Currency barrier option pricing with mean reversion

CURRENCY BARRIER

OPTION PRICING WITH

MEAN REVERSION

C. H. HUI*C. F. LO

This article develops a barrier option pricing model in which the exchangerate follows a mean-reverting lognormal process. The correspondingclosed-form solutions for the barrier options with time-dependent barriersare derived. The numerical results show that barrier option values and thecorresponding hedge parameters under the proposed model are differentfrom those based on the Black-Scholes model. For an up-and-out call, themean-reverting process keeps the exchange rate in a small range aroundthe mean level. When the mean level is below the barrier but above thestrike price, the risk of the call to be knocked out is reduced and its optionvalue is enhanced compared with the value under the Black-Scholesmodel. The parameters of the mean-reverting lognormal process thereforehave a material impact on the valuation of currency barrier options andtheir hedge parameters. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark26:939–958, 2006

The conclusions herein do not represent the views of the Hong Kong Monetary Authority.*Correspondence author, Research Department, Hong Kong Monetary Authority, 55/F, TwoInternational Finance Centre, 8 Finance Street, Central, Hong Kong, China; e-mail: [email protected]

Received October 2005; Accepted January 2006

■ C. H. Hui is in the Research Department at the Hong Kong Monetary Authorityin Hong Kong, China.

■ C. F. Lo is at the Institute of Theoretical Physics and Department of Physics, TheChinese University of Hong Kong in Hong Kong, China.

The Journal of Futures Markets, Vol. 26, No. 10, 939–958 (2006)© 2006 Wiley Periodicals, Inc.Published online in Wiley InterScience (www.interscience.wiley.com).DOI: 10.1002/fut.20223

940 Hui and Lo

Journal of Futures Markets DOI: 10.1002/fut

INTRODUCTION

Barrier options are path-dependent options that are usually structured asmodifications of simple European puts and calls. The existence of aEuropean barrier option depends upon whether the underlying assetprice has crossed a predetermined barrier prior to the exercise time. Forexample, an up-and-out call pays off the usual European-call payoff atexpiry, unless at any time before expiry the underlying asset has beentraded at or higher than the barrier level. In this example, it is said toknock out, becoming worthless. On the other side of the picture are “in”options, which actually commence when the underlying price touchesthe barrier. Barrier options are cheaper than regular European optionsbecause of these extinguishing and activating features. Thus they areattractive to investors who are averse to paying high premiums. In addi-tion, sellers of barrier options may be able to limit their downside risk.Barrier options have emerged as significant products for hedging andinvestment in foreign exchange, equity, and commodity markets sincethe late 1980s, largely in the over-the-counter markets and for structur-ing financial products (e.g., structured notes linked to exchange rates).The valuation of barrier options has been well covered in literature (seeHui, 1996, 1997; Kunitomo & Ikeda, 1992; Merton, 1973, Rich, 1994;Rubinstein & Reiner, 1991). The dynamics of the underlying asset pricein the valuation follow the lognormal process proposed by Black andScholes (1973).

Regarding currency options, Garman and Kohlhagen (1983) adaptthe Black-Scholes model to develop the currency option valuationmodel. However, it is not entirely satisfactory because the ordinaryBlack-Scholes model is for stock options and currencies are differentfrom stocks in important respects. In this connection, Sørensen (1997)and Ekvall, Jennergren, and Näslund (1997) present revised currencyoption pricing models in which the exchange rate follows a mean-reverting process (i.e., the logarithm of the exchange rate follows anOrnstein-Uhlenbeck process). Sørensen (1997) proposes an equilibriummodel that establishes a mean-reverting process for the exchange-ratedynamics through its effect on the dynamics in the domestic and foreignterm structures of interest rates. Ekvall et al. (1997) state several reasonswhy a mean-reverting process may be reasonable for exchange rates,even though these three reasons for mean reversion are not all in effectsimultaneously. First, in the long run the equilibrium exchange rate isdetermined by domestic money stock, interest-rate level, and nationalincome relative to the same quantities in the foreign economy. However,

Currency Barrier Option Pricing 941


1Examples are the currencies within the European Monetary System in 1990s, the Chineserenminbi, Hong Kong dollar, Singapore dollar and Malaysia ringgit.

in the short run it may deviate (or overshoot) from equilibrium, becauseof the sluggishness with which goods prices react to disturbances. Theresulting tendency of the exchange rate to return to an equilibrium canbe thought of as mean reversion. Such a sticky-price monetary approachincorporating overshooting has been discussed by Dornbusch (1976).

Second, mean reversion could come about because of interventionsin the foreign-exchange market by the central banks. Such interventionshave been shown in many studies (see MacDonald, 1988). An underly-ing feature of the interventions is that the authorities try to bring theexchange rate back to some normal or equilibrium level. A natural way tomodel this is through a mean-reverting process for the exchange rate.

Third, certain currencies are constrained to move inside targetzones or under a managed-floating regime.1 An important prediction ofthe theoretical literature on targeted exchange rates is that one mightexpect mean reversion of the exchange rate when the central banksengage in intramarginal intervention, and market participants expect theexchange-rate band to be fully credible and to engage in stabilising spec-ulation. This mean-reverting property is widely referred to in the litera-ture (see, for example, Anthony & MacDonald, 1998; Krugman, 1991;Rose & Svensson, 1994; Svensson, 1992, 1993). Several recent studieshave attempted to investigate this theoretical prediction empirically byexamining the time-series properties of the currencies participatingin the European Monetary System (see, for example, Anthony &MacDonald, 1998, 1999; Ball & Roma, 1993, 1994; Kanas, 1998;Nieuwland, Verschoor, & Wolff, 1994; Rose & Svensson, 1994;Svensson, 1993). Although their investigations had mixed results, theempirical results suggest that mean reversion is present.

Sørensen (1997) finds that the mean-reverting process has signifi-cant implications for the valuation of American currency options andoptimal exercise strategies. As barrier options are path-dependentoptions like American options, implications of mean reversion for thevaluation of currency barrier options could also be significant. This arti-cle develops a barrier option pricing model in which the exchange ratefollows a mean-reverting lognormal (MRL) process to study the implica-tions. The corresponding closed-form solutions for the barrier optionswith time-dependent barriers are derived. The pricing solutions are usedto examine the effects of the MRL dynamics on the values and hedgeparameters of European-style currency barrier options. The results are

942 Hui and Lo


2F0 can be interpreted as the historical mean instantaneous exchange rate.

compared with those generated from the Black-Scholes model. As meanreversion could be present in foreign exchange dynamics and the use ofcurrency barrier options for hedging foreign exchange exposures andstructuring financial products is popular in the financial market, thefindings in this article could provide an analytical valuation frameworkfor research in currency option pricing.

The model developed here may also be applicable for pricing barrieroptions on commodity products or other assets that follow a mean-reverting process.

The article is organized as follows. The next section presents thecurrency barrier option pricing model under the MRL process andderives the corresponding option pricing formulas. Next the effects ofthe MRL dynamics on the value of an up-and-out call and its hedgeparameters are examined. The results are compared with those under theBlack-Scholes model and those of a regular call under the MRL process.The final section summarizes the findings.

PRICING EUROPEAN BARRIER OPTIONS

In the MRL model, it is assumed that the exchange rate F (i.e., thedomestic currency value of a unit of foreign currency) evolves accordingto the diffusion process specified as

(1)

where F0 is the conditional mean exchange rate,2 k is the parametermeasuring the speed of reversion to this mean, s is the volatility of theexchange rate, m is the instantaneous return on F, and W is a standardWiener process so that dW is normally distributed.

It is assumed that option prices depend on F as the only statevariable. By the usual arbitrage-free argument for currency options, therisk-adjusted expected excess returns of holding a currency option andholding its underlying currency must be identical (see, for example,Garman & Kohlhagen, 1983). Applying Ito’s lemma, the partial differentialequation governing the option price P(F, t) with time-to-maturity of t basedon the model is

(2)� F0P(F, t)

0F� rP(F, t)

� [k(ln(F0 � ln F) � (r � r*)]0P(F, t)

0t�

12

s2F2 02P(F, t)0F2

dF � [k(ln F0 � ln F) � m]F dt � sF dW



where r and r* are the risk-free interest rates of the domestic and foreigncurrencies, respectively. By putting x � ln(F�H) and u � [ln(F0�H) �s2�2k � (r � r*)�k], where H is a level to determine the barrier,Equation (1) becomes

(3)

With further changing variables:

(4)

P~(x, t) satisfies the following equation:

(5)

Sørensen (1997) and Ekvall et al. (1997) derive currency regular call andput pricing formulas in which the exchange rate follows a mean-revertingprocess. The respective formulas are presented in Equations (A8) and(A14) in Appendix A. The solution of Equation (5) with an up-and-out3

option-payoff condition of P~(x, t � 0) at option maturity is

(6)

and the distribution function G is given by

(7)

where

(8)

(9) � [ln(F0�H) � s2�2k � (r � r*)�k](1 � e�kt)

c2 � u(1 � e�kt)

c1 �s2

4k(1 � e�2kt)

G(x, t; x�, t � 0) �124pc1(t)

exp •� [x� � x � c2(t)]2

4c1(t)¶

(x, t; x�, 0)

P~(x, t) � �

0

��

dx�[G(x, t; x�, t � 0) � G(x, t;�x�, t � 0)e�bx�]P~(x�, t � 0)

0P~(x, t)0t

�12

(e�kts)2 02P~(x, t)0x2 � e�kt

ku0P

~(x, t)0x

e�rtP~(xe�kt, t) � P(x, t)

0P(x, t)0t

�12s2 02P(x, t)

0x2 � k(u � x)0P(x, t)

0x� rP(x, t)

3It is a regular option that ceases to exist if the underlying asset price reaches a certain level, thebarrier. The barrier level is above the initial asset price.

944 Hui and Lo


4When the model parameters of barrier options are time dependent under the Black-Scholes model,there are no closed-form solutions (see Robert & Shortland, 1997 and Lo, Lee, & Hui, 2003).

and is a real number. The payoff condition of an up-and-out call ismax(F � K, 0) if F � h(t) and 0 if F � h(t), where h(t) is the movingbarrier and K is the strike price at option maturity, such that

(10)

Although closed-form solutions of fixed barrier options can beobtained under the Black-Scholes model, closed-form solutions ofbarrier options under the MRL model are obtained when the barrier istime dependent (i.e., moving with time).4 The trajectory of the movingbarrier under the MRL model is

or

(11)

The option vanishes when

(12)

The solution of a call is obtained by solving Equation (6) subject tothe payoff condition (10) and the boundary condition (12). After substi-tuting back the variables, if h(t) is greater than the strike price K, theup-and-out call value is

(13)

where

(14) F~

� a FHbexp(�kt)�1

ec1�c2 F

� KaH

F~ b

�b

eb(b�1)c1�rt[N(c) � N(d)]

� F~ aH

F~ b

2�b

e[(b�1)2�(b�1)]c1�rt[N(c �12c1) � N(d �12c1)]

� F~e�rtN(b � 12c1) � Ke�rtN(b)

Puo�call(F, t) � F~e�rtN(a � 12c1) � Ke�rtN(a)

P~

call(x � h~(t), t) � 0

h(t) � H exp5�ekt[c2(t) � bc1(t)]6

h~(t) � �c2(t) � bc1(t)

h~

(t)

P~

call(x, t � 0) � 0,�F � h(t)

P~

call(x, t � 0) � max(Hex � K, 0),�F � h(t)

b



5Rapisarda (2003) applied the results of Lo et al. (2003) to derive in an analytical fashion theapproximate prices of various types of barrier options, for example forward start/early expiry barriersand window barriers.6Based upon the method of multiple images proposed by Lo et al. (2006) for valuation of double-barrier options with time-dependent parameters, a similar solution in Equation (6) for double-barrier options with mean reversion can be obtained.7The derivation of the formulas under the Black-Scholes model can be found in Rubinstein andReiner (1991) and Rich (1994).

(15)

(16)

(17)

(18)

and N() is the cumulative normal distribution function.To obtain the value of a fixed barrier (i.e., time-independent con-

stant barrier) knock-out call, the parameter can be adjusted such thatthe solution in Equation (13) provides the best approximation to theexact value of the fixed barrier call by using a simple method developedby Lo et al. (2003) for solving barrier option values with time-dependentmodel parameters. The method is based upon simulating the fixedbarrier as a slowly fluctuating barrier with a small oscillating amplitudeby tuning the parameter . The upper and lower bounds (in closed form)provided by the method are also very tight for the exact fixed barrieroption prices. Because the bounds and estimates of the option priceappear in closed form, they can be computed very efficiently.Furthermore, the bounds can be improved systematically, and theseimproved bounds are again expressed (in closed form) in terms of themultivariate normal distribution functions.5 In the next section, Figure 1illustrates the movements of the time-dependent barriers with different

over option life.The derivations of option pricing formulas of a down-and-out call,

up-and-out and down-and-out puts, and knock-in options are similar tothat of the up-and-out call. They are presented in Appendix A.6

The resemblance between the barrier option pricing formulasderived above and the corresponding formulas with the exchange rateunder the Black-Scholes model is obvious.7 The random variable F

~

k

b

b

d �ln(H2�F

~K ) � c1 � 2(b �1)c112c1

c �ln(H�F

~) � c1 � 2(b � 1)c112c1

b �ln(F

~�H) � c112c1

a �ln(F

~�K) � c112c1

946 Hui and Lo


105

106

107

108

109

110

111

112

113

114

115

116

0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05

Time to maturity (t )

Bar

rier

(h

)k = 0

k = 0.5

k = 1

k = 2

FIGURE 1Barrier level with time to maturity.

specified in Equation (14) above is lognormal, resulting in similar for-mulas as those under the Black-Scholes model. , which is the vari-ance of the logarithm of the price , replaces of the Black-Scholesmodel, which has the same meaning. In other words, is lognormal withvar where

(19)

The resemblance is equivalent to the change of numeraire of F by Different magnitudes of the speed of reversion to this mean give someintuitive interpretations of . When the speed is very strong (i.e.,

converges to F0. This means that the dynamical process ofthe exchange rate is almost deterministic, such that it will stick to F0

with the effective volatility . Given an up-and-out call, its valueconverges to max if F0 is below the barrier h(t) or 0 other-wise (i.e., the call is knocked out). This illustrates that the presence ofmean reversion makes the associated barrier option values very differentfrom those in the Black-Scholes model. The differences will be shownnumerically in the following section. Conversely, when the speed is veryweak (i.e., ), the dynamical process specified in the model con-verges to a lognormal process where and such that the cor-responding barrier option values are those based on the Black-Scholes

s~ � sF~

� FkS 0

[(F0 � K), 0]e�rts~ S 0

F~

k W 1),F~

kF~.

s~ � 12c1�t �s12kt21 � e�2kt

[lnF~] � s~ 2�t

F~

s2F~

A s~ 2



model. Equation (19) and the limits of with different show that isa decreasing function with and is less than . The proof of being adecreasing function with is given in Appendix B.

EFFECTS OF MRL PROCESS ON BARRIEROPTION VALUES AND HEDGE PARAMETERS

When dealers trade any derivative instruments, they often need to hedgetheir positions. It is important to determine the hedge parameters ofthe derivative instruments. The following discussion is focused on theeffects of the mean-reverting process on the values and hedge parame-ters including delta, gamma, and vega (the effect of the volatility on theoption value) of the barrier options. Examples are using an up-and-outforeign currency call against domestic currency put option withexchange rate (F) � 100, strike price (K) � 100, barrier (H) � 115, timeto maturity (t) � 6 months, foreign currency interest rate (d) � 5%,domestic currency interest rate (r) � 1%, volatility ( ) � 15%, and �

0.5, 1 and 2. These values of are consistent with the estimates of (which are between 0.1 and 1.2) in Sørensen (1997) for the exchangerates of USD against GBP, DEM, and JPY in the period from January1985 to December 1994. Option values and hedge parameters under theMRL model based on Equation (13) are compared with those of an up-and-out call option under the Black-Scholes model (i.e., � 0). b is setto be �c2(t)�c1(t) for the initial value of the time-dependent barrier h(t)such that the moving barrier is close to 115 over time. Figure 1 illus-trates the movements of the barriers with different over option life. Itshows that the form of the barrier specified in Equation (11) ensuresthat the barrier converges to a fixed barrier when is reduced to zero.The values of the hedge parameters are measured as finite differenceapproximations to their continuous time equivalents.

Figure 2 illustrates the option value variations with different meanexchange rates F0 of the up-and-out calls. When F0 is below the strikeprice of 100, the barrier options under the MRL model, especially withhigh , cost less than the barrier option under the Black-Scholes modelbecause the mean-reverting process will push the exchange rate belowthe strike over time and make the option become out-of-the-money.Conversely, when F0 is higher than the strike price and will push theexchange rate towards higher values above the strike price during theoption life, the option values under the MRL model increase with F0 andare higher than those under the Black-Scholes model. In the case ofstrong mean reversion (i.e., � 2), the option values under the MRLk

k

k

k

k

kk

ks

k

s~ 2s2k

s~ 2ks~

948 Hui and Lo


0.0

0.5

1.0

1.5

2.0

2.5

3.0

81 86 91 96 101 106 111 116 121 126

Opt

ion

valu

e

k = 0.5

k = 2

k = 1

k = 0

Mean exchange rate (F0)

FIGURE 2Variation of value of 6-month up-and-out calls with the mean exchange rate.

8The delta of a regular at-the-money call/put under the Black-Scholes model is about 0.5.

model decreases when F0 is higher than 115, where risk of knock-outincreases. In the case of � 1, when F0 is higher than 122, the variationof the option values with the underlying price is small. It is because therisk of knock-out and the effect of in-the-moneyness due to the mean-reverting process cancel each other.

Compared with Figure 2, the option values are higher in Figure 3where the option maturity is reduced to 3 months. The shorter time tomaturity reduces the probability of knock-out and thus enhances the val-ues of the up-and-out calls. The effects of the mean-reverting process onthe option values in Figure 2 are in general similar to those in Figure 3.In the case of strong mean reversion (i.e., � 2), the option valuesunder the MRL model decrease when F0 is higher than 128, which isabove the barrier at 115. This reflects that the effect of the mean rever-sion on option values weakens with the shorter option maturity.

Dealers need accurate values for the delta and the delta behavior inorder to hedge their option portfolios effectively. The delta variationswith the exchange rate F in Figure 4 with F0 � 100 and the option matu-rity of 6 months show two typical features of the up-and-out calls. First,their deltas are smaller than those of the regular option because of thecheaper knock-out option values.8 Second, the deltas become negativewhen the underlying price is close to the knock-out barrier. This is due to

k

k



FIGURE 3Variation of value of 3-month up-and-out calls with the mean exchange rate.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

81 86 91 96 101 106 111 116 121 126


Opt

ion

valu

e

k = 2

k = 1

k = 0. 5

k = 0

FIGURE 4Variation of delta of 6-month up-and-out calls with the exchange rate, where F0 � 100.

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

81 86 91 96 101 106 111

Exchange rate (F )

Del

ta

k = 0

k = 0.5

k =1

k =2

the fact that the option values decrease when the underlying price moveshigher and closer to the barrier. Figure 4 also illustrates that the differ-ences in deltas between the MRL and Black-Scholes models are in gen-eral not significant. However, in Figure 5 where the mean exchange rate

950 Hui and Lo


FIGURE 5Variation of delta of 6-month up-and-out calls with the exchange rate, where F0 � 110.

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

81 86 91 96 101 106 111

Exchange rate (F)

Del

ta

k = 0k = 0.5k = 1k = 2

F0 is set to be 110, the changes in the deltas with F are quite differentamong the option values under the MRL and Black-Scholes models. Forinstance, when F is 104, the delta under the Black-Scholes model iszero, whereas the deltas under the MRL model with � 1 and 2 areabout �0.05 and �0.11, respectively. As the mean exchange rate F0 is at110, the mean-reverting process will push the exchange rate at 100toward higher values and increases the risk of knock-out during theoption life. On the other hand, when the options are out-of-the-money,the deltas under the MRL model are higher in values because the mean-reverting process will push the exchange rate toward higher values with-out significantly increasing the risk of knock-out.

Figure 6 illustrates the variations of gamma with the exchange rateF with the options in Figure 5. The gamma is defined as the change indelta per unit of the exchange rate. Similar to the gamma under theBlack-Scholes model, the gamma under the MRL model can be negativebecause of the negative slopes of the delta (see Figure 5) and hasthe lowest value at �0.025 where F � 106. In terms of magnitude, thegamma under the MRL model is in general lower than the gamma underthe Black-Scholes model, when F is above 103. This reflects that themean-reverting process reduces the gamma risk of the up-and-out call bypushing the exchange rate to be around the mean exchange rate F0 at110. Compared with the Black-Scholes model, the delta under the MRLmodel is relatively stable for effective hedging due to the lower gamma.

k



-0.030

-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

81 86 91 96 101 106 111

Exchange rate (F )

Gam

ma

0

0.5

k = k = k =

1

k = 2

FIGURE 6Variation of gamma of 6-month up-and-out calls with the exchange rate, where F0 � 110.

The vega variations in Figure 7 are the percentage differences of theoption value with 1% increase in the volatility. The vega of the up-and-outcalls turns from positive to negative when the exchange rate is close to thebarrier. It is a typical feature of an up-and-out call, where the higher

FIGURE 7Variation of vega of 6-month up-and-out calls with the exchange rate, where F0 � 110.

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

81 86 91 96 101 106 111

Exchange rate (F )

Veg

a

k = 0k = 0.5k = 1k = 2

952 Hui and Lo


FIGURE 8Variation of value of 6-month up-and-out and regular calls with

the mean exchange rate.

0

2

4

6

8

10

12

14

16

18

81 86 91 96 101 106 111 116 121 126


Op

tion

val

ue

Barrier k = 0.5

Barrier k = 2Regular k = 0.5

Regular k = 2

volatility makes the call easier to be knocked out near the barrier.Figure 7 shows that the vega risk in the MRL model is smaller than thatin the Black-Scholes model, and the vega decreases with an increasein k. This observation is consistent with the analysis that the effectivevolatility in Equation (19) under the mean-reverting process is lessthan s.

Figure 8 compares the option value variations with different meanexchange rates F0 of the up-and-out and regular calls. Similar to theprice relations between the barrier and regular options under the Black-Scholes model, the values of the up-and-out calls under the MRL modelare less than those of the regular calls. Because of no barrier stoppingthe increase in option values, the option values of the regular callsincrease with the mean exchange rate, as F0 will push the exchange ratetoward higher values above the strike price, in particular in the case ofstrong mean reversion (i.e., k � 2). When F0 is below 102, the regularcalls with k � 2 cost less than the regular calls with k � 0.5 because thestronger mean-reverting process of k � 2 will push the exchange ratebelow the strike over time and make the option become out-of-the-money. This characteristic is similar to that of the up-and-out calls.

In Figure 9, the delta variations with the exchange rate F ofthe regular calls with F0 � 100 show that their deltas are positivebecause there is no barrier to reduce the option values when theexchange rate increases. As reflected in the option values in Figure 8,their deltas are higher than those of the up-and-out calls. The deltas of

s~



FIGURE 9Variation of delta of 6-month up-and-out and regular calls with

the exchange rate, where F0 � 100.

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

81 86 91 96 101 106 111

Exchange rate (F )

Del

ta

Regular k = 0.5

Regular k = 2

Barrier k = 0.5

Barrier k = 2

the regular call under strong mean reversion (i.e., k � 2) are smallerthan those of the regular call with k � 0.5 when the exchange rate ishigher than 95 because the strong mean-reverting process will pushthe exchange rate toward the mean level of 100 and thus limit the up-sideof the call.

SUMMARY

This article presented a model for valuing currency barrier options whenthe foreign exchange rate follows a MRL process. The correspondingclosed-form solutions for the option valuation with time-dependent bar-riers are derived. The mean-reverting process keeps the exchange rate ina small range around the mean level over the option life and hence maylimit the uncertainty of the option to be knocked out. In the case of anup-and-out call, when the mean level is below the barrier but above thestrike price, the risk of the call to be knocked out is reduced and itsoption value is enhanced compared with the value under the Black-Scholes model. The numerical results show that the mean exchange rate(relative to the current exchange rate and barrier) and the speed of meanreversion have material impact on the valuation of currency barrieroptions and their hedge parameters.

As mean reversion could be present in foreign-exchange dynamicsand the use of currency barrier options for hedging foreign exchangeexposures and structuring financial products is popular in the financial

954 Hui and Lo


market, the findings here could provide an analytical valuation frame-work for research in currency option pricing. This framework can also beapplied to other barrier options in which the underlying assets follow amean-reverting process. For example, commodity prices such as energyprices seem to exhibit some mean reversion.

APPENDIX A

Regarding a down-and-out call, the solution of Equation (2) at optionmaturity is

(A1)

where

(A2)

If h(t) is less than or equal to K, the value of a down-and-out call is given by

(A3)

If h(t) � K, then

(A4)

Because the value of a regular call equals the value of a knock-in callplus the value of a knock-out call, the value of an up-and-in call is given by

(A5)

and the value of a down-and-in call is given by

(A6)Pdi�call(F, t) � Pcall(F, t) � Pdo�call(F, t)

Pui�call(F, t) � Pcall(F, t) � Puo�call(F, t)

� KaH

F~ b

�b

eb(b�1)c1�rtN(c)

� F~ aH

F~ b

2�b

e[(b�1)2�(b�1)]c1�rtN(c � 12c1)

Pdo�call(F, t) � F~e�rtN(b � 12c1) � Ke�rtN(b)

� KaHF~ b

�b

eb(b�1)c1�rtN(d)

� F~ aH

F~ b

2�b

e[(b�1)2�(b�1)]c1�rt N(d � 12c1)

Pdo�call(F, t) � F~e�rt N(a � 12c1) � Ke�rtN(a)

P~call(x, t � 0) � 0,�F h(t)

P~

call(x, t � 0) � max(Hex � K, 0),�F � h(t)

P~(x, t) � �

�

0

dx�[G(x, t; x�, t � 0) � G(x, t;�x�, t � 0)e�bx�]P~ (x�, t � 0)



The value of a regular call can be obtained by

(A7)

with the call payoff condition and is equal to

(A8)

Barrier put values are derived similarly to barrier call values byusing the payoff of (x, t � 0) � max(K � Hex, 0), if F < h(t) and zeroif F h(t) in Equation (6) for an up-and-out put and if F > h(t) and 0if F h(t) in Equation (A1) for a down-and-out put, respectively. Whenh(t) K, an up-and-out put value is

(A9)

If h(t) � K, an up-and-out put value is then

(A10)

The down-and-out put value is

(A11)N(�d � 12c1)� F~aH

F~ b

2�b

e[(b�1)2�(b�1)]c1�rt

� KaHF~ b

�b

eb(b�1)c1�rtN(�d)� F~ aH

F~ b

2�b

e[(b�1)2�(b�1)]c1�rtN(�c�12c1)

� K aH

F~ b

�b

eb(b�1)c1�rtN(�c)

� Ke�rtN(�b) � F~e�rtN(�b � 12c1)

Pdo�put(F, t) � Ke�rtN(�a) � F~e�rtN(�a � 12c1)

� F~aH

F~ b

2�b

e[(b�1)2�(b�1)]c1�rtN(�c � 12c1)

� KaH

F~ b

�b

eb(b�1)c1�rtN(�c)

Puo�put(F, t) � Ke�rtN(�b) � F~e�rtN(�b � 12c1)

� F~aH

F~ b

2�b

e[(b�1)2�(b�1)]c1�rtN(�d � 12c1)

� KaH

F~ b

�b

eb(b�1)c1�rtN(�d)

Puo�put(F, t) � Ke�rtN(�a � 12c1) � F~e�rtN(�a � 12c1)

�

�

P~

put

Pcall(F, t) � Fexp(�kt)ec1�c2�rtN(a � 12c1) � Ke�rtN(a)

P~(x, t) � �

�

��

dx�G(x, t; x�, t � 0)P~(x�, t � 0)

956 Hui and Lo


Similar to knock-in calls, the value of an up-and-in put is given by

(A12)

and the value of a down-and-in put is given by

(A13)

The value of a regular put can be obtained by solving Equation (A7) witha put payoff condition, and is equal to

(A14)

APPENDIX B

As specified in Equation (19), is

(B1)

By putting y � 2kt and squaring Equation (B1), it becomes

(B2)

The derivative of with respect to y is

(B3)

It is noted that

(B4)

By using Taylor expansion for ey, Equation (B4) becomes

(B5) �e�y

y

�e�y

y a1 �y

2!�

y2

3!� # # #b,�y � 0

1y2(1 � e�y) �

e�y

y �1y a1 � y �

y2

2!�

y3

3!� # # # �1b

1y2(1 � e�y) �

e�y

y �1y (ey � 1)

ds~2

dy� �

s2

y2 (1 � e�y) �s2

y e�y

s~ 2

s~ 2 �s2

y (1 � e�y)

s~ �s12kt21 � e�2kt

s~

Pput(F, t) � Ke�rtN(�a) � Fexp(�kt)ec1�c2�rtN(�a � 12c1)

Pdi�put(F, t) � Pput(F, t) � Pdo�put(F, t)

Pui�put(F, t) � Pput(F, t) � Puo�put(F, t)



From Equation (B3), because of the inequality in Equation (B5),

(B6)

This shows that is a decreasing function with y and thus with k.

BIBLIOGRAPHY

Anthony, M., & MacDonald, R. (1998). On the mean reverting properties oftarget zone exchange rates: Some evidence from the ERM. EuropeanEconomic Review, 42, 1493–523.

Anthony, M., & MacDonald, R. (1999). The width of the band and exchangerate mean reversion: Some further ERM-based results. Journal ofInternational Money and Finance, 18, 411–428.

Ball, C., & Roma, A. (1993). A jump diffusion model for the EuropeanMonetary System. Journal of International Money and Finance, 12,475–492.

Ball, C., & Roma, A. (1994). Target zone modelling and estimation forEuropean Monetary System exchange rates. Journal of Empirical Finance,1, 385–420.

Black, F., & Scholes, M. (1973). The pricing of options and corporate liability.Journal of Political Economics, 81, 637–654.

Dornbusch, R. (1976). Expectations and exchange rate dynamics. Journal ofPolitical Economics, 84, 1161–1176.

Ekvall, N., Jennergren, L. P., & Näslund, B. (1997). Currency option pricingwith mean reversion and uncovered interest parity: A revision of theGarman-Kohlhagen model. European Journal of Operational Research,100, 41–59.

Garman, M. B., & Kohlhagen, S. W. (1983). Foreign currency option values.Journal of International Money and Finance, 2, 231–237.

Hui, C. H. (1996). One-touch double barrier binary option values. AppliedFinancial Economics, 6, 343–346.

Hui, C. H. (1997). Time dependent barrier option values. Journal of FuturesMarkets, 17, 667–688.

Kanas, A. (1998). Testing for a unit root in ERM exchange rates in the presenceof structural breaks: Evidence from the boot-strap. Applied EconomicsLetters, 5, 407–410.

Krugman, P. (1991). Target zones and exchange rate dynamics. QuarterlyJournal of Economics, 106, 669–682.

Kunitomo, N., & Ikeda, M. (1992). Pricing options with curved boundaries.Mathematical Finance, 2, 275–298.

Lo, C. F., Lee, H. C., & Hui, C. H. (2003). A simple approach for pricing Black-Scholes barrier options with time-dependent parameters. QuantitativeFinance, 3, 98–107.

s~ 2

ds~ 2

dy� 0

958 Hui and Lo


Lo, C. F., Lee, H. C., & Hui, C. H. (2006). Valuing double barrier options withtime-dependent parameters by the method of multiple images (workingpaper). The Chinese University of Hong Kong.

MacDonald, R. (1988). Floating Exchange Rates (p. 293). London: UnwinHyman.

Merton, R. C. (1973). Theory of rational option pricing. Bell Journal ofEconomics and Management Science, 4, 141–183.

Nieuwland, F. G. M. C., Verschoor, W. F. C., & Wolff, C. C. P. (1994).Stochastic jumps in EMS exchange rates. Journal of International Moneyand Finance, 13, 699–727.

Rapisarda, F. (2003). Pricing barriers on underlyings with time-dependentparameters (working paper). Banca IMI.

Rich, D. R. (1994). The mathematical foundations of barrier option-pricingtheory. Advances in Futures and Options Research, 7, 267–311.

Robert, R. O., & Shortland, C. F. (1997). Pricing barrier options with time-dependent coefficients. Mathematical Finance, 7, 83–93.

Rose, A. K., & Svensson, L. E. O. (1994). European exchange rate credibilitybefore the fall. European Economics Review, 38, 1185–1216.

Rubinstein, M., & Reiner, E. (1991). Breaking down the barriers. RiskMagazine, 8, 28–35.

Sørensen, C. (1997). An equilibrium approach to pricing foreign currencyoptions. European Financial Management, 3, 63–84.

Svensson, L. E. O. (1992). An interpretation of recent research on exchangerate target zones. Journal of Economic Perspectives, 6, 119–144.

Svensson, L. E. O. (1993). Assessing target zone credibility. EuropeanEconomic Review, 37, 763–802.

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Currency barrier option pricing with mean reversion