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Math 3589 Introduction to Financial Mathematics Homework Assignment #8 Solutions Ohio State University £6.50   Add to cart

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Math 3589 Introduction to Financial Mathematics Homework Assignment #8 Solutions Ohio State University

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Introduction to Financial Mathematics Homework Assignment #8 Solutions Exercise 16. Consider the two period binomial model, with the stock price at time t = 0, S0 = 4, the “up factor” u = 2, “down factor” d = 1/2, and risk free interest rate r = 1/4 so that ˜p = 1/2. Assume in each per...

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  • April 3, 2023
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Math 3589
Introduction to Financial Mathematics

Homework Assignment #11



1. Prove that a symmetric random walk is a martingale.
Proof: Let {Mn }∞
n=0 be a symmetric random walk. Then

1 1
En [Mn+1 ] = (Mn + 1) + (Mn − 1) = Mn .
2 2



2. Prove that a symmetric random walk is a Markov process.
Proof: Let {Mn }∞ n=0 be a symmetric random walk and let f (x) be any
function of a dummy variable x. Then
1 1
En [f (Mn+1 )] = En [f (Mn + Xn+1 )] = f (Mn + 1) + f (Mn − 1) = g(Mn ).
2 2



3. Prove that the first passage time, τm , m = 0, 1, 2, . . . is a stopping time.
Proof: It’s clear from the definition that τm takes only the values 0, 1, . . .
and ∞. Furthermore, if τm = n for a sequence ω1 . . . ωj . . ., then this means
that the number of heads minus the number of tails in the first n coin-tosses
is m, and that this is not true for any smaller value of n. In particular, no
information about any coin-tosses after the n-th is used in determining τm ,
so τm satisfies the second condition for a stopping time:

τm (ω1 . . . ωn ωn+1 . . . ) = n ⇐⇒ τm (ω1 . . . ωn ω
en+1 . . . ) = n

for all sequences which agree on the first n tosses.

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