( t ) = f ( t, X ( t )) (4) Recall that a stochastic process X ( t ) is called an Itˆo process (sometimes called a diffusion process ) if X ( t ) satisfies a stochastic differential equation of the form dX ( t ) = μ ( t, X ( t )) dt + σ ( t, X ( t )) dW ( t ) (5) with the properties: (i) The functions μ ( t, x ) and σ ( t, x ) are twice continuously differentiable deterministic functions of t and x . (ii) The composite functions μ ( t, X ( t )) and σ ( t, X ( t )) are adapted processes (relative to the underlying Brownian filtration). (iii) The process X ( t ) is square-integrable, i.e. t 0 E [ X 2 s ] ds < for all 0 t T (6) These conditions are technical conditions which we will discuss later. Note: the last section of these notes is a review of deterministic approximations which will motivate the stochastic version.
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