1、Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.1,Interest Rate Derivatives: The Standard Market ModelsChapter 22,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.2,Why Interest Rate Derivatives are Much More Difficult to Value Than Stock Opti
2、ons,We are dealing with the whole term structure of interest rates; not a single variableThe probabilistic behavior of an individual interest rate is more complicated than that of a stock price,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.3,Why Interest Rate Derivativ
3、es are Much More Difficult to Value Than Stock Options,Volatilities of different points on the term structure are differentInterest rates are used for discounting as well as for defining the payoff,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.4,Main Approaches to Pric
4、ingInterest Rate Options,Use a variant of Blacks model Use a no-arbitrage (yield curve based) model,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.5,Blacks Model & Its Extensions,Blacks model is similar to the Black-Scholes model used for valuing stock optionsIt assumes
5、 that the value of an interest rate, a bond price, or some other variable at a particular time T in the future has a lognormal distribution,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.6,Blacks Model & Its Extensions(continued),The mean of the probability distribution
6、 is the forward value of the variableThe standard deviation of the probability distribution of the log of the variable is where s is the volatilityThe expected payoff is discounted at the T-maturity rate observed today,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.7,Bl
7、acks Model (Eqn 22.1 and 22.2, p 509),K : strike price F0 : forward value of variable,T : option maturity s : volatility,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.8,The Blacks Model: Payoff Later Than Variable Being Observed,K : strike priceF0 : forward value of va
8、riable s : volatility,T : time when variable is observed T * : time of payoff,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.9,Validity of Blacks Model,Blacks model appears to make two approximations: 1. The expected value of the underlying variable is assumed to be its
9、 forward price 2. Interest rates are assumed to be constant for discountingWe will see that these assumptions offset each other,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.10,European Bond Options,When valuing European bond options it is usual to assume that the futu
10、re bond price is lognormalWe can then use Blacks model (equations 22.1 and 22.2)Both the bond price and the strike price should be cash prices not quoted prices,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.11,Yield Vols vs Price Vols,The change in forward bond price i
11、s related to the change in forward bond yield bywhere D is the (modified) duration of the forward bond at option maturity,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.12,Yield Vols vs Price Volscontinued,This relationship implies the following approximationwhere sy is
12、 the yield volatility and s is the price volatility, y0 is todays forward yield Often sy is quoted with the understanding that this relationship will be used to calculate s,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.13,Theoretical Justification for Bond Option Model
13、,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.14,Caps,A cap is a portfolio of capletsEach caplet can be regarded as a call option on a future interest rate with the payoff occurring in arrearsWhen using Blacks model we assume that the interest rate underlying each cap
14、let is lognormal,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.15,Blacks Model for Caps(Equation 22.11, p. 517),The value of a caplet, for period tk, tk+1 is,Fk : forward interest rate for (tk, tk+1) sk : interest rate volatility,L: principal RK : cap ratedk=tk+1-tk,Op
15、tions, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.16,When Applying Blacks ModelTo Caps We Must .,EITHERUse forward volatilitiesVolatility different for each capletORUse flat volatilitiesVolatility same for each caplet within a particular cap but varies according to life of c
16、ap,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.17,Theoretical Justification for Cap Model,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.18,European Swaptions,When valuing European swap options it is usual to assume that the swap rate is
17、 lognormalConsider a swaption which gives the right to pay sK on an n -year swap starting at time T . The payoff on each swap payment date is where L is principal, m is payment frequency and sT is market swap rate at time T,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22
18、.19,European Swaptions continued(Equation 22.13, page 545),The value of the swaption is s0 is the forward swap rate; s is the swap rate volatility; ti is the time from today until the i th swap payment; and,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.20,Theoretical J
19、ustification for Swap Option Model,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.21,Relationship Between Swaptions and Bond Options,An interest rate swap can be regarded as the exchange of a fixed-rate bond for a floating-rate bondA swaption or swap option is therefore
20、 an option to exchange a fixed-rate bond for a floating-rate bond,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.22,Relationship Between Swaptions and Bond Options (continued),At the start of the swap the floating-rate bond is worth par so that the swaption can be viewe
21、d as an option to exchange a fixed-rate bond for parAn option on a swap where fixed is paid and floating is received is a put option on the bond with a strike price of parWhen floating is paid and fixed is received, it is a call option on the bond with a strike price of par,Options, Futures, and Oth
22、er Derivatives, 5th edition 2002 by John C. Hull,22.23,Convexity Adjustments,We define the forward yield on a bond as the yield calculated from the forward bond priceThere is a non-linear relation between bond yields and bond pricesIt follows that when the forward bond price equals the expected futu
23、re bond price, the forward yield does not necessarily equal the expected future yieldWhat is known as a convexity adjustment may be necessary to convert a forward yield to the appropriate expected future yield,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.24,Relationsh
24、ip Between Bond Yields and Prices (Figure 22.4, page 525),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.25,Analytic Approximation forConvexity Adjustment (Eqn 22.15, p. 525),Suppose a derivative depends on a bond yield, yT observed at time T . Define: G(yT) : price of
25、the bond as a function of its yield y0 : forward bond yield at time zerosy : forward yield volatilityThe convexity adjustment that should be made to the forward bond yield is,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.26,Convexity Adjustment for Swap Rate,The same f
26、ormula gives the convexity adjustment for a forward swap rate. In this case G(y) defines the relationship between price and yield for a bond that pays a coupon equal to the forward swap rate,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.27,Example 22.5 (page 526),An in
27、strument provides a payoff in 3 years equal to the 1-year zero-coupon rate multiplied by $1000Volatility is 20%Yield curve is flat at 10% (with annual compounding)The convexity adjustment is 10.9 bps so that the value of the instrument is 101.09/1.13 = 75.95,Options, Futures, and Other Derivatives,
28、5th edition 2002 by John C. Hull,22.28,Example 22.6 (Page 527),An instrument provides a payoff in 3 years = to the 3-year swap rate multiplied by $100Payments are made annually on the swapVolatility is 22%Yield curve is flat at 12% (with annual compounding)The convexity adjustment is 36 bps so that
29、the value of the instrument is 12.36/1.123 = 8.80,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.29,Timing Adjustments,When a variable is observed at time T1 and the resultant payoff occurs at time T2 rather than T1 , the growth rate of the variable should be increased
30、by where R is the forward interest rate between T1 and T2 expressed with a compounding frequency of m, sR is the volatility of R, R0 is the value of R today, F is the forward value of the variable for a contract maturing at time T1, sF is the volatility of F, and r is the correlation between R and F
31、,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.30,When is a Convexity or Timing Adjustment Necessary,A convexity or timing adjustment is necessary when the payoff from a derivative does not incorporate the natural time lags between an interest rate being set and the in
32、terest payments being madeThey are not necessary for a vanilla swap, a cap or a swap option,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,22.31,Deltas of Interest Rate Derivatives,Alternatives:Calculate a DV01 (the impact of a 1bps parallel shift in the zero curve)Calcula
33、te impact of small change in the quote for each instrument used to calculate the zero curveDivide zero curve (or forward curve) into buckets and calculate the impact of a shift in each bucketCarry out a principal components analysis. Calculate delta with respect to each of the first few factors factors,
