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# section10 - Derivative Securities Fall 2007 Section 10...

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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) Black’s model, and (ii) trees. Briefly: we’re 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; I’ll also take some examples from the book Implementing Derivatives Models by Clewlow and Strickland, Wiley, 1998. (Steve Allen’s version of these notes includes a few pages about how a Monte Carlo scheme based on the Heath-Jarrow-Morton theory works. I don’t attempt that here.) ************************ Black’s model. Recall the formulas derived in Section 5 for the value of a put or call on a forward price: c [ F 0 , T ; K ] = e rT [ F 0 N ( d 1 ) - KN ( d 2 )] p [ F 0 , T ; K ] = e rT [ KN ( - d 2 ) - F 0 N ( - d 1 )] where d 1 = 1 σ T bracketleftBig log( F 0 /K ) + 1 2 σ 2 T bracketrightBig d 2 = 1 σ T bracketleftBig log( F 0 /K ) - 1 2 σ 2 T bracketrightBig = d 1 - σ T. Black’s 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 Black’s model is this: consider an option with maturity T , whose payoff φ ( 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 φ ( V T ) = ( V T - K ) + . Black’s model stipulates that (a) the value of the option today is its discounted expected payoff. No surprise there – it’s the same principle we’ve 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 [ φ ( V T )] . We write E rather than E RN because in the stochastic interest rate setting this is not the risk-neutral expectation; we’ll explain why E is different from the risk-neutral expectation later on. For the moment however, we concentrate on making Black’s model computable. For this purpose we simply specify that (under the distribution associated with E ) 1

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(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 ). We have not specified the variance of X = log V T ; it must be given as data. It is customary to specify the “volatility of the forward price” σ , with the convention that log V T has standard deviation σ T .
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section10 - Derivative Securities Fall 2007 Section 10...

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