 Solution

ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2009/b/6/d/solution

The exchange option has terminal payoffs determined by the two successive up/down moves:

path uu u d, d u dd S_{1} (2) 19611264 S_{2} (2) 14410881 (S_{1} (2) - S_{2} (2))^{+} 5240

The two stocks span both one-step states, so this finite market is complete. The risk-neutral pricing uses the independent transition probabilities

q = 1/3

and

1 - q = 2/3

. Since the synthetic risk-free rate is zero,

V_{0} = \frac{1}{9} 52 + 2 (\frac{1}{3} \frac{2}{3}) 4 + \frac{4}{9} 0 = \frac{68}{9} .

The price is approximately 7.56 in the currency units of the stock prices.

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