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2015-5-11 Martingales and Measures Chapter 21

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

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2015-5-13 Forming a Riskless Portfolio

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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

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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

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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.

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

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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).

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

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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

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2015-5-111 Alternative Worlds

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2015-5-112 A Key Result

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

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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

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

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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

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2015-5-117 Money Market Account continued

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

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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.

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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

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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

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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

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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:

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

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2015-5-125 Extension to Several Independent Factors

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2015-5-126 Extension to Several Independent Factors continued

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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

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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.

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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,

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2015-5-130 Change of Numeraire

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

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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)

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2015-5-133 Quantos continued

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2015-5-134 Quantos continued

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2015-5-135 Siegel ’ s Paradox

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2015-5-136 Siegel ’ s Paradox(2) In the process of dS, the numeraire is the money market account in currency Y. In the second equation, the numeraire is also the money market account in currency Y. To change the numeraire from Y to X,the growth rate of 1/S increase by ρσ V σ S where V=1/S and ρ is the correlation between S and 1/S.In this case, ρ=-1, and σ v =σ S.It follows that the change of numeraire causes the growth rate of 1/S to increase – σ S 2.

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5.1 Option pricing: pre-analytics Lecture 5. 5.2 Notation c : European call option price p :European put option price S 0 :Stock price today X :Strike.

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