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# ch3 - Chapter 3 American Options in Discrete Time In this...

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Chapter 3 American Options in Discrete Time In this short chapter we’ll brieﬂy discuss the pricing and hedging of American options and the relationship with the “optimal stopping problem.” We will continue to work in a finite market model and we will assume that this is complete so that every ECC can be hedged. By the second fundamental theorem there is a unique martingale measure which is denoted by P . 3.1 The Price Process for an American Op- tion In Chapter 2, we have considered the pricing and hedging of a general ECC X in discrete time. So X is the payoff (or value) of an ECC at the terminal date T and it is an F T measurable random variable. Whereas an ECC can only be exercised at time T , a general ACC ( American contingent claim ) may be exercised at any time n = 1 , 2 , . . . , T . To model this we construct an adapted process ( X ( n ) , n ∈ T ) whereby each X ( n ) is an ECC with terminal date n , e.g. for an American put option with exercise price k , X ( n ) = max { k S ( n ) , 0 } . We call ( X ( n ) , n ∈ T ) the pay-off process . We want to price the option at any time n and so we want to construct a process ( U ( n ) , n N ) which we will call the price process . The trick with American options is to argue backwards in time starting at time T . Clearly we must have U ( T ) = X ( T ) . What is the price of the process at time T 1? At this time, the holder has two options - either exercise the option to earn X ( T 1) or wait until time 19

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T to earn X ( T ). In the second case, the option is equivalent to an ECC held over the time period T 1 to T and so the writer of the option needs to invest β ( T 1) 1 E ( β ( T ) X ( T ) |F T 1 ) in a replicating portfolio in order to generate X ( T ) at time T . Hence we can argue that the price of the option at time T 1 should be U ( T 1) = max { X ( T 1) , β ( T 1) 1 E ( β ( T ) X ( T ) |F T 1 ) } . Iterating the argument, we deduce that the price at time n 1 is U ( n 1) = max { X ( n 1) , β ( n 1) 1 E ( β ( n ) U ( n ) |F n 1 ) } , and if we discount both pay-off and price processes by defining U ( n ) = β ( n ) U ( n ) and X ( n ) = β ( n ) X ( n ) we find that for n = 1 , 2 , . . . , T U ( n 1) = max { X ( n 1) , E ( U ( n ) |F n 1 ) } . (3.1.1) This ensures that at each time n 1, the writer has enough funds to either cover the cost of the option being exercised at time n 1 or generate enough funding to cover all eventualities at time n . Mathematically, the structure obtained in (3.1.1) has been previously studied by probabilists who are interested in modelling optimal stopping prob- lems . We will look at this in the next section. Before we do that notice that there is a nice pattern in (3.1.1). In particular, if X ( n 1) E
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