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