 Solution

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

At time zero, exercise pays

115 - A_{0} = 15

, while continuation is worth

\frac{E _{Q} V _{1}}{R} = \frac{( 5 + 15 ) /2}{1.1} = \frac{100}{11} < 15.

Therefore exercise at time zero is optimal, and

V_{0} = 15

pounds.

The discounted unexercised option-value process is not a Q-martingale: already

E_{Q} [V_{1} / B_{1}] = 100/11 < 15 = V_{0} / B_{0}

. It is the Snell envelope of the discounted exercise payoff, hence a supermartingale in general, with strict decrease of expected value when immediate exercise beats continuation. Stopping at an optimal exercise time restores the stopped martingale property; that is a different assertion from the unexercised price being a martingale.

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