American Option Valuation under Stochastic Interest Rates

Review of Derivatives Research, 3, 283–307 (1999)c© 2000 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.

American Option Valuation under StochasticInterest Rates

SAN-LIN CHUNG* [email protected] of Finance, National Central University, Chung Li, 320, Taiwan

Abstract. By applying Ho, Stapleton and Subrahmanyam’s (1997, hereafter HSS) generalised Geske–Johnson(1984, hereafter GJ) method, this paper provides analytic solutions for the valuation and hedging of Americanoptions in a stochastic interest rate economy. The proposed method simplifies HSS’s three-dimensional solutionto a one-dimensional solution. The simulations verify that the proposed method is more efficient and accurate thanthe HSS (1997) method. We illustrate how the price, the delta, and the rho of an American option vary betweenthe stochastic and non-stochastic interest rate models. The magnitude of this effect depends on the moneyness ofthe option, interest rates, volatilities of the underlying asset price and the bond price, as well as the correlationbetween them.

Keywords: American option pricing, stochastic interest rates, Richardson extrapolation.

JEL classification: G13.

Over the last few years, more and more American-style long-term derivatives have becometraded in the exchange and over-the-counter markets. Equity options, currency warrants,1

swaptions, and commodity trust units all trade with maturities of two years or more. Forvaluing these options, interest rate risk is a significant factor since the term structure ofinterest rates affects the values of these options on each possible exercise date beforeexpiration. Furthermore, the payoff of many contingent claims, such as differential swapsand equity swaps, explicitly depends on interest rates and asset prices. Therefore, it isimportant to develop an option valuation model for explicitly combining stochastic interestrates and stochastic asset prices.

American option valuation is an important topic which has been studied widely in thefinance literature. Among others, Geske and Johnson (1984, hereafter GJ) provide anefficient and intuitively appealing technique for the valuation and hedging of American-style contingent claims. In their method, an American option value is approximated, usingRichardson extrapolation, by using a series ofn exercise-date Bermudan options2 (hereaftern-times exercisable options) values. Their analytic solution is very efficient since it cancompute prices and hedges very quickly. Moreover, their method can achieve arbitraryaccuracy by increasing the number of exercise-date. The GJ method has been extended,using different extrapolation techniques, by Omberg (1987), Bunch and Johnson (1992),

* The author would like to acknowledge the helpful comments of Richard Stapleton, Marti Subrahmanyam,Ren-Raw Chen (the referee), and Mark Shackleton. I thank the participants at the 24th EFA Annual Conferencein Vienna, and in particular to the discussant Ton Vorst.

284 SAN-LIN CHUNG

and Ho, Stapleton and Subrahmanyam (1994, hereafter HSS), and others. Unfortunately,all these methods assume that a constant risk-free rate prevails during the option’s life.

HSS (1997) generalise the GJ (1984) approach to a stochastic interest rate economy.They first establish a risk-neutral valuation relationship (RNVR) forn-times exercisableoptions in a stochastic interest rate economy. The derived risk neutral distributions for thestock price and zero-coupon bond price are then applied to price the European and twice-exercisable options. Unlike GJ, in a stochastic interest rate model, there is no unique criticalexercise price at each exercise date. The critical exercise price depends on the evolutionof stock prices and zero coupon bond prices at each exercise date. For this reason, HSSconstruct three binomial trees to approximate the joint distributions for the stock prices attwo exercise dates and bond price at the first exercise date. Their method can be regardedas a three-dimensional solution.

The HSS method is so far the most efficient method to price the American option understochastic interest rates. In this article, we make an improvement to reduce their three-dimensional solution to one-dimension. The main idea is to find theminimal set of assetprice(s) at the first exercise datethat allow(s) the critical exercise price (and thus the twice-exercisable option value) at the first exercise date areexplicit functionsof this set of assetprice(s). The current twice-exercisable option value is then calculated using the distribu-tion(s) for this set of asset price(s). The solution is in analytic (integral) form and thus veryefficient and easy to implement. Our method is attractive from a computational viewpointbecause the required computational time has the same order as that of GJ. Moreover, theproposed method can also provide analytic hedge ratios with high accuracy.

In addition to the HSS (1997), this article provides a thorough analysis on the effect ofstochastic interest rates. We show how option values, and delta and rho hedge ratios maydiffer between stochastic and non-stochastic interest rate models. Sensitivity analysis isalso made to see how the stochastic interest rate effects change with respect to the interestrates, moneyness, time to maturity, volatilities of the underlying asset and the bond prices,as well as the correlation between them.

The plan of this paper is as follows. Section 1 presents the valuation model for theAmerican-style contingent claims in a stochastic interest rate economy and shows the RNVRfor n-times exercisable options. Section 2 derives an analytic solution for the European andtwice-exercisable put options within the framework of HSS (1997). The delta and rho hedgeratios of the European, the twice-exercisable, and the American options are also shown.In Section 3, we derive the variance-covariance terms which are the inputs for pricing theEuropean and twice-exercisable options in the continuous time framework. Section 4 carriesout the simulations and compares our approach with the HSS (1997) method. We discusshow model parameters may affect the difference between the stochastic and non-stochasticinterest rate models, while Section 5 concludes.

1. The Valuation Model

As in HSS (1997), we consider an American-style contingent claim, on a nondividend payingasset, whose price at timet is S(t). The expiration date of the option is timeT and its payofffunction, if exercised at timet , is g(S(t)) ≥ 0, t ∈ [0, T ]. To fulfill the requirement of

AMERICAN OPTION VALUATION 285

RNVR, we assume that the pricing kernel, the underlying asset price, and the zero-couponbond prices, are joint lognormally distributed.3 Because the American call option has thesame value as its counterpart European option under our assumption that the underlyingasset has no dividends, this article studies the valuation of American put options only.

Following HSS, we also use two-point GJ approximation for stock options, that is

PA(0) = P2(0)+ (P2(0)− P1(0)), (1)

whereP1(0), P2(0), andPA(0) are the European option value, the twice-exercisable optionvalue, and the estimated American option value at time 0, respectively. Therefore, this paperwill focus on pricing the European and twice-exercisable options. In a stochastic interest rateeconomy, the European and twice-exercisable options can be priced by taking expectationsunder a two-period forward-risk-adjusted (FRA) measure4 respectively as follows,

P1(0) = E0

{Et [g(S(T))]B(t, T)

}B(0, t), (2)

P2(0) = E0

{max

[g(S(t)), Et [g(S(T))]B(t, T)

]}B(0, t), (3)

whereE(.) is the risk neutral expectation under the two-period FRA measure,B(t, τ ) isthe price at timet of a zero-coupon bond paying $1 at timeτ , and

E0[S(t)] = S(0)

B(0, t),

Et [S(T)] = S(t)

B(t, T),

E0[B(t, T)] = B(0, T)

B(0, t).

2. European and Twice-Exercisable Put Option Valuation in a Stochastic InterestRate Economy

In this section, we develop a one-dimensional, analytic solution to implement the HSSmodel. In the HSS method, the European put option and twice-exercisable put option arevalued respectively as follows:

P1(0) = E0

[Et [(K − S(T))+]B(t, T)

]B(0, t), (4)

P2(0) = E0

[max

{(K − S(t))+, Et [(K − S(T))+]B(t, T)

}]B(0, t), (5)

where(K − S(.))+ = max[K − S(.),0]. As mentioned before, their method is actually athree-dimensional solution.

It is not difficult to reduce the HSS method to a two-dimensional solution. At timet , S(t) and B(t, T) are realised, thus the holding (European option) value(Et [(K −S(T))+]B(t, T))5 and the early exercise value((K − S(t))+) are known. In other words,

286 SAN-LIN CHUNG

the European and twice-exercisable option values at timet are simply explicit functions ofS(t) andB(t, T),

P1(t) = P1(S(t), B(t, T)), (6)

P2(t) = P2(S(t), B(t, T)). (7)

Therefore the values of both options at time 0 can be presented as the following integrationsrespectively,

P1(0) = B(0, t)∫ ∞−∞

∫ ∞−∞

P1(t) f (ln S(t), ln B(t, T))d ln S(t)d ln B(t, T), (8)

P2(0) = B(0, t)∫ ∞−∞

∫ ∞−∞

P2(t) f (ln S(t), ln B(t, T))d ln S(t)d ln B(t, T), (9)

where f (ln S(t), ln B(t, T)) is the probability density function of lnS(t) and lnB(t, T)under the two-period FRA measure. This analytic solution is a two-dimensional integrationin practice.

To produce the most efficient method within the HSS framework, one should try to reducethe dimensionality as much as possible. It will also save computational time if the earlyexercise decision (or critical exercise price) and the option values can be obtained withoutusing a numerical method such as Newton–Raphson method. Thus we define the essentialasset price(s) as follows.

Definition: Essential Asset Price(s).The essential asset price(s) is (are) theminimal setof asset price(s) at the first exercise datesuch that the critical exercise price (and thus thetwice-exercisable option value) at the first exercise date areexplicit functionsof this set ofasset price(s).

To search for the most efficient method within the HSS framework is equivalent to findingthe essential asset price(s). It is not easy to know what the essential asset price(s) will beand one might need to guess by trial and error. As in GJ (1984), the critical exercise priceat t for the twice-exercisable option is the solution of the following equation:

K − S(t) = K B(t, T)N(−h)− S(t)N(−h− σ ), (10)

whereN(.) is the standard cumulative univariate normal distribution function, and

h =ln S(t)

K B(t,T) − 12 σ

2

σ,

σ =√

Var[ln S(T) | S(t), B(t, T)].

Obviously, if S(t) and B(t, T) are given, then the values of both sides in equation (10)are known as well. From the formula forh, we observe thath is an explicit function of

S(t)B(t,T) , which is the forward (delivery atT) price of the stock at timet, F(t, T). We tryto substituteF(t, T) into equation (10) and find that the critical exercise price is also anexplicit function of the forward price. The results are presented in Proposition 1.

AMERICAN OPTION VALUATION 287

Proposition 1 The critical stock price,S(t), and critical bond price,B(t, T), are explicitfunctions of F(t, T), that is

S(t) = K F(t, T)

K N(−h)+ F(t, T)N(h+ σ ) (11)

B(t, T) = K

K N(−h)+ F(t, T)N(h+ σ ) (12)

where

h = ln F(t,T)K − 1

2 σ2

σ

σ =√

Var[ln S(T) | F(t, T)].

Proof: The critical exercise price is the solution of the following equation,

K − S(t) = K B(t, T)N(−h)− S(t)N(−h− σ ). (13)

SubstitutingS(t) = F(t, T)B(t, T) into equation (13) yields

K − F(t, T)B(t, T) = K B(t, T)N(−h)− F(t, T)B(t, T)N(−h− σ ) (14)

Moving the second term of the left-hand side to the right-hand side will getB(t, T) andthenS(t) = F(t, T)B(t, T).

Note that the critical exercise price is an explicit function of the forward price. In asimilar manner, one can show that ifB(t, T) is given then the critical stock price att isalso determined. However, in that case, one needs to use numerical method to find outthe critical stock price.In other words, the critical stock price is an implicit function ofB(t, T).

It seems that the forward price is the essential asset price which gives us the most efficientmethod within the HSS framework. One needs to check whether the European and twice-exercisable option values att are explicit functions ofF(t, T) or not. If it is the case, thenthe option values at time 0 can be calculated using the integration over the distribution ofthe forward price. As shown in Proposition 2, the answer to the above question is positiveand then we can obtain an analytic solution for the option values.

Proposition 2 The value of European option and that of the twice-exercisable option attime 0 are respectively,

P1(0) = B(0, t)∫ ∞−∞

P1(F(t, T)) f (ln F(t, T))d ln F(t, T), (15)

P2(0) = B(0, t)∫ ∞−∞

P2(F(t, T)) f (ln F(t, T))d ln F(t, T), (16)

288 SAN-LIN CHUNG

where f(ln F(t, T)) is the probability density function ofln F(t, T) under the two-periodFRA measure, and

P1(F(t, T)) = [K N(−h)− F(t, T)N(−h− σ )] exp

(µ2+ 1

2σ 2

2

), (17)

P2(F(t, T)) = [K N(−h)− F(t, T)N(−h− σ )] exp

(µ2+ 1

2σ 2

2

)N(d1)

+ K N(−d2)− F(t, T)exp

(µ2+ 1

2σ 2

2

)N(−d1), (18)

where d2 = µ2−ln B(t,T)σ2

, d1 = d2 + σ2, h andσ are from Proposition 1, andµ2 andσ 22

are conditional mean and variance ofln B(t, T) given F(t, T) under the two-period FRAmeasure respectively.

Proof: See the Appendix.

Our analytic solution is only a one-dimensional integration which is obviously moreefficient than the two-dimensional integration solution mentioned above, and far moreefficient than HSS’s (1997) three-dimensional solution. Compared with HSS, the pricingerror of our method would be smaller because we only approximate the distribution of theforward price, while they need to approximate the joint distributions ofS(t), B(t, T), andS(T).

Note that one should be able to derive the Black–Scholes (1973) formula from equation(15), confirming that one can price a European option using the two-period FRA measure.The risk neutral distribution ofF(t, T), and the conditional risk neutral distributions ofS(T) and B(t, T) given F(t, T) under the two-period FRA measure, which are essentialinputs for applying Propositions 1 and 2, are shown in the Appendix.

One advantage of this analytic solution is that one could derive analytic hedge parametersfor the options as well. As an illustration we give the delta hedge ratios for the European, thetwice-exercisable, and the American options in Corollary 1. The proof is straightforwardfrom Proposition 2 and can be obtained from the author on request.

Corollary 1 The delta hedge ratios for the European, the twice-exercisable, and the Amer-ican options are respectively as follows,

11 = ∂P1(0)

∂S(0)

= B(0, t)

S(0)σ 2F(t,T)

∫ ∞−∞

P1(F(t, T)) f (ln F(t, T))

× [ln F(t, T)− µF(t,T) − σF(t,T),B(t,T)]d ln F(t, T),

12 = ∂P2(0)

∂S(0)

AMERICAN OPTION VALUATION 289

= B(0, t)

S(0)σ 2F(t,T)

∫ ∞−∞

{P2(F(t, T))[ln F(t, T)− µF(t,T) − σF(t,T),B(t,T)]

+ σF(t,T),B(t,T)

[Kn(−d2)− P1(F(t, T))n(d1)− F(t, T)exp(µ2+ 1

2σ22 )n(−d1)

σ2

+ K N(−d2)

]}f (ln F(t, T))d ln F(t, T),

1A = ∂ PA(0)

∂S(0)= 212−11

whereµF(t,T) is the mean ofln F(t, T) under the two-period FRA measure,σ 2F(t,T) is the

variance ofln F(t, T), σF(t,T),B(t,T) is the covariance betweenln F(t, T) and ln B(t, T),and n(.) is the standard normal density function.

Since stochastic interest rates have potential effect on option values, an interesting measureis “rho”, the hedge with respect to the interest rate changes. This Greek has different valuesunder stochastic and non-stochastic interest rate models. It would be interesting to see thehedging difference of a rho-neutral portfolio in the static model. Thus we also derive the rhohedge ratios, for the European, the twice-exercisable, the American options, in Corollary 2.6

Corollary 2 The rho hedge ratios for the European, the twice-exercisable, the Americanoptions are respectively as follows,

ρ1 = ∂P1(0)

∂r

= T B(0, t)

σ 2F(t,T)

∫ ∞−∞

P1(F(t, T)) f (ln F(t, T))

× [ln F(t, T)− µF(t,T) − σF(t,T),S(T)]d ln F(t, T),

ρ2 = ∂P2(0)

∂r

= T B(0, t)∫ ∞−∞

{T P2(F(t, T))

σ 2F(t,T)

(ln F(t, T)− µF(t,T) − σF(t,T),S(T))

+(

t− TσF(t,T),S(T)

σ 2F(t,T)

)[P1(F(t, T))n(d1)−Kn(−d2)+F(t, T)exp(µ2+ 1

2σ22 )n(−d1)

σ2

− K N(−d2)

]}f (ln F(t, T))d ln F(t, T),

ρA = ∂ PA(0)

∂r= 2ρ2− ρ1.

290 SAN-LIN CHUNG

It should be noted that the correct maturities to use in the rho hedge are the ones whichmatch the exercisable dates of the option. For instance, as argued in Merton (1973), if thepricesB(0, t) and B(0, T) remain fixed while the price of other maturities changes, theprice of an option exercisable att or T will remain unchanged.7

3. The Variance-Covariance Terms: A Continuous Time Case

The variance-covariance terms are the most important inputs in the option pricing formulae.For the valuation of the European and twice-exercisable options, one needs the variance-covariance matrix of the underlying asset prices at two exercise dates and theT-maturityzero-coupon bond price at the first exercise date. In HSS (1997), this variance-covariancematrix is exogenously given. However, this variance-covariance matrix can be endoge-nously determined if one assumes that the stock price and short rate follow some specifiedstochastic processes. In this section, we give an example to show the derivation of thevariance-covariance terms.

Following Rabinovitch (1989), assume that the stock price,S, follows a geometric Brow-nian motion,

dS= αS dt+ σS d Z1, (19)

whereα is the instantaneous expected return on the stock,σ 2 is the instantaneous varianceof the stock return, andd Z1 is a standard Wiener process. For the short rate process,r (t),unlike Rabinovitch (1989), we assume it follows Hull and White’s (1990) extended-Vasicekmodel

dr = (θ(t)− qr)dt + v d Z2, (20)

where(θ(t)−qr) is the instantaneous drift in the short rate,v2 is the interest rate’s instanta-neous variance, andd Z2 is another standard Wiener process. The instantaneous correlationcoefficient between these two Wiener processes isρ, i.e.,d Z1(t)d Z2(t) = ρ dt. The rele-vant variance-covariance terms for pricing the European and the twice-exercisable optionsare shown in Proposition 3.

Proposition 3 If the stock price and short rate processes follow equations (19) and (20),respectively, then the variance-covariance terms for pricing the European and the twice-exercisable options are

σ 2B(t,T) =

v2

2q(1− e−2qt)A(t, T)2

σ 2S(t) = σ 2t +

(t − 2A(0, t)+ 1− e−2qt

2q

)(v

q

)2

+ 2ρσ(t − A(0, t))v

q

σ 2s(T) = σ 2T +

(T − 2A(0, T)+ 1− e−2qT

2q

)(v

q

)2

+ 2ρσ(T − A(0, T))v

q

AMERICAN OPTION VALUATION 291

σS(t),B(t,T) = v2

2q2(1− e−2qt)A(t, T)−

(ρσv + v

2

q

)A(0, t)A(t, T)

σS(t),S(T) = σ 2t + ρσ (2t − (1+ e−q(T−t))A(0, t)) v

q

+(

t − (1+ e−q(T−t))A(0, t)+ e−q(T−t) − e−q(T+t)

2q

)(v

q

)2

σS(T),B(t,T) = v2

2q2(e−q(T−t) − e−q(T+t))A(t, T)−

(ρσv + v

2

q

)A(0, t)A(t, T)

where

A(m,n) = 1

q[1− e−q(n−m)]

Proof: See the Appendix.

In a similar manner to Proposition 3, one can derive the variance-covariance terms forother Gaussian term structure models such as the Ho and Lee (1986) model. The parameterscan be generalised to be time-varying functions as well, see for example, HSS (1995) andJamshidian (1993).

It should be noted that the varianceσ 2S(T) is the total volatility in the Black–Scholes

(1973) formula for pricing the European option. Therefore we have actually derived theclosed-form solution for the European option under the assumed processes. For example,the European call option price is

S(0)N(d1)− K B(0, T)N(d2), (21)

whered1 = ln S(0)K B(0,T)+ 1

2σ2S(T)

σS(T), andd2 = d1 − σS(T). Compared with Rabinovitch’s pricing

formula for the European option,8 the zero-coupon local priceB(0, T) in our pricing formulais exogenously given while theirs is endogenously determined by the parameters of the shortrate process.

4. Simulations

To implement the proposed analytic solutions (Proposition 2, Corollaries 1 and 2), wesuggest two methods. The first method is to do the numerical integration directly. Thesecond method is to build up a binomial tree to approximate the distribution of forwardprice and then use this tree to calculate the option values. Both methods are applied to testtheir accuracy and computational efficiency using the HSS method as the benchmark.

292 SAN-LIN CHUNG

Table 1.European put option prices: non-stochastic interest rates.

(1) (2) (3) (4) (4)−(3)(3) (5) (5)−(3)

(3) (6) (6)−(3)(3)

r σ accurate value HSS(200) % Chung(200) % Chung(NI) %

0.125 0.5 0.13271 0.13259 −0.09 0.13273 0.01 0.13271 0.000.080 0.4 0.11698 0.11688 −0.08 0.11699 0.01 0.11698 0.000.045 0.3 0.09591 0.09584 −0.08 0.09592 0.01 0.09591 0.000.020 0.2 0.06936 0.06931 −0.07 0.06937 0.01 0.06936 0.000.005 0.1 0.03733 0.03731 −0.07 0.03734 0.01 0.03733 0.000.090 0.3 0.07612 0.07619 0.09 0.07613 0.01 0.07612 0.000.040 0.2 0.06004 0.05999 −0.08 0.06005 0.01 0.06004 0.000.010 0.1 0.03490 0.03493 0.07 0.03491 0.01 0.03490 0.000.080 0.2 0.04417 0.04413 −0.10 0.04418 0.01 0.04417 0.000.020 0.1 0.03037 0.03039 0.08 0.03037 0.01 0.03037 0.000.120 0.2 0.03168 0.03164 −0.13 0.03169 0.01 0.03168 0.000.030 0.1 0.02626 0.02629 0.09 0.02627 0.01 0.02626 0.00

This table shows the pricing errors of using HSS’s (1997) and the proposed methods to price the European putoptions in a non-stochastic interest rate economy. Columns (1) to (2) are from GJ’s Table 1. Columns (1) and(2) represent the parameter input forr , the continuously compounded risk-free rate, andσ , the volatility of theunderlying asset. The strike price and the initial stock price in all cases are $1. The time to maturity of theoption is 1 year. Column (3) is the accurate value of the European put option which is calculated from Black–Scholes formula. Columns (4) to (6) are the European put values obtained from using HSS’s method with 200binomial steps, the proposed method with 200 binomial steps, and the proposed method with numerical integration,respectively. The % pricing errors for each method are also shown.

4.1. The Accuracy and Efficiency of the Proposed Method

We first test the accuracy of the numerical integration method, the binomial method, and theHSS (1997) method. In the numerical integration method, the range is from the mean minusfive standard deviations (lower limit) to the mean plus five standard deviations (upper limit),rather than from−∞ to∞.9 The range is divided into equally spaced sections where thelength of each section equals one tenth of the standard deviation. The numerical integrationmethod used is identical to the extended trapezoidal rule (see Presset al. (1994)) exceptthat the weights on the lower and upper limits are one.10

Tables 1 and 2 report the pricing errors of each method for pricing the European andtwice-exercisable put options in a non-stochastic interest rate economy,11 respectively. Thetwice-exercisable option can be exercised at timeT

2 or T , whereT , the time to maturityof the option, equals one year. The accurate values are calculated from the closed-formsolution for the European option and from the analytic solution in GJ (1984) method12

for the twice-exercisable option. Pricing error comparisons across Tables 1 and 2 showthat the proposed method implementations with numerical integration is the most accurateone. The maximum error is only 0.03 percent of the option value. On the other hand, itis not surprising that the HSS method is the worst since one has to use the approximatedistributions ofS(t), B(t, T), andS(T) to value the options.

Next, we show the computational efficiency of using the GJ method, the numerical inte-gration method, the binomial method, and the HSS (1997) method. Table 3 presents therequired CPU time for each method to calculate the prices of the twelve options reported in

AMERICAN OPTION VALUATION 293

Table 2.Twice-exercisable put option prices: non-stochastic interest rates.

(1) (2) (3) (4) (4)−(3)(3) (5) (5)−(3)

(3) (6) (6)−(3)(3)

r σ accurate value HSS(200) % Chung(200) % Chung(NI) %

0.125 0.5 0.14082 0.14075 −0.05 0.14080 −0.01 0.14083 0.010.080 0.4 0.12158 0.12152 −0.04 0.12161 0.03 0.12158 0.000.045 0.3 0.09806 0.09801 −0.05 0.09808 0.02 0.09806 0.000.020 0.2 0.07007 0.07003 −0.06 0.07008 0.02 0.07007 0.000.005 0.1 0.03744 0.03741 −0.07 0.03744 0.01 0.03744 0.000.090 0.3 0.08142 0.08148 0.07 0.08143 0.00 0.08143 0.000.040 0.2 0.06202 0.06198 −0.06 0.06203 0.02 0.06202 0.000.010 0.1 0.03523 0.03525 0.07 0.03523 0.02 0.03523 0.000.080 0.2 0.04873 0.04869 −0.08 0.04875 0.05 0.04873 0.000.020 0.1 0.03128 0.03131 0.08 0.03128 0.01 0.03128 0.000.120 0.2 0.03811 0.03806 −0.15 0.03814 0.07 0.03810 −0.030.030 0.1 0.02782 0.02784 0.07 0.02782 0.00 0.02782 0.00

This table shows the pricing errors of using HSS’s (1997) and the proposed methods to price the twice-exercisableput options in a non-stochastic interest rate economy. Columns (1) to (2) are from GJ’s Table 1. Columns (1) and(2) represent the parameter input forr , the continuously compounded risk-free rate, andσ , the volatility of theunderlying asset. The strike price and the initial stock price in all cases are $1. The time to maturity of the optionis 1 year. Column (3) is the accurate value of the twice-exercisable put option which is calculated from Geske–Johnson method. Columns (4) to (6) are the twice-exercisable put values obtained from using HSS’s method with200 binomial steps, the proposed binomial method with 200 steps, and the proposed numerical integration method,respectively. The % pricing errors for each method are also shown.

Table 2. The GJ method, the numerical integration method, and our binomial method withsmall number of time steps are all very efficient. Each of them takes less than 0.1 secondto calculate twelve option prices. In contrast, the HSS (1997) method with large number oftime steps is impracticable. For instance, it takes more than two hours to calculate twelveoption prices using the HSS method with 200 time steps.

In summary, the results in this subsection demonstrate that the proposed method is afurther improvement of the HSS method in terms of both the efficiency and accuracy. Sincethe numerical integration method is more accurate than the binomial method, we will applythis method in all remaining simulations.

4.2. The Effect of Stochastic Interest Rates on the Option Values

Using the proposed method, we re-examine the effect of stochastic interest rates reportedby HSS (1997).13 HSS show that the effect may depend on the moneyness and the time tomaturity of the option, the interest rate level, volatilities of the underlying asset and the bondprices, as well as the correlation between them. Thus we vary the parameters mentionedin the HSS to obtain a comprehensive analysis of the stochastic interest rate effect on theoption values. From Figures 1 to 3, we vary the underlying asset volatilities from 5 to 45percent, bond volatilities from 0.5 to 4.5 percent, maturities from 0.5 to 5 years, correlationsfrom −0.9 to 0.9, the moneyness (measured as the ratio of the exercise price to the assetprice) from 0.8 to 1.2, and interest rates from 1 to 10 percent.

294 SAN-LIN CHUNG

Figure 1. The stochastic interest rate effect on the option value. This figure provides sensitivity analysis ofstochastic interest effect on the value of a put option exercisable atT/2 or T . The effect is defined as percentagedifference between the stochastic and non-stochastic interest rate models. We vary two parameters each time, whileholding all others at the following benchmark values:K = S(0) = 1.0, r = 0.05, T = 1, σ = 0.3, σB(T/2,T) =0.03, andρB(T/2,T),S(T/2) = 0.3.

AMERICAN OPTION VALUATION 295

Figure 2. The stochastic interest rate effect on the delta hedging. This figure provides sensitivity analysis ofstochastic interest rate effect on the delta of a put option exercisable atT/2 or T . The effect is defined aspercentage difference between the stochastic and non-stochastic interest rate models. We vary two parameterseach time, while holding all others at the following benchmark values:K = S(0) = 1.0, r = 0.05, T = 1, σ =0.3, σB(T/2,T) = 0.03, andρB(T/2,T),S(T/2) = 0.3.

296 SAN-LIN CHUNG

Figure 3.The stochastic interest rate effect on the rho hedging. This figure provides sensitivity analysis of stochasticinterest effect on the rho of a put option exercisable atT/2 or T . The effect is defined as percentage differencebetween the stochastic and non-stochastic interest rate models. We vary two parameters each time, while holdingall others at the following benchmark values:K = S(0) = 1.0, r = 0.05, T = 1, σ = 0.3, σB(T/2,T) = 0.03, andρB(T/2,T),S(T/2) = 0.3.

AMERICAN OPTION VALUATION 297

Table 3. The CPU time required to valuetwelve options reported in Table 2.

Panel A: Geske and Johnson method

0:00.014

Panel B: Numerical integration method

0:00.05

Panel C: Binomial method

binomial steps Chung HSSN = 10 0:00.007 0:00.12N = 20 0:00.017 0:01.3N = 40 0:00.04 0:17.2N = 80 0:00.13 4:10.1N = 120 0:00.29 20:37.8N = 160 0:00.52 64:25.9N = 200 0:00.81 158:07.4

This table shows the CPU time (in minutes,seconds, hundred of seconds) required tovalue twelve options reported in Table 2 us-ing GJ’s, HSS’s (1997), and the proposedmethods. The proposed method is im-plemented using the numerical integrationmethod (Panel B) and binomial tree method(Panel C). The CPU time is based on theUNIX system in Lancaster University.

Figure 1 plots the stochastic interest rate effect on the twice-exercisable put option valuesas the parameter values are varied. Figure 1(a) indicates that the effect is higher if thevolatility of the underlying asset is lower. This may be due to the fact that the value is lowfor options on low volatility assets, and thus the percentage effect is high. On the other hand,the effect remains stable as the interest rates vary except for options on low volatility assets.Figure 1(b) suggests that the effect increases as the maturities increases for out-of-the-money options. The reason for this can be explained as follows. Out-of-the-money optionsare likely to be exercised late as the time to maturity increases. Since the present value ofthe exercise premium is more sensitive to the interest rate change if the option is exercisedlater, the effect is also likely to increase as the maturity rises. Unlike HSS, Figure 1(c)shows that the effect may be positive or negative. Moreover, the results indicate that thestochastic interest rate effect is likely to be negative (positive) if the correlation between thestock price and the zero-coupon bond price is negative (positive). The magnitude (absolutevalue) of the effect is generally higher for higher bond volatilities and correlations.

4.3. The Effect of Stochastic Interest Rates on Delta Hedging

The analytic solution in Corollary 1 is implemented over a range of put options to see theeffect of the stochastic interest rates on delta values. We first investigate the magnitude of the

298 SAN-LIN CHUNG

Table 4. The effect on the delta values for long-maturityoptions with two possible exercise points.

(1) (2) (3) (4) (5) (5)−(4)(4)

S(0) σ r 12(N SR) 12(SR) %

1.00 0.05 0.03 −0.1684 −0.1806 7.241.00 0.05 0.06 −0.0270 −0.0276 2.221.00 0.10 0.03 −0.2933 −0.3060 4.301.00 0.10 0.06 −0.1579 −0.1622 2.741.00 0.20 0.03 −0.3336 −0.3419 2.491.00 0.20 0.06 −0.2629 −0.2671 1.601.05 0.05 0.03 −0.0626 −0.0691 10.331.05 0.05 0.06 −0.0056 −0.0058 3.311.05 0.10 0.03 −0.2083 −0.2200 5.621.05 0.10 0.06 −0.0980 −0.1017 3.721.05 0.20 0.03 −0.2888 −0.2972 2.901.05 0.20 0.06 −0.2205 −0.2248 1.940.95 0.05 0.03 −0.3660 −0.3794 3.640.95 0.05 0.06 −0.0996 −0.1010 1.460.95 0.10 0.03 −0.3994 −0.4105 2.790.95 0.10 0.06 −0.2436 −0.2479 1.760.95 0.20 0.03 −0.3838 −0.3916 2.030.95 0.20 0.06 −0.3124 −0.3160 1.17

This table shows the effect on the delta values for long-maturity put options with two exercise pointsT2 andT .The time to maturity of the option is 5 years, the bondvolatility is 3 percent, the correlation coefficient betweenthe asset price and the bond price is 0.3, and the strike priceis $1. The first three columns are from HSS’s (1997) Ta-ble 2. Columns (1) to (3) represent the parameter input forS(0), the initial asset price,σ , the volatility of the underly-ing asset,r , the continuously compounded risk-free rate.Columns (4) and (5) show delta values, obtained from theanalytic solution in Corollary 1, with non-stochastic andstochastic interest rates respectively. The % change in thedelta value due to stochastic interest rates is shown in thelast column.

effect for options with varying volatility, interest rate, and moneyness. Table 4 presents theeffect on a range of long-maturity twice-exercisable put options reported in HSS’s Table 2.HSS have shown that the effect of stochastic interest rates on option values is generallysmall. Similarly, we find that the effect on delta values is also small even for long-maturityoptions. Moreover, the results indicate that the effect increases as the put option goes outof the money and as interest rates decrease.

The proposed solution (and the HSS method) can also be applied to the value interest ratesensitive securities such as bond options. In this case, the correlation coefficient betweenthe asset price and the zero-coupon bond price at timeT

2 is high. In Table 5, we showthe results of a simulation where the correlation ranges from−0.9 to 0.9. The effect ofstochastic interest rates on delta values is generally small. However, it is significantly higherin the case where the bond volatility and the magnitude of the correlation are high.

AMERICAN OPTION VALUATION 299

Table 5.The effect of the correlation between the asset price and the zero-coupon bond price on the delta valueof an option with two exercise dates.

Volatilityof zero-coupon bond Coefficient of correlation

−0.9 −0.6 −0.3 0 0.3 0.6 0.9

0.05 −0.4028 −0.4125 −0.4195 −0.4249 −0.4295 −0.4337 −0.43770.03 −0.4020 −0.4082 −0.4133 −0.4175 −0.4212 −0.4245 −0.42770.01 −0.4045 −0.4076 −0.4099 −0.4119 −0.4138 −0.4154 −0.4170

The option is a put at a strike priceK = 1 on an asset whose current price isS(0) = 1. Volatility is 30% and therisk-free rate is 3%. The maturity of the option isT = 1 year and the option is exercisable at either timeT

2 or attime T . The delta is−0.4112 in the static model. The delta values are calculated from the analytic solution inCorollary 1 using the numerical integration method.

We also vary the parameters mentioned in the above subsection to obtain a detailed analysisof the stochastic interest rate effect on the delta hedging. Figure 2(a) shows that the effecton the delta values is higher if the volatility of the underlying asset and the interest ratesare lower. Moreover, the effect can be as large as 16% in some case. For out-of-the-moneyoptions, delta is small and one would expect that the percentage effect is larger. Figure 2(b)confirms this expectation. Furthermore, the effect remains stable as the time to maturity ofthe option increases except for out-of-the-money options. Similar to Figure 1(c), Figure 2(c)also shows that the effect may be positive or negative. The results indicate that the stochasticinterest rate effect is likely to be negative (positive) if the correlation is negative (positive).As expected, the magnitude of the effects is generally higher for higher bond volatilitiesand correlations. When the correlation moves from extreme negative to extreme positive,it has a significant impact on the delta values.

4.4. The Effect of Stochastic Interest Rates on Rho Hedging

In this subsection, we apply Corollary 2 to calculate rho values for a range of put options,and to analyze the effect of the stochastic interest rates on the rho values. We first investigatethe magnitude of the effect for options with varying volatility, interest rate, and moneyness.Comparing Table 6 with Table 4, we find that the effect of stochastic interest rates on rhois usually larger than on delta. The results also indicate that the effects have no systematicpattern with varying volatility, interest rate, and depth-in-the-money. To investigate thepotential impact of the correlation, Table 7 reports the rho values of a simulation where thecorrelation ranges from−0.9 to 0.9. It shows that the effect is generally higher in the casewhere the bond volatility and the magnitude of the correlation are high.

Figure 3 plots the percentage difference of rho between stochastic and non-stochasticinterest rate models as the parameter values are varied. Comparing Figures 1 to 3, wefind that the effect on the rho is higher than on the price and on the delta. In a stochasticinterest rate model, the magnitude of rho increases nearly 20% in some cases. Nevertheless,no clear patterns can be found from Figures 3(a) and 3(b). In other words, the effect onthe rho values varies with varying asset volatilities, interest rates, depth-in-the-money,

300 SAN-LIN CHUNG

Table 6. The effect on the rho values for long-maturity options with twopossible exercise points.

(1) (2) (3) (4) (5) (5)−(4)(4)

S(0) σ r ρ2(N SR) ρ2(SR) %

1.00 0.05 0.03 −0.5422 −0.5965 10.031.00 0.05 0.06 −0.0700 −0.0759 8.371.00 0.10 0.03 −1.1351 −1.1983 5.571.00 0.10 0.06 −0.4991 −0.5404 8.291.00 0.20 0.03 −1.6385 −1.7015 3.841.00 0.20 0.06 −1.1274 −1.1365 0.811.05 0.05 0.03 −0.2274 −0.2553 12.281.05 0.05 0.06 −0.0148 −0.0171 15.731.05 0.10 0.03 −0.8849 −0.9364 5.821.05 0.10 0.06 −0.3332 −0.3669 10.111.05 0.20 0.03 −1.4818 −1.5716 6.061.05 0.20 0.06 −0.9775 −1.0197 4.320.95 0.05 0.03 −1.0018 −1.1363 13.420.95 0.05 0.06 −0.2643 −0.2609 −1.300.95 0.10 0.03 −1.3348 −1.4801 10.890.95 0.10 0.06 −0.7296 −0.7644 4.760.95 0.20 0.03 −1.7245 −1.8308 6.160.95 0.20 0.06 −1.2211 −1.2592 3.12

This table shows the effect on the rho values for long-maturity put optionswith two exercise pointsT2 andT . The time to maturity of the option is 5years, the bond volatility is 3 percent, the correlation coefficient between theasset price and the bond price is 0.3, and the strike price is $1. The firstthree columns are from HSS’s (1997) Table 2. Columns (1) to (3) representthe parameter input forS(0), the initial asset price,σ , the volatility of theunderlying asset,r , the continuously compounded risk-free rate. Columns (4)and (5) show rho values, obtained from the analytic solution in Corollary 2,with non-stochastic and stochastic interest rates respectively. The % changein the rho value due to stochastic interest rates is shown in the last column.

and maturities. On the other hand, Figure 3(c) shows that raising the bond volatility anddecreasing the asset-bond correlations yield a larger positive effect on the rho values.

5. Conclusion

In this paper, we proposed a one-dimensional analytic solution for applying HSS’s (1997)generalised Geske–Johnson (1984) method to value American options in a stochastic interestrate economy. Analytic hedge ratios are also available from our pricing formulae. Themethod is a further improvement of the HSS (1997) in terms of computational efficiencyand accuracy. Within the framework of HSS (1997), the idea of this article (essentialasset prices) can be applied to derive the most efficient valuation method for pricing otherkinds of American options under stochastic interest rates. For example, Chung (1998)has generalised the HSS method and the proposed approach to price American options oncurrencies and foreign assets.

AMERICAN OPTION VALUATION 301

Table 7.The effect of the correlation between the asset price and the zero-coupon bond price on the rho valueof an option with two exercise dates.

Volatilityof zero-coupon bond Coefficient of correlation

−0.9 −0.6 −0.3 0 0.3 0.6 0.9

0.05 −0.5012 −0.4811 −0.4679 −0.4583 −0.4509 −0.4452 −0.44140.03 −0.5017 −0.4806 −0.4660 −0.4554 −0.4475 −0.4417 −0.43780.01 −0.4749 −0.4608 −0.4531 −0.4478 −0.4439 −0.4411 −0.4388

The option is a put at a strike priceK = 1 on an asset whose current price isS(0) = 1. Volatility is 30% andthe risk-free rate is 3%. The maturity of the option isT = 1 year and the option is exercisable at either timeT

2or at timeT . The rho is−0.4443 in the static model. The rho values are calculated from the analytic solutionin Corollary 2 using the numerical integration method.

We applied the proposed method to simulate a range of American put prices reported byHSS (1997), providing a detailed analysis on the effect of stochastic interest rates. Thefindings are summarised as follows. (1) Our method is far more efficient and accurate thanthe HSS method. For example, our method is at least ten times faster than the HSS methodwhen both methods are implemented with the same number of binomial steps. (2) Themagnitude of the effect on the option prices and on the delta values are generally higher ifthe asset volatility and interest rates are low, the option is out-of-money, the bond volatilityis high, or the magnitude of the asset-bond correlation is high. (3) Increases in the bondvolatility and decreases in the asset-bond correlation increase the effect on the rho values.(4) The stochastic interest rate effect can be positive or negative.

In future research, we hope to extend the results reported here in two directions. First, wecould investigate how the yield curve shape would change the option values and hedges.14

For example, as an upward sloping yield curve becomes flat, it might increase the possibilityof early exercise and hence change the price and hedge ratio. Second, we can compare theeffect of stochastic interest rates using a one-factor model to using a multi-factor model.Such a comparison may be important to the valuation of interest rate sensitive securities,as in the case of convertible bonds.

Appendix

Proof of Proposition 2: Given the forward priceF(t, T), the European option value attime t is

P1(F(t, T)) =∫ ∞−∞

[K B(t, T)N(−h)− S(t)N(−h− σ )] f2(ln B(t, T))d ln B(t, T)

=∫ ∞−∞

[K B(t, T)N(−h)− F(t, T)B(t, T)N(−h− σ )] f2(.)d ln B(t, T)

= [K N(−h)− F(t, T)N(−h− σ )]∫ ∞−∞

B(t, T) f2(.)d ln B(t, T)

= [K N(−h)− F(t, T)N(−h− σ )] exp

(µ2+ 1

2σ 2

2

),

302 SAN-LIN CHUNG

where f2(.),µ2, andσ2 are conditional probability density function, mean, and variance ofln B(t, T) given F(t, T) under the two-period FRA measure, respectively.

GivenF(t, T), if S(t) is larger (smaller) thanS(t) (or on the other handB(t, T) is larger(smaller) thanB(t, T), then one should hold (exercise) the option. Thus the holding valueat t , givenF(t, T), is therefore equal to∫ ∞

ln B(t,T)[K B(t, T)N(−h)− S(t)N(−h− σ )] f2(ln B(t, T))d ln B(t, T)

=∫ ∞

ln B(t,T)[K B(t, T)N(−h)− F(t, T)B(t, T)N(−h− σ )] f2(ln B(t, T))d ln B(t, T),

= [K N(−h)− F(t, T)N(−h− σ )]∫ ∞

ln B(t,T)B(t, T) f2(ln B(t, T))d ln B(t, T),

= [K N(−h)− F(t, T)N(−h− σ )] exp

(µ2+ 1

2σ 2

2

)N(d1).

And the early exercise value att , givenF(t, T), is∫ ln B(t,T)

−∞[K − S(t)] f2(ln B(t, T))d ln B(t, T)

=∫ ln B(t,T)

−∞[K − F(t, T)B(t, T)] f2(ln B(t, T))d ln B(t, T)

= K N(−d2)− F(t, T)exp

(µ2+ 1

2σ 2

2

)N(−d1).

The twice-exercisable option value, givenF(t, T), is then the sum of the above twovalues.

Proof of Proposition 3: Because the short rate process follows the extended-Vasicek model,the zero-coupon bond price follows (see Hull and White (1990))

d ln B(t, T) = µ(.)dt − vA(t, T)d Z2,

whereµ(.) is drift and A(t, T) = 1−e−q(T−t)

q . Thus the instantaneous variance of the log

of theT-maturity zero-coupon bond priceB(t, T) equalsv2A(t, T)2. Following Merton’s(1973) approach, for example, the variance of the log of the underlying asset price att is

σ 2S(t) =

∫ t

0Var

(d ln

S(τ )

B(τ, t)

)=∫ t

0[σ 2+ v2A(τ, t)2+ 2ρσvA(τ, t)]dτ

= σ 2t +(

t − 2A(0, t)+ 1− e−2qt

2q

)(v

q

)2

+ 2ρσ(t − A(0, t))v

q,

AMERICAN OPTION VALUATION 303

the covariance between the underlying asset price and theT-maturity zero-coupon bondprice att is

σS(t),B(t,T) =∫ t

0Cov

(d ln

S(τ )

B(τ, t),d ln

B(τ, T)

B(τ, t)

)=∫ t

0{ρσv[ A(τ, t)− A(τ, T)] + v2[ A(τ, t)2− A(τ, t)A(τ, T)]}dτ

= v2

2q2(1− e−2qt)A(t, T)−

(ρσv + v

2

q

)A(0, t)A(t, T).

The other variance-covariance terms can be derived in a similar manner.

Risk Neutral Distributions of Asset Prices under the Two-Period FRA Measure

Lemma 1 The mean ofln S(t), ln B(t, T), andln S(T) under the two-period FRA measureare

µS(t) = lnS(0)

B(0, t)− 1

2σ 2

S(t) (22)

µB(t,T) = lnB(0, T)

B(0, t)− 1

2σ 2

B(t,T) (23)

µS(T) = lnS(0)

B(0, T)+ σ 2

B(t,T) − σS(t),B(t,T) − 1

2σ 2

S(T), (24)

whereµX, σ 2X, andσXY are respectively the mean, variance ofln X, and the covariance of

ln X andln Y .

Proof: The risk neutral distributions forS(t) and B(t, T) are shown in Section 1. Fromthe joint lognormality ofS(t) andB(t, T), we can write

µS(t) = ln E0(S(t))− 1

2σ 2

S(t)

= lnS(0)

B(0, t)− 1

2σ 2

S(t)

µB(t,T) = ln E0(B(t, T))− 1

2σ 2

B(t,T)

= lnB(0, T)

B(0, t)− 1

2σ 2

B(t,T).

304 SAN-LIN CHUNG

Applying one result of the HSS (1997) thatE0(S(T)) = S(0)B(0,T) exp(σ 2

B(t,T) − σS(t),B(t,T))

to the lognormality ofS(T) will yield

µS(T) = ln E0(S(T))− 1

2σ 2

S(T)

= lnS(0)

B(0, T)+ σ 2

B(t,T) − σS(t),B(t,T) − 1

2σ 2

S(T).

Lemma 2 The risk neutral distribution of F(t, T) under the two-period FRA measure is alognormal distribution with mean and variance

µF(t,T) = lnS(0)

B(0, T)− 1

2σ 2

S(t) +1

2σ 2

B(t,T), (25)

σ 2F(t,T) = σ 2

S(t) + σ 2B(t,T) − 2σS(t),B(t,T). (26)

Proof: The mean of lnF(t, T) under the two-period FRA measure is

µF(t,T) = E0

[ln

S(t)

B(t, T)

]= µS(t) − µB(t,T)

= lnS(0)

B(0, T)− 1

2σ 2

S(t) +1

2σ 2

B(t,T),

where the third equality comes from Lemma 1. The variance of lnF(t, T) is

Var[ln F(t, T)] = Var

[ln

S(t)

B(t, T)

]= Var[ln S(t)] + Var[ln B(t, T)] − 2Cov[ln S(t), ln B(t, T)]

= σ 2S(t) + σ 2

B(t,T) − 2σS(t),B(t,T).

Lemma 3 The conditional risk neutral distributions of S(T) and B(t, T) given F(t, T)under the two-period FRA measure are lognormal distributions with mean and variance asfollows,

E[ln S(T)|F(t, T)] = µS(T) + Cov(ln F(t, T), ln S(T))

Var(ln F(t, T))(ln F(t, T)− µF(t,T)),

Var[ln S(T)|F(t, T)] = σ 2S(T) −

Cov(ln F(t, T), ln S(T))2

Var(ln F(t, T)),

E[ln B(t, T)|F(t, T)] = µB(t,T)

+ Cov(ln F(t, T), ln B(t, T))

Var(ln F(t, T))(ln F(t, T)− µF(t,T)),

AMERICAN OPTION VALUATION 305

Var[ln B(t, T)|F(t, T)] = σ 2B(t,T) −

Cov(ln F(t, T), ln B(t, T))2

Var(ln F(t, T)),

Cov[ln S(T), ln B(t, T)|F(t, T)] = 0,

where Cov(ln F(t, T), ln B(t, T)) = σS(t),B(t,T) − σ 2B(t,T), and Cov(ln F(t, T), ln S(T)) =

σS(t),S(T) − σS(T),B(t,T).

Proof: Because lnS(t), ln S(T), and lnB(t, T) are trivariate normally distributed,ln F(t, T), ln S(T), and lnB(t, T) are also trivariate normally distributed. The condi-tional mean and variance of trivariate normal distribution, which can be found in manyprobability textbooks, are

E(ln X|Z) = µX + σX Z

σ 2Z

(ln Z − µZ),

Var(ln X|Z) = σ 2X −

σ 2X Z

σ 2Z

.

The conditional covariance is

Cov[ln S(T), ln B(t, T)|F(t, T)]= Cov[Et [ln S(T)], ln B(t, T)|F(t, T)]

= Cov

[ln F(t, T)− 1

2Vart [ln S(T)], ln B(t, T)|F(t, T)

]= 0

Note thatE[ln B(t, T)|F(t, T)] and Var[ln B(t, T)|F(t, T)] are the conditional meanand variance of lnB(t, T) given F(t, T) respectively. In other words, they areµ2 andσ 2

2in Proposition 2.

Notes

1. In 1993, the Philadelphia Stock Exchange (PHLX) began to trade currency options with maturity up to twoyears.

2. An n exercise-date Bermudan option allows the holder to exercise the option at one ofn exercise dates. Thisoption is also called ann-times exercisable option. In this paper, ann exercise-date Bermudan option and ann-times exercisable option are used interchangeably.

3. As pointed out by HSS (1997), if the pricing kernel, the underlying asset price, and the zero-coupon bondprice are joint lognormally distributed, risk neutral measures for pricing options exist. In other words, by ourassumption, risk neutral distributions for pricing options exist.

4. If an option’s payoff depends on the asset prices at various dates, it can be valued by taking the expectedvalue of its payoff, using a multi-period FRA measure, and discounting using the relevant zero-coupon bondprices. The distributions for the asset price at various dates under the multi-period FRA measure are identicalto the true distributions except for a mean shift which makes the conditional expected value of each of theprices equal to their respective (conditional) forward prices. For more details of FRA measure, one can referto Merton (1973), Jamshidian (1989, 1991, 1993), Satchell, Stapleton, and Subrahmanyam (1997), and HSS(1997).

306 SAN-LIN CHUNG

5. This is actually the Black-Scholes (1973) formula with an initial stock priceS(t), a time to maturityT − t ,and a risk-free rater = − ln B(t,T)

T−t .6. It should be noted that the rho hedge ratio of the American (and also twice-exercisable) option in GJ’s equation

(8) is not correct. The critical stock price is an implicit function of the interest rate, and thus the option value.Therefore it is not possible to directly derive the partial derivative of the option value with respect to the interestrate. In other words, the rho hedge ratio is a complex, implicit function of interest rate. However, Corollary 2can be applied to calculate the rho hedge ratios for a static model which is a limiting case as the bond volatilitytends to zero.

7. One related issue is that one might expect that the yield curve shape would change the hedges in a significantway for the American options. However, it may not be the casefor the twice-exercisable options. This is apotential problem of applying the HSS model to calculate the hedges for the American options.

8. We would like to point out two errors in Rabinovitch’s (1989) paper. First, note that the correlation coefficientin Merton (1973) is the instantaneous correlation coefficient between the stock price and the zero-couponbond price, instead of Rabinovitch’s instantaneous correlation coefficient between the stock price and shortrate. As a result,ρ should be replaced with−ρ in Rabinovitch’s equation (7). Therefore, in his equation (8),−2ρσ(τ − B) vq should change sign and then will be consistent with the results of our Proposition 3. Second,in his Table 1, some input parameters are not internally consistent. For example, a bond is implicated as theunderlying asset whenρ = −1.0 in Rabinovitch’s model because he assumes a one-factor term structure modelof interest rates. The instantaneous variance of the bond (underlying asset) price should be a deterministicfunction of time to maturity of the bond, rather than a constant.

9. The reason for this choice is because the probability density function outside the lower and upper limits issmall and thus negligible.

10. There could be other numerical integration methods which are more accurate and efficient than the chosenmethod.

11. In the simulations, we let the volatility of a12 year bond be 0.0002 percent and the correlation between the logstock price and the log zero-coupon bond price be zero in the non-stochastic interest rate case.

12. Remember that the GJ method can be applied to a non-stochastic interest rate economy, and one needs to usenumerical method to find out the critical exercise price,. We use Newton-Raphson method to search the criticalexercise price until the price difference between the last two searches is within 10−5% of the underlying assetprice.

13. As in HSS, the total volatility of the stock price in a stochastic interest rate economy is adjusted to be the sameas that in a non-stochastic interest rate economy. Therefore we investigate the pure effect of stochastic interestrates.

14. The author thanks the referee for pointing out this suggestion.

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