Lecture14 - Lecture 13 Mariana Olvera-Cravioto Columbia...

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Lecture 13 Mariana Olvera-Cravioto Columbia University [email protected] March 7th, 2012 IEOR 4404, Simulation Lecture 13 1/15
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A lookback option I Consider an option that at its expiration date T pays ( M T - K ) + where M T = max 0 t T S t . I Equivalent to allowing the owner to buy the stock at price K exactly on the day that it reached its highest price over the period [0 , T ] . I The price of this option is then E Q [( M T - K ) + ] I Suppose we model the price of the stock according to a GBM with risk neutral parameters ( r, σ ) , and we divide the interval [0 , T ] into N evenly spaced subintervals. Let Δ = T/N and set t n = n Δ for 0 n N . I Algorithm: I Generate S t 1 , S t 2 , . . . , S t N according to the GBM algorithm. I Set M T = max 0 n N S t n , and return ( M T - K ) + . IEOR 4404, Simulation Lecture 13 2/15
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American options I An American option may be exercised at any time before the expiration date of the option. I The payoffs of American options are the same as for Europeans, that is, Call option: ( S T - K ) + , Put option: ( K - S T ) + I The price of the options is given by Call option: sup τ E Q [( S τ - K ) + ] Put option: sup τ E Q [( K - S τ ) + ] where the supremum is taken over all the stopping times τ with 0 τ T . I Unlike European options, there is no closed form formula for their price. IEOR 4404, Simulation Lecture 13 3/15
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Pricing an American option using a suboptimal policy I Let S n , n 0 denote the price of a stock at the end of day n , and assume S n = S 0 exp { X 1 + · · · + X n } , n 0 , where X 1 , X 2 , . . . is a sequence of iid Normal ( ν, σ 2 ) r.v.’s. I We want to price an American put option with strike price K that expires in T days. I A policy is a rule that tells us when to exercise the option (defines the stopping time τ appearing in the price expressions). I Policy: exercise the option when there are m days to go whenever for each i = 1 , . . . , m , that action leads to a higher expected payoff than letting exactly i days go by and then either exercise it (if the price at that point is lower than K ) or give up on ever exercising.
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