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Martingales and Measures Chapter 21

Martingales and Measures Chapter 21. Derivatives Dependent on a Single Underlying Variable. Forming a Riskless Portfolio. Market Price of Risk. This shows that ( m – r )/ s is the same for all derivatives dependent only on the same underlying variable , q, and t.

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Martingales and Measures Chapter 21

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  1. Martingales and MeasuresChapter 21

  2. Derivatives Dependent on a Single Underlying Variable

  3. Forming a Riskless Portfolio

  4. Market Price of Risk • This shows that (m – r )/s is the same for all derivatives dependent only on the same underlying variable, q, and t. • We refer to (m – r )/s as the market price of risk for q and denote it by l

  5. Differential Equation for ƒ Using Ito’s lemma to obtain expressions for m and s in terms of m and s. The equation m-ls=r becomes

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

  7. Extension of the Analysisto Several Underlying Variables

  8. 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 mi-λisi rather than mi. • The volatility of the variables and the coefficient of the correlation between variables are not changed. (CIR,1985).

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

  10. Martingales • A martingale is a stochastic process with zero drfit • A martingale has the property that its expected future value equals its value today

  11. Alternative Worlds

  12. A Key Result

  13. 讨论 • f和g是否必须同一风险源？ • 推导过程手写。

  14. 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

  15. Aleternative Choices for the Numeraire Security g • Money Market Account • Zero-coupon bond price • Annuity factor

  16. Money Market Accountas 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

  17. Money Market Accountcontinued

  18. Zero-Coupon Bond Maturing at time T as Numeraire

  19. 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 ST-K at time T. Define f as the value of this forward contract. We have f0 equals 0 if F=K, So, F=ET(fT) F is the forward price.

  20. 利率 • T2时刻到期债券T1交割的远期价格F＝P(t, T2)/P(t, T1） • 远期价格F又可写为

  21. 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

  22. Annuity Factor as the Numeraire-1 • Let Sn(t)is the forward swap rate of a swap starting at the time T0, with payment dates at times T1, T2,…,TN. Then the value of the fixed side of the swap is

  23. Annuity Factor as the Numeraire-2 • If we add $1 at time TN, the floating side of the swap is worth $1 at time T0.So, the value of the floating side is: P(t,T0)-P(t, TN) • Equating the values of the fixed and floating side we obstain:

  24. Annuity Factor as the Numeraire-3

  25. Extension to Several Independent Factors

  26. Extension to Several Independent Factorscontinued

  27. Applications • Valuation of a European call option when interest rates are stochastic • Valuation of an option to exchange one asset for another

  28. Valuation of a European call option when interest rates are stochastic • Assume ST is lognormal then: • The result is the same as BS except r replaced by R.

  29. 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 fT=max(VT-UT,0), so,

  30. Change of Numeraire

  31. 证明 • 当记帐单位从g变为h时，V的偏移率增加了

  32. 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 ST– K dollars ($) where S is the Nikkei stock index (a yen number)

  33. Quantos continued

  34. Quantos continued

  35. Siegel’s Paradox

  36. 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 ρσVσS where V=1/S and ρ is the correlation between S and 1/S.In this case, ρ=-1, and σv =σS.It follows that the change of numeraire causes the growth rate of 1/S to increase – σS2.

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