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Foreign Exchange Rate Modeling Tom´ aˇs Tich´ y1

Abstract Foreign exchange rates market is probably the most liquid financial market segment. It plays very important role in financial risk management and economic activities of almost all subjects. The high liquidity and efficiency can lead to the presence of peak and skew in the distribution of returns. In this paper we look more closely at the evolution of three particular FX rates (EUR, GBP, USD) with respect to CZK. We estimate the density of empirical returns and compare it with normal and variance gamma density. We show, that although none of them fit the distribution perfectly, the variance gamma fits better especially the tails. Keywords Foreign exchange rate, stochastic processes, diffusion, jumps, L´evy models, Variance gamma.

1.

Introduction

Foreign exchange rates market belongs to the most efficient and liquid segments of financial markets. Each relevant information is adapted into the market price immediately after its observation. It can lead to excess kurtosis of the probability distribution of returns and presence of jumps. It is given by the fact, that the information come to the day light with various intensity. Moreover, since the information with either positive or negative price influence can be more frequent, also the skew of the returns can be significant. Standard approaches to model the evolution of financial quantities (such as stock prices, interest rates, foreign exchange rates) are based on normal distribution (through the application of Wiener process or Brownian motion) with possible combination with Poisson distribution (pure jump process). More advanced models belong to the family of L´evy process, processes with infinite activity. Up to now, there have been introduced many various types of such models, mostly defined as a subordinated (geometric) Brownian motion. Hence, the classical time is replaced by another suitable process to model the arrival of new information. In this paper, we try to examine if and how much the Variance gamma model – defined as a geometric Brownian motion subordinated by gamma-time process – overcomes the classical geometric Brownian motion in modeling the evolution of three particular foreign exchange rates. We proceed as follows. Section 2 is devoted to review of several suitable models of financial price evolution. Next, in Section 3 we briefly describes the evolution of selected ˇ Ing. Tom´ aˇs Tich´ y, Ph.D., Department of Finance, Faculty of Economics, VSB-TU Ostrava, Sokolsk´a 33, 701 21 Ostrava, Czech Republic. E-mail: [email protected]. ˇ (Czech Science Foundation – This paper is based on research activities sponsored through GACR ˇ e Republiky) project No. 402/05/2758. All support is greatly acknowledged. Grantov´ a Agentura Cesk´ 1

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exchange rates (levels and returns) during the period since 1 January 2000 to 30 September 2006. Finally, in Section 4 we estimate the parameters of the model. Subsequently, on the basis of Monte Carlo simulation we try to price the Asian option and compare the results with geometric Brownian motion and empirical distribution function.

2.

Suitable stochastic processes

The most standard models to describe the stochastic evolution of financial prices can consists of the following parts – a drift term (µdt), a diffusion term (σdZ), a jump term (dJ = kdq), and a mean reversion term (αβdt), where dt is infinitesimal time interval, µ is the long-run average of the price return, σ is a volatility (diffusion coefficient), k is a price increment resulting from jump occurrence (possibly of random value) and dq is a Poisson process, which gives value one with intenzity λ (if a jump occurs), α is the velocity of the mean mean reversion and β describes the tendency of mean reversion. It is based on the difference between (or ratio of) the actual price level and its long-run equilibrium. While the drift term is constant and linear in its basic form (hence, it is the riskless part of the overall evolution) for any future time period, the mean-reversion term is known in advance only for the next time moment. Opposite to that, the diffusion and jump parts reflect the randomness. The diffusion part is continuous in any time interval (even if nondifferentiable). By contrast, the jump part is neither continuous nor differentiable for the whole path. Denote the increment of an asset price within infinitesimal time interval as dS. Then, we can formulate either diffusion process: dS = µSdt + σSdZdt ,

(1)

dS = (µ − λk)Sdt + SdJdt ,

(2)

dS = (µ − λk)Sdt + σdZdt + SdJdt ,

(3)

µ ¶ L dS = α ln St dt + σSt dZdt . St

(4)

jump process: jump-diffusion process:

or mean-reverting process:

The latter one is an example of the model on the basis of continuously defined returns, so that β is defined as ln SLt , where L is long run equilibrium of the asset price and St is its actual value. Another feasible way of modeling price returns is to use an example of a L´evy model family. Under the family of L´evy models, so called in honor of Paul L´evy, are generally understood such processes that are of independent and stationary increments. These processes are also typical by the stochastic continuity – the probability of jump occurrence for given time t is zero, and exhibit infinite activity. As before, the L´evy process can be decomposed into a diffusion part and a jump part. Clearly, not all parts must be present. Very interesting property of several L´evy-type models is that the modeling of higher moments of the underlying distribution (more particularly, the skewness and kurtosis) is 373

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also feasible. Such group of models is defined on the basis of Brownian motion – the only difference is that the standard time t is replaced by an internal random time, modeled with respect to suitable distribution. In case of Variance gamma model (VG model) it is the gamma distribution. Hence, we get: · ¸ p dt (5) dX = θ˜ gdt + ϑ˜ ε g˜dt , g˜dt ∈ G ;ν . ν The term X can be put into any of models specified above or it can replace particular part of such model. Since the intensity of jumps is infinite, it is supposed, that it can replace both, the jump as well as the diffusion part of the model. However, we should respect the fact, that the expected value of X is non-zero, E[X ] = θdt. Thus, if we need the expected price increment to be based only on µ term, we must deduce θdt. Alternatively, if we model the process in the exponential form, we must deduce E[exp(X )]. In this paper we will use either the geometric Brownian motion (GBM): · ¸ √ σ2 St+dt = St exp µdt − dt + σ ε˜ dt , (6) 2 which is a diffusion model with deterministic drift, or Variance gamma model (VG model): h p i (7) St+dt = St exp µdt − ωdt + θ˜ gdt + ϑ˜ ε g˜dt . ¡ ¢ Here, ω is mean correcting parameter of the form ω = − ν1 ln 1 − θν − 12 ϑ2 ν . Note, that the pricing of financial derivatives is totally different issue. It is usually done within the risk-neutral world. Hence, each model must be reformulated so that the asset provides risk-less return, r. For example, denote statistical probabilities by P and equivalent risk neutral ones by Q: EP [St+dt ] = St exp[µ] ⇒ EQ [St+dt ] = St exp[r].

(8)

Of course, if the asset provides any other return, such as dividend yield, storage costs or convenience yield, the formulation must be adjusted. The big difference between GBM and VG model is that when we move from the real setting P to its risk-neutral counterpart Q, we should change also other parameters of VG model (θ, ϑ, ν). (Note, that within GBM, the volatility should stay the same.)

3.

Foreign exchange rate evolution

In this paper, we study the following foreign exchange rates: EUR/CZK, GBP/CZK, and USD/CZK. The time series consist of 1700 data – official exchange rates as published by ˇ CNB (Czech National Bank) over the period starting in January, 2000 up to September, 2006. The evolution of particular exchange rates is illustrated in Figure 1 (top). Relevant log-returns are depicted at the bottom of Figure 1. Each exchange rate seems to be decreasing in time, without any mean reversion tendency. During few intervals, abnormal movements of the rate can be identified, especially for EUR-rate. By contrast, for USDrate such movements seem to occur scarcely. 374

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Figure 1: Daily evolution of EUR, GBP, and USD exchange rates and the continuous returns We have also calculated several basic characteristics of returns – the mean, standard deviation, skewness and kurtosis, each for three distinct time intervals: 1-day, 5-days and 10-days,2 see Table 1. It is little surprising, that both, the mean and the volatility of exchange rate returns are relatively stable, regardless the length of the time interval. By contrast, it is not true for skewness and kurtosis (which would be more surprising). We can conclude, that the USD-rate is the most volatile, but the EUR-rate exhibit the highest kurtosis – the arrival of information with high and rapid impact on the exchange rate level is more frequent. Table 1: Basic parameters of exchange rate returns (p.a.) τ days 1 5 10

4.

µ −0.036 −0.035 −0.035

EUR/CZK σ skew 0.05 −0.24 0.05 −0.01 0.05 −0.39

kurt 7.28 4.43 4.12

GBP/CZK σ skew 0.08 −0.42 0.08 0.05 0.07 −0.20

µ −0.05 −0.05 −0.05

kurt 5.24 3.58 3.21

µ −0.07 −0.07 −0.07

USD/CZK σ skew 0.11 −0.14 0.11 −0.04 0.11 0.14

kurt 3.83 3.19 2.86

Underlying process

In order to identify the process which is followed by selected foreign exchange rates we first try to estimate the empirical distribution. In order to get its shape we apply so called Gaussian kernel function K(x). It is defined as follows: 2

exp[ −x ] K(x) = √ 2 . 2π

(9)

Since the kernel density estimator, N

1 X fˆh (x) = K Nh n=1 2

µ

xn − x h

¶ ,

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1

depends on bandwidth h, we set it as h = 1.06σN− 5 , where N is the number of observations. The results are depicted in Figure 2. The left part illustrates the empirical distribution of each foreign exchange rate. We can see that the EUR-rate is really highly peaked. It seems that the USD-rate is the most normal. In order to examine the distribution of tails, we have also evaluated the natural logarithm of the empirical density, see Figure 2, the right part. Clearly, neither EUR-rate nor GBP-rate resemble the normal distribution, especially the left tails seem to be very instable. By contrast, the decay of USD-rate tails seems to be rather quadratic, as supposed by geometric Brownian motion. Although we cannot expect that either GBM or VG model could fit the tails of the empirical distribution precisely, we hope that the VG model will do that at least partly and beat GBM in this way. Since each foreign exchange rate process exhibits various standard deviations, it should be better to interpret particular results separately. First, we examine the results for the exchange rate of EUR/CZK. According to empirical estimation of basic statistics, Table 1, we set these two models as follows, GBM(−0.00014, 0.0034) and VG(−0.0002, 0.0034, 1.4152).3 Following Figure 3 provides results of how good is the fitting of both models (GBM and VG) to the empirical distribution.4 The simple density function allows us to compare mainly the middle part of the distribution and the presence of peaks. By contrast, the log-densities are better when checking the distribution of tails. The results are little surprising. Although the empirical kurtosis is high, the peak is not so high as we expect (due to VG model). In other words it lies somewhere between normal and VG distribution. The log-densities of normal and VG distribution differs due to the type of decay – the normal distribution exhibit quadratic, the VG is rather linear. However, the empirical one seems to have two parts, quadratic around zero and the rest resemble linear. Especially the nonstability of left tails can substantially influence the overall peak and skew. Finally, if we must choose between normal and VG distribution, the latter seems to be a better choice. Furthermore, we have examine also the returns due to one-week and two-week period. 3

Now, we work with values per day for GBM(µ, σ) and VG(θ, ϑ, ν). In each case, solid grey line depicts the empirical distribution, solid black line the VG-density and dotted black one the normal density. 4

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It means, that we have worked with GBM(−0.0007, 0.0072)—VG(−0.00006, 0.0072, 0.48) and GBM(−0.0014, 0.0097)—VG(−0.0034, 0.0095, 0.37). Since the results do not differ significantly, we provide only the 1-week based (Figure 3, bottom). We can see that due to the simple density, the normal model fits the empirical distribution much better. As before, with log-densities, the distribution seems to be of quadratic decay around zero and linear decay otherwise. Hence, if we are much more interested in the tail distribution, we should prefer the VG model, although also in this case, through the instability of empirical tails, significant deviations can be present. Next, we examine the data of GBP-rate and USD-rate. This time, however, we graphically present only the density of daily returns, Figure 4 (GBP) and Figure 5 (USD). Note, that with lower frequency of data, the distribution starts to be almost normal and also the jumps (or instability of tails) are scarcely present. Due to the results, it is obvious, that the normal density fits the middle part of the distribution much better, since the VG model overestimates the peak. By contrast, the decay of the tails is clearly linear, which can be fitted only by the VG model. We end this part of the paper by conclusion that we did not identify any general and significant difference between particular series.

4.1

Asian option valuation

In this subsection we try to evaluate the Asian option with the following payoff function: # " 5 1X St − K; 0 , (11) Ψ = max 5 t=1 where K is the exercise price, the time to maturity is one week (five business days). We set the exercise price to be equal the initial exchange rate (of each period). First we evaluate the EUR-rate. Suppose that the rate at time zero is 28. We also suppose, for simplicity, that the interest rate difference is insignificant within this period. Moreover, we produce only the payoff and take it relatively to the initial foreign exchange rate. We run 100 000 scenarios within Geometric Brownian motion and Variance gamma model. Next, we evaluate the option payoff due to the time series. The results are included in Table 2. We can see that the VG model gives us almost the same result as the time series. By contrast, the GBM significantly overestimates the option price. Table 2: Asian option payoff (relative results) FX rate EUR GBP USD

VG 0.00127 0.0023 0.0030

GBM 0.00167 0.0026 0.0035

empirical 0.00129 0.0021 0.0031

We apply the same procedure to evaluate the GBP-rate (suppose the initial level to be 42) and the USD-rate (the initial level is 23). It is evident, that for both, the GBP-rate and the USD-rate, the VG model is again much close to the empirical payoff, while GBM is significantly overestimating the option payoff.

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Conclusions

In this paper we have examined the evolution of three distinct foreign exchange rates with respect to the CZK, more particularly the EUR, GBP, and USD. We have estimate the empirical density function and compare it to two stochastic models, standard one – the Geometric Brownian motion, based on normal density, and more advanced Variance gamma model, based on parametric variance gamma density. Although the returns exhibit significant kurtosis and some skewness, it is probably not true for middle returns – simple normal density fits the distribution better. By contrast, we have shown that the tails have linear rather than quadratic decay. It is therefore better to model it by VG process. Probably, it is main reason, why the VG model done much better in modeling the Asian option payoff. Anyway, in order to mimic the future evolution of each foreign exchange rate studied here, more complex model should be used. For example, variance gamma model including diffusion part or the model with two sources of jumps. Moreover, it was obvious that the skewness and kurtosis are not stable in time. Thus, adding of another parametr allowing to model their stochastic feature should improve the efficiency of applications. Concerning the pricing issue, opposite to the application provided here, we should be aware of applying the statistical parameters, since the risk-neutral ones should be used.

References [1] BATES, D.S. The crash premium: Option pricing under asymmetric processes, with application to options on Deutchemark futures, Working paper, University of Pennsylvania, 1988. [2] BLACK, F., SCHOLES, M. The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81 (May-June 1973), 637–659, 1973. [3] BOYLE, P. Options: a Monte Carlo approach. Journal of Financial Economics 4, 323–338, 1977. [4] CARR, P., GEMAN, H., MADAN, D.B., YOR, M. The fine structure of asset returns: An empirical investigation, Journal of Business 75, 305–332, 2002. [5] CARR, P., GEMAN, H., MADAN, D.B., YOR, M. Stochastic Volatility for L´evy Processes, Mathematical Finance 13, 345–382, 2003. [6] CONT, R., TANKOV, P. Financial Modelling with Jump Processes. Chapman & Hall/CRC press. 2004. ˇ A, ´ D. et al. New Approaches and Financial Intruments in Financial [7] DLUHOSOV Decision-Making – Nov´e pˇr´ıstupy a finanˇcn´ı n´astroje ve finanˇcn´ım rozhodov´ an´ı, ˇ VSB-TU Ostrava, 2004. [8] GLASSERMAN, P. Monte Carlo Methods in Financial Engineering, SpringerVerlag, 2004. [9] HULL, J.C., WHITE, A. The pricing of options on assets with stochastic volatility. Journal of Finance 42, 281–300, 1987.

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[10] MADAN, D.B., SENETA, E. The VG model for Share Market Returns, Journal of Business 63 (4), 511–524, 1990. [11] MADAN, D.B., CARR, P., CHANG, E.C. The variance gamma process and option pricing, European Finance Review 2, 79–105, 1998. ´ T. Finanˇcn´ı deriv´aty – typologie finanˇcn´ıch deriv´atu, [12] TICHY, ˙ podkladov´e procesy, ˇ oceˇ novac´ı modely, VSB-TU Ostrava, 2006. ´ T. Model Dependency of the Digital Option Replication: Replication under [13] TICHY, ´ er – Czech Journal of Economics and Finance 56 Incomplete Model. Finance a Uvˇ (7-8), 361–379, 2006. ˇ ˇ A, ´ D., TICHY, ´ T. Financial Models, VSB-TU ˇ [14] ZMESKAL, Z., DLUHOSOV Ostrava, 2004.

Summary Modelov´ an´ı mˇ enov´ eho kurzu Trh se zahraniˇcn´ımi mˇenami pˇredstavuje pravdˇepodobnˇe nejlikvidnˇejˇs´ı souˇca´st finanˇcn´ıho trhu. Mˇenov´e kurzy jsou z´aroveˇ n duleˇ ˙ zitou souˇc´ast´ı ˇr´ızen´ı finanˇcn´ıho rizika a ekonomick´e aktivity t´emˇeˇr vˇsech subjektu. ˙ Vysok´a likvidita a efektivnost muˇ ˙ ze v´est k pˇr´ıtomnosti ˇspiˇcatosti a ˇsikmosti v´ ynosu. ˙ V tomto ˇcl´anku je pozornost zmˇeˇrena na v´ yvoj tˇr´ı vybran´ ych kurzu˙ (EUR, GBP, USD) vzhledem k ˇcesk´e korunˇe (CZK). Nejprve je odhadnuta funkce hustoty v´ ynosu˙ na z´akladˇe ˇcasov´e ˇrady. Ta je pak srovn´ana s funkc´ı hustoty norm´aln´ı rozloˇzen´ı a Variance gamma procesu. Je uk´az´ano, ˇze i kdyˇz ˇza´dn´ y z porovn´avan´ ych modelu˙ zcela nevystihuje rozloˇzen´ı v´ ynosu, ˙ druh´ y jmenovan´ y vystihuje l´epe zejm´ena tˇeˇzk´e konce.

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