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# Options#2 - Let r be the zero-coupon yield for maturity T...

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Options#2 - 1 OPTIONS #2: INTEREST RATE DERIVATIVES: THE STANDARD MARKET MODELS (Hull Ch. 28, 7 th Ed.) Extensions of Black’s Model (Delayed Payoff) As we’ll see below, there are option contracts in which the decision to exercise occurs at date T , but the payoff doesn’t actually occur until a later date, T * > T . In this case we discount the payoff from time T * , but the probabilities in Black’s formula all depend on date T . Options#2 - 2 Let r * be the zero-coupon yield for maturity T * . Then, equations (28.1) and (28.2) 7 th Ed. [(26.1) and (26.2) 6 th Ed.] become c = * * T r e ± [ FN ( d 1 ) ± XN ( d 2 )], = b (0, T * )[ FN ( d 1 ) ± XN ( d 2 )] p = * * T r e ± [ XN ( ± d 2 ) ± FN ( ± d 1 )], = b (0, T * )[ XN ( ± d 2 ) ± FN ( ± d 1 )], where d 1 = T T X F V V 2 2 1 ) / ln( ² , Options#2 - 3 and d 2 = T T X F V V 2 2 1 ) / ln( ± = d 1 ± V T . and b (0, T * ) is the price of a zero coupon bond that pays \$1 at time T * . Interest Rate Caps (Hull § 26.3, 6 th Ed., §28.2, 7 th Ed.) Suppose you have a floating-rate loan based on six-month LIBOR. If the principal is \$100 million, and if Options#2 - 4 payments are made semi-annually, then you would normally pay 0.5 u 100 u R , where R is the six-month LIBOR (with semi-annual compounding) from the beginning of the six-month period. Now suppose a Financial Institution (FI) has offered you a cap on your payments of 10% per year (compounded semi-annually). In that case, your payment, PMT , is

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