Stochastic

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Unformatted text preview: ms and use the previous calculation. Condition (c) is a natural assumption for the physical system which motivated this deﬁnition: the erratic movement of a pollen grain in water as seen under a microscope by Brown in 1825. Using the new deﬁnition, our stock price model can be written as St = S0 · exp(µt + Bt ) (6.26) where Bt is a standard Brownian motion Here µ is the exponential growth rate of the stock, and is its volatility. If we also assume that the per period interest rate in the approximating model is rh, and recall that ✓ 1 1 + rh ◆t/h = 1 1 ! rt = e e (1 + rh)t/h rt then the discounted stock price is e rt St = S0 · exp((µ r)t + Bt ) By the formula for the moment generating function for the normal with mean 0 and variance 2 t, see (5.15), E exp( ( 2 /2)t + Bt ) = 1 Since Bt has independent increments, if we let µ=r 2 /2 (6.27) then reasoning as for the exponential martingale, Example 5.6, the discounted stock price, e rt St is a martingale. Extrapolating wildly from di...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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