8.3 Perpetual American Put

1. 8.3 Perpetual American Put Stochastic Calculus for Finance II : Continuous-Time Models Steven E. Shreve Presented by CaiYu-Hong(994208019) 2012/1/5

2. Perpetual American Put

3. Underlying Assets • The underlying asset in most of this chapter has the price process S(t) given by (8.3.1) • : interest rate(strictly positive constants) • : volatility(strictly positive constants) • : Brownian motion under the risk-neutral probability measure • The Perpetual American put pays if it is exercised at time t.

4. Definition 8.3.1 • The price of the perpetual American put is defined to be (8.3.2)

5. The idea behind Definition 8.3.1 is that the owner of the perpetual American put can choose an exercise time , subject only to the condition that she may not look ahead to determine when to exercise. This risk-neutral pricing definition of the perpetual American put price appears to differ from the construction of the price of a European call in Section 4.5. The defined above is the initial capital required for an agent to hedge a short position in the American put regardless of the exercise strategy used by the owner of the put(see Corollaries 8.3.6 and 8.3.7).

6. Exercise the put a s soon as S(t) falls to the level . There is no expiration date after which the put can no longer be exercised. This makes every date like every other date; the time remaining to expiration is always the same (i.e., infinity). Because every date is like every other date, it is reasonable to expect that the optimal exercise policy depends only on the value of S(t) and not on the time variable t. The owner of the put should exercise as soon as S(t) falls “far enough” below K. In other words, the optimal exercise policy is of the form

7. Two Questions to Answer

8. Thanks for your listening.