Pricing currency options in a fractional Brownian motion with jumps

Economic Modelling 27 (2010) 935–942

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Economic Modelling j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c m o d

Pricing currency options in a fractional Brownian motion with jumps Wei-Lin Xiao a, Wei-Guo Zhang a,⁎, Xi-Li Zhang a, Ying-Luo Wang b a b

School of Business Administration, South China University of Technology, Guangzhou, 510641, PR China School of Management, Xi'an Jiaotong University, Xi'an, 710049, PR China

a r t i c l e

i n f o

Article history: Accepted 20 May 2010 JEL classification: G11 G12 G21 Keywords: Currency options Fractional Brownian motion Poisson jump Option pricing

a b s t r a c t A new framework for pricing the European currency option is developed in the case where the spot exchange rate fellows a fractional Brownian motion with jumps. An analytic formula for pricing European foreign currency options is proposed using the equivalent martingale measure and the estimation method of parameters in the pricing model is given, enabling option prices to be computed efficiently and accurately. For the purpose of understanding the pricing model, some properties of this pricing model are discussed in the latter part of this paper. Finally, the numerical simulations illustrate that our model is flexible and easy to implement. © 2010 Elsevier B.V. All rights reserved.

1. Introduction A currency option is a contract which gives the owner the right, but not the obligation, to buy or sell the indicated amount of foreign currency at a specified price within a specified period of time (American Option) or on a fixed date (European Option). Since the currency option can be used as a tool for investment and hedging, it is one of the best ways for corporations or individuals to hedge against adverse movements in exchange rates and the theoretical models for pricing currency options have been carried out. The standard European currency option valuation model has been presented by Garman and Kohlhagen (1983). However, some papers (see, for instance, Cookson, 1992) have provided evidence of the mispricing for currency options by the Garman–Kohlhagen (hereafter G–K) model. The most important reason why this model may not be entirely satisfactory could be that currencies are different from stocks in important respects and the geometric Brownian motion cannot capture the behavior of currency return (Ekvall et al., 1997). Since then, many methodologies for currency option pricing have been proposed by using modifications of G–K model, such as Lim et al. (1998), Rosenberg (1998), Sarwar and Krehbiel (2000), Bollen and Rasiel (2003), Lim et al. (2006), and Yu et al. (2006). The papers above assume that the logarithmic returns of the exchange rate are independent identically distributed normal random variables. However, the empirical research on asset return indicates that discontinuities or “jumps” are believed to be an essential component of financial asset prices (see, Andersen et al., ⁎ Corresponding author. Tel.: +86 20 87114121. E-mail addresses: [email protected], [email protected] (W.-G. Zhang). 0264-9993/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.econmod.2010.05.010

2002; Chernov et al., 2003; Pan, 2002; Eraker, 2004). Fortunately, Merton (1976) proposed a jump–diffusion process with Poisson jump to match the abnormal fluctuation of stock price. Based on this theory, Kou (2002) and Cont and Tankov (2004) also considered the problem of pricing options under a jump–diffusion environment in a larger setting. Moreover, Chang et al. (2007) derived explicit pricing formulas for European foreign exchange options when the exchange rate dynamics were governed by jump–diffusion processes. Ma (2006) derived an option pricing formula in the presence of Lévy jumps. Indeed, the empirical researches are also shown that the distributions of the logarithmic returns in the financial market usually exhibit excess kurtosis with more probability mass near the origin and in the tails and less in the flanks than would occur for normally distributed data (see Lo, 1991). For example, Dai and Singleton (2000) refuted the assumption of independent state variables based on empirical findings. They stated that these time series illustrated significant autocorrelation between observations habitually broken up in time. That is to say the features of financial return series are non-normality, non-independence and nonlinearity. Furthermore, Berg and Lyhagen (1998), Hsieth (1991) and Huang and Yang (1995) showed that returns are of short-term (or long-term) dependency. To capture these non-normal behaviors, many scholars have considered other distributions with fat tails such as the Pareto-stable distribution and the Generalized Hyperbolic Distribution among others. Moreover, self-similarity and long-range dependence have become important concepts in analyzing the financial time series. There is strong evidence that the stock return has little or no autocorrelation. Since fractional Brownian motion has two important properties called self-similarity and long-range dependence, it has the ability to capture the typical tail behavior of stock prices or indexes.

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Fractional Brownian motion is a family of Gaussian processes that is indexed by the Hurst parameter H in the interval (0, 1). Since fractional Brownian motion is neither a Markov process nor a semimartingale (except in the geometric Brownian motion case), we cannot use the usual stochastic calculus to analyze it. Fortunately, the research interest in this field was re-encouraged by new insights in stochastic analysis based on the Wick integration (see Hu and Øksendal, 2003) called the fractional-Itô-integral. Using this type of stochastic integration Hu and Øksendal (2003) proofed that the fractional Black–Scholes market presents “no arbitrage opportunity and is complete”. However, Björk and Hult (2005) argued that the use of fractional Brownian motion in this context does not make much economic sense because, while Wick integration leads to no arbitrage, the definition of the corresponding self-financing trading strategies is quite restrictive and, for example, in the setup of Elliott and Van der Hoek (2003), the simple buy-and-hold strategy is not self-financing. We noted that this arbitrage example in discrete-time does not, however, rule out the use of fractional Brownian motion in finance. For example, Bender et al. (2007) showed that the existence of arbitrage opportunities depends very much on the definition of the admissible trading strategies. Furthermore, Bender et al. (2008) stated that the financial market does not admit arbitrage opportunities in a class of trading strategies if a continuous price process has the conditional small ball property and pathwise quadratic variation. Hence it is not too hard to accept this idea: some restrictions are sufficient to exclude arbitrage in the fractional Brownian market. Indeed, some authors have used the geometric fractional Brownian motion to capture the behavior of underlying asset and to obtain fractional Black–Scholes formulas for pricing options, including Necula (2002), Bayraktar et al. (2004), and Meng and Wang (2010). To capture the behaviors of exchange rate, the combination of Poisson jumps and fractional Brownian motion is introduced in this paper. The jump fractional Brownian motion is based on the assumption that exchange rate returns are generated by a two-part

stochastic process: (1) small, continuous price movements are generated by a fractional Wiener process, and (2) large, infrequent price jumps are generated by a Poisson process. This two-part process is intuitively appealing, as it is consistent with an efficient market in which major information arrives infrequently and randomly. In addition, this process may provide an explanation for empirically observed distributions of exchange rate changes that are skewed, leptokurtic, long memory and have fatter tails than comparable normal distributions, and for the apparent nonstationarity of variance. Despite several models having been used in a currency option pricing context, applying fractional Brownian motion with jumps to currency option pricing model has not been studied. Moreover, a strong rationale exists on both theoretical and empirical grounds for using the jump fractional Brownian motion model to value currency options. In this paper, we capture the behavior of exchange rates using the fractional Brownian motion with jumps. Then we show how to price European currency options using the G–K type model derived in a jump fractional Brownian environment. The comparative results of our model and other available valuation models show that our model is easy to implement. Furthermore, the numerical computations also show our model has a good explanation of the “volatility smile”. The rest of this paper is organized as follows. After giving the assumptions of pricing environment in Section 2, we present an analytic pricing formula for the European foreign currency option based on the principle of option pricing. Furthermore, the estimation of parameters required in our pricing model is proposed. In Section 3, we study some special properties of this currency pricing formula. In Section 4, we show how to use our model to price currency options by numerical simulations. The comparison of our jump fractional Brownian motion model and traditional models is undertaken in this section. Moreover, some influences of the jump parameters are also reported in this section. Finally, Section 5 draws the concluding remarks.

2. Pricing model for currency option in a jump fractional environment Since a financial system is a complex system with great flexibility, investors do not make their decisions immediately after receiving the financial information, but rather wait until the information reaches to its threshold limit value. This behavior can lead to the features of “asymmetric leptokurtic” and “long memory”. The fractional Brownian motion may be a useful tool for capturing this phenomenon. In addition, to capture the abnormal fluctuation of exchange rate, the Poisson jumps are also introduced in this paper. 2.1. The assumptions of the jump fractional Brownian market Consider a probability space (Ω, FH, PH) on which all the random variables and processes below are defined. A fractional Brownian motion BH = {BH(t, ω), t N 0}, with Hurst parameter H ∈ (0, 1) is a centered Gaussian process with mean zero and covariance EPH ½BH ðtÞBH ðsÞ =

o 1 n 2H 2H 2H þ ; s; t∈R jt j + jsj −jt−sj 2

where the parameter H is the self-similarity index. For α N 0, H

LawðBH ðαÞj PH Þ = Lawðα BH ð⋅Þj PH Þ; and FtH = σ {BH(s), 0 ≤ s ≤ t} with FTH = F H. To derive the currency option pricing formula in a jump fractional market, we shall make the following assumptions: (1) (2) (3) (4) (5)

There are no transaction costs or taxes and all securities are perfectly divisible; Security trading is continuous; The short-term domestic interest rate rd and foreign interest rate rf are known and constant through time; There are no riskless arbitrage opportunities; The spot exchange rate follows a fractional Brownian motion with random jumps under the probability measure PH. Thus   JðtÞ dSðtÞ = SðtÞ μ−λμ JðtÞ dt + SðtÞσdBH ðtÞ + SðtÞðe −1ÞdNðtÞ; 0≤t≤T; Sð0Þ = S;

ð1Þ

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where S(t) denotes the spot exchange rate at time t of one unit of the foreign currency measured in the domestic currency; the drift μ and volatility σ are assumed to be constants; BH(t) is a fractional Brownian motion; N(t) is a Poisson process with rate λ; J(t) is jump size percent at time t which is a sequence of independent identically distributed, and (e J(t) − 1) ∼ N(μJ(t), δ2(t)). In addition, all three sources of randomness, the fractional Brownian motion BH(t), the Poisson process N(t), and the jump size e J(t) − 1, are assumed to be independent. 2.2. The pricing model The profit of the holder of the currency option at time T is (S(T) − K)+ (the call) or (K − S(T))+ (the put). Hence, the fair price to hold a currency option at any time before maturity T is given by the discounted conditional expected value of this profit based on a suitable measure. To obtain the value of a currency option in a fractional jump environment, we recall some basic results that will be used in this paper. We refer to Necula (2002) for further reading. 2 ̂ Lemma 2.1. (Necula, 2002) The price at every t ∈ [0, T] of a bounded FH T -measurable claim Z(T,ω) ∈ L (PH) is given by

−rðT−tÞ

Zðt; ωÞ = e

H E˜Pˆ ½ZðT; ωÞj Ft ; H

μ−λμ

+ rf −rd

JðtÞ ̂ is a probability measure under (Ω, F H) such that t + BH ðtÞ is a new fractional Brownian motion; EP̃ Ĥ [·|FtH] denotes the quasiwhere PH σ H ̂ conditional expectation with respect to Ft under the probability measure PH.

Theorem 2.1. For a bounded FTH-measurable claim Z(T,ω) ∈ L2(P̂H), let Z̃(t,ω) = e− rtZ(t,ω). Then Z̃(t,ω) is a quasi-martingale, and the jump fractional currency market does not have arbitrage. Proof. The proof is obvious from the independence of jumps.

□

Lemma 2.2. (Necula, 2002) Let f be a function such that E[ f(BH(T))] b ∞. Then for every t ≤ T ! 1 ðx−BH ðtÞÞ2 f ðxÞdx: E½f ðBH ðTÞÞjFt  = ∫ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp − R 2ðT 2H −t 2H Þ 2πðT 2H −t 2H Þ Let V(S(t), K, T, t, σ, rd, rf, φ) be the price of an European currency option at time t with a strike price K that matures at time T. Then we obtain the following theorem. Theorem 2.2. Suppose an exchange rate S(t) defined by Eq. (1), then the valuation of the currency option V(S(t), K, T, t, σ, rd, rf, φ) at time t (t ∈ [0, T]) can be given by 8  ∞   n n n −λðT−tÞ λ ðT−tÞ Jðti Þ −ðrf + λμ JðtÞ ÞðT−tÞ −rd ðT−tÞ > > E ; r ; φÞ = φ ∑ e SðtÞ ∏ e e Nðφd Þ−Ke Nðφd Þ ; VðSðtÞ; K; T; t; σ ; r > d f n 1 2 > > n! n=0 i=1 < 2 n ð2Þ Jðt Þ 2H 2H σ > > lnðSðtÞ ∏ e i = KÞ + ðrd −rf −λμ JðtÞ ÞðT−tÞ + ðT −t Þ > p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 > i=1 : d1 = pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; d2 = d1 −σ T 2H −t 2H : 2H 2H σ

T

−t

where, φ = +1 for call options while φ = −1 for put options; E n denotes the expectation operator over the distribution of ∏ ni = 1eJ(ti); N( ⋅ ) is the cumulative normal distribution function. The proof of Theorem 2.2 is provided in Appendix A. 2.3. The estimation method of parameters Comparing G–K model with formula (2), we have the conclusion that the G–K pricing model is a function of six parameters, while our jump fractional model requires ten. Fortunately, six of the required parameters are usually known: the spot price of the underlying exchange rate, the domestic (riskless) interest rate, the foreign (riskless) interest rate, the volatility of the spot currency price and the remaining time to option expiration. In the following part, we estimate the remaining parameters, which describe the intertemporal movements and the long memory in the spot exchange rate price, of our jump fractional Brownian process using a technique introduced by Kendall and Stuart (1977) and Beckers (1981). Although there exist many methods for estimating Hurst parameter (see, for example, Faÿ et al., 2009), we shall use the so-called R/S method which is the most famous approach and seems to be the simplest one to us. The idea of parameters estimation, which consists of two parts, is disarmingly simple and extremely powerful. To be specific we show the procedure as follows: (1) Using R/S analysis methodology to estimate the Hurst exponent H. (2) Obtaining the jump parameters by calculating the currency returns over the estimation period. This algorithm consists of three steps. First, we obtain the first six sample moments, ms, s = 1 to 6, ms =

1 T s ∑t = 1 ½ΔZðtÞ ; T

ð3Þ

where ΔZ(t) is the change in the natural log of the currency price during time t, and T is the number of days in the estimation period. Next, using the sample moments, the required sample cumulates are determined. The relationships between the sample moments and cumulants are given by Kendall and Stuart (1977, P.72), K1 = m1 ;

ð4Þ

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W.-L. Xiao et al. / Economic Modelling 27 (2010) 935–942 2

K2 = m2 −m1 ;

ð5Þ 2

2

4

K4 = m4 −4m3 m1 −3m2 + 12m2 m1 −6m1 ; 2

ð6Þ 2

K6 = m6 −6m5 m1 −15m4 m2 + 30m4 m1 −10m3 + 12m3 m2 m1 3

3

2

2

4

ð7Þ

6

−120m3 m1 + 30m2 −270m2 m1 + 360m2 m1 −120m1 :

ð8Þ

Finally, using the sample cumulants, the estimating parameters are obtained as 3

2

ð9Þ

2

ð10Þ

λ = 25K4 = 3K1 ; μ JðtÞ = K2 −5K4 = 3K6 ; 2

δ ðtÞ = lnðK6 = 5K4 + 1Þ:

ð11Þ

3. Properties of pricing formula In the previous sections, we derive the formula to calculate the valuation of currency options. In this section, we will discuss the properties of n

this pricing formula. For the sake of convenience, we define the forward price of an exchange rate as f ≜SðtÞ ∏ e Jðti Þ eðrd −rf −λ μ JðtÞ ÞðT−tÞ and σ

2

lnðf = KÞ  ðT 2H −t 2H Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . σ T 2H −t 2H

d ≜

i=1

We are now ready to return to discuss some symmetrical properties of our pricing formula.

Remark 3.1. The relationship of call–put parity can be written as VðSðtÞ; K; T; t; σ ; rd ; rf ; + 1Þ−VðSðtÞ; K; T; t; σ ; rd ; rf ; −1Þ = SðtÞe

−ðrf + λμ JðtÞ ÞðT−tÞ

−Ke

−rd ðT−tÞ

:

This is just a more complicated way to write the trivial equation S(t) = S(t)+ − S(t)−. Remark 3.2. The relationship of put–call parity satisfies ∂VðSðtÞ; K; T; t; σ ; rd ; rf ; + 1Þ ∂SðtÞ

−

∂VðSðtÞ; K; T; t; σ; rd ; rf ; −1Þ ∂SðtÞ

−ðrf + λμ JðtÞ ÞðT−tÞ

=e

:

This relationship is also named put–call parity delta. It should be pointed out that the absolute value of a put delta and a call delta are not exactly adding up to one, but only to a positive number e−(rf + λμ J(t))(T − t). They add up to one approximately if either the time to expiration is very short or if the foreign interest rate is close to the product of λ and μJ(t). Remark 3.3. The delta of the spot strike price has a space-homogeneity property, such that for any a N 0, aVðSðtÞ; K; T; t; σ ; rd ; rf ; φÞ = VðaSðtÞ; aK; T; t; σ ; rd ; rf ; φÞ: Moreover, differentiating both sides with respect to a and then setting a = 1 yields ′ ðSðtÞ; K; T; t; σ; r ; r ; φÞ + KV ′ ðSðtÞ; K; T; t; σ; r ; r ; φÞ: VðSðtÞ; K; T; t; σ ; rd ; rf ; φÞ = SðtÞVSðtÞ d f d f K

Indeed, this formula is another version of the valuation of currency option, when we wish to measure the value of the underlying in a different ′ and VK′ can be obtained by comparing this formula with Eq. (2). This formula gives us a new method to solve delta. unit. Furthermore, VS(t) Remark 3.4. For any a N 0, we can perform a similar computation for the time-affected parameters and obtain the time-homogeneity equation T t pffiffiffi VðSðtÞ; K; T; t; σ ; rd ; rf ; φÞ = VðSðtÞ; K; ; ; aσ ; ard ; arf ; φÞ: a a Differentiating both sides with respect to a and then setting a = 1 yields 0 = ðT−tÞVt′ +

1 σ Vσ′ + rd Vr′d + rf Vr′f : 2

This formula can also be verified by direct computation. The overall use of such equations is to generate double checking benchmarks when computing Greeks. These homogeneity methods can easily be extended to other more complex options. Remark 3.5. The put–call symmetry can be expressed as

VðSðtÞ; K; T; t; σ ; rd ; rf ; + 1Þ =

! K f2 V SðtÞ; ; T; t; σ; rd ; rf ; −1 : K f

W.-L. Xiao et al. / Economic Modelling 27 (2010) 935–942

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This relationship shows that the strike of the put and the strike of the call result in a geometric mean equal to the forward f. The forward can be interpreted as a geometric mirror reflecting a call into a certain number of puts. Note that for at-the-money options (K = f) the put–call symmetry coincides with the special case of the put–call parity where the call and the put have the same value. Remark 3.6. The rate symmetry can be written as ∂VðSðtÞ; K; T; t; σ ; rd ; rf ; φÞ ∂rd

+

∂VðSðtÞ; K; T; t; σ ; rd ; rf ; φÞ ∂rf

= VðSðtÞ; K; T; t; σ ; rd ; rf ; φÞðt

2H

−T

2H

Þ:

This relationship, in fact, holds for all European options and a wide class of path-dependent options. Remark 3.7. The foreign–domestic symmetry can be expressed as

1 1 1 VðSðtÞ; K; T; t; σ; rd ; rf ; φÞ = KV ; ; T; t; σ ; rd ; rf ; −φ : SðtÞ SðtÞ K This equality can be viewed as one of the faces of put–call symmetry. The reason is that the value of an option can be computed both in a domestic as well as in a foreign scenario. We consider the example of S(t) modelling the exchange rate of CNY/USD. In New York, the call option K, T, t, σ, rusd, rcny, 1)/S(t) (ST − K)+ costs V(S(t), K, T, t, σ, rusd, rcny, 1) US dollars and hence we have V(S(t),

CNY. This CNY-call option can also be   1 1 þ 1 1 1 . This option costs KSðtÞ ; ; T; t; σ; rcny ; rusd ; −1 CNY in Shanghai, because S(t) and viewed as a USD-put option with payoff K − K

ST

SðtÞ K

SðtÞ

have the same volatility. Of course, the New York value and the Shanghai value must agree, which leads to Equation of Remark 3.7. 1 2

Remark 3.8. The jump–diffusion model is a special case of our jump fractional Brownian motion model when H = . It is clear that fractional type pricing model of a European currency call option no longer only depends on T − t with respect to the G–K model. This may be because the fractional Brownian motion has long memory property. Hence the price of an option at a moment t ∈ [0, T] will depend on the price of underlying asset and will take into consideration the evolution of the financial asset in the period [0, t], despite the classical G–K model. This influence is reflected in the jump fractional model by the Hurst parameter H. Consider three moments t1 ≤ t2 ≤ t ≤ T and two options with maturity T, one of them written on t1 and the other one on t2. In the classical G–K model the prices of the two options at the moment t are equal (all other things being equal). However, the prices of these two options at the moment t are no longer equal in the fractional case. Due to the long memory property, the price of the first option is also influenced by the evolution of the stock price in the period [t1, t2]. 4. Simulation studies The aim of this section is to show how to implement our jump fractional Brownian motion model and to present the effects of jump parameters of our pricing model. For these purposes, we report on two sets of numerical experiments. In the first set, we compare the theoretical prices of some hypothetical options among the following models: the G–K, the pure fractional Brownian motion (hereafter PFBM) and our jump fractional Brownian motion (hereafter JFBM). These tests will not be based on empirical data, but they will consist of some simulations of different pricing models with some chosen parameters. With the second set, we report the influences of different parameters of JFBM. The code lines are written in Matlab. All experiments are performed using the same IBM Intellistation Z-Pro with a 3.8 GHz Xeon processor running Windows XP.

4.1. Comparison of option prices Now, for an illustration of the differences among these models: the G–K, the PFBM and our JFBM, we report the theoretical prices of some hypothetical options using different methods. Table 1 presents the parameters for computing the hypothetical currency call options. The first row displays the parameters for calculating the prices by the G–K model. The second row presents the parameters for calculating the prices Table 1 The valuations of the chosen parameters used in these models. Model type

rd (%)

rf (%)

σ (%)

K

H

G–K PFBM JFBMa JFBMb

3.10 3.10 3.10 3.10

4.20 4.20 4.20 4.20

10.91 10.91 10.91 10.91

100 100 100 100

0.618 0.618 0.618

J

λ

μJ

by the PFBM (see Meng and Wang, 2010). The third row, which has low jump parameters, provides the parameters for calculating the prices by the JFBM. The forth row, which has high jump parameters, also provides the parameters for calculating the prices by the JFBM. Other parameters, including the exchange rate and the remaining times to maturity, vary in Table 2. The prices computed by different models are also presented in Table 2, where S denotes the prices of the exchange rate; PG–K denotes the prices computed by the G–K model; PP–F denotes the price simulated by the PFBM; and PJ–F denotes the price computed according to JFBM. ⁎,a in Table 2 for the low- and By comparing columns PG–K, PP–F and PJ–F high-maturity cases, we have the conclusion that the call option prices obtained by three valuation methods are close to each other. This is mainly because that the jump parameters are very low. Meanwhile, we can investigate that the prices given by the PFBM are smaller than the ⁎,a in prices given by the G–K. Now, we take a look at columns PG–K and PJ–F Table 2. Apparently, we can also investigate that the prices obtained by the JFBM are smaller than the price obtained by the G–K. The main reason is also that the chosen jump parameters are very low. Next, we want to investigate whether they are close for hypothetical options with high jump parameters. As time to maturity increases, the magnitude of the difference between option prices computed by these three methods increases in the high jump parameters case. This can be seen by ⁎,b in Table 2 for the low and comparing columns PG–K, PP–F and PJ–F high time to maturity cases. When combining the findings from ⁎,b with S ∈ [80, 140], we arrive at the following columns PG–K, PP–F and PJ–F conclusion: the magnitude of the difference ratio in prices is higher for out-of-the-money options in the high time to maturity case and this difference ratio decreases with the increase in exchange rate.

δ

4.2. The influence of the parameters 0.0091 0.0026

1.25 6.25

0.00068 − 0.00068

0.0010 0.0013

In what follows, we will present the values of currency call options using JFBM model for different parameters. For the sake of simplicity, we

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Table 2 Pricing results by different pricing models. S

80 85 90 95 98 100 102 110 120 130 140

Low time to maturity, T = 0.5, t = 0

High time to maturity, T = 2, t = 0

PG–K

PP–F

⁎ ,a PJ–F

⁎,b PJ–F

PG–K

PP–F

⁎,a PJ–F

⁎,b PJ–F

0.0029 0.0363 0.2426 0.9913 1.9063 2.7589 3.8128 9.7313 19.0754 28.8375 38.6287

0.0011 0.0195 0.1691 0.8189 1.6850 2.5222 3.5776 9.6120 19.0602 28.8368 38.6286

0.0011 0.0194 0.1684 0.8164 1.6807 2.5165 3.5704 9.6004 19.0472 28.8237 38.6156

0.0013 0.0225 0.1887 0.8886 1.8026 2.6758 3.7671 9.9044 19.3838 29.1624 38.9543

0.3054 0.7466 1.5607 2.8701 3.9281 4.7519 5.6706 10.2369 17.5556 25.9564 34.8617

0.4410 0.9777 1.8965 3.2956 4.3913 5.2318 6.1598 10.6948 17.8783 26.1371 34.9466

0.4291 0.9546 1.8571 3.2360 4.3183 5.1495 6.0682 10.5677 17.7168 25.9547 34.7538

0.8459 1.7223 3.0983 5.0391 6.4798 7.5512 8.7069 14.0698 21.9944 30.6683 39.6734

⁎The maximum number of iterations is 100. The parameters for calculations are in the third line of Table 1. b The parameters for calculations are in the fourth line of Table 1. a

will just consider the in-the-money case. Indeed, using the same method, one can also discuss the remaining cases: out-of-the-money and at-themoney. We take a first look at the values of our jump fractional currency options for different Hurst parameters H and then consider the values for different jump parameters. Fig. 1 displays the values of a European currency call option versus its parameters, H, λ, μJ and δ. The defaulting parameters are S=120, K=100, rd =0.031, rf =0.042, σ=0.01091, t=0, T=2, H=0.618, J=0.0026, λ=6.25, μJ =−0.0068 and δ=0.0013. It only takes less than 3 s to generate all the pictures in Fig. 1 on a highperformance computer. Not surprisingly, Fig. 1 indicates that: (1) the option value is a decreasing function of λ; and (2) increasing parameters of H, μJ and δ come along with a decrease of the option value. A similar phenomenon was also pointed out in Merton (1976). As we know, one of the properties of the classical currency option pricing model of the G–K that is often criticized, is the fact that real market prices of derivatives do not—as stated by this model—imply constant volatility for different maturities. Indeed, when evaluating the quality of a new model concerning this aspect, the usual testing procedure is to calculate the implied volatilities of option prices derived by this new model. In the sequel one checks whether this model is able to provide the respective shape of the term structure observed for real market data. In what follows, we consider the implied volatility surface of JFBM for different maturity and moneyness (S/K). Now, using our JFBM model we can calculate simultaneously currency call option prices for various

maturities. Then using an iterative search procedure like the Newton– Raphson method, we can easily find the implied volatilities for every option price and plot an implied volatility surface. In Fig. 2 we plot the volatility surfaces generated with our JFBM. The chosen parameters are S=100, rd =0.031, rf =0.042, t=0, H=0.618, J=0.0026, λ=11.12, μJ =0.0035 and δ=0.0013. The strike prices vary from 90 to 110. As can be seen, the model allows us to discuss the “smile” effect. That is, the implied volatility is not a constant, but varies with moneyness and maturity. 5. Conclusions Currency options are popular financial derivatives that play essential roles in financial markets. How to price them efficiently and accurately is very important both in theory and practice. This theory has been developed and improved on the basis of the G–K log-normal stochastic differential models since the mid-1980s. However, high frequency financial returns display potential jumps, long memory, volatility clustering, skewness, and excess kurtosis. These cannot be fully described by geometric Brownian motion. The goal of this paper is to propose a parsimoniously parameterized model that captured the essential features in the data. The main results of the paper are: (1) capturing these features of currency financial datum requires a specification that allows both jumps and fractional Brownian motion; (2) a specification that only allows for jumps or only allows for geometric Brownian motion, seriously

Fig. 1. Value of the currency call options.

W.-L. Xiao et al. / Economic Modelling 27 (2010) 935–942

941

This paper uses a jump fractional Brownian motion to capture the behavior of the spot exchange rate price and deduces the European currency option pricing model in this jump fractional environment. Furthermore, some special properties of currency pricing formula are also presented. Our numerical examples in Section 4 show that it is necessary to use our jump fractional Brownian motion model when the time to maturity is large enough. Meanwhile, our jump fractional Brownian motion model is easy to implement and has the potential to explain the phenomenon of volatility smile. In conclusion, since fractional Brownian motion is a well-developed mathematical model of strongly correlated stochastic processes and jumps are very important events in financial markets, jump fractional Brownian model will be a more efficient model for pricing currency options. Fig. 2. Smile surface generated with the jump fractional Brownian model.

Acknowledgements We would like to thank editors and referees for their penetrating remarks and suggestions concerning the earlier version of this paper. This research was supported by a grant from National Science Fund for Distinguished Young Scholars of China (No. 70825005), Humanities and Social Sciences Foundation of China (No. 07JA630048), and NCET (No. 06-0749).

misrepresents the data; and (3) the estimation of these parameters for jump fractional pricing model has a great influence on the pricing results. Usually, reasonably accurate estimates of the parameters require a very large sample and a convergent algorithm.

Appendix A This Appendix provides the proof of Theorem 2.2. Proof of Theorem 2.2. In a risk-neutral world, from Lemma 2.1 and Theorem 2.1 it follows that the valuation of a European put currency option can be represented as h i  h i h i −r ðT−tÞ ˜ þ H −r ðT−tÞ ˜ H H EPˆ ðK−SðTÞÞ j Ft = e d EPˆ SðTÞ1½−∞bSðTÞ≤K  ðBH ðTÞÞjFt −K⋅E˜Pˆ 1½−∞bSðTÞ≤K  ðBH ðTÞÞj Ft ; VðSðtÞ; K; T; t; σ ; rd ; rf ; −1Þ = e d H

H

where 1[−∞ b S(T) ≤ K] is an indicator function. Let BˆH ðtÞ = N

SðTÞ = SðtÞ∏i T−t = 1e

Jðti Þ

μ−λ μ JðtÞ + rf −rd σ

H

t + BH ðtÞ. Then we obtain 2

σ 2H 2H ðT −t ÞÞ: expððμ−λμ JðtÞ ÞðT−tÞ + σðBˆH ðTÞ−BˆH ðtÞÞ− 2

σ Let Sn ðTÞ = SðtÞ∏ni = 1 e Jðti Þ expððμ−λ μ JðtÞ ÞðT−tÞ + σðBˆH ðTÞ−BˆH ðtÞÞ− ðT 2H −t 2H ÞÞ. Using the independence of NT − t and J(ti) and the theory 2

2

∞

∞

λn ðT−tÞn

S n ðTÞ. of Poisson distribution with intensity λ(T − t), we have SðTÞ = ∑ PðNt = nÞSn ðTÞ = ∑ e−λðT−tÞ n! n=0 n=0 h It is clear that B̂H(t) is a new fractional Brownian motion under P̂H. Hence, setting d⁎2 = lnðK = SðtÞ∏ni = 1 e Jðti Þ Þ−ðrd −rf −λμ JðtÞ ÞðT−tÞ + σ 2 2H ðT −t 2H Þ + σ BˆH ðtÞ = σ and using Lemma 2.1, we deduce that 2

E˜Pˆ ½1½−∞bSðTÞ≤K ðBH ðTÞÞj FtH  = E˜Pˆ ½1½−∞b Bˆ H

⁎ H ðTÞ≤d2 

H

ðBˆH ðTÞÞj FtH 

d⁎2

=

1 ffi exp ∫−∞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

−

2πðT 2H −t 2H Þ

! ðx−BˆH ðtÞÞ2 dx 2ðT 2H −t 2H Þ

= Nð−d2 Þ where d2 =

Bˆ ðtÞ−d⁎2 pHffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2H −t 2H

σ

=

2

lnðSðtÞ∏ni= 1 e Jðti Þ = KÞ + ðrd −rf −λμ JðtÞ ÞðT−tÞ− ðT 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ T 2H −t 2H

2H

−t

2H

Þ

.

Girsanov Setting BH⁎(t) = B̂H(t) − σ t 2H, from the fractional   formula, we know that there exists a probability measure PH⁎ in which BH⁎(t) is a σ2 fractional Brownian motion. Let ZðtÞ = exp σ BˆH ðtÞ− t 2H . Then S(t) = Se(rd− rf)tZ(t), and from Lemma 2.1 and Lemma 2.2, we have 2

H ðr −r ÞT H E˜Pˆ ½SðTÞ1½SðTÞ≤K ðBH ðTÞÞj Ft  = Se d f E˜Pˆ ½ZðTÞ1½SðTÞ≤K ðBH ðTÞÞjFt  H

H

ðrd −rf ÞT

= Se

ZðtÞE˜P⁎ ½1½Bˆ

ðrd −rf ÞðT−tÞ

=e

⁎ H ðTÞ≤d2 

H

ðBˆH ðTÞÞ j Ft 

SðtÞE˜P⁎ ½1½SðTÞ≤K ðBH ðTÞÞjFt  H

H

= eðrd −rf ÞðT−tÞ SðtÞNð−d1 Þ: σ2

where d1 =

H

lnðSðtÞ∏ni = 1 eJðti Þ = KÞ + ðrd −rf −λ μ JðtÞ ÞðT−tÞ + ðT 2H −t 2H Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . σ T 2H −t 2H

942

W.-L. Xiao et al. / Economic Modelling 27 (2010) 935–942

From the analysis above, it is concluded that the price of a European call currency option can be written as   n λn ðT−tÞn −r ðT−tÞ Jðt Þ −ðr + λμ JðtÞ ÞðT−tÞ E n Ke d Nð−d2 Þ−SðtÞ ∏ e i e f Nð−d1 Þ : n! n=0 i=1 Expression for the case of call option can also be derived in a similar manner or be obtained by using put–call parity relationship. Thus the price of a European call currency option is given by ∞

−λðT−tÞ

VðSðtÞ; K; T; t; σ ; rd ; rf ; −1Þ = ∑ e

−rd ðT−tÞ

VðSðtÞ; K; T; t; σ ; rd ; rf ; 1Þ = e

ðE˜Pˆ ½SðTÞ1½SðTÞ N K ðBH Þj Ft −K⋅E˜Pˆ ½1½SðTÞ N K ðBH Þj Ft  H

H

∞

−λðT−tÞ

= ∑ e n=0

H

H

n

n

n λ ðT−tÞ Jðt Þ −ðr + λμ JðtÞ ÞðT−tÞ −r ðT−tÞ E n ½SðtÞ ∏ e i e f Nðd1 Þ−Ke d Nðd2 Þ: n! i=1

This completes the proof.

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