 Solution

ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2010/b/4/c/solution

Retain the whole price history when calculating the arithmetic average. The Asian option payoffs at expiry are

path uu u d d u dd (S_{0}, S_{1}, S_{2}) (100, 120, 144) (100, 120, 120) (100, 100, 120) (100, 100, 100) A_{2} 364/3 340/3 320/3 100 (115 - A_{2})^{+} 0 5/3 25/3 15

In particular

u d

and

d u

have the same terminal stock price but different averages and payoffs. The average is an additional state variable; the terminal stock nodes cannot be merged for this claim.

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