Past exam of the martingales and financial mathematics course of the University of Oxford / 2010 / B / 4

Short one share at time zero and invest the one hundred pounds received in the bank asset. The initial net investment is zero. After one period, repurchasing the share leaves $100e^{0.2}-S_1$. Since $S_1\le 120$ and $e^{0.2}>1+0.2=1.2$, this payoff is strictly positive in both states. Hence this is an arbitrage opportunity.

The risk-neutral probability must make the expected stock multiplier equal the bank multiplier:Both up and down paths have positive probability under this measure. A self-financing portfolio's discounted value is then a Q-martingale, since the discounted bank asset is constant and the discounted stock is a martingale. An arbitrage with zero initial wealth would have nonnegative terminal value and positive value in some possible state, giving a strictly positive Q-expectation and contradicting its zero discounted initial value. Thus the market with $e^r=1.1$ is arbitrage free.

Retain the whole price history when calculating the arithmetic average. The Asian option payoffs at expiry areIn particular $ud$ and $du$ 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.

Use $q=1/2$ and discount one step by $R=1.1$. At the up node, $A_1=110$ and immediate exercise pays $5$. Continuation is worth $(0+5/3)/(2R)=25/33<5$. At the down node, $A_1=100$ and immediate exercise pays $15$, while continuation is $(25/3+15)/(2R)=350/33<15$. Thus the American put option should be exercised at either time-one node, and

At time zero, exercise pays $115-A_0=15$, while continuation is worthTherefore exercise at time zero is optimal, and $V_0=15$ pounds.

The discounted unexercised option-value process is not a Q-martingale: already ${\mathbb{E}}_Q[V_1/B_1]=100/11<15=V_0/B_0$. 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.