1. Ch. 19 J. Hull, Options, Futures and Other Derivatives Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html Framework for pricing derivatives

2. Zvi WienerHull - 19 slide 2 The Market Price of Risk Observable underlying process, for example stock, interest rate, price of a commodity, etc. Here dz is a Brownian motion. We assume that m(x, t) and s(x, t). x is not necessarily an investment asset!

3. Zvi WienerHull - 19 slide 3 Suppose that f 1 and f 2 are prices of two derivatives dependent only on x and t. For example options. We assume that prior to maturity f 1 and f 2 do not provide any cashflow. The same!

4. Zvi WienerHull - 19 slide 4 Form an instantaneously riskless portfolio consisting of  2 f 2 units of the first derivative and  1 f 1 units of the second derivative.

5. Zvi WienerHull - 19 slide 5 An instantaneously riskless portfolio must earn a riskless interest rate.

6. Zvi WienerHull - 19 slide 6 This implies or is called the market price of risk of x.

7. Zvi WienerHull - 19 slide 7 If x is a traded asset we must also have But if x is not a financial asset this is not true. For all financial assets depending on x and time a similar relation must held.

8. Zvi WienerHull - 19 slide 8 Volatility Note that  can be positive or negative, depending on the correlation with x. |  | is called the volatility of f. If s>0 and f and x are positively correlated, then  >0, otherwise it is negative.

9. Zvi WienerHull - 19 slide 9 Example 19.1 Consider a derivative whose price is positively related to the price of oil. Suppose that it provides an expected annual return of 12%, and has volatility of 20%. Assume that r=8%, then the market price of risk of oil is:

10. Zvi WienerHull - 19 slide 10 Example 19.2 Consider two securities positively dependent on the 90-day IR. Suppose that the first one has an expected return of 3% and volatility of 20% (annual), and the second has volatility of 30%, assume r=6%. What is the market price of interest rate risk? What is the expected return from the second security?

11. Zvi WienerHull - 19 slide 11 Example 19.2 The market price of IR risk is: The expected return from the second security:

12. Zvi WienerHull - 19 slide 12 Differential Equation Finally leading to: f is a function of x and t, we can get using Ito’s lemma:

13. Zvi WienerHull - 19 slide 13 Differential Equation Comparing to BMS equation we see that it is similar to an asset providing a continuous dividend yield q=r-m+ s. Using Feynman-Kac we can say that the expected growth rate is r-q and then discount the expected payoff at the risk-free rate r.

14. Zvi WienerHull - 19 slide 14 Risk-neutral approach True dynamics Risk-neutral dynamics

15. Zvi WienerHull - 19 slide 15 Example 19.3 Price of copper is 80 cents/pound. Risk free r=5%. The expected growth rate in the price of copper is 2% and its volatility is 20%. The market price of risk associated with copper is 0.5. Assume that a contract is traded that allows the holder to receive 1,000 pounds of copper at no cost in 6 months. What is the price of the contract?

16. Zvi WienerHull - 19 slide 16 Example 19.3 m=0.02, =0.5, s=0.2, r=0.05; the risk-neutral expected growth rate is The expected (r-n) payoff from the contract is Discounting for six months at 5% we get

17. Zvi WienerHull - 19 slide 17 Derivatives dependent on several state variables State variables (risk factors): Traded security

18. Zvi WienerHull - 19 slide 18 Multidimensional Risk Here i is the market price of risk for x i. This relation is also derived in APT (arbitrage pricing theory), Ross 1976 JET.

19. Zvi WienerHull - 19 slide 19 Pricing of derivatives To price a derivative in the case of several risk factors we should change the dynamics of x i to risk neutral derive the expected (r-n) discounted payoff If r is deterministic

20. Zvi WienerHull - 19 slide 20 Example 19.5 Consider a security that pays off $100 at time T if stock A is above X A and stock B is above X B. Assume that stocks A and B are uncorrelated. The payoff is $100 Q A Q B, here Q A are Q B are r-n probabilities of stocks to be above strikes.

21. Zvi WienerHull - 19 slide 21 Example 19.5

22. Zvi WienerHull - 19 slide 22 Derivatives on Commodities The big problem is to estimate the market price of risk for non investment assets. One can use futures contracts for this. Assume that the commodity price follows (no mean reversion and constant volatility)

23. Zvi WienerHull - 19 slide 23 Derivatives on Commodities The expected (r-n) future price of a commodity is its future price F(t).

24. Zvi WienerHull - 19 slide 24 Example 19.6 Futures prices August 9962.20 Oct 9960.60 Dec 9962.70 Feb 0063.37 Apr 0064.42 Jun 0064.40 The expected (r-n) growth rate between Oct and Dec 99 is ln(62.70/60.60)=3.4%, or 20.4% annually.

25. Zvi WienerHull - 19 slide 25 Convenience Yield y - convenience yield u - storage costs, then then r-n growth rate is m - s = r - y + u

26. Zvi WienerHull - 19 slide 26 Martingales and Measures A martingale is a zero drift stochastic process for example: dx = s dz an important propertyE[x T ] = x 0, fair game.

27. Zvi WienerHull - 19 slide 27 Martingales and Measures Real world Risk-neutral world In the risk-neutral world the market price of risk is zero, while in the real world it is

28. Zvi WienerHull - 19 slide 28 Martingales and Measures By making other assumptions we can define other “worlds” that are internally consistent. In a world with the market price of risk * the drift (expected growth rate)  * must be

29. Zvi WienerHull - 19 slide 29 Equivalent Martingale Measures Suppose that f and g are price processes of two traded securities dependent on a single source of uncertainty. Define x=f/g. This is the relative price of f with respect to g. g is the numeraire.

30. Zvi WienerHull - 19 slide 30 Equivalent Martingale Measures The equivalent martingale measure result states that when there are no arbitrage opportunities, x is a martingale for some choice of market price of risk. For a given numeraire g the same market price of risk works for all securities f and the market price of risk is equal to the volatility of g.

31. Zvi WienerHull - 19 slide 31 Equivalent Martingale Measures Using Ito’s lemma

32. Zvi WienerHull - 19 slide 32 Equivalent Martingale Measures A martingale

33. Zvi WienerHull - 19 slide 33 Forward risk neutral wrt g Since f/g is a martingale (19.19) in Hull

34. Zvi WienerHull - 19 slide 34 Money market as a numeraire Money market accountdg = rgdt zero volatility, so the market price of risk will be zero and we arrive at the standard r-n world. g 0 =1 and

35. Zvi WienerHull - 19 slide 35 Zero-Coupon Bond as a Numeraire Define P(t,T) the price at time t of a zero- coupon bond maturing at T. Denote by E T the appropriate measure. g T = P(T,T)=1, g 0 = P(0,T) we get

36. Zvi WienerHull - 19 slide 36 Zero-Coupon Bond as a Numeraire Define F as the forward price of f for a contract maturing at time T. Then In a world that is forward risk neutral with respect to P(t,T) the forward price is the expected future spot price.

37. Zvi WienerHull - 19 slide 37 Important Conclusion We can value any security that provides a payoff at time T by calculating its expected payoff in a world that is forward risk neutral with respect to a bond maturing at time T and discounting at the risk-free rate for maturity T. In this world it is correct to assume that the expected value of an asset equals its forward value.

38. Zvi WienerHull - 19 slide 38 Interest Rates With a Numeraire Define R(t, T 1, T 2 ) as the forward interest rate as seen at time t for the period between T 1 and T 2 expressed with a compounding period T 1 - T 2. The forward price of a zero coupon bond lasting between T 1 and T 2 is

39. Zvi WienerHull - 19 slide 39 Interest Rates With a Numeraire A forward interest rate implied for the corresponding period is

40. Zvi WienerHull - 19 slide 40 Interest Rates With a Numeraire Setting We get that R(t, T 1, T 2 ) is a martingale in a world that is forward risk neutral with respect to P(t,T 2 ).

41. Zvi WienerHull - 19 slide 41 Annuity Factor as a Numeraire Consider a swap starting at time T n with payment dates T n+1, T n+2, …, T N+1. Principal $1. Denote the forward swap rate S n,N (t). The value of the fixed side of the swap is

42. Zvi WienerHull - 19 slide 42 Annuity Factor as a Numeraire The value of the floating side is The first term is $1 received at the next payment date and the second term corresponds to the principal payment at the end. The swap rate can be found as

43. Zvi WienerHull - 19 slide 43 Annuity Factor as a Numeraire We can apply an equivalent martingale measure by setting P(t,T n )-P(t,T N+1 ) as f and A n,N (t) as g. This leads to For any security f we have

44. Zvi WienerHull - 19 slide 44 Multiple Risk Factors

45. Zvi WienerHull - 19 slide 45 Multiple Risk Factors Equivalent world can be defined as Where i * are the market prices of risk

46. Zvi WienerHull - 19 slide 46 Multiple Risk Factors Define a world that is forward risk neutral with respect to g as a world where i *=  g,i. It can be shown from Ito’s lemma, using the fact that dz i are uncorrelated, that the process followed by f/g in this world has zero drift.

47. Zvi WienerHull - 19 slide 47 An Option to Exchange Assets Consider an option to exchange an asset worth U to an asset worth V. Assume that the correlation between assets is  and they provide no income. Setting g=U, f T =max(V T -U T,0) in 19.19 we get.

48. Zvi WienerHull - 19 slide 48 An Option to Exchange Assets The volatility of V/U is This is a simple option.

49. Zvi WienerHull - 19 slide 49 An Option to Exchange Assets Assuming that the assets provide an income at rates q U and q V.

50. Zvi WienerHull - 19 slide 50 Change of Numeraire Dynamics of asset f with forward risk neutral measure wrt g and h we have When changing numeraire from g to h we update drifts by

51. Zvi WienerHull - 19 slide 51 Change of Numeraire Set v be a function of traded securities. Define  v,i as the i-th component of v volatility. The rate of growth of v responds to a change of numeraire in the same way. Define q=h/g, then  h,i -  g,i is the i-th component of volatility of q. Thus the drift update of v is

52. Zvi WienerHull - 19 slide 52 Quantos Quanto provides a payoff in currency X at time T. We assume that the payoff depends on the value of a variable V observed in currency Y at time T. F(t) - forward value of V in currency Y P X (t,T) - value (in X) of 1 unit of X paid at T P Y (t,T) - value (in X) of 1 unit of Y paid at T G(T) forward exchange rate units of Y per X G(t)= P X (t,T)/P Y (t,T)

53. Zvi WienerHull - 19 slide 53 Quantos In equation 19.19 set g=P Y (t,T), f - be a security that pays V T units of currency Y at time T. f T =V T /S T, and g T =1/S T f 0 = P Y (t,T)E Y (V T ), no arbitrage means F(0)=f 0 /P Y (0,T), hence E Y (V T )=F(0) When we move from X world to Y world the expected growth rate increases by  F  G

54. Zvi WienerHull - 19 slide 54 Quantos This means that approximately Since V(T)=F(T) and E Y (V T )=F(0)

55. Zvi WienerHull - 19 slide 55 Example 19.7 Nikkei = 15,000 yen dividend yield = 1% one-year USD risk free rate = 5% one-year JPY risk free rate = 2% The forward price of Nikkei for a contract denominated in yen is 15,000e (0.02-0.01)1 =15,150.75

56. Zvi WienerHull - 19 slide 56 Example 19.7 Suppose that the volatility of the one-year forward price of the index is 20%, the volatility of the one-year forward yen/USD exchange rate is 12%. The correlation of one- year forward Nikkei with the one-year forward exchange rate  =0.3. The forward price of the Nikkei for a contract that provides a payoff in dollars is 15,150.75 e 0.3*0.2*0.12 =15,260.2

57. Zvi WienerHull - 19 slide 57 Siegel’s paradox Consider two currencies X and Y. Define S an exchange rate (the number of units of currency Y for a unit of X). The risk-neutral process for S is By Ito’s lemma the process for 1/S is

58. Zvi WienerHull - 19 slide 58 Siegel’s paradox The paradox is that the expected growth rate of 1/S is not r y - r X, but has a correction term. If we change numeraire from currency X to currency Y the correction term is  2 =-  2. The process in terms of currency Y becomes