is forward risk neutral with respect to a zero-coupon bond maturing at time 7}. This result,when combined with the result in equation (21.20), will be critical to our understanding of thestandard market model for interest rate caps in the next chapter.
Annuity Factor as the Numeraire
For our next application of equivalent martingale measure arguments, we consider a swap startingat a future time T with payment dates at times T,, T>,..., Ty. Define T) = T. Assume that theprincipal underlying the swap is $1. Suppose that the forward swap rate (ie., the interest rate on492 CHAPTER 21
the fixed side that makes the swap have a value of zero) is s(t) at time t (t < T). The value of thefixed side of the swap iss(QNA(D)
where
N-1AC) = SOs = Ti) PU Tr)i=0
We showed in Chapter 6 that, when the principal is added to the payment on the last payment dateswap, the value of the floating side of the swap on the initiation date equals the underlyingprincipal. It follows that if we add $1 at time 7 then the floating side is worth $1 at time Ty. Thevalue at time ¢t of $1 received at time Ty is P(t, Ty). The value at time t of $1 at time Ty is P(t, 79).The value of the floating side at time ¢ is therefore
P(t, To) — P(t, Ty)Equating the values of the fixed and floating sides, we obtain
SAW = PCt, Te) — PCL, Tw)° P(t, Ty) — P(t, Ty)
s(t) = A
(21.23)We can apply the equivalent martingale measure result by setting f equal to P(t, Ty) — P(t, Ty) andg equal to A(t). This leads to
s(t) = E,ls(T)] (21.24)
where E, denotes the expectation in a world that is forward risk neutral with respect to A(¢). Inaworld that is forward risk neutral with respect to A(t), the expected future swap rate is therefore thecurrent swap rate.
For any security, f, the result in equation (21.15) shows
 
fy = AOE, ( ts) (21.25)
This result, when combined with the result in equation (21.24), will be critical to our understandingof the standard market model for European swap options in the next chapter.
21.5 EXTENSION TO MULTIPLE INDEPENDENT FACTORS
The results presented in Sections 21.3 and 21.4 can be extended to cover the situation when thereare many independent factors.® Assume that there are n independent factors and that the processesfor f and g in the traditional risk-neutral world are
df=rfdi+ > opi fay
i=
© The independence condition is not critical. If factors are not independent, they can be orthogonalized.Martingales and Measures 493
and
dg =rgdt+ SS o,:9 dz;
i=l
It follows from Section 21.2 that we can define other worlds that are internally consistent by setting
nh nt ,df= (« + Sores) fat + drops fdai=l i=land
it itdg = (r+ Dy) a + Yo 9,19 dzi=} i=l
where the A; (1 <i <n) are the m market prices of risk. One of these other worlds is the realworld.
We define a world that is forward risk neutral with respect to g as a world where 4; = o,;. It canbe shown from Ité’s lemma, using the fact that the dz; are uncorrelated, that the process followedby f/g in this world has zero drift (see Problem 21.12). The rest of the results in the last twosections (from equation (21.15) onward) are therefore still true.
21.6 APPLICATIONS
 
 
In this section we provide two applications of the forward risk-neutral valuation argument. Severalothers are in Chapters 22 and 24.
The Black-Scholes Result
We can use forward risk-neutral arguments to extend the Black-Scholes result to situations whereinterest rates are stochastic. Consider a European call option maturing at time T on a non-dividend-paying stock. From equation (21.20), the call option’s price is given by
c= P(0, T)E7imax(S, — K, 0)) (21.26)
where Sp is the stock price at time T, K is the strike price, and Ey denotes the expectation in aworld that is forward risk neutral with respect to a zero-coupon bond maturing at time T. Define Ras the zero rate for maturity 7, so thatP(0,T) =e *and equation (21.26) becomesce =e" E,[max(S; — K, 0)] (21.27)
If we assume that S; is lognormal in the forward risk-ncutral world we are considering, with thestandard deviation of In(S;) equal to s, then Appendix 12A shows that
Er{max(S; — K, 0)] = E7(Sp)N(d,) — KN(d2) (21.28)494 CHAPTER 21
where
_ WnlEr(Sr)/K] + 8°/2
s
_ InlEr(S7)/K] = s°/2
s
dyd,
From equation (21.21), E7(S;) is the forward stock price for a contract maturing at time T. Hence,E7(Sp) = Sye*7 (21.29)Equations (21.27), (21.28), and (21.29) give
c = SyN(d,) — Ke“®" N(dp)where__ In(Sg/K) + RT + s°/2~ Ss_ In(Sy/K) + RT — 8? /2s
a,dy
If the stock price volatility a is defined so that o/T = s, then the expressions for d, and d, become
In(So/K) + (R + 0° /2)T
d, =—“SoVT
gy = InSo/K) + (R — o /2)T° oVT
showing that the call price is given by the Black-Scholes formula with r replaced by R. Similarresults can be produced for European put options.
Option to Exchange One Asset for Another
Consider next an option to exchange an investment asset worth U for an investment asset worth V.This has already been discussed in Section 19.11. We suppose that the volatilities of U and V areoy and oy and the coefficient of correlation between them is p.
Suppose first that the assets provide no income. We choose the numeraire security g to be U.Setting f = V in equation (21.15), we obtain
Vo = UvEy (=) (21.30)
where Ey denotes the expectation in a world that is forward risk neutral with respect to U.Next we set f in equation (21.15) as the value the option under consideration, so thatfr = max(V; — Up, 0). It follows that
max(V; — U;, 0fi = 0) By (OU)TMartingales and Measures 495
or
fo =Uo Fy] max( 7 —1, 0) (21.31)UrThe volatility of V/U is 6, where
6 = 04) +04 — 2payoy
From Appendix 12A, equation (21.31) becomes
fo = Up [Ee (=) M(dy) - ma]
where1 _ In(Vo/Up) + FT/2i éVTd, =d,-—6VT
Substituting from equation (21.30), we obtainfo = VoN(d,) ~ UpN(da) (21.32)Problem 21.8 shows that, when f and g provide income at rates qg, and q,, equation (21.15)becomesfo = get OE, (2)Or,
This means that equations (21.30) and (21.31) become
Ey ea) _ eiaumwort foT 0
ande _ ,-gul sp lr (Vr 4 NV]fo=et UoFu| max| 7 — 1, oF
and equation (21.32) becomesfo = € 7 VoN(di) — €" UgN(d2)
with d, and d, being redefined as
a, — Vol Uo) + Gu = 4 + &/2)T‘ 6VTdy =d, -6VT
This is the result given in equation (19.3).
21.7_ CHANGE OF NUMERAIRE
 
In this section we consider the impact of a change in numeraire on the process followed by amarket variable. In a world that is forward risk neutral with respect to g, the process followed by a496 CHAPTER 21
traded security f isdf= (- +> cuits) fdt +) op; faz;i=l i=l
Similarly, in a world that is forward risk neutral with respect to another security h, the process
followed by f is= (« + yy oust) Fa + Sop sf az;i=l i=l
where o;, ; is the ith component of the volatility of A.
The effect of moving from a world that is forward risk neutral with respect to g to one that isforward risk neutral with respect to A (.e., of changing the numeraire from g to h) is therefore toincrease the expected growth rate of the price of any traded security f by
Yoni — Oy i)O pii=
Consider next a variable v that is a function of the prices of traded securities. (The variable v is notnecessarily the price of a traded security itself.) Define o,; as the ith component of the volatility of
necessaril the price of a traded security sei, enne o, ; as tne ith component oF tne volatnty
v. From Ité’s lemma i in Appendix 21A, we can calculate what happens to the process followed by vwhen there is a change in numeraire causing the expected growth rate of the underlying tradedsecurities to change. It turns out that the expected growth rate of v responds to a change innumeraire in the same way as the expected growth rate of the prices of traded securities (seeProblem 11.6 for the situation where there is only one stochastic variable and Problem 21.13 forthe general case). It increases by
Ot, = Yn = Fya)%o3 (21.33)i=l
Define w=h/g and o,,; as the ith component of the volatility of w. From itd’s lemma (seeProblem 21.14),
wi = Cn — Feso that equation (21.33) becomes& = ow ier (21.34)i=l
We will refer to w as the numeraire ratio. Equation (21.34) is equivalent toAy = PTO y (21.35)
where o, is the total volatility of v, o,, is the total volatility of w, and p is the instantaneouscorrelation between v and w.”
7 To see this, note that the changes év and dw in v and w in a short period of time ét are given by
7 Soa, V8, dws + So oygwe V8rBecause the dz, are uncorrelated, E(e; €;) = 0 when i # j. Also, from the definition of p,U0, WI, = E(dvdw) — E(dv) E(w)When terms of order higher than dt are ignored, this leads to
poy Ow = » Opi Ty,Martingales and Measures 497
This is a surprisingly simple result. The adjustment to the expected growth rate of a variable vwhen we change from one numeraire to another is the instantaneous covariance between thepercentage change in v and the percentage change in the numeraire ratio. We will apply the resultto quantos in this chapter and to what are termed timing adjustments in Chapter 22.
21.8. QUANTOS
 
A quanto of cross-currency derivative is an instrument where two currencies are involved. Thepayoff is defined in terms of a variable that is measured in one of the currencies and the payoff ismade in the other currency. One example of a quanto is the CME futures contract on the Nikkeimentioned in Section 3.10. The market variable underlying this contract is the Nikkei 225 index(which is measured in yen), but the contract is settled in U.S. dollars.
Consider a quanto that provides a payoff in currency X at timc 7. Wo assume that the payoffdepends on the value of a variable V that is observed in currency ¥ at time 7’. Define:
F(t): Forward value of V at time ¢ for a contract denominated in currency Y and maturingat time T
Vr: Value of V at time TPy(t,T): Value at time ¢ in currency X of a zero-coupon bond paying off | unit of currency Xat time 7Py(t,7): Value at time t in currency X of a zero-coupon bond paying off 1 unit of currency ¥at time TEy(-): Expectation at time zero in a world that is forward risk neutral with respect to Py (t, T)Ey(+): Expectation at time zero in a world that is forward risk neutral with respect to Py(t, T)G(t): Forward exchange rate at time t (number of units of currency Y¥ that equal one unitof currency X) in a forward contract maturing at time Tof: Volatility of F(t)6g: Volatility of GZ)p: Instantaneous correlation between F(t) and G(t)S;: Spot exchange rate at time T (= G(T))
We know thatEy(Vr) = Fo
and are interested in Ey(V;). When we change the numeraire from Py(t, T) to P(t, 7) world, thenumeraire ratio isPxQT)Py(t, T) ~
 
Gi)
It follows from equation (21.35) that the numeraire change leads to the expected growth rate ofF(t) increasing byporog498 CHAPTER 21
This means that it is approximately true thatEx(F(7)] = EylF(r)jer7e"or because V; = F(T) and Ey(Vr) = F(0)Ex(Vz) = Ferre" (21.36)This relationship is in turn approximately the same as
Ex(Vp) = FlO)(1 + popogT) (21.37)
Example 21.4 Suppose that the current value of the Nikkei stock index for a one-year contract is15,000 yen, the one-year dollar risk-free rate is 5%, the one-year yen risk-free rate is 2%, and the yendividend yield is 1%. The forward price of the Nikkei for a contract denominated in yen can becalculated in the usual way from equation (3.12) as
15,000¢0?-0-D*! = 1550.75
Suppose that the volatility of the one-year forward price of the index is 20%, the volatility of the one-year forward yen per dollar exchange rate is 12%, and the correlation of the one-year forward Nikkeiwith the one-year forward exchange rate is 0.3. In this case, F(0) = 15,150.75, o, = 0.20, og = 0.12,and p = 0.3. From equation (21.36), the expected value of the Nikkei in a world that is forward riskneutral with respect to a dollar bond maturing in one year is
15,150,75e03%02%012x1 = 15,960.23
This is the forward price of the Nikkei for a contract that provides a payoff in dollars rather than yen.(As an approximation, it is also the futures price of such a contract.)
We will apply equation (21.37) to the valuation of what are known as diff swaps in Chapter 25.
Using Traditional Risk-Neutral Measures
The forward risk-neutral measure we have been using works well when payoffs occur at only onetime. In other situations it is sometimes more appropriate to use the traditional risk-neutralmeasure. Suppose we know the process followed by a variable V in the traditional currency-Yrisk-neutral world and we wish to estimate its process in the traditional currency-X risk-neutralworld. Define:
S: Spot exchange rate (units of ¥ per unit of X)os: Volatility of Soy: Volatility of Vp: Instantaneous correlation between S and V.In this case, the change of numeraire is from the money market account in currency Y to themoney market account in currency X (with both money market accounts being denominated in
currency X). As indicated by equation (21.16), each money market account has a stochasticgrowth rate, but zero volatility. It can be shown from Ité’s lemma that the volatility of theMartingales and Measures 499
numeraire ratio is os (see Problem 21.15). The change of numeraire therefore involves increasingthe expected growth rate of V byPoyos
The market price of risk changes from zero to pos.
Example 21.5 A two-year American option provides a payoff of max(S — K, 0) pounds sterling,where S is the level of the S&P 500 at the time of exercise and K is the strike price. The current levelof the S&P 500 is 1,200. The risk-free rates in sterling and dollars are both constant at 5% and 3%,respectively, the correlation between the dollars-per-sterling exchange rate and the S&P 500 is 0.2, thevolatility of the S&P 500 is 25%, and the volatility of the exchange rate is 12%. The dividend yield onthe S&P 500 is 1.5%.
We can value this option by constructing a binomial tree for the S&P 500 using as the numerairethe money market account in the U.K. (i.e., using the traditional risk-neutral as seen from theperspective of a U.K. investor). We have just shown that the change in numeraire leads to an increase
in the expected growth rate of0.2 x 0.25 x 0.12 = 0.006
or 0.6%. The growth rate of the S&P 500 using a U.S. dollar numeraire is 3— 1.5 = 1.5%. Thegrowth rate using the sterling numeraire is therefore 2.1%. The risk-free interest rate in sterling is 5%.The S&P 500 therefore behaves like an asset providing a dividend yield of 5 — 2.1 = 2.9% using thesterling numeraire. Using the parameter values of S = 1,200, K = 1,200, r=0.05, q =0.029,o = 0.25, and T =2 with 100 time steps, DerivaGem estimates the value of the option as £179.83.
21.9 SIEGEL’'S PARADOX
 
 
An interesting application of the results in the last two sections is to what has become known asSiegel’s paradox. Consider two currencies, X and Y. Define S as the number of units of currency Yper unit of currency X. Because an exchange rate behaves like a stock paying a dividend yield equal
ty @ foreien 4 ee interac a e ition risk-neutral process fo 1.to the foreign risk-free interest rate, the traditional risk-neutral process for S is
dS = (ry — ry)Sdt + 05S dz (21.38)
where ry and ry are the interest rates (assumed constant) in currencies X and Y.From [té’s lemma, equation (21.38) implies that the process for 1/S is
d(1/S) = (ry — ry + 08)C/S) dt — o5(1/S) dz (21.39)
This leads to what is known as Siege/’s paradox. Since the expected growth rate of S is ry — ry,symmetry suggests that the expected growth rate of 1/S should be ry —ry. It appears to be aparadox that the expected growth rate in equation (21.39) is ry — ry +o% rather than ry — ry.To understand Siegel’s paradox it is necessary to appreciate that equation (21.38) is the risk-neutral process for S$ in a world where the numeraire is the money market account in currency Y.Equation (21.39), because it is deduced from equation (21.38), gives the risk-neutral process for |/Swhen this is the numeraire. Because 1/S is the number of units of X per unit of Y, to besymmetrical we should measure its process in a world where the numeraire is the money marketaccount in currency X. The previous section shows that when we change the numeraire, from themoney market account in currency Y to the money market account in currency X, the growth rate500 CHAPTER 21
of 1/S, increases by poyos, where V = 1/S and p is the correlation between S and 1/S. In this case,p=~l and o, =as. It follows that the change of numeraire causes the growth rate of 1/5 toincrease by —oy. From equation (21.39), therefore, the growth rate of 1/S in a world where thenumeraire is the money market account in currency X rather than currency F is
d(1/S) = (rx — ry\(L/S) dt — o5(1/S) dz (21.40)
This is symmetrical with the process for $ in equation (21.38).
SUMMARY
 
The market price of risk of a variable defines the tradeoffs between risk and return for tradedsecurities dependent on the variable. When there is one underlying variable, a derivative’s excessreturn over the risk-free rate equals the market price of risk multiplied by the variable’s volatility.When there are many underlying variables, the excess return is the sum of the market price of riskmultiplied by the volatility for each variable.
A powerful tool in the valuation of derivatives is risk-neutral valuation. This was introduced inChapters 10 and 12. The principle of risk-neutral valuation shows that, if we assume that the worldis risk neutral when valuing derivatives, we get the right answer—not just in a risk-neutral world,but in all other worlds as well. In the traditional risk-neutral world, the market price of risk of allvariables is zero. Furthermore, the expected price of any asset in this world is its futures price.
In this chapter we have extended the principle of risk-neutral valuation. We have shown that,when interest rates are stochastic, there are many interesting and useful alternatives to thetraditional risk-neutral world. When there is only one stochastic variable, a world is defined asforward risk neutral with respect to a security price if the market price of risk for the variable is setequal to the volatility of the security price. A similar definition applies when there are manystochastic variables. We have shown that, in a world that is forward risk neutral with respect to asecurity price g, the ratio f/g is a martingale for all other security prices f. A martingale is a zero-drift stochastic process. Any variable following a martingale has the simplifying property that itsexpected value at any future time equals its value today. It turns out that by appropriatelychoosing the numeraire security, g, we can simplify the valuation of many interest rate dependentderivatives.
In this chapter we have shown how our extensions of risk-neutral valuation enable quantos andoptions to exchange one asset for another to be valued. In Chapters 22 and 24 the extensions willbe useful in valuing interest rate derivatives.
SUGGESTIONS FOR FURTHER READING
 
 
Baxter, M., and A. Rennie, Financial Calculus, Cambridge University Press, Cambridge, 1996.
Cox, J. C., J. E. Ingersoll, and S. A. Ross, “An Intertemporal General Equilibrium Model of AssetPrices,” Econometrica, 53 (1985), 363-84.
Duffie, D., Dynamic Asset Pricing Theory, Princeton University Press, Princeton, NJ, 1992.
Garman, M., ‘A General Theory of Asset Valuation under Diffusion State Processes,” WorkingPaper 50, University of California, Berkeley, 1976.Martingales and Measures 501
Harrison, J. M., and D. M. Kreps, “Martingales and Arbitrage in Multiperiod Securities Markets,”Journal of Economic Theory, 20 (1979), 381-408.
Harrison, J. M., and S. R. Pliska, “Martingales and Stochastic Integrals in the Theory of ContinuousTrading,” Stochastic Processes and Their Applications, \! (1981), 215-260.
Hull, J. C., and A. White, “An Overview of the Pricing of Contingent Claims,’ Canadian Journal ofAdministrative Sciences, 5 (September 1988), 55-61.
Jamshidian, F., “Corralling Quantos,” RISK, March 1994, pp. 71-75.Reiner, E., “Quanto Mechanics,” RIS‘K, March 1992, pp. 59-63.
QUESTIONS AND PROBLEMS(ANSWERS IN THE SOLUTIONS MANUAL)
 
 
21.1,21.2.
21.3.
21.4.
21.5.
21.6.
21.8.
How is the market price of risk defined for a variable that is not the price of an investment asset?
Suppose that the market price of risk for gold is zero. If the storage costs are 1% per annum andthe risk-free rate of interest is 6% per annum, what is the expected growth rate in the price ofgold? Assume that gold provides no income.
A security’s price is positively dependent on two variables: the price of copper and the yen-dollarexchange rate. Suppose that the market price of risk for these variables is 0.5 and 0.1,respectively. If the price of copper were held fixed, the volatility of the security would be8% per annum; if the yen-dollar exchange rate were held fixed, the volatility of the securitywould be 12% per annum. The risk-free interest rate is 7% per annum. What is the expected rateof return from the security? If the two variables are uncorrelated with each other, what is thevolatility of the security?An oil company is set up solely for the purpose of exploring for oil in a certain small area ofTexas. [ts value depends primarily on two stochastic variables: the price of oil and the quantityof proven oil reserves. Discuss whether the market price of risk for the second of these twovariables is likely to be positive, negative, or zero.Deduce the differential equation for a derivative dependent on the prices of two non-dividend-paying traded securities by forming a riskless portfolio consisting of the derivative and the twotraded securities.Suppose that an interest rate x follows the process
dx = a(xy — x) dt +o/xdz
where a, Xp, and ¢ are positive constants. Suppose further that the market price of risk for x is A.What is the process for x in the traditional risk-neutral world?
. Prove that, when the security f provides income at rate q, equation (21.9) becomes
+q—r=ho. (Hint: Form a new security, f* that provides no income by assuming that allthe income from f is reinvested in f.)
Show that when f and g provide income at rates qy and q,, respectively, equation (21.15)becomes
9T
(Hint: Form new securities f* and g* that provide no income by assuming that all the incomefrom f is reinvested in f and all the income in g is reinvested in g.)
fo = ge" E, (“)502
21.9.
21.10.
21.11.
NofadbenN
21.13.21.14.
21.15.
CHAPTER 21
“The expected future value of an interest rate in a risk-neutral world is greater than it is in thereal world.” What does this statement imply about the market price of risk for (a) an interest rateand (b) a bond price. Do you think the statement is likely to be true? Give reasons.
The variable S is an investment asset providing income at rate q measured in currency A. Itfollows the processdS = fusS dt + 05S dz
in the real world. Defining new variables as necessary, give the process followed by S, and thecorresponding market price of risk, ina. A world that is the traditional risk-neutral world for currency Ab. A world that is the traditional risk-neutral world for currency Bc. A world that is forward risk neutral with respect to a zero-coupon currency-A bondmaturing at time Td. A world that is forward risk neutral with respect to a zero-coupon currency-B bondmaturing at time T
A call option provides a payoff at time 7 of max(S; — K, 0) yen, where S; is the dollar price ofgold at time 7 and K is the strike price. Assuming that the storage costs of gold are zero anddefining other variables as necessary, calculate the value of the contract.
Dp at mitt tay Seti OF Sahat wl» FLOVe Ube feSull HI SeCUOL 21.) Ubal WILCKE
at itdf=[r+ > doz fdt + doy, f dr;i=] i=]
na ntdg =[r-+Y Ao, lade + oy ig dz)i=l i=l
with thtr
Prove equation (21.33) in Section 21.7.
Show that, when w=//g and h and g are each dependent on n Wiener processes, theith component of the volatility of w is the ith component of the volatility of # minus the ithcomponent of the volatility of g.
Suppose that # is the money market account in currency X and g is the money market account incurrency ¥ with both money market accounts being denominated in currency X. Show that thevolatility of h/g is the volatility of the exchange rate (units of Y per unit of X). Assume that thestochastic process for the exchange rate is geometric Brownian motion.
ASSIGNMENT QUESTIONS
 
 
21.16.
 
 
Suppose that an index of Canadian stocks currently stands at 400. The Canadian dollar iscurrently worth 0.70 U.S. dollars. The risk-free interest rates in Canada and the U.S. areconstant at 6% and 4%, respectively. The dividend yield on the index is 3%. Define Q as thenumber of Canadian dollars per U.S dollar and S$ as the value of the index. The volatility of S$ is20%, the volatility of Q is 6%, and the correlation between S and Q is 0.4. Use DerivaGem todetermine the value of a two year American-style call option on the index if
a. It pays off in Canadian dollars the amount by which the index exceeds 400
b. It pays off in U.S. dollars the amount by which the index exceeds 400.Martingales and Measures 503
21.17.
21.18.
Suppose that the price of a zero-coupon bond maturing at time T follows the processdP(t, T) = upP(t, T) dt + op P(t, T) dzand the price of a derivative dependent on the bond follows the processdf =pu,sfdtt+o;fdz
Assume only one source of uncertainty and that f provides no income.a. What is the forward price, F, of f for a contract maturing at time T?b. What is the process followed by F in a world that is forward risk neutral with respectto P(t, T)?c. What is the process followed