Martingales and measures Chapter 21 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 21.1 Martingales and Measures Chapter 21
212 Derivatives Dependent on a Single Underlying variable Consider a variable, 0, not necessarily the price of a traded security) that follows the process d e =mdt+s dz 0 Imagine two derivative s dependent on e with prices f, and f2. Suppose dt+o. dz =u, dt+o, dz Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 21.2 Derivatives Dependent on a Single Underlying Variable μ dt σ dz ƒ d? μ dt σ dz ƒ d ? ƒ ƒ m dt s dz d 2 2 2 2 1 1 1 1 . = + = + = + with prices and Suppose Imagine two derivative s dependent on of a traded security) that follows the process Consider a variable, ,(not necessarily the price 1 2
21.3 Forming a riskless portfolio We can set up a riskless portfolio il, consisting of +o,f2 of the 1st derivative and o,f, of the 2nd derivative ∏=(2f2)f-(o,f1)f2 6I=(02f12-201ff2)t Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 21.3 Forming a Riskless Portfolio = μ σ ƒ ƒ μ σ ƒ ƒ t σ ƒ ƒ σ ƒ ƒ σ ƒ σ ƒ − = − − ( ) ( ) ( ) 1 2 1 2 2 1 1 2 2 2 1 1 1 2 1 1 2 2 of the 2nd derivative + of the 1st derivative and We can set up a riskless portfolio , consisting of
214 Market Price of Risk(Page 485) Since the portfolio is riskless:δnI=rIδt This gives: u,02-u,0=ro,-ro 2 This shows that (H-r)o is the same for all derivatives dependent on the same underlying variable,θ We refer to(u-r)/o as the market price of risk for e and denote it by 2 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 21.4 or This gives : Since the portfolio i s riskless : 2 2 1 1 1 2 2 1 2 1 σ μ r σ μ r μ σ μ σ r σ r σ =r t − = − − = − Market Price of Risk (Page 485) • This shows that (m – r )/s is the same for all derivatives dependent on the same underlying variable, • We refer to (m – r )/s as the market price of risk for and denote it by l
Extension of the Analysis 21.5 to Several Underlying Variables (Equations 21.12 and 21.13, page 487) if f depends on several underlying variables d+∑od then ∑ Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 21.5 Extension of the Analysis to Several Underlying Variables (Equations 21.12 and 21.13, page 487) then with If depends on several underlying variables μ r λ σ μ dt σ dz ƒ d? f n i i i n i i i = = − = = + 1 1
21.6 Martingales(Page 488) A martingale is a stochastic process With zero drift A variable following a martingale has the property that its expected future value equals Its value today Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 21.6 Martingales (Page 488) • A martingale is a stochastic process with zero drift • A variable following a martingale has the property that its expected future value equals its value today
21.7 Alternative Worlds In the traditional risk -neutral world df=rfdt+ofda In a world where the market price of risk is入 df =(r+no)fdt +of dz Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 21.7 Alternative Worlds df r f dt f dz df rf dt σf dz = + ls + s l = + ( ) i s In a world where the market price of risk In the traditionalrisk -neutral world
218 A Key result (Page 489) if we set n equal to the volatility of a security g, then ito's lemma shows that f g is a martingale for all derivative security prices f (f and g are assumed to provide no income during the period under consideration Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 21.8 A Key Result (Page 489) consideration) no income during the period under and are assumed to provide all derivative security prices shows that i s a martingale for a security , then Ito' s lemma If we set equal to the volatility of f g f f g g ( l
219 Forward risk neutrality We refer to a world where the market price of risk is the volatility of g as a world that is forward risk neutral with respect to g If E, denotes a world that is Frn wrt g g 0 g Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 21.9 Forward Risk Neutrality We refer to a world where the market price of risk is the volatility of g as a world that is forward risk neutral with respect to g. If Eg denotes a world that is FRN wrt g f g E f g g T T 0 0 =
21.10 Alternative choices for the Numeraire security g Money Market Account Zero-coupon bond price Annuity factor Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 21.10 Aleternative Choices for the Numeraire Security g • Money Market Account • Zero-coupon bond price • Annuity factor