# 2015-5-11 Martingales and Measures Chapter 21. 2015-5-12 Derivatives Dependent on a Single Underlying Variable.

## Presentation on theme: "2015-5-11 Martingales and Measures Chapter 21. 2015-5-12 Derivatives Dependent on a Single Underlying Variable."— Presentation transcript:

2015-5-11 Martingales and Measures Chapter 21

2015-5-12 Derivatives Dependent on a Single Underlying Variable

2015-5-13 Forming a Riskless Portfolio

2015-5-14 Market Price of Risk This shows that (  – r )/  is the same for all derivatives dependent only on the same underlying variable,  and t. We refer to (  – r )/  as the market price of risk for  and denote it by

2015-5-15 Differential Equation for ƒ Using Ito’s lemma to obtain expressions for  and  in terms of m and s  The equation  =r becomes

2015-5-16 Risk-Neutral Valuation This analogy shows that we can value ƒ in a risk-neutral world providing the drift rate of  is reduced from m to m – s Note: When  is not the price of an investment asset, the risk-neutral valuation argument does not necessarily tell us anything about what would happen with  in a risk-neutral world.

2015-5-17 Extension of the Analysis to Several Underlying Variables

2015-5-18 Traditional Risk-Neutral Valuation with Several Underlying Variables A derivative can always be valued as if the would is risk neutral, provided that the expected growth rate of each underlying variable is assumed to be m i -λ i s i rather than m i. The volatility of the variables and the coefficient of the correlation between variables are not changed. (CIR,1985).

2015-5-19 How to measure λ? For a nontraded securities(i.e.commodity),we can use its future market information to measure λ.

2015-5-110 Martingales A martingale is a stochastic process with zero drfit A martingale has the property that its expected future value equals its value today

2015-5-111 Alternative Worlds

2015-5-112 A Key Result

2015-5-113 讨论 f 和 g 是否必须同一风险源？ 推导过程手写。

2015-5-114 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 g denotes a world that is FRN wrt g

2015-5-115 Aleternative Choices for the Numeraire Security g Money Market Account Zero-coupon bond price Annuity factor

2015-5-116 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=rgdt This has zero volatility. Using the money market account as the numeraire leads to the traditional risk-neutral world

2015-5-117 Money Market Account continued

2015-5-118 Zero-Coupon Bond Maturing at time T as Numeraire

2015-5-119 Forward Prices Consider an variable S that is not an interest rate. A forward contract on S with maturity T is defined as a contract that pays off S T -K at time T. Define f as the value of this forward contract. We have f 0 equals 0 if F=K, So, F=E T (f T ) F is the forward price.

2015-5-120 利率 T 2 时刻到期债券 T 1 交割的远期价格 F ＝ P(t, T 2 )/P(t, T 1 ） 远期价格 F 又可写为

2015-5-121 Interest Rates In a world that is FRN wrt P(0,T 2 ) the expected value of an interest rate lasting between times T 1 and T 2 is the forward interest rate

2015-5-122 Annuity Factor as the Numeraire-1 Let S n (t) is the forward swap rate of a swap starting at the time T 0, with payment dates at times T 1, T 2,…,T N. Then the value of the fixed side of the swap is

2015-5-123 Annuity Factor as the Numeraire-2 If we add \$1 at time T N, the floating side of the swap is worth \$1 at time T 0. So, the value of the floating side is: P(t,T 0 )-P(t, T N ) Equating the values of the fixed and floating side we obstain:

2015-5-124 Annuity Factor as the Numeraire-3

2015-5-125 Extension to Several Independent Factors

2015-5-126 Extension to Several Independent Factors continued

2015-5-127 Applications Valuation of a European call option when interest rates are stochastic Valuation of an option to exchange one asset for another

2015-5-128 Valuation of a European call option when interest rates are stochastic Assume S T is lognormal then: The result is the same as BS except r replaced by R.

2015-5-129 Valuation of an option to exchange one asset(U) for another(V) Choose U as the numeraire, and set f as the value of the option so that f T =max(V T -U T,0), so,

2015-5-130 Change of Numeraire

2015-5-131 证明 当记帐单位从 g 变为 h 时， V 的偏移率增加了

2015-5-132 Quantos Quantos are derivatives where the payoff is defined using variables measured in one currency and paid in another currency Example: contract providing a payoff of S T – K dollars (\$) where S is the Nikkei stock index (a yen number)

2015-5-133 Quantos continued

2015-5-134 Quantos continued