Stochastic

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

Ask a homework question - tutors are online