771-818.
Rubinstein, M., “Return to Oz,” RISK, November 1994, pp. 67-70.
Schroder, M., “Computing the constant Elasticity of Variance Option Pricing Formula,” Journal ofFinance, 44, no. 1 (1989), 211-18.
Stutzer, M., “A Simple Nonparametric Approach to Derivative Security Valuation,” Journal ofFinance, 51 (December 1996), 1633-52.
Tilley, J. A., “Valuing American Options in a Path Simulation Model,” Transactions of the Society ofActuaries, 45 (1993), 83-104.
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
 
20.1. Confirm that the CEV model formulas satisfy put-call parity.
20.2. Explain how you would use Monte Carlo simulation to sample paths for the asset price whenMerton’s jump diffusion model is used.
20.3. Confirm that Merton’s jump diffusion model satisfies put-call parity when the jump size islognormal. :
20.4. Suppose that the volatility of an asset will be 20% from month 0 to month 6, 22% from month 6to month 12, and 24% from month 12 to month 24. What volatility should be used in Black-Scholes to value a two-year option?
20.5. Consider the case of Merton’s jump diffusion model where jumps always reduce the asset price tozero. Assume that the average number of jumps per year is 2. Show that the price of a Europeancall option is the same as in a world with no jumps except that the risk-free rate is r+ ratherthan r. Does the possibility of jumps increase or reduce the value of the call option in this case?(Hint: Value the option assuming no jumps and assuming one or more jumps. The probability ofno jumps in time T is e~*7).
20.6. At time zero the price of a non-dividend-paying stock is Sp. Suppose that the time intervalbetween 0 and T is divided into two subintervals of length t; and t,. During the first subinterval,the risk-free interest rate and volatility are r, and o), respectively. During the second subinterval,they are r. and o>, respectively. Assume that the world is risk neutral.
a. Use the results in Chapter 11 to determine the stock price distribution at time 7 in termsof r1, rz, 01, 02, fj, t2, and Sp.
b. Suppose that 7 is the average interest rate between time zero and T and that V is the averagevariance rate between times zero and 7. What is the stock price distribution as a functionof T in terms of 7, V, T, and Sp?More on Models and Numerical Procedures 481
20.9.
20.10.
20.11.20.12.
20.13.
20.14.20.15.
20.16.
c. What are the results corresponding to (a) and (b) when there are three subintervals withdifferent interest rates and volatilities?
d. Show that if the risk-free rate, r, and the volatility, 7, are known functions of time then thestock price distribution at time T in a risk-neutral world is
In Sp ~ gin Sy + F-4V)T, VVT]
where ; is the average value of r, V is equal to the average value of o”, and Sp is the stockprice today.
. Write down the equations for simulating the path followed by the asset price in the stochastic
volatility model in equation (20.2) and (20.3).
. “The IVF model does not necessarily get the evolution of the volatility surface correct.” Explain
this statement.
“When interest rates are constant the [VF model correctly values any derivative whose payoffdepends on the value of the underlying asset at only one time.” Explain this statement.
Use a three-time-step tree to value an American lookback call option on a currency when theinitial exchange rate is 1.6, the domestic risk-free rate is 5% per annum, the foreign risk-freeinterest rate is 8% per annum, the exchange rate volatility is 15%, and the time to maturity is18 months. Use the approach in Section 20.5,
Repeat Problem 20.10 using the approach in Section 20.6.
Use a three-time-step tree to value an American put option on the geometric average of the priceof a non-dividend-paying stock when the stock price is $40, the strike price is $40, the risk-freeinterest rate is 10% per annum, the volatility is 35% per annum, and the time to maturity is threemonths. The geometric average is measured from today until the option matures.
Can the approach for valuing path-dependent options in Section 20.5 be used for a two-yearAmerican-style option that provides a payoff equal to max(Saye — K, 0), where Sy, is the averageasset price over the three months preceding exercise. Explain your answer.
Verify that the 6.492 number in Figure 20.2 is correct.
Examine the early exercise policy for the eight paths considered in the example in Section 20.9.What is the different between the early exercise policy given by the least-squares approach andthe exercise boundary parametrization approach? Which gives a higher option price for the pathssampled?
Consider a European put option on a non-dividend-paying stock when the stock price is $100,the strike price is $110, the risk-free rate is 5% per annum, and the time to maturity is one year.Suppose that the average variance rate during the life of an option has a 0.20 probability ofbeing 0.06, a 0.5 probability of being 0.09, and a 0.3 probability of being 0.12. The volatility isuncorrelated with the stock price. Estimate the value of the option. Use DerivaGem.
ASSIGNMENT QUESTIONS
 
 
20.17.
A new European-style lookback call option on a stock index has a maturity of nine months. Thecurrent level of the index is 400, the risk-free rate is 6% per annum, the dividend yield on theindex is 4% per annum, and the volatility of the index is 20%. Use the approach in Section 20.6to value the option and compare your answer to the result given by DerivaGem using theanalytic valuation formula.482
20.18.
20.19.
20.20.
CHAPTER 20
Suppose that the volatilities used to price a six-month currency option are as in Table 15.2.Assume that the domestic and foreign risk-free rates are 5% per annum and the currentexchange rate is 1.00. Consider a bull spread that consists of a long position in a six-month calloption with strike price 1.05 and a short position in a six-month call option with a strikeprice 1.10.a. What is the value of the option?b. What single volatility if used for both options gives the correct value of the bull spread?(Use the DerivaGem Application Builder in conjunction with Goal Seek or Solver.)c. Does your answer support the assertion at the beginning of the chapter that the correctvolatility to use when pricing exotic options can be counterintuitive?d. Does the TVF model give the correct price for the bull spread?
Repeat the analysis in Section 20.9 for the put option example on the assumption that the strikeprice is 1.13. Use both the least-squares approach and the exercise boundary parametrizationapproach.
Consider the situation in Merton’s jump diffusion model where the underlying asset is a non-dividend-paying stock. The average frequency of jumps is one per year. The average percentagejump size is 2% and the standard deviation of the logarithm of the percentage jump size is 20%.The stock price is 100, the risk-free rate is 5%, the volatility, 0, provided by the diffusion part ofthe process is 15%, and the time to maturity is six months. Use the DerivaGem ApplicationBuilder to calculate an implied volatility when the strike price is 80, 90, 100, 110, and 120. Whatdoes the volatility smile of skew that you obtain imply about the probability distribution of thestock price?CHAPTER 21MARTINGALES AND MEASURES
 
Up to now we have assumed that interest rates are constant when valuing options. In thischapter we relax this assumption in preparation for valuing interest rate derivatives inChapters 22 to 24.
The risk-neutral valuation principle we have used up to now states that a derivative can be valuedby (a) calculating the expected payoff on the assumption that the expected return from theunderlying asset equals the risk-free interest rate and (b) discounting the expected payoff at therisk-free interest rate. When interest rates are constant, risk-neutral valuation provides a well-defined and unambiguous valuation tool. When interest rates are stochastic, it is less clear-cut.What does it mean to assume that the expected return on the underlying asset equals the risk-freerate? Does it mean (a) that each day the expected return is the one-day risk-free rate, or (b) that eachyear the expected return is the one-year risk-free rate, or (c) that over a five-year period the expectedreturn is the five-year rate at the beginning of the period? What does it mean to discount expectedpayoffs at the risk-free rate? Can we, for example, discount an expected payoff realized in year 5 attoday’s five-year risk-free rate?
In this chapter we explain the theoretical underpinnings of risk-neutral valuation when interestrates are stochastic and show that there are many different risk-neutral worlds that can beassumed in any given situation. We first define a parameter known as the market price of riskand show that the excess return over the risk-free interest rate earned by any derivative in a shortperiod of time is linearly related to the market prices of risk of the underlying stochastic variables.What we will refer to as the traditional risk-neutral world assumes that all market prices of risk arezero, but we will find that other assumptions about the market price of risk are useful in somesituations.
Martingales and measures are critical to a full understanding of risk-neutral valuation. Amartingale is a zero-drift stochastic process. A measure is the units in which we value securityprices. A key result in this chapter will be the equivalent martingale measure result, This states thatif we use the price of a traded security as the unit of measurement then there is some market priceof risk for which all security prices follow martingales.
In this chapter we apply the equivalent martingale measure result to stock options when interestrates are stochastic and to the valuation of options to exchange one asset for another. We alsodiscuss quantos, which are derivatives where an interest rate defined in one currency is applied to aprincipal amount in another currency. In Chapter 22 we use the result to understand the standardmarket models for valuing interest rate derivatives and in Chapter 24 it will assist us in developingthe LIBOR market model.
483484 CHAPTER 21
21.1 THE MARKET PRICE OF RISK
 
We start by considering the properties of derivatives dependent on the value of a single variable, 6.We will assume that the process followed by 6 isa8a>
~BS=
mdt + sdzwhere dz is a Wiener process.
The parameters m and s are the expected growth rate in 6 and the volatility of 0, respectively. Weassume that they depend only on 6 and time t. The variable @ need not be the price of aninvestment asset. It could be something as far removed from financial markets as the temperaturein the center of New Orleans.
Suppose that f; and f, are the prices of two derivatives dependent only on @ and t. These can beoptions or other instruments that provide a payotl equal to some function of @ at some future time.We assume that during the time period under consideration f; and fy provide no income.
Suppose that the processes followed by f, and f; are
d4h a dt-toidefi
and
d2 <a att ondefr
where f41, {42, 0}, and o> are functions of 6 and t. The “dz” is the same Wiener process as inequation (21.1) because it is the only source of the uncertainty in the prices of f, and fh.The discrete versions of the processes are
bf, = ey f t+o) fi oz (21.2)5 fo = by fy St + 07 fy 82 (21.3)
We can eliminate the 6z by forming an instantaneously riskless portfolio consisting of of) of thefirst derivative and —o, f; of the second derivative. If 1 is the value of the portfolio, then
T= (mf)fi-MAdh (214)Th =m frsfi—a fishSubstituting from equations (21.2) and (21.3), we obtain
OM = (uy 02 f, fr — Mami fi fr) ot (21.5)
Because the portfolio is instantaneously riskless, it must earn the risk-free rate. Hence,
and
él] = rfl stSubstituting into this equation from equations (21.4) and (21.5) gives1192 — 120] = FO, — roy
' The analysis can be extended to derivatives that provide income (see Problem 21.7).Martingales and Measures 485
or
Moh _ daar (21.6)1 o% ,
Define 4 as the value of each side in equation (21.6), so that
MITP MTP,a Oo
Dropping subscripts, we have shown that if f is the price of a derivative dependent only on 6 and twith= watt ad: (21.7)
then
 
=h (21.8)
The parameter 4 is known as the market price of risk of @. It can be dependent on both @ and t, butit is not dependent on the nature of the derivative f. At any given time, (4 — r)/o must be the samefor all derivatives that are dependent only on @ and t.
It is worth noting that o, which we are referring to as the volatility of f, is defined as thecoefficient of dz in equation (21.7). It can be either positive or negative. If the volatility, s, of 4 ispositive and f is positively related to @ (so that 0f/90 is positive), then o is positive. But if f isnegatively related to 6, then o is negative. The volatility of /, as it is traditionally defined, is |o|.
The market price of risk of @ measures the tradeoffs between risk and return that are made forsecurities dependent on 6. Equation (21.8) can be written
b—-r=ho (21.9)
For an intuitive understanding of this equation, we note that the variable o can be looselyinterpreted as the quantity of @-risk present in f. On the right-hand side of the equation, therefore,we are multiplying the quantity of -risk by the price of @-risk. The left-hand side is the expectedreturn in excess of the risk-free interest rate that is required to compensate for this risk. Equation(21.9) is 2stock to its risk.
We will not be concerned with the measurement of the market price of risk in this chapter. Thiswill be discussed in Chapter 28 in the context of the evaluation of real options.
In Chapter 3, we distinguished between investment assets and consumption assets. An invest-ment asset is an asset that is bought or sold purely for investment purposes by a significant numberof investors. It can be used as part of a trading strategy to set up a riskless portfolio. Consumptionassets are held primarily for consumption and cannot be used in this way. If the variable 4 is theprice of an investment asset, it must be true that
 
sset pricing model, which relates the expected excess return on a
 
But if @ is the price of a consumption asset, this relationship is not, in general, true.
Example 21.1 Consider a derivative whose price is positively related to the price of oil anddepends on no other stochastic variables. Suppose that it provides an expected return of 12% per486 CHAPTER 21
 
annum and has a volatility of 20% per annum. Assume that the risk-free interest rate is 8% perannum. It follows that the market price of risk of oil is
0.12 — 0.08——_———_ = 0.20.2 0
Note that oil is a consumption asset rather than an investment asset. Therefore, its market price ofrisk cannot be calculated from equation (21.8) by setting 4% equal to the expected return from aninvestment in oil and @ equal to the volatility of oil prices.
Example 21.2 Consider two securities, both of which are positively dependent on the 90-dayinterest rate. Suppose that the first one has an expected return of 3% per annum and a volatility of20% per annum, and the second one has a volatility of 30% per annum. Assume that theinstantaneous risk-free rate of interest is 6% per annum. The market price of interest rate risk is,using the expected return and volatility for the first security,
0.03 — 0.06———_ = -0.150.2From a rearrangement of equation (21.9), the expected return from the second security is therefore0.06 — 0.15 x 0.3 = 0.015or 1.5% per annum.
Alternative WorldsThe process followed by derivative price f isdf=pfdt+ofdz
The value of 2 depends on the risk preferences of investors. In a world where the market price of
risk is zero, A equals zero. From equation (21.9), =r, so th
  
  
he process followed by f isdf=rfdt+ofdz
We will refer to this as the traditional risk-neutral world.By making other assumptions about the market price of risk, we define other worlds that areinternally consistent. In general, when the market price of risk is 4, equation (21.9) shows that
b=rthoso thatdf =(r+ho)fdt+of dz (21.10)
The market price of risk of a variable determines the growth rates of all securities dependent on thevariable. As we move from one market price of risk to another, the expected growth rates ofsecurity prices change, but their volatilities remain the same. We illustrated this in Section 10.7.Choosing a particular market price of risk is also referred to as defining the probability measure.For some value of the market price of risk, we obtain the ‘real world” and the growth rates ofsecurity prices that are observed in practice., Martingales and Measures 487
21.2 SEVERAL STATE VARIABLES
 
 
 
Suppose that ” variables, 6), @),..., On, follow stochastic processes of the form
a6;
= =m; dt + 5, dz; Q1.11)
VEfor i=1,2,...,, where the dz; are Wiener processes. The parameters m; and s; are expectedgrowth rates and volatilities and may be functions of the 6; and time. Appendix 21A provides aversion of Ité’s lemma that covers functions of several variables. It shows that the process for theprice, f, of a security that is dependent on the 6; has the form
df uop matt Yio des (21.12)i=t
In this equation, 4 is the expected return from the security and o; dz; is the component of the riskof this return attributable to 6;.Appendix 21B shows that
° w—r= Ao; (21.13)i=t
where 4; is the market price of risk for 6; This equation relates the expected excess return thatinvestors require on the security to the A; and o;. Equation (21.9) is the particular case of thisequation when n = |. The term A;o; on the right-hand side measures the extent that the excessreturn required by investors on a security is affected by the dependence of the security on 6.If 4,0; = 0, there is no effect; if 4,0; > 0, investors require a higher return to compensate them forthe risk arising from 6;; if A;0; < 0, the dependence of the security on 6; causes investors to requirea lower return than would otherwise be the case. The A,o; < 0 situation occurs when the variablehas the effect of reducing rather than increasing the risks in the portfolio of a typical investor.
Example 21.3 A stock price depends on three underlying variables: the price of oil, the price ofgold, and the performance of a stock index. Suppose that the market prices of risk for these variablesare 0.2, -0.1, and 0.4, respectively. Suppose also that the o; factors in equation (21.12) correspondingto the three variables have been estimated as 0.05, 0.1, and 0.15, respectively. The excess return on thestock over the risk-free rate is
0.2 x 0.05 — 0.1 x 0.1 + 0.4 x 0.15 = 0.06
or 6.0% per annum. If variables other than those considered affect the stock price, this result is stilltrue provided that the market price of risk for each of these other variables is zero.
Equation (21.13) is closely related to arbitrage pricing theory, developed by Stephen Ross in 1976.The continuous-time version of the capital asset pricing model (CAPM) can be regarded as aparticular case of the equation. CAPM argues that an investor requires excess returns tocompensate for any risk that is correlated to the risk in the return from the stock market, butrequires no excess return for other risks. Risks that are correlated with the return from the stock
2 See S. A. Ross, “The Arbitrage Theory of Capital Asset Pricing,” Journal of Economic Theory, 13 (December1976), 343-62.488 CHAPTER 21
market are referred to as systematic; other risks are referred to as nonsystematic. If CAPM is true, A;is proportional to the correlation between changes in 6, and the return from the market. When @, isuncorrelated with the return from the market, A; is zero.
21.3_ MARTINGALES
 
A martingale is a zero-drift stochastic process. A variable @ follows a martingale if its process hasthe form
d@=adz
where dz is a Wiener process. The variable o may itself be stochastic. It can depend on 6 and otherstochastic variables.
A martingale has the convenient property that its expected value at any future time is equal to itsvalue today. This means that
E(6r) = O%
where 6) and 6; denote the values of @ at times zero and 7, respectively. To understand this result,we note that over a very small time interval the change in @ is normally distributed with zero mean.The expected change in @ over any very small time interval is therefore zero. The change in 6between time zero and time 7 is the sum of its changes over many small time intervals. It followsthat the expected change in @ between time zero and time T must also be zero.
The Equivalent Martingale iVieasure Result
Suppose that f and g are the prices of traded securities dependent on a single source ofuncertainty. We assume that the securities provide no income during the time period underconsideration.” We define
o=
& Is
The variable @ is the relative price of f with respect to g. It can be thought of as measuring theprice of f in units of g rather than in dollars. The security price g is referred to as the numeraire.
The equivalent martingale measure result shows that, when there are no arbitrage opportunities,@ is a martingale for some choice of the market price of risk. What is more, for a given numerairesecurity g, the same choice of the market price of risk makes ¢ a martingale for all securities f.This choice of the market price of risk is the volatility of g. In other words, when the market priceof risk is set equal to the volatility of g. the ratio f/g is a martingale for all security prices f.
To prove this result, we suppose that the volatilities of f and g are a, and o,. From equation(21.10), in a world where the market price of risk is o,, we have
df =(r+o,o7)f dt + apf dzdg = (r+ oayg dt + a,gdz
* Problem 21.8 extends the analysis to situations where the securities provide income.Martingales and Measures 489
 
Using It6’s lemma givesdin f= + ogo, — o7/2)dt + opdz
ding =(r +0, /2)dt +0, dz
so thatd(n f — Ing) = (a,05 — 0/2 ~ 02/2) dt + (a7 — oy) azorf (of - 94)"a(n“) = Se dt + (0; —6,)dz
Using Itd’s lemma to determine the process for f/y from the process for in(f/g), we obtain
a(f\= (a, — oy dz (21.14)\g/ ° “9g
showing that f/g is a martingale.
This provides the required result. We refer to a world where the market price of risk is o, as aworld that is forward risk neutral with respect to g.
Because f/g is a martingale in a world that is forward risk neutral with respect to g, it follows
thati. p,(S)go “AGTorfo = HE, (2) (21.15)IT.
where E, denotes the expected value in a world that is forward risk neutral with respect to g.
21.4 ALTERNATIVE CHOICES FOR THE NUMERAIRE
 
We now present a number of examples of the equivalent martingale measure result. Our firstexample shows that it is consistent with the traditional risk-neutral valuation result we have usedup to now. The other examples prepare the way for the valuation of bond options, interest-ratecaps, and swap options in Chapter 22.
Money Market Account as the Numeraire
The dollar money market account is a security that is worth $1 at time zero and earns theinstantaneous risk-free rate r at any given time.* The variable r may be stochastic. If we set gequal to the money market account, it grows at rate 7, so that
dg =rgdt (21.16)The drift of g is stochastic, but the volatility of g is zero. The world that is forward risk neutral with
4 The money market account is the limit as 6 approaches zero of the following security. For the first short period oftime of length 61, it is invested at the initial 5t period rate; at time 4, it is reinvested for a further period of time 4t at thenew &t period rate; at time 2 8t, it is again reinvested for a further period of time 4t at the new 6t period rate; and so on.The money market accounts in other currencies are defined analogously to the dollar money market account.490 CHAPTER 21
respect to g is therefore a world where the market price of risk is zero. This is the world we definedearlier as the traditional risk-neutral world. It follows from equation (21.15) that
fo= wé(4) (21.17)GT.
where £ denotes the expectation in the traditional risk-neutral world.In this case, gg = 1 and
ror = elo rdtso that equation (21.17) reduces to :fa = Ele” fr) (21.18)orfo= Ele’ fr) (21.19)
This equation shows that one way of valuing an interest rate derivative is to simulate the short-terminterest rate r in the traditional risk-neutral world. On each trial we calculate the expected payoffand discount at the average value of the short rate on the sampled path.
When the short-term interest rate r is assumed to be constant, equation (21.19) reduces to
fo = TES)
or the risk-neutral valuation relationship we used in earlier chapters.
Zero-Coupon Bond Price as the Numeraire
Define P(t, T) as the price at time t of a zero-coupon bond that pays off $1 at time T. We nowexplore the implications of setting g equal to P(t, T). We use E; to denote the expectation in aworld that is forward risk neutral with respect to P(t,T). Because gp = P(T,T)=1 andgo = P(O, T), equation (21.15) gives
fo = PO, T)Er( fr) (21.20)
Notice the difference between equations (21.20) and (21.19). In equation (21.19), the discounting isinside the expectation operator. In equation (21.20), the discounting, as represented by the P(0, T)term, is outside the expectation operator. By using a world that is forward risk neutral with respectto P(t, T), we considerably simplify things for a security that provides a payoff solely at time T.Define F as the forward price of f for a contract maturing at time T. Arbitrage arguments
similar to those in Chapter 3 show that
_ fo
~ P(0, T)
 
This equation and equation (21.20) jointly imply thatF = Er(Sr) (21.21)
showing that, in a world that is forward risk neutral with respect to P(t, T), the forward price of fis ils expected future spot price.°
> This can be contrasted with the traditional risk-neutral world where the expected future spot price of a variable isits futures price in a continuously settled futures contract.Martingales and Measures 491
 
Equation (21.20) shows that we can value any security that provides a payoff at time T bycalculating its expected payoff in a world: that is forward risk neutral with respect to a bond
maturing at time T and discounting at the risk-free rate for maturity T. Equation (21.21) showsmaturing at time 7 and discounting at the risk iree rate for maturity 7. Equation (21.21) snows
that it is correct to assume that the expected value of the underlying asset equals its forward valuewhen computing the expected payoff. These results will be critical to our understanding of the. Standard market model for bond options in the next chapter.
Interest Rates When a Bond Price Is the Numeraire
For our next result, we define R(t, T,, Tz) as the forward interest rate as seen at time t for the periodbetween 7; and 7 expressed with a compounding period of T, — 7; (e.g., if T, — 7; = 0.5, theinterest rate is expressed with semiannual compounding; if T, — T,; = 0.25, it is expressed withquarterly compounding; and so on). The forward price, as seen at time t, of a zero-coupon bondlasting between times 7, and 7> is
P(t, Ty)
P(t. 7)
Because a forward interest rate is the interest rate implied by the corresponding forward bondprice, it follows thatl _ P(t, Tr)1+(%)—T)Rt,T,T2) P(t, T1)
a) (PGT)ROT.) = 55 (FET ')
so that
 
or
ROT. Lh) = 5
1 (= T\) ~ P(t, »)2-7
P(t, Tr)If we set
f=
 
Bop PT) ~ PT
and g = P(t, 7), then the equivalent martingale measure result shows that R(t, 7), 7) is amartingale in a world that is forward risk neutral with respect to P(t, 7). This means that
ROO, T, Th) = £1 R(T, T), Tr) . (21.22)
where E, denotes the expectation in a world that is forward risk neutral with respect to P(t, T)).
We have shown that the forward interest rate equals the expected future interest rate in a worldthat