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Unformatted text preview: Continuous Time Finance Notes, Spring 2004 Section 6, March 3, 2004 Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use in connec- tion with the NYU course Continuous Time Finance. The first bit of this section addresses the pricing of swaptions something we didnt get around to before. The rest is a supplement to Hulls discussion of the trinomial tree version of Hull-White. ***************** Pricing swaptions using Hull-White. We explained in Section 4 how a caplet is equiv- alent to a put option on a zero coupon bond. A similar argument shows that a floorlet is equivalent to a call option on a zero coupon bond. So we can easily derive formulas for the prices of caplets and floorlets using Blacks formula (problem 1 of HW 3 covers the case of a caplet). Caps and floors are simply portfolios of caplets and floorlets, so weve priced them too. But what about swaptions? The first observation is general: the task of pricing a swaption is identical to that of pricing a suitable option on a coupon bond. To be specific, lets suppose the underlying swap exchanges the floating rate for fixed rate k , the interest payments being at times T 1 , . . . , T N with a return of principal at T N . (The holder of the swap receives the fixed rate and pays the floating rate.) Consider the associated swaption, which gives the holder the right to enter into this swap at time T . For simplicity assume the time intervals T j- T j- 1 are all the same length t . The value of the underlying swap at time T is then P ( T , T N ) + k t N X j =1 P ( T , T j )- 1 times the notional principal. Indeed, the first term is the value at time T of the principal payment at T N ; the second term is the value at time T of the coupon payments; and the third term is the value at time T of a (short position in a) bond which pays the floating rate. Therefore the payoff of the swaption at time T is ( P ( T , T N ) + k t N X j =1 P ( T , T j )- 1) + . This is identical to the payoff of a call option on a coupon bond (with interest rate k and payments at times T j ) with strike 1. The next observation is special to Hull-White (well, its a bit more general than that: the argument works for any one-factor short-rate model). We claim that a call or put on a coupon bond is equivalent to a suitable portfolio of calls or puts options on zero coupon bonds. To explain why, lets focus on the case of a call. Recall that P ( t, T ) = A ( t, T ) exp[- B ( t, T ) r ( t )]. The key point is that P ( t, T ) is really a function of three variables: 1 t, T , and r ( t ), and it is monotone in the third argument r ( t ). So there is a unique critical value r * for the short rate at time T such that an option with payoff X = P ( T , T N ) + k t N X j =1 P ( T , T j )- K + is in the money (at time T ) precisely if r ( T ) < r * . Moreover we can write X as a sum of call payoffs on zero-coupon bonds, X = P ( T , T N )- A ( T , T N...
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