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Unformatted text preview: Derivative Securities Fall 2007 Section 8 Notes by Robert V. Kohn, extended and improved by Steve Allen. Courant Institute of Mathematical Sciences. American and exotic options. We have thus far focused on European options. This weeks topic is the valuation and hedging of American and exotic options. This short document (4 pages) discusses only American options. Please also read Steve Allens Section 8 notes (posted on Blackboard); they focus mainly on (a) the numerical valuation of path-dependent options, and (b) creation of a binomial tree thats consistent with an observed volatility skew/smile. ************************ American options . American options are different in that they permit early exercise: the holder of an American option can exercise it at any time up to the maturity T . Of the options actually traded in the market, the majority are American rather than European. Clearly an American option is at least as valuable as the analogous European option, since the holder has the option to keep it to maturity. Fact: An American call written on a stock that earns no dividend has the same value as a European call; early exercise is never optimal. To see why, suppose the strike price is K and consider the value of the American option now, at some time t < T . Exercising the option now achieves a value at time t of s t- K . Holding the option to maturity achieves a value at time t equal to that of a European call, c [ s t , K, T- t ]. Without using the Black- Scholes formula (thus without assuming lognormal stock dynamics) we know the value of a European call is at least that of a forward with the same strike and maturity. Thus holding the option to maturity achieves a value at time t of at least s t- e- r ( T- t ) K . If r > 0 this is larger than s t- K . So early exercise is suboptimal, as asserted. The preceding is in some sense a fluke. When the underlying asset pays a dividend early exercise of a call can be optimal. But the simplest example where early exercise occurs is that of a put on a non-dividend-paying stock: Fact: An American put written on a stock that earns no dividend can have a value greater than that of the associated European put; early exercise can be optimal. To see why, consider once again the value of the American option now, at some time t < T . Exercising....
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