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


Problem Statement

A stock price is currently $50. it is known that at the end of 6 months, it will either be $60 or $42. The risk-free rate of interest with continuous compounding on a $1 bond is 12% per annum. Calculate the value of a 6-month European call option on the stock with strike price $48 and find the replicating portfolio.

This problem is adapted from Hull, page 240, problem 10.16

Solution

The payoffs are f(SU) = max(SU - 50, 0) = 10 and F(SD) = max(SD - 50, 0) = 0.

A replicating portfolio must satisfy:

60f + exp(0.12 × 0.5)y = 10 42f + exp(0.12 × 0.5)y = 0.

Solving this system we obtain that f = 0.556 and y = -21.97, so we own 0.556 shares of stock, and short (borrow) 21.97 bonds. The value of the portfolio and therefore the option is then phiS + y = 5.80.

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

 exp(0.06) - 0.84 p = ---------------- = 0.616 1.2 - 0.84

and

1 - p = 0.384

and so

V = exp(- 0.06)(p10 + (1 - p)0) = 5.80.

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