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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 Girsanovs 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 Siegels paradox. For topics (a) and (b) see also Baxter and Rennies sections 4.1 and 4.2 which are parallel to my discussion, but different enough to be well worth reading and comparing to whats here. For a discussion of Siegels 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 and the riskfree rate is (constant) r then the riskneutral process is dS = ( r q ) S dt + S dw , and the time0 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 putcall parity). What does it mean that the riskneutral process is dS = ( r q ) S dt + S 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 Girsanovs 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 riskfree moneymarket account, i.e. it solves dB = rB dt with B (0) = 1. Put differently: V/B is a Qmartingale....
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