Martingales and Measures Chapter 27

1. Martingales and MeasuresChapter 27

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. Extension of the Analysisto Several Underlying Variables

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

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

8. Alternative Worlds

9. A Key Result

10. f和g是否必须同一风险源？ • 设: • 在以g为记账单位的风险中性世界中：

11. f和g是否必须同一风险源？ • 令

12. 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

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

14. 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

15. Money Market Accountcontinued

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

17. 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.

18. 利率 求终值 S • 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

19. 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

20. Annuity Factor as the Numeraire-1 • Let s(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

21. 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:

22. Annuity Factor as the Numeraire-3

23. Extension to Several Independent Factors

24. Extension to Several Independent Factorscontinued

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

26. 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.

27. 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,

28. Change of Numeraire

29. 证明 • 当记帐单位从g变为h时，V的偏移率增加了