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

2. BTM
For European vanilla options
For American vanilla options

3. Continuous-time pricing model for American put

4. American call
What’s the solution?

5. Continuous dividend payments

6. American put

7. American call

8. A free boundary problem

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

10. continued

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

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

13. Put-call symmetry
The symmetry holds also for American options

14. 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

15. Continued

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

18. 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