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Unformatted text preview: Derivative Securities Fall 2007 Section 10 Notes by Robert V. Kohn, extended and improved by Steve Allen. Courant Institute of Mathematical Sciences. Options on interest-based instruments: pricing of bond options, caps, floors, and swaptions. This section provides an introduction to valuation of options on interest rate products. We focus on two approaches: (i) Blacks model, and (ii) trees. Briefly: were taking the same methods developed earlier this semester for options on a stock or forward price, and applying them to interest-based instruments. The material discussed here can be found in chapters 26, 27, and 28 of Hull; Ill also take some examples from the book Implementing Derivatives Models by Clewlow and Strickland, Wiley, 1998. (Steve Allens version of these notes includes a few pages about how a Monte Carlo scheme based on the Heath-Jarrow-Morton theory works. I dont attempt that here.) ************************ Blacks model. Recall the formulas derived in Section 5 for the value of a put or call on a forward price: c [ F ,T ; K ] = e rT [ F N ( d 1 )- KN ( d 2 )] p [ F ,T ; K ] = e rT [ KN (- d 2 )- F N (- d 1 )] where d 1 = 1 T bracketleftBig log( F /K ) + 1 2 2 T bracketrightBig d 2 = 1 T bracketleftBig log( F /K )- 1 2 2 T bracketrightBig = d 1- T. Blacks model values interest-based instruments using almost the same formulas, suitably interpreted. One important difference: since the interest rate is no longer constant, we replace the discount factor e rT by B (0 ,T ). The essence of Blacks model is this: consider an option with maturity T , whose payoff ( V T ) is determined by the value V T of some interest-related instrument (a discount rate, a term rate, etc). For example, in the case of a call ( V T ) = ( V T- K ) + . Blacks model stipulates that (a) the value of the option today is its discounted expected payoff. No surprise there its the same principle weve been using all this time for valuing options on stocks. If the payoff occurs at time T then the discount factor is B (0 ,T ) so statement (a) means option value = B (0 ,T ) E [ ( V T )] . We write E rather than E RN because in the stochastic interest rate setting this is not the risk-neutral expectation; well explain why E is different from the risk-neutral expectation later on. For the moment however, we concentrate on making Blacks model computable. For this purpose we simply specify that (under the distribution associated with E ) 1 (b) the value of the underlying instrument at maturity, V T , is lognormal; in other words, V T has the form e X where X is Gaussian. (c) the mean E [ V T ] is the forward price of V (for contracts written at time 0, with delivery date T )....
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