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Math 489/Math 889
Stochastic Processes and
Advanced Mathematical Finance
Dunbar, Fall 2010


Problem Statement

Consider a stock whose price today is $50. Suppose that over the next year, the stock price can either go up by 10%, or down by 3%, so the stock price at the end of the year is either $55 or $48.50. The interest rate on a $1 bond is 6%. If there also exists a call on the stock with an exercise price of $50, then what is the price of the call option? Also, what is the replicating portfolio?

Solution

The payoffs are f(SU) = max(SU - 50, 0) = 5 and F(SD) = max(SD - 50, 0) = 0.

A replicating portfolio must satisfy:

 55f + exp(0.06)y = 5 48.50f + exp(0.06)y = 0.

Solving this system we obtain that f = 0.769 and y = -35.14, so we own 0.769 shares of stock, and short (borrow) 35.14 bonds. The value of the portfolio and therefore the option is then fS + y = 3.32.

We can also figure the value from the risk-neutral measure, namely

 exp(0.06)- 0.97 p = ----------------= 0.706434 1.1 - 0.97

and

1- p = 0.293565

and so

V = exp(- 0.06)(p5 + (1 - p)0) = 3.32.

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