# U of M MATH 5076 - Lecture 22 Interest Rate Derivatives (38 pages)

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Lecture 22 Interest Rate Derivatives Options Futures Derivatives April 21 2008 1 Black s Model Consider a European call option on a variable whose value is V which does not have to be the price of a traded security Define T Time to maturity of the option F Forward price of V for a contract with maturity T F0 Value of F at time zero K Strike price of the option P t T Price at time t of a zero coupon bond paying 1 at time T VT Value of V at time T Volatility of F We value the option by 1 Assuming ln VT is normal with mean F0 and standard deviation T 2 Discounting the expected payoff at the T year rate equivalent to multiplying the expected payoff by P 0 T Options Futures Derivatives April 21 2008 2 The payoff from the option at time T is max VT K 0 The lognormal assumption for VT implies that the expected payoff is E VT N d1 KN d2 where E VT is the expected value of VT and d1 d2 ln E VT K 2 2 T ln E VT K 2 2 T T T Because we are assuming that E VT F0 the value of the option is c P 0 T F0N d1 KN d2 where d1 d2 Options Futures Derivatives April 21 2008 ln F0 K 2 ln F0 K 2 2 T T 2 T T 3 This is the Black model and importantly it does not assume geometric Brownian motion for the evolution of either V or F All that we require is that VT be lognormal at time T The parameter is usually referred to as the volatility of F or the forward volatility of V Its only role is to define the standard deviation of ln VT by means of the relationship Standard deviation of ln VT T The volatility parameter does not necessarily say anything about the standard deviation of ln V at times other than T Delayed Payoff We can extend Black s model to allow for the situation where the payoff is calculated from the value of the variable V at time T but the payoff is actually made at some later time T The expected payoff is discounted from time T instead of time T so that c P 0 T F0N d1 KN d2 where d1 d2 Options Futures Derivatives April 21 2008 ln F0 K 2 ln F0 K 2 2 T T 2 T T 4 Validity of Black s

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