Bounds and prices of currency cross-rate options
San-Lin Chung, Yaw-Huei Wang
*Department of Finance, National Taiwan University, No. 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan, ROC Received 27 February 2006; accepted 30 January 2007
Available online 14 September 2007
Abstract
This paper derives the pricing bounds of a currency cross-rate option using the option prices of two related dollar rates via a copula theory and presents the analytical properties of the bounds under the Gaussian framework. Our option pricing bounds are useful, because (1) they are general in the sense that they do not rely on the distribution assumptions of the state variables or on the selection of the copula function; (2) they are portfolios of the dollar-rate options and hence are potential hedging instruments for cross-rate options; and (3) they can be applied to generate bounds on deltas. The empirical tests suggest that there are persistent and stable rela-tionships between the market prices and the estimated bounds of the cross-rate options and that our option pricing bounds (obtained from the market prices of options on two dollar rates) and the historical correlation of two dollar rates are highly informative for explain-ing the prices of the cross-rate options. Moreover, the empirical results are consistent with the predictions of the analytical properties under the Gaussian framework and are robust in various aspects.
Ó 2007 Elsevier B.V. All rights reserved. JEL classification: F3; F4; G1
Keywords: Option pricing; Option bounds; Exchange rates; Cross-rate; Correlation; Copulas
1. Introduction
In the option pricing literature, researchers are not only interested in pricing, but also in bounding the option val-ues. There are many useful techniques that can be employed to derive option pricing bounds. For example,
Merton (1973), Garman (1976), Levy (1985), and Grundy
(1991) use the arbitrage-free approach to derive option
pricing bounds. Ritchken (1985), Ritchken and Kuo (1989), Basso and Pianco (1997), Mathur and Ritchken
(2000), and Ryan (2003)use linear programming methods
to derive option pricing bounds. In addition to the above two types of techniques, some other approaches, such as optimization methods and restrictions on the volatility of the pricing kernel, have also been used in the literature.
Most, if not all, of the previous studies derive option pricing bounds by directly using the price information (such as the price distribution or price process) of the underlying asset. In contrast to the previous literature, this study uses the option prices of the related dollar rates to derive the pricing bounds for the cross-rate option. In other words, we bound cross-rate option values using the market prices of the dollar-rate options.1In this sense, the idea of this paper is close to that in the static hedge literature (see
Carr et al., 1998), whereby the exotic options are priced
(and hedged) in terms of the prices of standard options. Since there is a triangular relationship between the for-eign exchange rates among three currencies, Taylor and
0378-4266/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2007.01.027
*
Corresponding author. Tel.: +886 2 83695581.
E-mail address:[email protected](Y.-H. Wang).
1 The motivation for doing this is as follows. It is generally observed that
options on dollar-denominated exchange rates are traded under satisfac-tory liquidity, while cross-rate option markets are much less liquid. Thus, the pricing bounds obtained from the liquid market prices of dollar-rate options are useful for pricing, hedging, and arbitraging.
www.elsevier.com/locate/jbf Journal of Banking & Finance 32 (2008) 631–642
Wang (2005)show that it is plausible to estimate risk-neu-tral densities (RNDs) and option prices of a cross-rate under the US dollar measure2 using the market prices of two related dollar-rate options. Instead of directly explor-ing the option pricexplor-ing formula, this study (also usexplor-ing the dollar risk-neutral measure) derives the pricing bounds for cross-rate options by utilizing the exchange option pric-ing bounds implied in the copula theory.3Some analytical properties of the bounds under the Gaussian framework are also presented in this paper. Compared with the previ-ous studies, our objective is not to derive tight bounds, but rather to generate informative bounds from useful market price information (dollar-rate option prices).
Our option pricing bounds contribute to the literature in at least three aspects. First of all, although our pricing bounds are not tight, they are general in the sense that they do not rely on the distribution assumptions of the state variables or on the selection of a copula function. Sec-ondly, our pricing bounds have economic meanings, because they are portfolios composed of the dollar-rate options (and sometimes also composed of spot dollar rates) and hence provide potential hedging instruments for cross-rate options. Finally, our pricing bounds are also useful for generating bounds on deltas.
The empirical tests of our pricing bounds are conducted using the prices of options on foreign exchange rates among the US dollar, euro, and pound sterling. We first show that there are strong and stable relationships between the market prices of cross-rate options and the pricing bounds obtained from the market prices of options on the two dollar rates. Both of the above finding and the ana-lytical analysis in Section2 motivate us to run the regres-sion models to measure the extent where the cross-rate option prices can be explained by our pricing bounds and the correlation between the two dollar rates.
Our empirical results indicate that the pricing bounds estimated from option prices of two dollar rates and the correlation of two dollar rates can provide highly signifi-cant information (about 85%) for explaining the cross-rate option prices across deltas. Our results are immune to the assumption of the RND distribution for the dollar rates, the market volatility level, and the change in the curvature of the implied volatility function. Finally, we demonstrate how to calculate bounds on deltas using our pricing bounds.
The remainder of this paper is organized as follows. Sec-tion 2 derives option pricing bounds for the cross-rate option, presents their analytical properties under the Gaussian framework, and shows how to bound the delta of the cross-rate option using the derived pricing bounds. Data and the empirical methodologies for generating the risk-neutral densities and option pricing bounds are pre-sented in Section 3. Section 4 discusses the empirical results, while Section5 concludes the paper.
2. Bounds of the price and delta of the cross-rate option 2.1. Bounds of the price of the cross-rate option
By applying the Fre´chet bounds in the copula theory,
Cherubini et al. (2004) show that the super-replication
bounds of the option to exchange one asset for the other asset are composed of the prices of the univariate options on the two individual exchanged assets.4We first show that the payoff of a cross-rate option under the dollar measure is equivalent to that of an exchange option where the two underlying risky assets are the corresponding dollar rates. Following the same logic as inCherubini et al. (2004), we use the risk-neutral pricing approach to derive the pricing bounds for the cross-rate option.
Consider options whose payoffs depend on the exchange rates among the following three currencies: US dollars ($, USD), British pounds (£, GBP), and euros (€, EUR). We denote the dollar price of one pound at time t by S$=£t and likewise the dollar price of one euro at the same time is denoted by S$=€t . The cross-rate price of one pound in euros is then given by S€=£
t ¼ S $=£ t =S
$=€
t under the
no-arbi-trage argument.
Now consider a European call option where the holder has the right to buy £1 for €K at time T. Under the dollar measure (or from the viewpoint of US residents), the above option is identical to an option to exchange KS$=€T dollars for S$=£T dollars at time T. Hence, a cross-rate call option under the dollar measure is equivalent to an option to exchange one asset for the other asset and its dollar payoff equals max S $=£T KS$=€T ;0. This payoff can be rear-ranged as follows: S$=£T max min S$=£T ; KS $=€ T ;0 h i : ð1Þ
Hence, the current dollar price of an exchange option is determined by the risk-neutral pricing approach as follows: Call€=£$ ¼ S$=£
t e
r£ðT tÞ Call
minðS$=£; KS$=€;0; t; TÞ; ð2Þ
where Callmin(S1, S2, 0, t, T) represents the price at time t of
a call option on the minimum of S1and S2with strike price
0 and maturity time T. Applying the Fre´chet bounds in the 2
BothTaylor and Wang (2005)and this paper commence the analyses
under the foreign (dollar) risk-neutral measure to price the cross-rate options. As suggested byBakshi et al. (forthcoming), it may be better to specify the generic pricing kernels in each country to derive the prices of derivatives. Nonetheless,Taylor and Wang (2005)analytically show that the prices of derivatives under different measures are equivalent when the law of one price holds. In order to utilize the concept of the exchange option, this article formulates the pricing problem under the dollar measure.
3 The details of the copula theory can be found inJoe (1997) and Nelsen
(1999).Cherubini et al. (2004)first apply the copula theory to derive the
pricing bounds for the exchange options.
4 However,Cherubini et al. (2004)only derive the lower bound for one
particular probability condition. There should be an alternative formula applied to the other probability condition. See our Eq.(4)for these two probability conditions.
copula theory, we are able to derive the upper (lower) bound of the minimum call option price and thus the lower (upper) bound of the cross-rate option price as follows. Proposition 1. The upper bound of the cross-rate option price in dollars is
Call€=£$ þ¼ CallðS$=£; K; t; TÞ þ KPutðS$=€; K00; t; TÞ; ð3Þ
where K** is a constant satisfying that F
S$=£ðKÞþ FKS$=€ðKÞ ¼ 1; FiðxÞ ¼ 1 FiðxÞ; FiðxÞ is the cumulative
distribution function, and K00= K**/K. Let K*be a constant
which solves FS$=£ðKÞ ¼ FKS$=€ðKÞ. The lower bound of the cross-rate option price in dollars is thus
Call€=£$ ¼
CallðS$=£; K; t; TÞ KCallðS$=€; K0; t; TÞ
if FS$=£ðuÞ < FKS$=€ðuÞ for u < K; S$=£t er£ðT tÞ KS$=€ t er€ðT tÞ þKCallðS$=€; K0; t; TÞ CallðS$=£; K; t; TÞ otherwise; 8 > > > > > > < > > > > > > : ð4Þ where K0= K*/K.
Proof. Please seeAppendix A. h
From Eqs. (3) and (4), we observe that our pricing bounds for cross-rate options are portfolios of the corre-sponding dollar-rate options (and may also be of the spot assets). Therefore, different from most option pricing bounds in the literature, the derived pricing bounds have economic meanings. Moreover, the derivation of our cross-rate option pricing bounds does not rely on the distri-bution assumption of two dollar rates and the selection of an appropriate copula function. Therefore, one can apply the technique utilized here to derive the price bounds for any European-style derivatives whose payoffs can be rear-ranged as the same type as that of an exchange option.
Since the cross-rate is completely determined by the other two dollar rates under the triangular arbitrage rela-tion, a natural question to ask is how the payoff of the cross-rate option is related to the payoffs of the other two dollar-rate options. If this relationship can be speci-fied, one is able to apply the spanning approach ofBakshi
and Madan (2000)to price (or to provide pricing bounds
for) the cross-rate options using the prices of two dollar-rate options. InCorollary 1 we show that the correlation options considered in Bakshi and Madan (2000) provide a lower bound for the cross-rate option price. The proof
ofCorollary 1is available from the authors upon request.
Corollary 1. The price of a cross-rate call option with strike price K is bounded below by the price of a correlation option with the following payoff:
max S$=£T Kx;0 max 1=S$=€T 1=Ky;0 ; where K = Kx/Ky.
Note thatCorollary 1relates the lower pricing bound of a cross-rate call option to the price of a dollar-rate call (with a payoff of maxðS$=£T Kx;0Þ) and the price of a
dol-lar-rate put (with a payoff of maxðKy S $=€
T ;0Þ). Therefore,
Corollary 1suggests that the dollar-rate option prices (and
hence our pricing bounds) may be informative for explain-ing the cross-rate option prices. Later in Section3 we will run a regression model to investigate the explanatory power of our pricing bounds.
2.2. Analytical properties of the pricing bounds under the Gaussian framework
When the two dollar rates follow a bivariate lognormal distribution, then under the triangular arbitrage relation the cross-rate also follows a lognormal distribution. Thus, there exist closed-form solutions for the option prices of two dollar rates and our pricing bounds. To have some insights on our pricing bounds, we investigate the analyti-cal properties of these bounds when the two dollar rates follow a bivariate lognormal distribution.
Denote the closed-form solutions of two dollar-rate option prices at time t, denominated in US dollars, as CBS S$=£t ; K; r$; r£;r$=£;s
and CBS S$=€t ; K; r$; r€;r$=€;s
, respectively.Proposition 2shows that the upper and lower bounds have closed-form solutions under the bivariate log-normal distribution assumption.
Proposition 2. Assume that two dollar rates S$/£ and S$/€ follow a bivariate lognormal distribution with a correlation coefficient of q and volatilities per year of r$/£ and r$/€,
respectively. The upper and lower bounds, denominated in euros, of the cross-rate call option with a strike price of K thus have closed-form solutions of CBSðS€=£t ; K; r€; r£;r$=£þ
r$=€;sÞ and CBSðS€=£t ; K; r€; r£;jr$=£ r$=€j; sÞ, respectively.
Proof. Please see Appendix B. h
Under the bivariate lognormal distribution assumption, the triangular arbitrage relation implies that the cross-rate option price, denominated in euros, is CBS S€=£t ; K; r€; r£;
r€=£;sÞ, where r2€=£¼ r 2 $=£þ r 2 $=€ 2qr$=£r$=€. Therefore,
Proposition 2 is intuitively true, because jr$/£ r$/€j 6
r€/£6r$/£+ r$/€. When the correlation between two dollar
rates is higher (lower), the cross-rate option price is closer to the lower (upper) bound. Moreover, Proposition 2
implies that when the implied volatility curves of two dollar-rate option prices are flat, then the implied volatility curves of our upper and lower bounds for cross-rate options are also flat.
2.3. Bounds on the delta of the cross-rate option
Given the estimated pricing bounds in terms of implied volatilities, it is plausible to derive bounds on the cross-rate option’s delta using Proposition 5 ofBergman et al. (1996). Assume that the volatility function, r(s, t), is a function of the underlying asset price s and time T only. Let r and r,
respectively, denote the lower and upper bounds on volatil-ity, c(s, t) and c1(s, t), respectively, represent the market (or
accurate) call price and its delta, and cbs(r) and cbsðrÞ1 , respectively, stand for the Black–Scholes call price and its delta.Bergman et al. (1996)derive bounds on the option’s delta as follows.
Proposition 5 of Bergman et al. (1996). If for all s and t;rðtÞ 6 rðs; tÞ 6 rðtÞ, then cbsðrÞ1 ðs00; tÞ 6 c 1ðs; tÞ 6 c bsðrÞ 1 ðs0; tÞ, where s00 solves cbsðrÞðs; tÞ ¼ cbsðrÞðs00; tÞ þ cbsðrÞ 1 ðs00; tÞðs s00Þ and s0 solves cbsðrÞðs; tÞ ¼ cbsðrÞðs0; tÞ cbsð1 rÞðs0; tÞðs0 sÞ. The delta bounds of Bergman et al. (1996)are true for general Markovian diffusion processes. When the cross-rate option’s value today is known,Bergman et al. (1996)
show that the bounds on its delta can be strengthened as follows.
Proposition 6 of Bergman et al. (1996). If for all s and t; rðs; tÞ 6 rðtÞ, then for any s and t such that one knows cðs; tÞ; cbsð1 rÞðs00; tÞ 6 c 1ðs; tÞ 6 cbsð1 rÞðs0; tÞ, where s00 solves cðs; tÞ ¼ cbsðrÞðs00; tÞ þ cbsðrÞ 1 ðs00; tÞðs s00Þ and s0 solves cðs; tÞ ¼ cbsðrÞðs0; tÞ cbsðrÞ 1 ðs0; tÞðs0 sÞ.
Our pricing bounds are directly applicable to Proposi-tion 5 of Bergman et al. (1996), and thus the bounds on the deltas can be obtained straightforward. For instance, the implied volatility of our lower (upper) bound provides an estimate of r(t) ðrðtÞÞ for applying Proposition 5 of
Bergman et al. (1996). When the market price of the
cross-rate option is known, our upper bound can be used in conjunction with Proposition 6 of Bergman et al.
(1996) to obtain tighter bounds for deltas. Later, we will
calculate bounds on the deltas when our pricing bounds are applied to Propositions 5 and 6 of Bergman et al. (1996).
3. Data and empirical methodologies 3.1. Data
The primary data used in this article are daily option prices that are quoted as Black–Scholes implied volatilities for three currency options ($/£, $/€, and €/£). We make use of a confidential file of OTC option price mid-quotes, sup-plied by the trading desk of an investment bank in Lon-don.5Our currency option data cover the period from 15 March 1999 to 11 January 2001. The OTC quotes are for all three foreign exchange options, recorded at the end of the day in London. The data include option prices for seven exercise prices, based upon ‘‘deltas’’ equal to 0.1, 0.25, 0.37, 0.5, 0.63, 0.75, and 0.9. The time to maturity of the options
is one month, with which options in the OTC market are most frequently traded. We also use the spot exchange rates of $/£, $/€, and €/£ and the euro-currency interest rates (proxies of risk-free rates) of $, £, and € recorded by Data-Stream as the inputs for all relevant calculations.
The summary statistics of the quoted implied volatilities show that all implied volatility functions exhibit a smile shape with the level for the $/€ options being the highest, while the level for the $/£ options are the lowest. The low standard deviations of the quotes imply that the levels of implied volatilities for these three exchange rate options do not change much as time goes. The skewness is positive and the kurtosis is close to 3, which does not depend on the moneyness of the options.
3.2. Empirical methodologies for generating the bounds Because K*and K** are determined by the risk-neutral
densities of two dollar rates, we use the observed market prices of European call options on $/£ and $/€ and a parametric distribution specification to estimate their risk-neutral densities. Once the risk-risk-neutral densities are obtained, K* and K** can be calculated with a numerical
method (such as the Newton–Raphson method) to solve FS$=£ðKÞ ¼ FKS$=€ðkÞ and FS$=£ðKÞ þ FKS$=€ðKÞ ¼ 1, respectively. We are then able to price dollar-rate options with all strikes and get pricing bounds of the cross-rate options using Eqs.(3) and (4).
In this paper we use the generalized beta density of the second kind (GB2) to estimate the RNDs of two dollar rates.6 The GB2 density has few parameters, but it pre-serves many desirable properties: general levels of skewness and kurtosis are allowed, the shapes of the tails are fat rel-ative to the lognormal density, and there are analytic for-mulae for the density, its moments, and the prices of options. Furthermore, the parameter estimation of the GB2 density is easy and the estimated densities are never negative. The details about the estimation of the GB2 den-sity can be found inBookstaber and McDonald (1987). 4. Empirical results
The empirical studies in this article contain four parts. We first analyze the properties of our pricing bounds and their relationships with the market prices of the cross-rate options. Second, we investigate the explanatory powers of the pricing bounds and the correlation between two dollar
5 Some settlement prices are available for cross-rate options traded in
the Chicago Mercantile Exchange, but they correspond to almost no trading volume. Consequently, we rely on over-the-counter (OTC) option prices, with which we have the same time-to-maturity option data every day. To the best of our knowledge, such prices are not available in the public domain.
6
Many types of univariate RNDs have been proposed, including lognormal mixtures (Ritchey, 1990; Melick and Thomas, 1997), general-ized beta densities (Bookstaber and McDonald, 1987), multi-parameter discrete distributions (Jackwerth and Rubinstein, 1996), and densities derived from fitting spline functions to implied volatilities (Bliss and
Panigirtzoglou, 2002). Providing that options are traded for a range of
exercise prices that encompass most areas of the risk-neutral distribution, it is documented that several flexible density families provide similar empirical estimates. The bounds estimated with the lognormal mixtures RNDs are compared with for the robustness check.
rates for the market prices of the cross-rate options. Third, some robust analyses for the accuracy of our results are pro-vided. Finally, given the estimated price bounds of the cross-rate options, we demonstrate how to bound their del-tas using the approach proposed byBergman et al. (1996). 4.1. Empirical pricing bounds of the cross-rate options
In order to have a standardized comparison, all the mar-ket prices and pricing bounds are converted into the Black– Scholes implied volatilities. The pricing bounds of the one-month cross-rate options with seven different strike prices (deltas) are estimated every day. All lower bounds are determined by the second alternative of Eq. (4), because the implied volatilities of $/€ are always larger than those of $/£ during our sample period.7The descriptive statistics of the estimated pricing bounds and the market implied volatilities across deltas are shown in Table 1. We also show the evolution of the estimated pricing bounds and the market implied volatilities in Fig. 1. As the patterns across deltas are very similar,Fig. 1presents the result with a delta of 0.5 only.
As shown inFig. 1, the market implied volatility always lies within the estimated bounds, and the evolution of the market implied volatility of the cross-rate (€/£) option exhibits a similar pattern to those of the estimated bounds. As the foreign exchange market became more volatile from 1999 to 2000, the bound range, defined as the difference between the upper bound and the lower bound, turned wider as time went by during the period.
Table 1suggests that the level, the mean, and the
vola-tility of the upper bounds are almost the same across deltas with an extremely shallow smile. In contrast, the lower bound and the market implied volatilities exhibit clearer smile shapes across deltas with the lower bound smile being deeper than the market implied smile.
To explore the relationships between the option market prices and the estimated bounds, we further look at the
behavior of the difference between the upper bound and the market implied (upper range) and the difference between the lower bound and the market implied (lower range). Their descriptive statistics across deltas are illus-trated inTable 2. Both the level and variation of the upper ranges are larger than those of the lower ranges across del-tas. We also investigate the relationships between the ranges and the correlation of two dollar rates as
Proposi-tion 2suggests that the higher the correlation is, the closer
the market implied volatility will be to the lower bound. We regress the upper range and the lower range, respec-tively, on the correlation and report the slope coefficient estimates in Table 2.8The results clearly indicate that the
Delta: 0.5 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 03/15/99 05/15/99 07/15/99 09/15/99 11/15/99 01/15/00 03/15/00 05/15/00 07/15/00 09/15/00 11/15/00
Upper Market Lower
Fig. 1. The implied volatilities from market prices and the estimated bounds for cross-rate options. This figure shows the evolution of the Black–Scholes implied volatilities from the market prices and the estimated bounds of the cross-rate (€/£) options with the delta of 0.5. The cross-rate option pricing bounds are estimated by calibrating Eqs.(3)
and (4)using the option prices of two dollar rates, $/£ and $/€.
Table 1
Summary statistics of the implied volatilities from the market prices and the estimated bounds
Delta 0.9 0.75 0.63 0.5 0.37 0.25 0.1
Panel 1: Upper bounds
Mean 0.2087 0.2061 0.2056 0.2058 0.2064 0.2077 0.2118
Standard deviation 0.0325 0.0329 0.0330 0.0331 0.0332 0.0333 0.0334
Panel 2: Market implieds
Mean 0.1054 0.0996 0.0980 0.0972 0.0979 0.0995 0.1050
Standard deviation 0.0169 0.0173 0.0176 0.0176 0.0180 0.0180 0.0183
Panel 3: Lower bounds
Mean 0.0405 0.0319 0.0302 0.0315 0.0352 0.0396 0.0478
Standard deviation 0.0095 0.0117 0.0124 0.0122 0.0113 0.0108 0.0110
This table consists of the summary statistics of the implied volatilities from the market prices and estimated upper and lower bounds of the cross-rate €/£ options across deltas. The option bounds are estimated by calibrating Eqs.(3) and (4)with the option prices of two dollar rates, $/£ and $/€.
7 The analytical properties of our lower bound under the Gaussian
framework suggest that if the volatility of $/€ is greater than that of $/£, then FS$=£ðuÞ > FKS$=€ðuÞ for u < K*and vice versa.
8
The correlation coefficients are estimated using the dynamic condi-tional correlation (DCC) multivariate GARCH model proposed byEngle
(2002)with the historical time series data of two dollar spot rates. The fact
that correlations between financial assets are usually time-varying has important implications in many ways such as portfolio hedging and multivariate asset pricing. This model overcomes the complexity of conventional multivariate GARCH models in computation by directly modeling the time-varying correlation as a conditional process. The procedure of using the DCC GARCH model to generate the time-varying correlation series is detailed inEngle (2002).
upper (lower) range is significantly positively (negatively) associated with the correlation of two dollar rates across deltas; i.e. the higher the correlation is, the closer the mar-ket implied will be to the lower bound. This finding is con-sistent with the analytical properties of our pricing bounds. In summary, the lower bounds exhibit a smile shape while the upper bounds and the market implied volatilities are relatively flat across deltas. Both the upper and lower bounds exhibit tractable and persistent relationships with the market prices of cross-rate options.
4.2. Pricing bounds, correlation, and the cross-rate option prices
Since it has been found both analytically and empirically that there are persistent relationships between the market prices of the cross-rate options and our pricing bounds, we further use a regression model to measure the extent where the cross-rate option prices can be explained by our pricing bounds. We regress the market implied volatil-ities on the upper and lower bounds.9The regression model is specified as
Model 1 : MIVt¼ c þ b1UBtþ b2LBtþ et; ð5Þ
where MIVt, UBt, and LBt, respectively, denote the market
implied volatility of the one-month cross-rate option on €/£, the upper bound, and the lower bound at day t, and et is the residual term.10 The estimates for this model are
shown in Panel 1 ofTable 3.
From Panel 1 of Table 3, we find highly significant regression coefficients of b1and b2. The adjusted R2s are
very high and range from 0.72 to 0.77 across deltas. It is noticeable that b1adheres to a small range (between 0.33
and 0.39) while b2ranges from 0.26 to 0.59. In other words,
the upper bound contains almost the same level of informa-tion content for the cross-rate opinforma-tions across delta, while the lower bound contains different levels of information content across deltas. In short, we confirm that there are strong and stable relationships between the market prices of cross-rate options and the pricing bounds estimated from the market prices of the options on two dollar rates. From the analytical properties inProposition 2, we find that no correlation information is used in the calculation of the pricing bounds of the cross-rate options, for which we only utilize the price information of the options on two dollar rates individually. However, Driessen et al. (2006)
analyze the relationship between the prices of stock index options and the prices of individual stock options included in the index, and they find the relevance of correlation risk and the associated premium for stock index options pric-ing. Inspired by their results, this paper includes an extra explanatory variable, the historical correlation of two dol-lar rates, into Model 1 to see whether the correlation is able to provide additional explanatory power. Thus, the regres-sion model is modified as
Table 2
Summary statistics of the estimated upper ranges and lower ranges
Delta 0.9 0.75 0.63 0.5 0.37 0.25 0.1
Panel 1: Upper ranges
Mean 0.1033 0.1065 0.1076 0.1086 0.1085 0.1082 0.1069
Standard deviation 0.0208 0.0208 0.0206 0.0207 0.0205 0.0206 0.0206
b 0.1431 0.1478 0.1468 0.1454 0.1435 0.1405 0.1320
(8.02) (8.35) (8.37) (8.24) (8.18) (7.96) (7.44)
Panel 2: Lower ranges
Mean 0.0648 0.0678 0.0678 0.0656 0.0627 0.0599 0.0571
Standard deviation 0.0125 0.0149 0.0144 0.0131 0.0124 0.0120 0.0116
b 0.0855 0.1121 0.1049 0.0958 0.0905 0.0869 0.0818
(7.98) (8.93) (8.59) (8.65) (8.62) (8.50) (8.24)
This table consists of the summary statistics of the estimated upper ranges and lower ranges of the cross-rate (€/£) options across deltas. The upper ranges and lower ranges are the distances between the upper bounds and market implieds and between the lower bounds and market implieds, respectively. In addition, the parameter estimates of the following regression model are provided.
Rt¼ a þ bCorrtþ et;
where Rtis the upper or lower range, Corrtdenotes the correlation between the two dollar rates, and etis the residual term at time t. The correlations are
generated by the DCC model ofEngle (2002). The numbers in the parentheses are t-statistics.
9
The information content of our pricing bounds for cross-rate options is similar to that of the prices of options on the corresponding two dollar rates. Therefore, the implied volatilities of the two dollar rates have the potential to provide similar explanatory power for the cross-rate implied volatility as our bounds do. However, according to our analysis, the implied volatilities of $/£ and $/€ are highly correlated (about 0.8). Thus, a serious multicollinearity problem occurs when directly regressing the implied volatility of €/£ on those of $/£ and $/€ although its adjusted R2is
just slightly lower than Model 1. As our pricing bounds for cross-rate options are linear combinations of the prices of two dollar-rate options with particular strike prices, our bounds provide a solution to the multicollinearity problem by transforming two highly correlated implied volatilities to two less associated bounds. As a result, using the bounds instead of the dollar-rate implied volatilities enables our analysis to be more valid and reliable.
10As the bounds are estimated every day from the one-month options,
Model 2 : MIVt¼ c þ b1UBtþ b2LBtþ b3Corrtþ et;
ð6Þ where Corrtis the DCC correlation coefficients of two
dol-lar rates at day t. When the correlation of two doldol-lar rates increases, the variance of the cross-rate decreases and thus the cross-rate option price also decreases. Therefore, the regression coefficient of the historical correlation (b3) is
ex-pected to be negative.
The regression results for Model 2 are shown in Panel 2
of Table 3. It is clearly seen that the correlations of two
dollar rates provide incremental information in explaining cross-rate option prices as all adjusted R2s increase by about 10% in comparison to Model 1. The regression coef-ficients for the correlation across deltas are significantly negative and consistent with our expectation. Furthermore, our results are in line with the analyses and findings of Dri-essen et al. (2006).
As volatility is usually highly persistent, it is expected that including the one-period lagged volatility as an inde-pendent variable will increase the goodness-of-fit. How-ever, in this study what we intend to investigate is how well the cross-rate option price can be explained by the dol-lar-rate price information only (i.e. without previous cross-rate option price information), rather than how well the model can be specified. Therefore, we only use the esti-mated upper and lower bounds as the explanatory vari-ables in this study. For comparison, we also include the one-period lagged cross-rate implied volatility as an addi-tional explanatory variable in Model 2. The unreported results show that even with the additional explanatory var-iable which is highly correlated with the dependent vari-able, the coefficients of the upper and lower bounds and
the correlation are still significant at the 1% level and their signs are still consistent with the theoretical expectations.11 Due to the significant in-sample explanatory power of the estimated bounds and correlation to the market prices of cross-rate options, we are interested in the performance of our empirical models in the out-of-sample prediction. Given the estimated parameters of the previous model (Model 2), we predict the current implied volatility for the cross-rate (€/£) options from the current market prices of the dollar-rate options and the historical correlation. The actual and predicted implied volatilities of the cross-rate options for the delta of 0.5 are shown in Fig. 2. The results for other moneyness are very similar and thus omitted.
The prediction errors, defined as the absolute values of the actual values minus the predicted values of implied vol-atilities, from Model 2 are generally small. The average errors across deltas are about 0.3%, which is smaller than the bid-ask spread in the OTC market. In addition, the vol-atilities of the prediction errors are very small as well (about 0.3%), implying that the model performs consis-tently well across time.
In summary, the pricing bounds estimated from option prices of two dollar rates and the correlation of two dollar rates can provide highly significant information for explaining the cross-rate option prices across deltas. Our results are valuable since our pricing bounds (which are portfolios of dollar rate options) are applicable to practical usage not only for price explanation, but also for hedging, Table 3
Explanatory power of estimated bounds and correlation to market implied volatility
Delta 0.9 0.75 0.63 0.5 0.37 0.25 0.1 Panel 1: Model 1 b1 0.3388 0.3948 0.3871 0.3594 0.3409 0.3245 0.3088 (22.79) (27.01) (26.93) (24.60) (22.17) (19.22) (15.64) b2 0.5279 0.2611 0.3229 0.4180 0.5248 0.5730 0.5886 (10.40) (6.34) (8.41) (10.50) (11.61) (11.06) (9.82) Adjusted R2 0.7421 0.7162 0.7400 0.7576 0.7706 0.7636 0.7543 Panel 2: Model 2 b1 0.3720 0.4079 0.4033 0.3815 0.3700 0.3618 0.3598 (32.43) (38.10) (38.30) (35.19) (31.53) (26.93) (21.72) b2 0.3872 0.2506 0.2967 0.3629 0.4354 0.64470 0.4133 (9.82) (8.32) (10.57) (12.30) (12.62) (10.81) (8.17) b3 0.1127 0.1256 0.1225 0.1174 0.1131 0.1100 0.1055 (18.56) (20.38) (20.43) (19.97) (18.88) (17.23) (15.17) Adjusted R2 0.8502 0.8483 0.8613 0.8680 0.8687 0.8542 0.8341
This table consists of the regression results of the following two models: Model 1: MIVt= c + b1UBt+ b2LBt+ et.
Model 2: MIVt= c + b1UBt+ b2LBt+ b3Corrt+ et.
Here, MIVt, UBt, LBt, and Corrtdenote, respectively, the market implied volatility of an option on €/£, the upper bound, the lower bound, and the
historical DCC correlation between S/€ and $/£ at day t, and etis the residual term. The numbers in the parentheses are t-statistics.
11As expected, the result has an almost perfect goodness-of-fit
(R2
’ 0.96) owing to correcting the considerable first-order serial corre-lation in the residuals of Model 2.
particularly when the real-time cross-rate option prices are unobservable.
4.3. Robustness analysis
To investigate whether our results are robust, we first check whether the estimated bounds rely on the assump-tion of the RND for the dollar rates, and then we analyze whether the out-of-sample prediction errors from Model 2 are sensitive to sample selection, the implied volatility level, and changes in the curvature of the implied volatility function.
To check whether the estimated bounds depend on the distribution assumption of the dollar rate, we assume that the RNDs of two dollar rates follow the lognormal mix-tures distribution and then compare the bounds estimated under this assumption with those under the GB2 distribu-tion assumpdistribu-tion. As shown in Table 4, the differences between the bounds estimated using these two different RND assumptions are statistically insignificant across del-tas at the 10% significance level although the differences of
the lower bounds are larger than those of the upper bounds.
To check whether sample selection affects our findings, we re-do the out-of-sample prediction of Model 2 for two evenly divided sub-samples. Although the prediction errors are slightly higher in the second sub-periods (0.31% vs. 0.35% on average), the patterns across deltas are basically the same. In other words, our finding does not depend on the sample selection.
As the volatility of exchange rates increases over our sample period, it is natural to check whether the increasing volatility affects the accuracy of information provided by our pricing bounds. Moreover, although the average implied volatilities of all exchange rates exhibit a smile shape, the slopes of the implied volatility curves vary from being negatively sloped to positively sloped during our sample period. This implies that risk-neutral skewness and kurtosis change substantially every day. Therefore, to further check whether the out-of-sample prediction error of Model 2 depends on the volatility level or the change in the curvature of the implied volatility function, we first cal-culate the implied volatility, skewness, and kurtosis using Theorem 1 ofBakshi et al. (2003)and then run the follow-ing regression model:
Et¼ c þ aEt1þ bXt; ð7Þ
where Et denotes the percentage prediction error and Xt
represents the implied volatility, skewness, or kurtosis level estimated using the approach ofBakshi et al. (2003)at time t. The AR(1) specification is motivated by the high first-or-der autocorrelation of prediction errors. The estimates are reported inTable 5. All b coefficients are insignificant un-der the 10% significance level.
Fig. 3indicates that the RNDs of the cross-rates are
fat-tailed (average kurtosis equals 3.31) and slightly negatively skewed (average skewness equals0.13).Fig. 3also shows that the implied skewness changes noticeably over time. Nevertheless, Panel 2 of Table 5 suggests that there is no clear evidence supporting that the prediction errors across deltas are affected even though implied skewness changes much. Similarly, Panel 3 ofTable 5shows that the implied Table 4
Robustness analysis for option bounds
Delta 0.9 0.75 0.63 0.5 0.37 0.25 0.1
Panel 1: Upper bounds
GB2 0.2087 0.2061 0.2056 0.2058 0.2064 0.2077 0.2118
Mixtures 0.2086 0.2060 0.2056 0.2058 0.2065 0.2078 0.2119
p-Value 0.9741 0.9811 0.9925 0.9996 0.9936 0.9883 0.9763
Panel 2: Lower bounds
GB2 0.0405 0.0319 0.0302 0.0315 0.0352 0.0396 0.0478
Mixtures 0.0412 0.0325 0.0309 0.0316 0.0353 0.0399 0.0476
p-Value 0.3566 0.4237 0.3274 0.9205 0.8876 0.5895 0.7957
This table consists of the means of the upper and lower bounds of the cross-rate (€/£) options across deltas, which are estimated using two different RND assumptions, GB2 and lognormal mixtures, for the dollar rates. In addition, the p-values of the mean equality tests for the two assumed distributions are provided. Delta: 0.5 0.00 0.05 0.10 0.15 0.20 04/14/99 06/14/99 08/14/99 10/14/99 12/14/99 02/14/00 04/14/00 06/14/00 /14/0008 10/14/00 12/14/00 Actual Predicted
Fig. 2. Actual and predicted implied volatilities for cross-rate options. This figure consists of the evolution of the actual and predicted Black– Scholes implied volatilities of the cross-rate (€/£) options with the delta of 0.5. The actual implied volatilities are backed out from the market prices of options. The predicted implied volatilities are obtained from Model 2 in Section4using the market prices of options on two dollar rates, $/£ and $/ €, and the historical DCC correlation of two dollar rates.
kurtosis has little impact on the prediction errors across deltas.
In summary, the accuracy of information provided by our pricing bounds is immune to the market volatility level and the change in the curvature of the implied volatility function. Other proxies of the market volatility level, such as the implied volatilities for different moneyness levels, are
also used and the results (not reported here) are almost unchanged.
4.4. Bounds on delta of cross-rate options
Given the estimated pricing bounds in terms of implied volatility, it is plausible to bound the cross-rate option’s delta using our pricing bounds with Propositions 5 (or Proposition 6) ofBergman et al. (1996)when the call price of the cross-rate is unknown (or known).
We take the at-the-money (ATM) cross-rate call option traded on June 29, 1999 as an example and depict its delta bounds inFig. 4. The solid lines in the descending order are the Black–Scholes prices computed as a function of the underlying asset price using volatilities of the upper bound, the market implied volatility, and the lower bound, respec-tively. When the cross-rate call price is unknown, its delta is bounded between 0.0261 and 0.9704 (dashed lines). Table 5
Robustness analysis for out-of-sample prediction errors
Delta 0.9 0.75 0.63 0.5 0.37 0.25 0.1
Panel 1: Regression of errors on implied volatility
b 0.0133 0.0651 0.1138 0.1174 0.0495 0.0128 0.0990
(0.10) (0.55) (1.00) (1.02) (0.46) (0.12) (0.91)
Panel 2: Regression of errors on implied skewness
b 0.0099 0.0027 0.0104 0.0119 0.0242 0.0399 0.0612
(0.40) (0.12) (0.48) (0.54) (1.18) (1.96) (3.07)
Panel 3: Regression of errors on implied kurtosis
b 0.0027 0.0121 0.0153 0.0149 0.0018 0.0136 0.0264
(0.12) (0.58) (0.75) (0.73) (0.09) (0.71) (1.39)
This table consists of the parameter estimates of the following regression model used to analyze whether the out-of-sample prediction errors for Model 2 depend on volatility, skewness or kurtosis.
Et¼ c þ aEt1þ bXtþ et;
where Etdenotes the prediction error in percentage and Xtis the implied volatility, implied skewness, or implied kurtosis at time t estimated using
Theorem 1 ofBakshi et al. (2003). The numbers in the parentheses are t-statistics.
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 04/14/99 06/14/99 08/14/99 10/14/99 12/14/99 02/14/00 04/14/00 06/14/00 08/14/00 10/14/00 12/14/00 Kurtosis 06 /1 4/ 99 08 /1 4/ 99 10 /1 4/ 99 12 /1 4/ 99 02 /1 4/ 00 04 /1 4/ 00 06 /1 4/ 00 08 /1 4/ 00 10 /1 4/ 00 12 /1 4/ 00 3.1 3.2 3.3 3.4 3.5 3.6 3.7 04 /1 4/ 99 Skewness
Fig. 3. Implied skewness and kurtosis for cross-rate options. This figure consists of the evolutions of the implied skewness and kurtosis of the cross-rate (€/£) options. The implied skewness and kurtosis are calculated using Theorem 1 ofBakshi et al. (2003). The results indicate that the risk-neutral distributions of the cross-rates are fat-tailed (average kurtosis equals 3.31) and slightly negatively skewed (average skewness equals 0.13).
Fig. 4. Delta bounds of a cross-rate option. This figure consists of the upper and lower bounds on the delta of the ATM cross-rate (€/£) call option traded on June 29, 1999 when the cross-rate option price is unknown (dashed lines) or known (dotted lines). The delta bounds are calculated using the Propositions 5 and 6 ofBergman et al. (1996).
When the call price is known, the delta bounds become tighter and ranges from 0.1248 to 0.8775 (dotted lines).12 5. Concluding remarks
Instead of pricing cross-rate options directly, this study relates the option pricing bounds to the prices of the corre-sponding dollar-rate options. Our pricing bounds are derived from a general result of the copula theory and thus do not rely on the distribution assumptions of state vari-ables. Different from most option pricing bounds in the lit-erature, our cross-rate option bounds are functions of the option prices (and sometimes also the spot prices) of two dollar rates.
Using the prices of options on foreign exchange rates among US dollar, euro, and pound sterling for the empir-ical tests, we show the persistent relationships between the market prices of the cross-rate (€/£) options and our pric-ing bounds. Our pricpric-ing bounds and the correlation between two dollar rates provide 85% of the information in explaining the prices of the cross-rate options. There-fore, our results are useful for risk management and deriv-ative pricing, particularly for those having cross-rate risk exposures and when only the current option prices of two dollar rates are available.
The technique utilized to derive our cross-rate option pricing bounds can be applied to any European derivative security whose payoff can be rearranged as the same type as that of an exchange option. For example, one can derive the pricing bounds for quanto options using the copula approach applied in this paper.13Further analyses related to the pricing bounds of the other type of exchange options are left to interested readers for future research.
Acknowledgements
We are indebted to anonymous referees, Joao Amaro de Matos, Antonio Camara and Bing-Huei Lin for helpful comments and suggestions; and also to the seminar partic-ipants at National Chengchi University, National Chiao Tung University, National Taiwan University, National Tsing Hua University, Yuan Ze University, the European Financial Management Association Annual Meeting 2006, the 4th Bachelier Finance Society World Congress, the Finance Management Association Annual Meeting 2006, the Taiwan Finance Association Annual Meeting 2006, and the National Taiwan University International Conference on Finance 2006. We thank the National Sci-ence Council of Taiwan for financial support and acknowl-edge the research assistance of Der-Fong Chen.
Appendix A
Derivation of the price bounds for the cross-rate option Let Pr, Fi(x), and r be the probability, the cumulative
dis-tribution function, and the dollar risk-free interest rate, respectively. Due toBreeden and Litzenberger (1978), the price of an option on the minimum of two risky assets can be expressed as CallminðS$=£; KS$=€;0; t; TÞ ¼ erðT tÞZ 1 0 PrðminðS$=£; KS$=€Þ > xÞdx ¼ erðT tÞZ 1 0 PrðS$=£> x; KS$=€> xÞdx ¼ erðT tÞ Z 1 0 CðFS$=£ðxÞ; FKS$=€ðxÞÞdx; ðA:1Þ
where C is a survival copula14 and FiðxÞ ¼ 1 FiðxÞ.
According to the Fre´chet bounds in the copula theory, it is true that maxðu þ v 1; 0Þ 6 Cðu; vÞ 6 minðu; vÞ since Cðu; vÞ is a copula. Consequently, the upper and lower bounds of the minimum option are given as the following, respectively: CallþminðS$=£; KS$=€; 0; t; TÞ ¼ erðT tÞZ 1 0 minðFS$=£ðxÞ; FKS$=€ðxÞÞdx; CallminðS$=£; KS$=€;0; t; TÞ ¼ erðT tÞ Z 1 0 maxðFS$=£ðxÞ þ FKS$=€ðxÞ 1; 0Þdx: ðA:2Þ Since FiðuÞ is a decreasing function of u and K**is a
con-stant which solves FS$=£ðKÞ þ FKS$=€ðKÞ ¼ 1, it is true that FS$=£ðuÞ þ FKS$=€ðuÞ P 1 for u 6 K**. Therefore, the lower bound of the minimum option is
CallminðS$=£; KS$=€;
0; t; TÞ ¼ erðT tÞZ 1
0
maxðFS$=£ðuÞ þ FKS$=€ðuÞ 1; 0Þdu
¼ erðT tÞZ K 0 FS$=£ðuÞdu þ erðT tÞ Z K 0 FKS$=€ðuÞdu erðT tÞ Z K 0 du ¼ erðT tÞZ 1 0 FS$=£ðuÞ du erðT tÞ Z 1 K FS$=£ðuÞdu þ erðT tÞ Z 1 0 FKS$=€ðuÞdu erðT tÞ Z 1 K FKS$=€ðuÞdu erðT tÞ Z K 0 du 12The results for other moneyness levels (defined as the strike prices
divided by the forward price) are similar. For example, the delta of the call option with the moneyness level of 1.0325 (OTM) ranges from 0.0003 to 0.5986 (from 0.0127 to 0.5719) when the option price is unknown (known).
13The formulae of quanto option pricing bounds are also derived by the
authors and are available upon request.
14If two uniform variables U and V are jointed with a copula function C,
then the joint probability that U and V are greater than u and v, respectively, is given by a survival function:
¼ S$=£ t e r£ðT tÞ CallðS$=£; K; t; TÞ þ KS$=€ t e r€ðT tÞ CallðKS$=€; K; t; TÞ erðT tÞK: ðA:3Þ
Substituting Eq. (A.3)into Eq. (2) and applying the put– call parity, we obtain the upper bound of the cross-rate op-tion price as Call€=£$ þ¼ S$=£ t e r£ðT tÞ Call minðS $=£; KS$=€; 0; t; TÞ ¼ CallðS$=£; K; t; TÞ þ PutðKS$=€; K; t; TÞ
¼ CallðS$=£; K; t; TÞ þ KPutðS$=€; K00; t; TÞ; ðA:4Þ
where K00= K**/K.
Assume that there exists a constant K* such that
FS$=£ðKÞ ¼ FKS$=€ðKÞ. If FS$ =£ðuÞ < FKS$=€ðuÞ for u < K*, then it is straightforward to show that the upper bound of the minimum option is
CallþminðS$=£; KS$=€;0; t; TÞ
¼ erðT tÞ Z 1
0
minðFS$=£ðuÞ; FKS$=€ðuÞÞdu
¼ erðT tÞ Z K 0 FS$=£ðuÞdu þ erðT tÞ Z 1 K FKS$=€ðuÞdu ¼ erðT tÞ Z 1 0 FS$=£ðuÞdu erðT tÞ Z 1 K FS$=£ðuÞdu þ erðT tÞK Z 1 K=K FS$=€ðuÞdu ¼ S$=£ t e r£ðT tÞ CallðS$=£; K; t; TÞ þ KCallðS$=€; K0; t; TÞ; ðA:5Þ where K0= K*/K. Substituting Eq.(A.5)into Eq.(2)yields
the lower price bound of the cross-rate option as Call€=£$ ¼ S$=£
t e
r£ðT tÞ Callþ
minðS
$=£; KS$=€;0; t; TÞ
¼ CallðS$=£; K; t; TÞ KCallðS$=€; K0; t; TÞ: ðA:6Þ
Similarly, if FS$=£ðuÞ > FKS$=€ðuÞ for u < K*, then one can derive that CallþminðS $=£; KS$=€; 0; t; TÞ ¼ KS$=€ t e r€ðT tÞ KCallðS$=€; K0; t; TÞ þ CallðS$=£; K; t; TÞ; Call€=£$ ¼ S$=£ t er£ðT tÞ KS $=€ t er€ðT tÞ þ KCallðS$=€; K0; t; TÞ CallðS$=£; K; t; TÞ: ðA:7Þ Appendix B Proof ofProposition 2
When the two dollar rates follow a bivariate lognormal distribution, FS$=£ðKÞ is actually the risk-neutral probability that the European call option on $/£ with a strike price of K will be exercised, because
FS$=£ðKÞ ¼ 1 FS$=£ðKÞ ¼ 1 Pr S $=£ T < K ¼ N d2 S$=£t ; K; r$; r£;r$=£;s ; where s = T t and d2ðÞ ¼ ln Sð$=£t =KÞþ r$r£12r2$=£ s r$=£ ffiffis p .
There-fore, K**inProposition 1 is a constant satisfying that
N d2 S$=£t ; K; r$; r£;r$=£;s þ N d2 KS$=€t ; K ; r $; r€;r$=€;s ¼ 1:
Since NðxÞ þ N ðxÞ ¼ 1 is true, one can show that d2 S$=£t ; K ; r $; r£;r$=£;s ¼ d2 KS$=€t ; K ; r $; r€;r$=€;s : ðA:8Þ Thus, the solution of for K**is
K¼ KS$=€ t r$=£ r$=£þr$=€ S$=£ t r$=€ r$=£þr$=€exp r $ r$=€ r$=£þ r$=€ r£ r$=£ r$=£þ r$=€ r€ r$=£r$=€s 2 : ðA:9Þ
From Eqs. (3) and (A.8), the upper bound is given by S$=£t er£sN d 1 S$=£t ; K ; r $; r£;r$=£;s Ker$sN d 2 S$=£t ; K; r$; r£;r$=£;s þ Ker$sN d 2 KS$=€t ; K ; r $; r€;r$=€;s KS$=€ t e r€sN d 1 KS$=€t ; K ; r $; r€;r$=€;s ¼ S$=£ t e r£sN d 1S$=£t ; K; r$; r£;r$=£;s KS$=€ t e r€sN d 1 KS$=€t ; K ; r $; r€;r$=€;s :
Note that the above upper bound is denominated in US dollars and its value in euros is
S€=£t er£sN d 1 S$=£t ; K; r$; r£;r$=£;s Ker€sN d 1 KS$=€t ; K ; r $; r€;r$=€;s : ðA:10Þ
Substituting Eq. (A.9) into Eq. (A.10) yields the upper bound, i.e. S€=£ t e r£sN d 1 S€=£t ; K; r€; r£;r$=£þ r$=€;s Ker€sN d 2S€=£t ; K; r€; r£;r$=£þ r$=€;s ¼ CBS S€=£t ; K; r€; r£;r$=£þ r$=€;s :
For brevity, we derive the lower bound only for the case where r$/£> r$/€. Since K* satisfies that N d2 S$=£t ;
K; r$; r£;r$=£;sÞÞ ¼ N d2 KS$=€t ; K ; r $; r€;r$=€;s , one can show that its solution is
K¼ KS$=€ t r$=£ r$=£r$=€ S$=£ t r$=€ r$=£r$=€exp r $þ r$=€ r$=£ r$=€ r£ r$=£ r$=£ r$=€ r€þ r$=£r$=€s 2 : ðA:11Þ
Since r$/£> r$/€, it is straightforward to show that when
N d2 S$=£t ; u; r$; r£;r$=£;s ¼ N d2 S$=£t ; K ; r $; r£;r$=£;s þlnðK =uÞ r$=£ ffiffiffis p < N d2ðKS$=€t ; K ; r $; r€;r$=€;sÞ þ lnðK=uÞ r$=€ ffiffiffis p ¼ N d2 KS$=€t ; u; r$; r€;r$=€;s :
Therefore, the lower bound is determined by the first alter-native of Eq.(4),15i.e.
S$=£t er£sN d 1 S$=£t ; K; r$; r£;r$=£;s Ker$sN d 2 S$=£t ; K ; r $; r£;r$=£;s KS$=€t e r€sN d 1 KS$=€t ; K ; r $; r€;r$=€;s þ Ker$sN d 2 KS$=€t ; K; r$; r€;r$=€;s ¼ S$=£ t e r£sN d 1 S$=£t ; K ; r $; r£;r$=£;s KS$=€ t e r€sN d 1 KS$=€t ; K ; r $; r€;r$=€;s :
The above lower bound is denominated in US dollars and its value in euros is
S€=£t er£sN d 1 S$=£t ; K ; r $; r£;r$=£;s Ker€sN d 1 KS$=€t ; K ; r $; r€;r$=€;s : ðA:12Þ
Substituting Eq. (A.11) into Eq. (A.12) yields the lower bound, i.e. S€=£t er£sN d 1 S€=£t ; K; r€; r£;r$=£ r$=€;s Ker€sN d 2 S€=£t ; K; r€; r£;r$=£ r$=€;s ¼ CBS S€=£t ; K; r€; r£;r$=£ r$=€;s : References
Bakshi, G., Madan, D., 2000. Spanning and derivative-security valuation. Journal Financial Economics 55, 205–238.
Bakshi, G., Kapadia, N., Madan, D., 2003. Stock return characteristics, skew laws, and the differential pricing of individual equity options. Review of Financial Studies 16, 101–143.
Bakshi, G., Carr, P., Wu, L., forthcoming. Stochastic risk premiums, stochastic skewness in currency options, and stochastic discount factors in international economies. Journal of Financial Economics. Basso, A., Pianco, P., 1997. Decreasing absolute risk aversion and option
pricing bounds. Management Science 43, 206–216.
Bergman, T., Grundy, B., Weiner, Z., 1996. General properties of option prices. Journal of Finance 51, 1573–1610.
Bliss, R.R., Panigirtzoglou, N., 2002. Testing the stability of implied probability density functions. Journal of Banking and Finance 26, 381–422.
Bookstaber, R.M., McDonald, J.B., 1987. A general distribution for describing security price returns. Journal of Business 60, 401–424. Breeden, D., Litzenberger, R., 1978. Prices of state-contingent claims
implicit in options prices. Journal of Business 51, 621–651.
Carr, P., Ellis, K., Gupta, V., 1998. Static hedging of exotic options. Journal of Finance 53, 1165–1190.
Cherubini, U., Luciano, E., Vecchiato, W., 2004. Copula Methods in Finance. John Wiley and Sons.
Driessen, J., Maenhout, P., Vilkov, G., 2006. Option-implied correlations and the price of correlation risk. Working paper, University of Amsterdam and INSEAD.
Engle, R.F., 2002. Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business and Economic Statistics 20, 339–350. Garman, M., 1976. An algebra for evaluating hedging portfolios. Journal
of Financial Economics 3, 403–428.
Grundy, B.D., 1991. Option prices and the underlying asset’s return distribution. Journal of Finance 46, 1045–1070.
Jackwerth, J.C., Rubinstein, M., 1996. Recovering probability distribu-tions from option prices. Journal of Finance 51, 1611–1631. Joe, H., 1997. Multivariate Models and Dependence Concepts.
Chap-man & Hall.
Levy, H., 1985. Upper and lower bounds of call and put option value: Stochastic dominance approach. Journal of Finance 40, 1197– 1217.
Mathur, K., Ritchken, P., 2000. Minimum option prices under decreasing absolute risk aversion. Review of Derivative Research 3, 135–156. Melick, W., Thomas, C., 1997. Recovering an asset’s implied PDF from
option prices: An application to crude oil during the Gulf crisis. Journal of Financial and Quantitative Analysis 32, 91–115.
Merton, R.C., 1973. Theory of rational option pricing. Bell Journal of Economics and Management Science 4, 141–183.
Nelsen, R.B., 1999. An Introduction to Copulas. Springer, New York. Ritchey, R., 1990. Call option valuation for discrete normal mixtures.
Journal of Financial Research 13, 285–295.
Ritchken, P., 1985. On option pricing bounds. Journal of Finance 40, 1219–1233.
Ritchken, P., Kuo, S., 1989. On stochastic dominance and decreasing absolute risk averse option pricing bounds. Management Science 35, 51–59.
Ryan, P., 2003. Progressive option bounds from the sequence of currently expiring options. European Journal of Operation Research 151, 193– 223.
Taylor, S.J., Wang, Y.-H., 2005. Option prices and risk neutral densities for currency cross-rates. Working paper, Lancaster University and National Central University.
15On the other hand, if the volatility of $/£ is smaller than that of $/€,
