Lecture 22Interest Rate DerivativesOptions, Futures, Derivatives / April 21, 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 21, 2008 2The payoff from the option at time T is max{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 21, 2008 3This is the Black model and, importantly, it does not assume geomet ric 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 21, 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 21, 2008 5Embedded Bond OptionsBond option is an option to buy or sell a particular bond by a particular date for a praticular price.In addition to trading in the OTC market, bond options are frequently embedded in bonds whenthey are issued to make them more attractive to either the issuer or potential purchasers.Embedded Bond OptionsOne example of a bond with an embedded bond option is a callable bond• Here a bond contains a provision allowing the issuing firm to buy back the bond at apredetermined price at certain times in the future.• The holder of such a bond has sold a call option. The strike price or call price in the option isthe predetermined price that must be paid by the issuer to the holder.• Callable bonds cannot usually be called for the first few years of their life (lockout period).• After that the call price is usually a decreasing function of time.– For example in a 10-year callable bond, there might be no call privileges for the first 2 years.– The issuer might have the right to buy the bond back at a price of 110 in years 3 and 4 ofits life, at a price of 107.5 in years 5 and 6, at a price of 106 in years 7 and 8, and at a priceof 103 in years 9 and 10.• The value of the call option is reflected in the quoted yields on bonds. Bonds with call featuresgenerally offer higher yields than bonds with no call features.Options, Futures, Derivatives / April 21, 2008 6Another example of an embedded bond option is a puttable bond.• This type of bond contains a provision that allow the holder to demand early redemption at apredetermined price at certain times in the future.• The holder of the bond has purchased a put option on the bond as well as the bond itself.• Because the put option increases the value of the bond to the holder, bonds with put featuresprovide lower yields than bonds with no put features.• Example of a puttable bond is a 10-year bond where the holder has the right to be repaid atthe end of 5 years. (also called retractable bond).Options, Futures, Derivatives / April 21, 2008 7European Bond OptionsMany OTC bond options and some embedded bond options are European. The assumption madeis that the bond price at the maturity of the option is lognormal. Then our model can be used toprice the option with F0equal to the forward bond price FB. The variable σ is set equal to theforward bond price volatility, σB. Note σBis defined so that σB√T is the standard deviation ofthe logarithm of the bond price at the maturity of the option. The equations for pricing aEuropean bond option arec = P (0, T ) [FBN(d1) − KN(d2)]p = P (0, T ) [KN(−d2) − FBN(−d1)]whered1=lnFBK+σ2BT2σB√Td2= d1− σB√TWe note that we can compute FBasFB=B0− IP (0, T )where B0is the bond price at time zero and I is the present value of the coupons that will be paidduring the life of the option.Options, Futures, Derivatives / April 21, 2008 8Example: Consider a 10-month European call option on a 9.75-year bond with a face value of$1000.When the option matures, the bond will have 8 years and 11 months remaining.Suppose that the current cash bond price is $960, the strike price is $1000, the 10-m onth risk-freeinterest rate is 10% per annum, the volatility of the forward bond price in 10 months is 9% perannum.The bond pays a semiannual coupon of 10% and coupon payments of $50 are expected in 3months and 9 months (the accrued interest is $25 and the quoted bond price is $935). Supposethat the 3-month and 9-month risk-free interest rates are 9.0% and 9.5% per annum, respectively.The present value of the coupon payments is, therefore,50e−0.25×0.09+ 50e−0.75×0.095= 95.45The bond forward price is thereforeFB= (960 − 95.45 ) e0.1×0.8333=