Lecture 23Interest Rate Derivatives: Short RatesOptions, Futures, Derivatives / April 23, 2008 1Black’s ModelConsider a European call option on a variable whose value is V - which does not have to be theprice of a traded security. DefineT : Time to maturity of the optionF : Forward price of V for a contract with maturity TF0: Value of F at time zeroK : Strike price of the optionP (t, T ) : Price at time t of a zero-coupon bond paying $1 at time TVT: Value of V at time Tσ : Volatility of FWe value the option by:1. Assuming ln VTis normal with mean F0and standard deviation σ√T2. Discounting the expected payoff at the T -year rate (equivalent to multiplying the expectedpayoff by P (0, T ). )Options, Futures, Derivatives / April 23, 2008 2The payoff from the option at time T is ma x{VT− K, 0}. The lognormal assumption for VTimplies that the expected payoff isE(VT)N(d1) − KN(d2)where E( VT) is the expected value of VTandd1=lnE(VT)K+σ22Tσ√Td2=lnE(VT)K−σ22Tσ√TBecause we are assuming that E(VT) = F0, the value of the option isc = P (0, T ) [F0N(d1) − KN(d2)]whered1=lnF0K+σ22Tσ√Td2=lnF0K−σ22Tσ√TOptions, Futures, Derivatives / April 23, 2008 3This is the Black model and, importantly, it does not assume geometric Brownian motion for theevolution of either V or F . All that we require is that VTbe lognormal at time T . The parameterσ is usually referred to as the volatility of F or the forward volatility of V . Its only role is to definethe standard deviation of ln VTby means of the relationshipStandard deviation of ln VT= σ√TThe volatility parameter does not necessarily say anything about the standard deviation of ln V attimes other than T .Delayed Payoff:We can extend Black’s model to allow for the situation where the payoff is calculated from thevalue of the variable V at time T , but the payoff is actually made at some later time T∗. Theexpected payoff is discounted from time T∗instead of time T so thatc = P (0, T∗) [F0N(d1) − KN(d2)]whered1=lnF0K+σ22Tσ√Td2=lnF0K−σ22Tσ√TOptions, Futures, Derivatives / April 23, 2008 4Validity of Black’s Model:Black’s model is appropriate when interest rates are assumed to be either constant or deterministic.In this case, the forward price of V equals its future price and E(ST) = F0in a risk neutral world.When interest rates are stochastic, there are two aspects of the derivation of the formulas that areopen to question1. Why do we set E(VT) equal to the forward price F0of V ? This is not the same as the futuresprice.2. Why do we ignore the fact that interest rates are stochastic when discounting?These two assumptions offset each other. Black’s model has a sounder basis and widerapplicability than first guessed.Options, Futures, Derivatives / April 23, 2008 5Interest Rate Caps and FloorsAnother interest rate option offered in the OTC market is an interest rate cap.• Consider first a floating-rate note where the interest rate is reset periodically equal to LIBOR.• The time between resets is known as the tenor. Suppose the t enor is 3 months. The interestrate on the note for the first 3 months is the initial 3-month LIBOR rate; the interest rate forthe next 3 months is set equal to the 3-month LIBOR rate prevailing in the market at the3-month point; and so forth.An interest rate cap is designed to provide insurance against the rate of interest on thefloating-rate note rising above a certain level, known as the cap rate.• Suppose that the principal amount is $10 million, the tenor is 3 months, the life of the cap is 3years, and the cap rate is 4%.• The cap provides insurance against the interest on the floating rate note rising above 4%.Options, Futures, Derivatives / April 23, 2008 6Assume that there are no day-count issues and there is exactly 0.25 year between each paymentdate.• Suppose that on a particular reset date the 3-month LIBOR interest rate is 5%. The floatingrate note would require0.25 × 0.05 × $10, 000, 000 = $125, 00 0of interest to be paid 3 months later.• With a 3-month LIBOR rate of 4% the interes t payment would be0.25 × 0.04 × $10, 000, 000 = $100, 00 0• Therefore, the cap provides a payoff of $25,000.• At each reset date during the life of the cap we observe LIBOR. If LIBOR is less than 4%, thereis no payoff from the cap three months later.• If LIBOR is greater than 4%, the payoff is one quarter of the excess applied to the principal of$10 million. Note that caps are usually defined so that the initial LIBOR rate, even if it isgreater than the cap rate, does not lead to a payoff on the first reset date.Options, Futures, Derivatives / April 23, 2008 7Cap as a Portfolio of Interest Rate Options:Consider a cap with a total life of T , a principal of L, and a cap rate of RK.Suppose that the reset dates are t1, . . . , tnand define tn+1= T . Define Rkas the interest ratefor the period between time tkand tk+1observed at time tk. The cap leads to a payoff at timetk+1ofLδkmax{Rk− RK, 0} (1)where δk= tk+1− tk. Both Rkand RKare expressed with a compounding frequencey equal tothe frequency of resets.Equation (1) is a call option on the LIBOR rate observed at time tkwith the payoff occurring attime tk+1. The cap is a portfolio of n such options. LIBOR rates ar eobserved at times t1, . . . , tnand the corresponding payoffs occur at times t2, t3, . . . , tn+1. The n call options underlying thecap are known as caplets.Options, Futures, Derivatives / April 23, 2008 8Cap as a Portfolio of Bond Options:An interest rate cap can also be characterized as a portfolio of put options on zero-coupon bondswith payoffs on the puts occurring at the time they are calculated.The payoff in (1) at time tk+1is equivalent toLδk1 + Rkδkmax{Rk− RK, 0}at time tk. This reduces tomax{L −L(1 − RKδk)1 + Rkδk, 0} (2)The expressionL(1 + RKδk)1 + Rkδkis the value at time tkof a zero-coupon bond that pays off L(1 + RKδk) at time tk+1. Theexpression in (2) therefore the payoff from a put option with maturity tkon a zero-coupon bondwith maturity tk+1when the face value of the bond is L(1 + RKδk)and the strike price is L. Itfollows that an interest rate cap can be regarded as a portfolio of European put options onzero-coupon bonds.Options, Futures, Derivatives / April 23, 2008 9Floors and CollarsInterest rate floors and interest rate collars (sometimes called floor-ceiling agreements) are definedanaloguously to caps.• A floors provides a payoff when the interest rate on the