Chapter 3: American options and early exercise Two questions: (1) option price (2) what’s the optimal exercise policy Probabilistic approach

BTM For European vanilla options For American vanilla options

Continuous-time pricing model for American put

American call What’s the solution?

Continuous dividend payments

American put

American call

A free boundary problem

Perpetual American options Can be exercised at any time, no expiry

continued

Optimal exercise boundary (as a function of time to expiry (tau))

Put-Call parity The parity doesn’t hold for American options

Put-call symmetry The symmetry holds also for American options

Continued Call option C(S1,t;X1, r,q) max(S1-X1,0) to exchange one asset X with dividend yield r for another asset S1 with dividend yield q Put option P(S2,t;X2, r2,q2) max(X2-S2,0) to exchange one asset S2 with dividend yield q2 for another asset X2 with dividend yield r2 If we assume r2=q, q2=r, S2=X, X2=S (other parameters, such as maturity and volatility, are same), then C(S,t;X, r,q)= P(S2,t;X2, r2,q2) =P(X,t;S,q,r) Early exercise doesn’t affect this result

Continued

Bermudan options Early exercise is allowed only at certain dates. BTM

Model-independent and model-dependent Price of forward contract put-call parity, … American call should never be exercised early if no dividend payments Based on the no-arbitrage principle only Model-dependent Black-Scholes equations Black-Scholes prices Put-call symmetry Based on the assumption of geometric Brownian motion of underlying asset as well as the no-arbitrage principle All model-independent results must be valid in the Black-Scholes framework