Ch27HullOFOD7thEd by Salim Kasap

Martingales and Measures Chapter 27

Options, Futures, and Other Derivatives, 7th Edition, Copyright ÂŠ John C. Hull 2008

Derivatives Dependent on a Single Underlying Variable Consider a variable θ (not necessarily the price of a traded security) that follows the process dθ = m dt + s dz θ Imagine two derivative s dependent on θ with prices ƒ1 and ƒ2 . Suppose d ƒ1 = μ1 dt + σ1 dz ƒ1 dƒ2 = μ2 dt + σ 2 dz ƒ2 Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008

Forming a Riskless Portfolio We can set up a riskless portfolio Π consisting of + σ 2 ƒ2 of the 1st derivative and − σ1ƒ1 of the 2nd derivative Π = (σ 2 ƒ2 ) ƒ1 − (σ1ƒ1 ) ƒ2 ∆Π=( μ1σ 2 ƒ1ƒ2 − μ2 σ1ƒ1ƒ2 ) ∆t Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008

Market Price of Risk (Page 624) Since the portfolio is riskless : ∆Π=r Π∆t This gives : μ1σ 2 − μ2 σ1 = r σ 2 − r σ1 μ1 − r μ2 − r or = σ1 σ2  This shows that (µ – r )/σ is the same for all derivatives dependent on the same underlying variable, θ  We refer to (µ – r )/σ as the market price of risk for θ and denote it by λ Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008

Extension of the Analysis to Several Underlying Variables (Equations 27.12 and 27.13, page 627)

If f depends on several underlying variables with n dƒ = μ dt + ∑ σ i dzi ƒ i =1 then n

μ − r = ∑ λ iσi i =1

Martingales (Page 628)  

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

Alternative Worlds In the traditional risk - neutral world df = rf dt + σf dz In a world where the market price of risk is λ df = (r + λσ) f dt + σf dz Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008

The Equivalent Martingale Measure Result (Page 628-629) If we set 位 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

Options, Futures, and Other Derivatives, 7th Edition, Copyright 漏 John C. Hull 2008

Forward Risk Neutrality We will 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  fT  f0 = Eg   g0  gT  Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008

Alternative Choices for the Numeraire Security g   

Money Market Account Zero-coupon bond price Annuity factor

Money Market Account as the Numeraire 

 

The money market account is an account that starts at $1 and is always invested at the short-term risk-free interest rate The process for the value of the account is dg=rg dt This has zero volatility. Using the money market account as the numeraire leads to the traditional risk-neutral world where λ=0 Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008

Money Market Account continued T

rdt ∫ 0 Since g 0 = 1 and gT = e , the equation  fT  f0 = E g   g0  gT  becomes T − rdt   ∫ 0 ˆ f 0 = E e fT    where Eˆ denotes expectations in the traditional risk - neutral world Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008

Zero-Coupon Bond Maturing at time T as Numeraire The equation  fT f0 = E g  g0  gT

  

becomes f 0 = P(0, T ) ET [ fT ] where P(0,T ) is the zero - coupon bond price and ET denotes expectations in a world that is FRN wrt the bond price Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008

Forward Prices In a world that is FRN wrt P(0,T), the expected value of a security at time T is its forward price

Options, Futures, and Other Derivatives, 7th Edition, Copyright ÂŠ John C. Hull 2008

Interest Rates In a world that is FRN wrt P(0,T2) the expected value of an interest rate lasting between times T1 and T2 is the forward interest rate

Options, Futures, and Other Derivatives, 7th Edition, Copyright ÂŠ John C. Hull 2008

Annuity Factor as the Numeraire The equation  fT f0 = E g  g0  gT

  

Annuity Factors and Swap Rates Suppose that s(t) is the swap rate corresponding to the annuity factor A. Then: s(t)=EA[s(T)]

Options, Futures, and Other Derivatives, 7th Edition, Copyright ÂŠ John C. Hull 2008

Extension to Several Independent Factors (Page 633) In the traditional risk - neutral world m

df (t ) = r (t ) f (t )dt + ∑ σ f ,i (t ) f (t )dzi i =1 m

dg (t ) = r (t ) g (t )dt + ∑ σ g ,i (t ) g (t )dzi i =1

For other worlds that are internally consistent m m   df (t ) = r (t ) + ∑ λ i σ f ,i (t ) f (t )dt + ∑ σ f ,i (t ) f (t )dzi i =1 i =1   m m   dg (t ) = r (t ) + ∑ λ i σ g ,i (t ) g (t )dt + ∑ σ g ,i (t ) g (t )dzi i =1 i =1   Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008

Extension to Several Independent Factors continued We define a world that is FRN wrt g as world where λi = σ g ,i As in the one - factor case, f g is a martingale and the rest of the results hold.

Applications  

Extension of Black’s model to case where inbterest rates are stochastic Valuation of an option to exchange one asset for another

Black’s Model (page 634) By working in a world that is forward risk neutral with respect to a P(0,T) it can be seen that Black’s model is true when interest rates are stochastic providing the forward price of the underlying asset is has a constant volatility c = P(0,T)[F0N(d1)−KN(d2)] p = P(0,T)[KN(−d2) − F0N(−d1)] ln( F0 / K ) + σ 2F T d1 = σF T

Option to exchange an asset worth U for one worth V  

This can be valued by working in a world that is forward risk neutral with respect to U See equations 27.30, 27.31, and 27.32

Change of Numeraire (Section 27.8, page 636)

When we change the numeraire security from g to h, the drift of a variable v increases by ρσ vσ w where σ v is the volatility of v, w = h g , σ w is the volatility of w, and ρ is the correlation between v and w