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Unformatted text preview: Continuous Time Finance Notes, Spring 2004 – Section 2, Jan. 28, 2004 Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use in connec- tion with the NYU course Continuous Time Finance. In Section 1 we discussed how Girsanov’s theorem and the martingale representation the- orem tell us how to price and hedge options. This short section makes that discussion concrete by applying it to (a) options on a stock which pays dividends, and (b) options on foreign currency. We close with a brief discussion of Siegel’s paradox. For topics (a) and (b) see also Baxter and Rennie’s sections 4.1 and 4.2 – which are parallel to my discussion, but different enough to be well worth reading and comparing to what’s here. For a discussion of Siegel’s paradox and some related topics, see chapter 1 of the delightful book Puzzles of Finance: Six Practical Problems and their Remarkable Solutions by Mark Kritzman (J. Wiley & Sons, 2000, available as an inexpensive paperback). ******** Options on a stock with dividend yield. You probably already know from Derivative Securities how to price an option on a stock with continuous dividend yield q . If the stock is lognormal with volatility σ and the risk-free rate is (constant) r then the “risk-neutral process” is dS = ( r- q ) S dt + σS dw , and the time-0 value of an option with payoff f ( S T ) and maturity T is e- rT E RN [ f ( S T )]. From the SDE we get dE RN [ S ] /dt = ( r- q ) E RN [ S ], so E RN [ S ]( T ) = e ( r- q ) T S . This is the forward price , i.e. the unique choice of k such that a forward with strike k and maturity T has initial value 0. (Proof: apply the pricing formula to f ( S T ) = S T- k .) When the option is a call, we get an explicit valuation formula using the fact that if X is lognormal with mean E [ X ] = F and volatility s (defined as the standard deviation of log X ), then E [( X- K ) + ] = F N ( d 1 )- KN ( d 2 ) (1) with d 1 = ln( F/K ) + s 2 / 2 s , d 2 = ln( F/K )- s 2 / 2 s . There is of course a similar formula for a call (easily deduced by put-call parity). What does it mean that the “risk-neutral process is dS = ( r- q ) S dt + σS dw ?” What happens when the dividends are paid at discrete times? We can clarify these points by using the framework of Section 1. Remember the main points: (a) there is a unique “equivalent martingale measure” Q , obtained by an application of Girsanov’s theorem; and (b) this Q is characterized by the property that the value V t of any tradeable asset satisfies V t /B t = E [ V T /B T |F t ] where B t is the value of a risk-free money-market account, i.e. it solves dB = rB dt with B (0) = 1. Put differently: V/B is a Q-martingale....
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This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.
- Fall '11