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 =