Past exam of the martingales and financial mathematics course of the University of Oxford / 2007 / B / 7

Use the usual nondegenerate positive-rate case $r>0$, $\sigma >0$, $K>0$. Define the Black-Scholes equationThe American value dominates immediate exercise. Its discounted value is a supermartingale under the risk-neutral measure, so $\mathcal{L}V\le 0$, with equality where it is optimal to continue. The obstacle problem isEquivalently $\max (\mathcal{L}V,g-V)=0$.

For the put, write the exercise boundary as $S=b(t)$. On $S>b(t)$ solve $\mathcal{L}V=0$; on $0<S\le b(t)$ set $V=K-S$. Value matching and smooth pasting requireThe remaining conditions are $V(t,0)=K$, $V(t,S)\to 0$ as $S\to \infty$, and the terminal contingent claim payoff, with $b(T-)=K$ in this positive-rate setting. The boundary is an unknown to be found with the price, which is what makes this a free boundary problem.

put-call parity and nonnegativity of the European put option giveFor $r\ge 0$ this is at least $S-K$, and the call value is also nonnegative, so $C^E$ dominates the immediate call exercise contingent claim payoff. Its discounted price is a martingale. At any admissible exercise time, optional stopping theorem therefore bounds the expected discounted call option by the European call option price. Exercising at maturity attains that price. Hence American and European calls coincide for this non-dividend model with nonnegative rates.

For the strict put comparison assume $r>0$, $\sigma >0$, $S>0$ and $t<T$. Choose $u$ strictly between $t$ and $T$. A European put option at $u$ has value at most $K{e}^{-r(T-u)}$. On the eventimmediate exercise gives $K-S_u>K{e}^{-r(T-u)}\ge P^E(u,S_u)$. This event has positive probability because the lognormal stock density is positive on $(0,\infty )$. Exercise there at time $u$, and otherwise hold to maturity. This strategy strictly improves the European put option expected value, provingunder these positive-rate, pre-maturity assumptions.

The source's strict statement needs those qualifications. At maturity the prices agree, and at $r=0$ the put option is a convex function of the martingale stock and therefore is a submartingale, so waiting to maturity is optimal and the American and European puts coincide. Negative rates also invalidate the nonnegative-rate call argument. These are mathematical assumptions of the comparison, rather than changes to the printed question.

LetThe homogeneous function of the Black-Scholes formula gives its at-the-money value at the start date as $S_{T_1}c(T-T_1)$. Before that date, risk-neutral valuation and the discounted stock martingale giveAfter the start date the strike price is the realized value $S_{T_1}$, so the complete answer isAt maturity the second line is $(S_T-S_{T_1}{)}_{+}$. After entry the price depends on the historical strike price as well as the current stock, so that history cannot be discarded when giving the answer for all times.

At an entry date the holder chooses between entering the at-the-money call immediately and retaining the right to choose at the next date. The Snell envelope therefore gives, under risk-neutral valuation,with terminal value zero if entry has never occurred.

homogeneous function makes $B^{FSC}(T_i,s)=s{b}_i$. The discounted stock expected value is $s$, so the recursion reduces toIn the positive-rate case in the hint, put $\gamma =r/\sigma$ and $\beta =\sigma /2$. Then $c(u)$ is exactly its increasing function. The largest remaining maturity is $T-T_i$, so $b_i=c(T-T_i)$: immediate entry is optimal at every date while unentered.

At time zero this yieldsThus the Bermudan choice of start date adds no value over the ordinary at-the-money European call option in this setting. The maximum formula remains the general recursion; using monotonicity to select immediate entry uses the rate assumption supplied with the hint. This is immediate entry for a Bermudan forward-start call.