1. 2016-7-81 Martingales and Measures Chapter 27

2. 2016-7-82 Derivatives Dependent on a Single Underlying Variable

3. 2016-7-83 Forming a Riskless Portfolio

4. 2016-7-84 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

5. 2016-7-85 Extension of the Analysis to Several Underlying Variables

6. 2016-7-86 How to measure λ? For a nontraded securities(i.e.commodity),we can use its future market information to measure λ.

7. 2016-7-87 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. 2016-7-88 Alternative Worlds

9. 2016-7-89 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： 2016-7-810

11. f 和 g 是否必须同一风险源？ 令 2016-7-811

12. 2016-7-812 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

13. 2016-7-813 Aleternative Choices for the Numeraire Security g Money Market Account Zero-coupon bond price Annuity factor

14. 2016-7-814 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

15. 2016-7-815 Money Market Account continued

16. 2016-7-816 Zero-Coupon Bond Maturing at time T as Numeraire

17. 2016-7-817 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.

18. 2016-7-818 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 S 求终值

19. 2016-7-819 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

20. 2016-7-820 Annuity Factor as the Numeraire-1 Let s(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

21. 2016-7-821 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:

22. 2016-7-822 Annuity Factor as the Numeraire-3

23. 2016-7-823 Extension to Several Independent Factors

24. 2016-7-824 Extension to Several Independent Factors continued

25. 2016-7-825 Applications Valuation of a European call option when interest rates are stochastic Valuation of an option to exchange one asset for another

26. 2016-7-826 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.

27. 2016-7-827 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,

28. 2016-7-828 Change of Numeraire

29. 2016-7-829 证明 当记帐单位从 g 变为 h 时， V 的偏移率增加了