12 when valuingthe instrument. The instrument is worth
100 x 0.1236ee = 8.80112
or $8.80. (This compares with a price of 8.54 obtained without any convexity adjustment.)
22.7 TIMING ADJUSTMENTS
 
In this section we examine the situation where a derivative provides a payoff at time 7, based onthe value of a variable v observed at an earlier time T,. Define:
° elee eeF: Forward value of v for a contract maturing at time 7;
E,(v,): Expected value of v;, in a world that is forward risk neutral with respect to P(t, T;)528 CHAPTER 22
E,(v,): Expected value of v, in a world that is forward risk neutral with respect to P(t, T)G: Forward price of a zero-coupon bond lasting between T; and T,R: Forward interest rate for period between 7, and 7>, expressed with a compoundingfrequency of mRo: Value of R todayo,: Volatility of Fog: Volatility of Gor: Volatility of Rp: Instantaneous correlation between F and R.
When we move from a world that is forward risk neutral with respect to P(t, T,) to one that isforward risk neutral with respect to P(t, T)), the numeraire ratio is
_ P(t.Ts)~ P(t, T;)
 
From Section 21.7, the growth rate of v increases by
hy = —pogoy (22.17)(The minus sign reflects the fact that, because G and R are instantaneously perfectly negativelycorrelated, the correlation between G and F is —p.)Because
16 =(al + Rimy"2-1)
the relationship between the voiatility of G and the volatility of R can be caiculated from It6’slemma asOG = oRR(T, —T))1+R/mHence equation (22.17) becomes_ porogR(Tz — T))1+ R/m
a, =
As an approximation we can assume that R remains constant at Ro to get
_ _ pororRoT (Tr — Ti)Ex) = Evenyexn( 1+ Rj/m (22.18)This equation shows how we can adjust the forward value of a variable to allow for a delaybetween the variable being observed and the payoff being made.
Example 22.7 Consider a derivative that provides a payoff in six years equal to the value of astock index observed in five years. Suppose that 1,200 is the forward value of the stock index for acontract maturing in five years. Suppose that the volatility of the forward value of the index is 20%,the volatility of the forward interest rate between years five and six is 18%, and the correlationbetween the two is —0.4. Suppose further that the zero curve is flat at 8% with annual compounding.interest Rate Derivatives: The Standard Market Models 529
We apply the results we have just produced to the situation where v is equal the value of the index,T, =5, Tr = 6, m=1, Ry = 0.08, p = —0.4, of = 0.20, and op = 0.18, so that
 
—0.4 x 0.20 x 0.18 x 0.08 x 5 x *)
E,(v) = Ex(opexr(- T4008
or E,(v,;) = 1.00535 £,(v,). From the forward risk-neutral arguments in Chapter 21, we know thatE,(v)) is the forward price of the index, or 1,200. It follows thatE (vr) = 1,200 x 1.00535 = 1,206.42
The value of the derivative is 1,206.42 x P(0, 6). In this case P(0, 6) = 1/1.08° = 0.6302, so that thevalue of the derivative is 760.25.
Application 1 Revisited
The analysis just given provides a different way of producing the result in Application | of Section22.6. Using the notation from that application, we define Ry as the interest rate between T and T*and R, as the forward rate for the period between time T and T*. In a world that is forward risk
 
eritend th eaceant of OY T*\ the eynecte aly Tation (22 125 a4feutia. with TOspeer ur Fy, a ). toe expected vac of Rr is Ro- Equation (4-4. 1o) givesopRyTtRo = E7(Rr) exp| —1+ Rot
where tr = T* — T. Equivalently,
2ORRyTtE7(Rr) = Ro exp (Fe vr)
Approximating the exponential function, we see that
 
This is the same result as equation (22.16).
22.8 = NATURAL TIME LAGS
 
The results in Sections 22.2 to 22.5 showed:
1, When the payoff from a derivative depends on the price of a traded security in a certaincurrency at time 7 and is made at time 7 in the same currency, we can value the derivative byassuming that the expected future value of the traded security equals its forward price. Theexpected payoff from the derivative is discounted at the risk-free rate observed in the markettoday for a maturity of T.
2. When the payoff from a derivative depends on the 7;-maturity interest rate observed attime 7, in a certain currency and is made at time 7) in the same currency, we can value thederivative by assuming that the expected future value of the interest rate equals the forward
‘ In this application of equation (22.18), 7, = T, T = T*, F = R, so that op = og, and p= 1. Also, m= I/t.530 CHAPTER 22
interest rate. The expected payoff from the derivative is discounted at the risk-free rateobserved in the market today for a maturity of 7.
3. When the payoff from a derivative depends on the swap rate at time T in a certain currencyand is an annuity in the same currency starting at time T, with the annuity payment datesbeing the swap payment dates, we can value the derivative by assuming that the expectedfuture swap rate equals the forward swap rate. The expected payoffs from the derivatives arediscounted using today’s zero curve.
Tn each of these cases, we can say that the payoffs from the derivative incorporate natural time lagsand are made in the natural currency. When we buy a traded security, we are expected to pay for itimmediately. There is no time lag. When we borrow or lend money, we observe the interest rate atone time and the interest is paid (in the same currency) at the maturity of the interest rate. Whenwe enter into a swap, the payments corresponding to the swap rate are made in the form of anannuity. In general, when a derivative is structured so that it incorporates natural time lags, noadjustments to forward rates and prices are required to value the derivative.
In Sections 22.6 and 22.7 we covered situations where payoffs from derivatives do not incorporatenatural time lags and showed that adjustments to forward rates and forward prices are necessarywhen the derivatives are valued. This material complements the material in Section 21.8, whichshows that forward rates and forward prices have to be adjusted when payoffs are not in the naturalcurrency.
22.9 HEDGING INTEREST RATE DERIVATIVES
 
 
This section discusses how the material on Greek letters in Chapter 14 can he extended to coverinterest rate derivatives.
In the context of interest rate derivatives, delta risk is the risk associated with a shift in the zerocurve. Because there are many ways in which the zero curve can shift, many alternative deltas canbe calculated. Some alternatives are:
1. Calculate the impact of a one-basis-point parallel shift in the zero curve. This is sometimestermed a DVO1.
2. Calculate the impact of small changes in the quotes for each of the instruments used toconstruct the zero curve.
3. Divide the zero curve (or the forward curve) into a number of sections (or buckets).Calculate the impact of shifting the rates in one bucket by one basis point, keeping the rest ofthe initial term structure unchanged.
4. Carry out a principal components analysis as outlined in Section 16.9. Calculate a delta withrespect to changes in each of the first few factors. The first delta then measures the impact ofa small, approximately parallel, shift in the zero curve; the second delta measures the impactof a small twist in the zero curve; and so on.
In practice, traders tend to prefer the second approach. They argue that the only way the zerocurve can change is if the quote for one of the instruments used to compute the zero curvechanges. They therefore feel that it makes sense to focus on the exposures arising from changes inthe prices of these instruments.Interest Rate Derivatives: The Standard Market Models 531
When several delta measures are calculated, there are many possible gamma measures. Supposethat ten instruments are used to compute the zero curve and that we measure deltas with respect tochanges in the quotes for each of these..Gamma is a second partial derivative of the formaT /ax; ax;, where II is the portfolio value. We have 10 choices for x; and 10 choices for x;and a total of 55 different gamma measures. This may be more than the trader can handle. Oneapproach is ignore cross-gammas and focus on the 10 partial derivatives where i = j. Another is tocalculate a single gamma measure as the second partial derivative of the value of the portfolio withrespect to a parallel shift in the zero curve. A further possibility is to calculate gamma with respectto the first two factors in a principal components analysis.
The vega of a portfolio of interest rate derivatives measures its exposure to volatility changes.One approach is to calculate the impact on the portfolio of the making the same small change tothe Black volatilities of all caps and European swap options. However, this assumes that onefactor drives all volatilities and may be too simplistic. A better idea is to carry out a principalcomponents analysis on the volatilities of caps and swap options and calculate vega measurescorresponding to the first two or three factors.
SUMMARY
 
Black’s model provides a popular approach for valuing European-style interest rate options. Theessence of Black’s model is that the value of the variable underlying the option is assumed to belognormal at the maturity of the option. In the case of a European bond option, Black’s modelassumes that the underlying bond price is lognormal at the option’s maturity. For a cap, the modelassumes that the interest rate underlying each of the constituent caplets is lognormaily distributed.In the case of a swap option, the model assumes that the underlying swap rate is lognormallydistributed. Each of these models is internally consistent, but they are not consistent with eachother.
Black’s model involves calculating the expected payoff based on the assumption that theexpected value of a variable equal its forward price and then discounting the expected payoff atthe zero rate observed in the market today. When the payoff does not incorporate a natural timelag, a convexity or timing adjustment to the forward price of the variable is necessary. Forexample, if an interest rate or a swap rate is observed at time T and leads to a payoff at time T,a convexity adjustment is necessary. If a value of an investment asset is observed at time T andpayment is made at a later time, a timing adjustment is necessary.
SUGGESTIONS FOR FURTHER READING
 
Black, F., “The Pricing of Commodity Contracts,” Journal of Financial Economics, 3 (March 1976),167-79,
Brotherton-Ratcliffe, R., and B. Iben, “Yield Curve Applications of Swap Products,” in AdvancedStrategies in Financial Risk Management, ed. R. Schwartz and C. Smith, New York Institute ofFinance, New York, 1993.532
CHAPTER 22
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
 
22.2.22.3.
22.7.
22.8.
22.9.
22.10.
22.11.
22.12.22.13.
. A company caps three-month LIBOR at 10% per annum. The principal ameunt is $20 million.
On a reset date, three-month LIBOR is 12% per annum. What payment would this lead to underthe cap? When would the payment be made?
Explain why a swap option can be regarded as a type of bond option.Use Black’s model to value a one-year European put option on a 10-year bond. Assume that thecurrent value of the bond is $125, the strike price is $110, the one-year interest rate is 10% per
annum, the bond’s price volatility is 8% per annum, and the present value of the coupons to bepaid during the life of the option is $10.
. Explain how you would value a derivative that pays off 100R in five years, where R is the one-
year interest rate (annually compounded) observed in four ycars. What difference would it makeif the payoff were in four years? What difference would it make if the payoff were in six years?
. Explain whether any convexity or timing adjustments are necessary when:
a. We wish to value a spread option that pays off every quarter the excess (if any) of the five-year swap rate over the three-month LIBOR rate applied to a principal of $100. The payoffoccurs 90 days after the rates are observed.
b. We wish to value a derivative that pays off every quarter the three-month LIBOR rateminus the three-month Treasury bill rate. The payoff occurs 90 days after the rates areobserved.
. Explain carefully how you would use (a) spot volatilities and (b) flat volatilities to value a five-
year cap.Calculate the price of an option that caps the three-month rate, starting in 15 months time, at13% (quoted with quarterly compounding) on a principal amount of $1,000. The forwardinterest rate for the period in question is 12% per annum (quoted with quarterly compounding),the 18-month risk-free interest rate (continuously compounded) is 11.5% per annum, and thevolatility of the forward rate is 12% per annum.
A bank uses Black’s model to price European bond options. Suppase that an implied pricevolatility for a five-year option on a bond maturing in 10 years is used to price a nine-year optionon the bond. Would you expect the resultant price to be too high or too low? Explain.
Calculate the value of a four-year European call option on a five-year bond using Black’smodel. The five-year cash bond price is $105, the cash price of a four-year bond with the samecoupon is $102, the strike price is $100, the four-year risk-free interest rate is 10% per annumwith continuous compounding, and the volatility for the bond price in four years is 2% perannum.
If the yield volatility for a five-year put option on a bond maturing in 10 years time is specified as22%, how should the option be valued? Assume that, based on today’s interest rates, themodified duration of the bond at the maturity of the option will be 4.2 years and the forwardyield on the bond is 7%.
What other instrument is the same as a five-year zero-cost collar where the strike price of the capequals the strike price of the floor? What does the common strike price equal?
Derive a put-call parity relationship for European bond options.
Derive a put—call parity relationship for European swap options.Interest Rate Derivatives: The Standard Market Models 533
22.14.
22.15.22.16.
22.17.
22.18.
22.19.
22.20.
22.21,
22.22.
22.23.
22.24,
Explain why there is an arbitrage opportunity if the implied Black (flat) volatility of a cap isdifferent from that of a floor. Do the broker quotes in Table 22.1 present an arbitrageopportunity? :
When a bond’s price is lognormal can the bond’s yield be negative? Explain your answer.
Suppose that in Example 22.3 of Section 22.3 the payoff occurs after one year (i.e., when theinterest rate is observed) rather than in 15 months. What difference does this make to the inputsto Black’s models?
The yield curve is flat at 10% per annum with annual compounding. Calculate the value of aninstrument where, in five years’ time, the two-year swap rate (with annual compounding) isreceived and a fixed rate of 10% is paid. Both are applied to a notional principal of $100.Assume that the volatility of the swap rate is 20% per annum. Explain why the value of theinstrument is different from zero.
What difference does it make in Problem 22.17 if the swap rate is observed in five years, but theexchange of payments takes place in (a) six years, and (b) seven years? Assume that thevolatilities of all forward rates are 20%. Assume also that the forward swap rate for the periodbetween years 5 and 7 has a correlation of 0.8 with the forward interest rate between years 5 and 6and a correlation of 0.95 with the forward interest rate between years 5 and 7.
What is the value of a European swap option that gives the holder the right to enter into a three-year annual-pay swap in four years where a fixed rate of 5% is paid and LIBOR is received. Theswap principal is $10 million, Assume that the yield curve is flat at 5% per annum with annualcompounding and the volatility of the swap rate is 20%. Compare your answer with that givenby DerivaGem.
Suppose that the yield, R, on a zero-coupon bond follows the processdR = pdt+odz
where yt and o are functions of R and ¢, and dz is a Wiener process. Use It6’s lemma to show thatthe volatility of the zero-coupon bond price declines to zero as it approaches maturity.
The price of a bond at time T, measured in terms of its yield, is Gyr). Assume geometricBrownian motion for the forward bond yield, vy, in a world that is forward risk neutral withrespect to a bond maturing at time T. Suppose that the growth rate of the forward bond yield is@ and its volatility o,.a. Use Ité’s lemma to calculate the process for the forward bond price in terms of a, o,, y,and G(y).b. The forward bond price should follow a martingale in the world considered. Use this factto calculate an expression for a.c. Show that the expression for a is, to a first approximation, consistent with equation (22.15).Carry out a manual calculation to verify the option prices in Example 22.2.Suppose that the 1-year, 2-year, 3-year, 4-year and 5-year zero rates are 6%, 6.4%, 6.7%, 6.9%,and 7%. The price of a 5-year semiannual cap with a principal of $100 at a cap rate of 8% is $3.Use DerivaGem to determine:a. The 5-year flat volatility for caps and floorsb. The floor rate in a zero-cost 5-year collar when the cap rate is 8%
Show that V, + f = V2, where V, is the value of a swap option to pay a fixed rate of sx andreceive LIBOR between times 7, and 7>, f is the value of a forward swap to receive a fixed rateof sx and pay LIBOR between times T; and T, and V; is the value of a swap option to receive a534
22.25.
22.26.
CHAPTER 22
fixed rate of sx between times 7, and T,. Deduce that V; = V2 when sx equals the currentforward swap rate.
Suppose that zero rates are as in Problem 22.23. Use DerivaGem to determine the value of anoption to pay a fixed rate of 6% and receive LIBOR on a five-year swap starting in one year.Assume that the principal is $100 million, payments are exchanged semiannually, and the swaprate volatility is 21%.
Describe how you would (a) calculate cap flat volatilities from cap spot volatilities and(b) calculate cap spot volatilities from cap flat volatilities.
ASSIGNMENT QUESTIONS
 
22.27.
22.28.
22.29,
22.30.
22.31.
22.32.
 
Consider an eight-month European put option on a Treasury bond that currently has 14.25 yearsto maturity. The current cash bond price is $910, the exercise price is $900, and the volatility forthe bond price is 10% per annum. A coupon of $35 will be paid by the bond in three months.The risk-free interest rate is 8% for all maturities up to one year. Use Black’s model to determinethe price of the option. Consider both the case where the strike price corresponds to the cashprice of the bond and the case where it corresponds to the quoted price.
Calculate the price of a cap on the 90-day LIBOR rate in nine months’ time when the principalamount is $1,000. Use Black’s model and the following information:a. The quoted nine-month Eurodollar futures price = 92 (ignore differences between futuresand forward rates)b. The interest rate volatility implied by a nine-month Eurodollar option = 15% per annumc. The current 12-month interest rate with continuous compounding = 7.5% per annumd. The cap rate = 8% per annum. (Assume an actual/360 day count.)
Suppose that the LIBOR yield curve is flat at 8% with annual compounding. A swaption givesthe holder the right to receive 7.6% in a five-year swap starting in four years. Payments are madeannually. The volatility for the swap rate is 25% per annum and the principal is $1 million. UseBlack’s model to price the swaption. Compare your answer with that given hy DerivaGem.
Use the DerivaGem software to value a five-year collar that guarantees that the maximum andminimum interest rates on a LIBOR-based loan (with quarterly resets) are 5% and 7%,respectively. The LIBOR zero curve (continuously compounded) is currently flat at 6%. Use aflat volatility of 20%. Assume that the principal is $100.
Use the DerivaGem software to value a European swap option that gives you the right intwo years to enter into a 5-year swap in which you pay a fixed rate of 6% and receive floating.Cash flows are exchanged semiannually on the swap. The 1-year, 2-year, 5-year, and 10-yearzero-coupon interest rates (continuously compounded) are 5%, 6%, 6.5%, and 7%, respectively.Assume a principal of $100 and a volatility of 15% per annum. Give an example of how theswap option might be used by a corporation. What bond option is equivalent to the swapoption?
Suppose that the LIBOR yield curve is flat at 8% (with continuous compounding). The payofffrom a derivative occurs in four years. It is equal to the five-year rate minus the two-year rate atthis time, applied to a principal of $100 with both rates being continuously compounded. (Thepayoff can be positive or negative.) Calculate the value of the derivative. Assume that thevolatility for all rates is 25%, What difference does it make if the payoff occurs in five yearsinstead of four years? Assume all rates are perfectly correlated.Interest Rate Derivatives: The Standard Market Models 535
22.33,
22.34.
Suppose that the payoff from a derivative will occur in ten years and will equal the three-yearU.S. dollar swap rate for a semiannual-pay swap observed at that time applied to a certainprincipal. Assume that the yield curve is flat at 8% (semiannually compounded) per annum indollars and 3% (semiannually compounded) in yen. The forward swap rate volatility is 18%, thevolatility of the ten-year “‘yen per dollar” forward exchange rate is 12%, and the correlationbetween this exchange rate and U.S. dollar interest rates is 0.25.a. What is the value of the derivative if the swap rate is applied to a principal of $100 millionso that the payoff is in dollars?b. What is its value of the derivative if the swap rate is applied to a principal of 100 millionyen so that the payoff is in yen?
The payoff from a derivative will occur in 8 years. It will equal the average of the 1-year interestrates observed at times 5, 6, 7, and 8 years applied to a principal of $1,000. The yield curve is flatat 6% with annual compounding and the volatilities of all rates are 16%. Assume perfectcorrelation between all rates. What is the value of the derivative?536 CHAPTER 22
APPENDIX 22A
Proof of the Convexity Adjustment FormulaThis appendix calculates a convexity adjustment for forward bond yields. Suppose that the payofffrom a derivative at time T depends on a bond yield observed at that time. Define:
yo: Forward bond yield observed today for a forward contract with maturity Tyr: Bond yield at time TBy: Price of the bond at time To,: Volatility of the forward bond yieldWe suppose thatBy = G(yr)Expanding G(yr) in a Taylor series about yr = yo yields the approximation
Br = G(yq) + O'r — ¥0)G'(0) + 9.5097 — YoY GE" (v0)
where G’ and G” are the first and second partial derivatives of G. Taking expectations in a worldthat is forward risk neutral with respect to a zero-coupon bond maturing at time 7, we obtain
 
Ey(Br) = G(y9) + Err — YoG'(N0o) + LEO — oY IG" On)
where Ey denotes expectations in this world. The expression G(yo) is by definition the forwardbond price. Also, because of the particular world we are working in, E7(Bry) equals the forwardbond price. Hence E7(By) = G(yo), so that
Oly CMs 2 ynu Tt 4 YO) mY
VU ife TeV Va b eyETT JOY WOIT JETRIT JO) I
The expression E;{(yr — yo)"] is approximately onyeT. Hence it is approximately true that
G'(y)1,22 0E. ~y_—l T
r(¥r) = Yo 7 0Fy G'(v9)
This shows that, to obtain the expected bond yield in a world that is forward risk neutral withrespect to a Zero-coupon bond maturing at time T, we should add
G" (yo)1,2 2 9—3)09yT
27" G0)
to the forward bond yield. This is the result in equation (22.15). For an alternative proof, see
Problem 22.21.
 
 CHAPTER 23
r 7a
NTEREST RATE DERIVATIVES:MODELS OF THE SHORT RATE
The models for pricing interest rate options presented in Chapter 22 make the assumption that theprobability distribution of an interest rate, a bond price, or some other variable at a future point intime is lognormal. They are widely used for valuing instruments such as caps, European bondoptions, and European swap options. However, they have limitations. They do not provide adescription of how interest rates change through time. Consequently, they cannot be used forvaluing interest rate derivatives such as American-style swap options, callable bonds, andstructured notes.
This chapter and the next discuss alternative approaches for overcoming these limitations. Theseinvolve building what is known as a term structure model. This is a model describing the evolutionof the zero curve through time. In this chapter we focus on term structure models that areconstructed by specifying the behavior of the short-term interest rate, r.
23.1 EQUILIBRIUM MODELS
 
 
Equilibrium models usually start with assumptions about economic variables and derive a processfor the short rate, r. They then explore what the process for r implies about bond prices and optionprices. The short rate, r, at time t is the rate that applies to an infinitesimaily short period of time attime t. It is sometimes referred to as the instantaneous short rate. It is not the process for 7 in the realworld that matters. Bond prices, option prices, and other derivative prices depend only on theprocess followed by r in a risk-neutral world. The risk-neutral world we consider here will be thetraditional risk-neutral world, where, in a very short time period between ¢ and ¢ + dt, investors earnon average r(t) dt. All processes for r that we present will be processes in this risk-neutral world.From equation (21.19), the value at time ¢ of an interest rate derivative that provides a payoff of
fr at time T isEe" fr) (23.1)
where 7 is the average value of r in the time interval between ¢ and 7, and E denotes the expectedvalue in the traditional risk-neutral world.
As usual, we define P(t, T) as the price at time ¢ of a zero-coupon bond that pays off $1 at time T.From equation (23.1),
PU, T) = Ee) (23.2)
537538 CHAPTER 23
If R(t, T) is the continuously compounded interest rate at time t for a term of T —1t, then
 
 
P(t, T) = eT REME-9 (23.3)so thatRUT) =-5 . ; In Pt, 7) (23.4)and, from equation (23.2),RUT) = 7 L In Ee) (23.5)
This equation enables the term structure of interest rates at any given time to be obtained from thevalue of r at that time and the risk-neutral process for r. It shows that, once we have fully definedthe process for r, we have fully defined everything about the initial zero curve and its evolutionthrough time.
23.2 ONE-FACTOR EQUILIBRIUM MODELS
 
 
In a one-factor equilibrium model, the process for r involves only one source of uncertainty.Usually the risk-neutral process for the short rate is described by an It6 process of the form
dr = m(r) dt + s(r) dz
‘The instantaneous drift, m, and instantaneous standard deviation, s, are assumed to be functionsof r, but are independent of time. The assumption of a single factor is not as restrictive as it mightappear. A one-factor model implies that all rates move in the same direction over any short timeinterval, but not that they all move by the same amount. The shape of the zero curve can thereforechange with the passage of time.
The next three sections consider three one-factor equilibrium models:
mr) =r, s(r)=or (Rendleman and Bartter model)mr)=ab—r), sr)=o (Vasicek model)mr) =alb—r), sir) =a Vr — (Cox, Ingersoll, and Ross model)
23.3 THE RENDLEMAN AND BARTTER MODEL
 
 
In the Rendleman and Bartter model, the risk-neutral process for r is!dr = pr dt + ar dz
where jz and o are constants. This means that r follows geometric Brownian motion. The processfor ris of the same type as that assumed for a stock price in Chapter 12. It can be represented usinga binomial tree similar to the one used for stocks in Chapter 18.7
' See R. Rendieman and B. Bart