1. Primbs, MS&E 345 1 More Applications of Linear Pricing

2. Primbs, MS&E 345 2 Exchange one asset for another A generalization of Black-Scholes Black’s model with stochastic interest rates Bond options Caplets, etc. Futures, forwards, forward rates, and swap rates Swaptions Interest rate derivatives

3. Primbs, MS&E 345 3 Exchange one asset for another This is equivalent to the payoff at time T. Then our pricing formula is where from previous calculations To value this, we will let S 1 be the numeraire. Consider two assets: and the option to exchange asset 2 for asset 1 at time T.

4. Primbs, MS&E 345 4 where Note that: So letting: Hence, letting we just need to evaluate: where Exchange one asset for another

5. Primbs, MS&E 345 5 where Evaluating gives where: Substituting in terms of S 1 and S 2 gives the final answer: where Exchange one asset for another

6. Primbs, MS&E 345 6 Exchange one asset for another A generalization of Black-Scholes Black’s model with stochastic interest rates Bond options Caplets, etc. Futures, forwards, forward rates, and swap rates Swaptions Interest rate derivatives

7. Primbs, MS&E 345 7 Futures contracts and the risk neutral measure: time 0 time T time/position mark to market (at time t+dt) time T value t t (start) t+dt

8. Primbs, MS&E 345 8 Futures contracts and the risk neutral measure: time 0 time T time/position mark to market (at time t+dt) time T value t (start) t T-dt Total Cost: = 0 Total Payoff: Since f T T =S T t+dt

9. Primbs, MS&E 345 9 Plug into our risk neutral pricing formula: The futures price is the expected price of the stock at time T in a risk neutral world. In particular:

10. Primbs, MS&E 345 10 Exchange one asset for another A generalization of Black-Scholes Black’s model with stochastic interest rates Bond options Caplets, etc. Futures, forwards, forward rates, and swap rates Swaptions Interest rate derivatives

11. Primbs, MS&E 345 11 Tt0 payoff occurs Forward contracts and the bond forward risk neutral world Forward prices are a tradable (S t ) divided by B(t;T) In particular: The forward price is the expected price of the stock at time T in a bond (B(t;T)) forward risk neutral world. Hence, under B(t;T) as the numeraire, forward prices are martingales. Recall:

12. Primbs, MS&E 345 12 Exchange one asset for another A generalization of Black-Scholes Black’s model with stochastic interest rates Bond options Caplets, etc. Futures, forwards, forward rates, and swap rates Swaptions Interest rate derivatives

13. Primbs, MS&E 345 13 Spot and Forward Rates time spot rates spot rate curve t1t1 t2t2 0 Rates are quoted as yearly rates. Hence, the actual rate applied over time t 1 is t 1 R(0;t 1 ). If P is the principal on a zero coupon bond: Bond Price: Forward Bond Price: Forward Rates:

14. Primbs, MS&E 345 14 Tt0 payoff occurs Forward interest rates and the bond forward risk neutral world T+  The forward rate is tradables divided by B(t;T+  ). A forward bond price Hence, forward rates are martingales under the numeraire B(t;T+  ).

15. Primbs, MS&E 345 15 Tt0 payoff occurs Forward interest rates and the bond forward risk neutral world T+  In particular: in a forward risk neutral world with respect to B(t;T+  ).

16. Primbs, MS&E 345 16 Exchange one asset for another A generalization of Black-Scholes Black’s model with stochastic interest rates Bond options Caplets, etc. Futures, forwards, forward rates, and swap rates Swaptions Interest rate derivatives

17. Primbs, MS&E 345 17 Swaps and Swap rates An interest rate swap is an agreement to exchange a fixed rate of interest S for a floating rate of interest on the same notional principal P. The swap rate S is the fixed rate that makes a swap have zero value. 0 33 44 55  22 Fixed 0 33 44 55  22 Float Equating present values at time 0: where A(0) is the value of an annuity

18. Primbs, MS&E 345 18 Tt0 Forward swap rates and the annuity forward risk neutral world T+  T+2  T+n  To be a bit more precise, let S(t|T) be a forward swap rate where the swap begins at time T. Hence: where the Floating side is tradable (for instance as coupons on a floating rate bond). Therefore, forward swap rates are martingales under the numeraire A(t|T). Then the forward swap rate is defined by the relationship: where A(t|T) is the value of an annuity starting at time T.

19. Primbs, MS&E 345 19 Tt0 Forward swap rates and the annuity forward risk neutral world T+  T+2  T+n  In particular: in a forward risk neutral world with respect to A(t).

20. Primbs, MS&E 345 20 Summary: Forward Rates: Numeraire: B(t;T+  ) Forward Prices: Numeraire: B(t;T) Futures Prices: Numeraire: Money Market Swap Rates: Numeraire: A(t|T)

21. Primbs, MS&E 345 21 Exchange one asset for another A generalization of Black-Scholes Black’s model with stochastic interest rates Bond options Caplets, etc. Futures, forwards, forward rates, and swap rates Swaptions Interest rate derivatives

22. Primbs, MS&E 345 22 The set-up: Tt0 The option allows me to purchase the asset at time T for the strike price K Exercise decision Payoff occurs If the forward price at time t is greater than K, then this option is worth F t T -K at time T. Otherwise, the option is worthless because the forward price is less than the strike. (F t T -K) + Hence, the payoff is at time T. I have a European option on a forward contract with delivery date T, but the option expires at time t.

23. Primbs, MS&E 345 23 The set-up: Tt0 Exercise decision Payoff occurs (F t T -K) + To derive the price of this option, I will use the bond of maturity T as the numeraire: Hence, I just need to calculate in a bond forward risk neutral world.

24. Primbs, MS&E 345 24 The set-up: Tt0 Exercise decision Payoff occurs (F t T -K) + Hence, I just need to calculate in a bond forward risk neutral world. Assume F t T is log-normal:Then we need its mean and volatility: It’s mean in a forward risk neutral world is: Let denote the volatility of F t T at time t. (recall that the volatility is the same in every risk neutral world)

25. Primbs, MS&E 345 25 The set-up: Tt0 Exercise decision Payoff occurs (F t T -K) + Hence, I just need to calculate in a bond forward risk neutral world. So Black’s Formula is: where

26. Primbs, MS&E 345 26 Exchange one asset for another A generalization of Black-Scholes Black’s model with stochastic interest rates Bond options Caplets, etc. Futures, forwards, forward rates, and swap rates Swaptions Interest rate derivatives

27. Primbs, MS&E 345 27 A simple generalization of the Black-Scholes formula We can use Black’s model to generalize Black-Scholes by thinking of Black- Scholes as an option on a forward contract where the delivery of the forward, and expiration of the option are at the same time. Hence, we can use Black’s formula with t=T and S T =F T T. Tt0 Exercise decision Payoff occurs (F t T -K) + Black’s Model T0 Exercise decision Payoff occurs (S T -K) + =(F T T -K) + + Black-Scholes

28. Primbs, MS&E 345 28 A simple generalization of the Black-Scholes formula T0 Exercise decision Payoff occurs (S T -K) + =(F T T -K) + + Black-Scholes Plugging into Black’s formula gives: where But, I want the solution in terms of S 0 : We can use that: Additionally, let R(T) be the time T interest rate: (Here, I am using continuous compounding!) Then: (Substitute in)

29. Primbs, MS&E 345 29 A simple generalization of the Black-Scholes formula T0 Exercise decision Payoff occurs (S T -K) + =(F T T -K) + + Black-Scholes where: distribution function for a standard Normal (i.e. N(0,1)) It looks like standard Black-Scholes, just use the interest rate corresponding to the expiration date!

30. Primbs, MS&E 345 30 Exchange one asset for another A generalization of Black-Scholes Black’s model with stochastic interest rates Bond options Caplets, etc. Futures, forwards, forward rates, and swap rates Swaptions Interest rate derivatives

31. Primbs, MS&E 345 31 A Bond Option T0 Exercise decision Payoff occurs (B c (T)-K) + + Consider an option on a bond B c (t) with strike K and expiration T. (This bond could pay coupons and have any maturity). Then Black’s model says: where Let B c (0|T) be the forward price of the bond. (This could be calculated using the term structure). Let  denote the volatility of the bond price at time T.

32. Primbs, MS&E 345 32 Exchange one asset for another A generalization of Black-Scholes Black’s model with stochastic interest rates Bond options Caplets, etc. Futures, forwards, forward rates, and swap rates Swaptions Interest rate derivatives

33. Primbs, MS&E 345 33 Picture of a cap RKRK Interest Rate A cap is an upper limit on the interest payments corresponding to the life of a loan. A caplet is an upper limit on a single interest payment in a loan. Hence a cap corresponds to a caplet for every interest payment.

34. Primbs, MS&E 345 34 T0 An interest rate caplet: T+  natural time lag  Interest rate set R(T;T+  ) Interest payment occurs Use B(t;T+  ) as the numeraire: Then: where We know: Assume: R(T;T+  ) is log-normal with volatility 

35. Primbs, MS&E 345 35 T0 An interest rate caplet (a second approach): T+  natural time lag  Interest rate set R(T;T+  ) Interest payment occurs A put option on a bond. We can use Black’s model to value it: (In practice, the first approach is used more often.) Since this payoff is known at time T, we can discount it back to time T But:

36. Primbs, MS&E 345 36 Black’s Model is used to price a number of interest rate derivatives: We showed how to price a caplet, which puts an upper limit on interest rates at a specific time in the future. When we place an upper limit on all the interest payments for a loan, this is called a cap. Hence, a cap is just a portfolio of caplets. We price it by linearity. Price each caplet and add them together. A floor is a lower limit on interest rates over the life of a loan. A floorlet is for a single interest payment. A collar is a cap and a floor.

37. Primbs, MS&E 345 37 Exchange one asset for another A generalization of Black-Scholes Black’s model with stochastic interest rates Bond options Caplets, etc. Futures, forwards, forward rates, and swap rates Swaptions Interest rate derivatives

38. Primbs, MS&E 345 38 Tt0 A swap option (swaption) T+  T+2  T+n  We have an option to enter into a swap at time T, where we pay a fixed swap rate S K. Assume the swap rate at time T is S(T|T). If this is greater than S K, then we exercise the option. This is worth P  S(T|T)-S K ) at each swap date. total payoff Since the swap rate is known at time T, we can discount the payoff back to time T. Use an annuity as the numeraire:

39. Primbs, MS&E 345 39 Tt0 We know: Assume: S(T|T) is log-normal with volatility  Then: where T+  T+2  T+n  total payoff A swap option (swaption)

40. Primbs, MS&E 345 40 Tt0 T+  T+2  T+n  A swap option (swaption) (a second approach) Since a swap can be thought of as exchanging a bond with fixed interest for a bond with floating interest, a swaption can be thought of as an option on a bond. payoff (fixed bond - float bond) + The strike price is the value of the floating rate bond, which is always its principal P at every reset date. =(fixed bond-P) + Black’s model for a bond option can be used to price this. However, this approach is not used often in practice.

41. Primbs, MS&E 345 41 Appendix: Alternate proofs of martingale property for forwards, forward rates, and forward swap rates.

42. Primbs, MS&E 345 42 Forwards Forward Rates Swap Rates

43. Primbs, MS&E 345 43 Tt0 payoff occurs total payoff Buy a forward total cost Forward contracts and the bond forward risk neutral world Consider the bond forward risk neutral world with B(t;T) as the numeraire In particular: The forward price is the expected price of the stock at time T in a bond (B(t;T)) forward risk neutral world.

44. Primbs, MS&E 345 44 Forwards Forward Rates Swap Rates

45. Primbs, MS&E 345 45 Spot and Forward Rates time spot rates spot rate curve t1t1 t2t2 0 Rates are quoted as yearly rates. Hence, the actual rate applied over time t 1 is t 1 R(0;t 1 ). If P is the principal on a zero coupon bond: Bond Price: Forward Bond Price: Forward Rates:

46. Primbs, MS&E 345 46 Tt0 payoff occurs Forward interest rates and the bond forward risk neutral world T+  total cost total payoff Buy $1 worth of bond forward B(t|T;T+  ) Sell $1 worth of bond B(T|T;T+  ) Use B(t;T+  ) as the numeraire:

47. Primbs, MS&E 345 47 Tt0 payoff occurs T+  In particular: in a forward risk neutral world with respect to B(t;T+  ). Forward interest rates and the bond forward risk neutral world

48. Primbs, MS&E 345 48 Forwards Forward Rates Swap Rates

49. Primbs, MS&E 345 49 Swaps and Swap rates An interest rate swap is an agreement to exchange a fixed rate of interest S for a floating rate of interest on the same notional principal P. The swap rate S is the fixed rate that makes a swap have zero value. 0 33 44 55  22 Fixed 0 33 44 55  22 Float

50. Primbs, MS&E 345 50 Buy a swap to pay fixed S(T|T) Buy a forward swap to receive fixed S(t|T) Tt0 Forward swap rates and the annuity forward risk neutral world annuity total payoff T+  T+2  T+n  total cost at time T discount

51. Primbs, MS&E 345 51 Forward swap rates and the annuity forward risk neutral world at time T total payoff total cost Use the annuity A(t) as the numeraire: In particular: in a forward risk neutral world with respect to A(t). Tt0 T+  T+2  T+n 