1. Zvi WienerContTimeFin - 9 slide 1 Financial Engineering Risk Neutral Pricing Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049

2. Zvi WienerContTimeFin - 9 slide 2 Equivalent Martingale Measure and Risk-Neutral Pricing Rangarajan K. Sundaram New York University Journal of Derivatives

3. Zvi WienerContTimeFin - 9 slide 3 Replication A contingent claim is replicable if it is possible to construct a portfolio of other securities with two properties: F The value of the portfolio at maturity is identical in all circumstances to the value of the contingent claim. F Once the portfolio is set up, there are no other cash flows (self-financing).

4. Zvi WienerContTimeFin - 9 slide 4 Pricing by Arbitrage In a complete financial market one can price all securities from prices of a small set of securities used in replication. Otherwise there would be an arbitrage. Assumption: no taxes, transactions costs or short restrictions.

5. Zvi WienerContTimeFin - 9 slide 5 Settings Bond - default free money market account. Risk-free rate is the return on this bond. A contingent claim X with a known payoff at maturity depending on other securities.

6. Zvi WienerContTimeFin - 9 slide 6 The Binomial Model There are two assets: bond and stock: 1 S r r uS dS q 1-q q q + (1-q)=1, 0 < q < 1. d < r < u Arbitrage?

7. Zvi WienerContTimeFin - 9 slide 7 The Binomial Model assets: bond, stock, and an option 1 S r r uS dS q 1-q q X XuXu XdXd q Note that here r is what we usually denote by 1+r.

8. Zvi WienerContTimeFin - 9 slide 8 Pricing by Arbitrage The Binomial model is an example of a complete market. All claims can be priced by arbitrage. Fix any contingent claim (i.e. fix X u and X d ). Consider a portfolio consisting of units of the bond units of a stock

9. Zvi WienerContTimeFin - 9 slide 9 Pricing by Arbitrage Value of this portfolio at maturity is

10. Zvi WienerContTimeFin - 9 slide 10 Replication in a Binomial Model Replicating portfolio consists of  shares and a bonds. Thus the price of the contingent claim must be equal: Note that we did not use probabilities of up and down (they are hidden in prices already).

11. Zvi WienerContTimeFin - 9 slide 11 Risk-Neutral Probabilities 1. Identify a new probability measure, called risk-neutral probability, or an equivalent martingale measure. 2. Compute the expected discounted payoff from the contingent claims, where the expectations are taken under the risk-neutral measure.

12. Zvi WienerContTimeFin - 9 slide 12 Equivalent Measures Two probability measures are equivalent if and only if any event with positive probability under one measure has positive probability under the second measure (and vice versa).

13. Zvi WienerContTimeFin - 9 slide 13 Martingale A stochastic process is martingale if its expected change is always zero. A more precise definition is

14. Zvi WienerContTimeFin - 9 slide 14 Discounting Discounting is necessary, since otherwise the bond is not risky, but grows (at the risk free rate).

15. Zvi WienerContTimeFin - 9 slide 15 Existence and Uniqueness There is no risk-neutral probability measure if and only if there exists an arbitrage*. Multiple risk-neutral probability measures can occur if and only if there are contingent claims that can not be replicated. However they predict the same prices for all replicable claims.

16. Zvi WienerContTimeFin - 9 slide 16 Binomial Case 1 S r r uS dS q 1-q q Since d < r < u, the probability p is 0 < p <1

17. Zvi WienerContTimeFin - 9 slide 17 The Risk-Neutral Price of a Claim Consider a contingent claim paying X u and X d. Then its price can be found as Arbitrage free price coincide with the risk- neutral price.

18. Zvi WienerContTimeFin - 9 slide 18 Arrow Securities – State Prices A contingent claim that pays $1 if and only if a particular state of the world occurs. K State price = discounted probability of the state.

19. Zvi WienerContTimeFin - 9 slide 19 State Prices and Risk-Neutral Probabilities

20. Zvi WienerContTimeFin - 9 slide 20 Example Binomial model with two risky assets and a bond. Let S 1 and S 2 denote the initial prices of the risky assets. Let the possible prices be S1S1 u1S1u1S1 d1S1d1S1 S2S2 u2S2u2S2 d2S2d2S2

21. Zvi WienerContTimeFin - 9 slide 21 Example We must have Assume that this equation does not hold.

22. Zvi WienerContTimeFin - 9 slide 22 Example Consider a portfolio: $a in bonds $b in S 1 $c in S 2 its cost now is a + b + c. At maturity it gives: If the equality does not hold there is an arbitrage.

23. Zvi WienerContTimeFin - 9 slide 23 Completeness 3 variables, 2 equations, there are many solutions. S uS dS mS 1 r r r ququ qmqm qdqd

24. Zvi WienerContTimeFin - 9 slide 24 Continuous Time Models Note that the first equation describes a bond: B(t)=B 0 e rt. The second equation will correspond to a stock.

25. Zvi WienerContTimeFin - 9 slide 25 The Discounted Price Process Denote by Q the discounted price process

26. Zvi WienerContTimeFin - 9 slide 26 Girsanov’s Theorem If we wish to change the drift of the process to . F Define a new process Z using , , and . F Redefine Q using the new process Z.

27. Zvi WienerContTimeFin - 9 slide 27 Step 1 Define  by the solution to  –  =  Note that  is not constant. This equation can always be solved if  -1 exists. Define a new process Z such that dZ =  dt + dW

28. Zvi WienerContTimeFin - 9 slide 28 Step 2 By definition

29. Zvi WienerContTimeFin - 9 slide 29 Black-Merton-Scholes Model Try There is NO solution.

30. Zvi WienerContTimeFin - 9 slide 30 Black-Merton-Scholes Model Try There is a solution!

31. Zvi WienerContTimeFin - 9 slide 31 Financial Engineering Interest Rate Derivatives Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049 following Hull and White Hull and White

32. Zvi WienerContTimeFin - 9 slide 32 Black’s Model F Similar to the Black-Scholes model. F Assumes that the future value of interest rate, a bond price or some other variable is lognormal. F The mean of the probability distribution is the forward value of the variable. F The standard deviation is defined by a volatility as in BS. Hull and White

33. Zvi WienerContTimeFin - 9 slide 33 Black’s Model for Caps The value of a caplet corresponds to the time period between t 1 and t 2 is F - forward IR for t 1,t 2 X - the cap rate R - risk free yield to t 2 A - principal  - forward rate volatility Hull and White

34. Zvi WienerContTimeFin - 9 slide 34 Swap Options and Bond Options F An IR swap can be regarded as an exchange of a fixed-rate bond for a floating rate bond. F A swaption is an option to exchange these two bonds. F Floating rate bond is worth par, so the swaption is an option to exchange a fixed rate bond for par. F An option on a swap where fixed is paid and floating is received is a put option on the bond with a strike price of par. Hull and White

35. Zvi WienerContTimeFin - 9 slide 35 Assumptions of Black’s Model F Assume bond price is lognormal F Assume bond yield is lognormal Hull and White

36. Zvi WienerContTimeFin - 9 slide 36 If Bond Prices Are Lognormal A European call on a bond is priced with B - price of the forward bond B* - price of a discount bond maturing at T  B - volatility of the forward bond price Hull and White

37. Zvi WienerContTimeFin - 9 slide 37 Using Duration to Convert Yield Volatilities to Price Volatilities D - is the modified duration Hull and White

38. Zvi WienerContTimeFin - 9 slide 38 If Bond Yield Is Lognormal max[B-X,0] = max[  B,0]  max[XD(Y X -Y F ),0] Y F - the forward bond yield Y X - yield at which B=X Hull and White

39. Zvi WienerContTimeFin - 9 slide 39 Drawbacks of Black’s Model F Can be used when derivative depends on a single interest rate observed at a single time. F Provides no linkage between different interest rates and their volatilities. F Cannot be used for valuing long-dated American options and other complex derivatives. Hull and White

40. Zvi WienerContTimeFin - 9 slide 40 Yield Curve Based Models A no-arbitrage yield-curve-based model designed in such a way that it is automatically consistent with the current term structure and permits no arbitrage opportunities. Hull and White

41. Zvi WienerContTimeFin - 9 slide 41 Risk-Neutral Valuation The risk-neutral valuation for equity assumes that we get the right answer if we: F Assume that the expected return on the equity is the risk free rate. F Discount payoffs at the risk free rate. Hull and White

42. Zvi WienerContTimeFin - 9 slide 42 Risk-Neutral Valuation The risk-neutral valuation principle can be extended to value interest rate derivatives. We assume that the expected return on all bond prices is the risk-free rate and discount payoffs at the risk free rate. Hull and White

43. Zvi WienerContTimeFin - 9 slide 43 Risk-Neutral World or Real World In the two worlds variables have the same volatilities, but different drifts. RN - is a rough approximation, this is a world where there is no liquidity premium, so that forward rates equal expected future spot rates. All our models are valid in RN world only! Hull and White

44. Zvi WienerContTimeFin - 9 slide 44 Valuation A derivative security paying off f T at maturity is worth today Hull and White Here * means in the risk-neutral world.

45. Zvi WienerContTimeFin - 9 slide 45 Valuation of Bonds A discount bond maturing at time T is: Hull and White Here * means in the risk-neutral world.

46. Zvi WienerContTimeFin - 9 slide 46 Approaches Hull and White F Model Bond Prices: Ho and Lee 1986. F Model Forward Rates: Heath, Jarrow and Morton 1987. F Model Short Rate: Black, Derman and Toy 1990, Hull and White 1990.