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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 + rh ◆t/h = 1
! rt = 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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