Question

Please prove American inequality boundary in the options market.

Answer 1

All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These are, by and large, relatively simple to price and hedge, at least under the hypotheses of the Black-Sholes model, as pricing entails only the evaluation of a single expectation. American options, which may be exercised at any time up to expiration, are considerably more complicated, because to price or hedge these options one must account for many (infinitely many!) different possible exercise policies. For certain options with convex payoff functions, such as call options on stocks that pay no dividends, the optimal policy is to exercise only at expiration. In most other cases, including put options, there is also an optimal exercise policy; however, this optimal policy is rarely simple or easily computable. We shall consider the pricing and optimal exercise of American options in the simplest nontrivial setting, the Black-Sholes model, where the underlying asset Stock pays no dividends and has a price process St that behaves, under the risk-neutral measure Q, as a simple geometric
Brownian motion: (1) St = S0 exp σWt + (r − σ 2 /2)t
Here r ≥ 0, the riskless rate of return, is constant, and
Wt is a standard Wiener process under Q.
For any American option on the underlying asset Stock, the admissible exercise policies must be stopping times with respect to the natural filtration (Ft)0≤t≤T of the Wiener process Wt .
If F(s) is the payoff of an American option exercised when the stock price is s, and if T is the expiration date of the option, then its value Vt at time t ≤ T is (2) Vt = sup τ :t≤τ≤T E(F(Sτ )e −r(τ−t) | Ft).

2. Call Options Recall that a call option has payoff (s − K)+ if exercised when the stock price is s.
Here, as always, K is the strike price of the option. Note that, for each fixed K, the payoff function (s−K)+ is convex in the argument.
1. The optimal exercise policy for the owner of an American call option is to hold the option until expiration, that is, τ = T. Proof. Let τ ≤ T be any stopping time. If the American option were exercised at time τ , the payoff would be (Sτ − K)+, and so the value at time zero to a holder of the option planning to exercise at the stopping time τ would be E(Sτ − K)+e −rτ ≤ E(Sτ e −rτ − Ke −rT )+, using the fact that τ ≤ T.

Now recall that the discounted price process :
Ste −rt t≥0 is a martingale. Since the function x 7→ (x − C)+ is convex, the following lemma implies that E(Sτ e −rτ − Ke −rT )+ ≤ E(ST e −rT − Ke −rT )+.