section3 (1)

# section3 (1) - Continuous Time Finance Notes, Spring 2004...

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Continuous Time Finance Notes, Spring 2004 – Section 3, Feb. 4, 2004 Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use in connec- tion with the NYU course Continuous Time Finance. Announcement: There will be no class on Wednesday, February 11. These notes wrap up our treatment of the probabilistic framework for option pricing. We conclude the discussion of problems with one source of randomness by discussing the “mar- ket price of risk” and the use of alternative numeraires (leading to consideration of equivalent martingale measures other than the risk-neutral one). Then we discuss the analogous theory for complete markets with multiple sources of randomness. Finally we discuss two impor- tant examples where there are two sources of randomness: (a) exchange options, and (b) quanto options. Except for exchange options, this material is covered in Sections 4.4-4.5 and corresponding part of Hull (5th edition) is Chapter 21 (my discussion of exchange options is from there). Hull is relatively terse, but well worth reading. ***************** The market price of risk. We continue a little longer the hypothesis that the market has just one source of randomness (a scalar Brownian motion w ). Recall what we achieved in Section 1: let r t be the risk-free rate, and B the associated money-market instrument (characterized by dB = r t B dt with B (0) = 1); let S be tradeable, and assume S solves an SDE of the form dS = μ t S dt + σ t S dw . We showed that if S t /B t is a martingale relative to Q then any payoﬀ X at time T can be replicated by self-ﬁnancing portfolio which invests in just S and B , and the time- t value of this portfolio V t is characterized by V t = B t E Q [ X/B T |F t ]. We also showed that Q exists (and is unique), by an application of Girsanov’s theorem. Reviewing this: we need S/B to be a Q -martingale. Under the original probability we have d ( S/B ) = ( μ t - r t )( S/B ) dt + σ t ( S/B ) dw. According to Girsanov’s theorem changing the measure (without changing the sets of mea- sure zero) can change the drift but not the volatility. So Q is characterized by d ( S/B ) = σ t ( S/B ) d ˜ w where ˜ w is a Q -Brownian motion. Comparing these equations we see that dw + μ t - r t σ t dt = d ˜ w. The ratio λ t = μ t - r t σ t 1

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is called the market price of risk , by analogy with the Capital Asset Pricing Model (notice that λ is the ratio of excess return to volatility). It is important because it determines Q in terms of P , namely: dQ/dP = exp ± - Z T 0 λ t dw - 1 2 Z T 0 λ 2 t dt ! . This Q is called the “risk-neutral measure.” Finally, we observed that if S is any tradeable in this market, its value S t must have the property that S t /B t is a Q -martingale (for the same risk-neutral measure Q ). This shows that all tradeables must have the same market price of risk . It’s enlightening to give another proof that all tradeables have the same market price of
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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.

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section3 (1) - Continuous Time Finance Notes, Spring 2004...

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