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hw4 - Derivative Securities Fall 2007 Homework 4...

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Derivative Securities, Fall 2007 – Homework 4. Distributed 10/17/07, due 10/31/07. A solution sheet to HW3 will be posted 10/25; a solution sheet to HW4 will be posted 11/8; no late HW’s will be accepted once the corresponding solution sheet has been posted. 1. Let F solve the SDE dF = μF dt + σF dw , where μ and σ are constant and w is Brownian motion. Find the SDE solved by (a) V ( F ) = AF , where A is constant (b) V ( F ) = F (c) V ( F ) = cos F (d) V ( F, t ) = F 3 t 2 . 2. We continue to assume that F solves dF = μF dt + σF dw , where μ and σ are constant and w is Brownian motion. Find a function V ( F ) such that the process t V ( F ( t )) is a martingale (i.e. the SDE describing V ( F ) has no dt term). 3. This problem should help you understand Ito’s formula. If w is Brownian motion, then Ito’s formula tells us that z = w 2 satisfies the stochastic differential equation dz = 2 wdw + dt . Let’s see this directly: (a) Suppose a = t 0 < t 1 < . . . < t N - 1 < t N = b . Show that w 2 ( t i +1 ) - w 2 ( t i ) = 2 w ( t i )( w ( t i +1 ) - w ( t i )) + ( w ( t i +1 ) - w ( t i )) 2 , whence w 2 ( b ) - w 2 ( a ) = 2 N - 1 i =0 w ( t i )( w ( t i +1 ) - w (
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