Subdiffusive fractional Black–Scholes model for pricing currency options under transaction costs

A new framework for pricing European currency option is developed in the case where the spot exchange rate follows a subdiffusive fractional Black– Scholes. An analytic formula for pricing European currency call option is proposed by a mean self-financing delta-hedging argument in a discrete time setting. The minimal price of a currency option under transaction costs is obtained as time-step Δt 1⁄4 tα 1 ΓðαÞ 1 2 π 1 2H k σ 1 , which can be used as the actual price of an option. In addition, we also show that time-step and long-range dependence have a significant impact on option pricing. Subjects: Science; Mathematics & Statistics; Applied Mathematics; Statistics & Probability


Introduction
The standard European currency option valuation model has been presented by Garman and Kohlhagen ðG À KÞ (Garman & Kohlhagen, 1983). However, some papers have provided evidence of the mispricing for currency options by the G À K model.

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Subdiffusion refers to a well-known and established phenomenon in statistical physics. One description of subdiffusion is related to subordination, where the standard diffusion process is time-changed by the so-called inverse subordinator. According to the features of the subdiffusion process and the fractional Brownian motion, we propose the new model for pricing European currency options by using the fractional Brownian motion, subdiffusive strategy, and scaling time in discrete time setting, to get behavior from financial markets. Motivated by this objective, we illustrate how to price a currency options in discrete time setting for both cases: with and without transaction costs by applying subdiffusive fractional Brownian motion model. By considering the empirical data, we will demonstrate that the proposed model is further flexible in comparison with the previous models and it obtains suitable benchmark for pricing currency options. Additionally, impact of the parameters on our pricing formula is investigated. 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, Jennergren, & Näslund, 1997). Since then, many methodologies for currency option pricing have been proposed by using modifications of the G À K model (Garman & Kohlhagen, 1983;Ho, Stapleton, & Subrahmanyam, 1995).
All this research above assumes that the logarithmic returns of the exchange rate are independent identically distributed normal random variables. However, in general, the assumptions of the Gaussianity and mutual independence of underlying asset log returns would not hold. Moreover, the empirical research has 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 (Dai & Singleton, 2000). That is to say the features of financial return series are non-normality, non-independence, and nonlinearity. To capture these non-normal behaviors, many researchers have considered other distributions with fat tails such as the Pareto-stable distribution and the Generalized Hyperbolic Distribution. 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. As fractional Brownian motion ðFBMÞ 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 (Borovkov, Mishura, Novikov, & Zhitlukhin, 2018;Shokrollahi & Sottinen, 2017).
The fractional Black-Scholes ðFBSÞ model is an extension of the Black-Scholes ðBSÞ model, which displays the long-range dependence observed in empirical data. This model is based on replacing the classic Brownian motion by the fractional Brownian motion ðFBMÞ in the Black-Scholes model. That iŝ (1:1) where μ; and σ are fixed, andB H ðtÞ is a FBM with Hurst parameter H 2 ½ 1 2 ; 1Þ. It has been shown that the FBS model admits arbitrage in a complete and frictionless market (Cheridito, 2003;Sottinen & Valkeila, 2003;Wang, Zhu, Tang, & Yan, 2010;Xiao, Zhang, Zhang, & Wang, 2010). Wang (2010) resolved this contradiction by giving up the arbitrage argument and examining option replication in the presence of proportional transaction costs in discrete time setting (Mastinšek, 2006). Magdziarz (2009a) applied the subdiffusive mechanism of trapping events to describe properly financial data exhibiting periods of constant values and introduced the subdiffusive geometric Brownian motion V α ðtÞ ¼ VðT α ðtÞÞ; (1:2) as the model of asset prices exhibiting subdiffusive dynamics, where V α ðtÞ is a subordinated process (for the notion of subordinated processes please refer to Refs. Weron (1993, 1995), Kumar, Wyłomańska, Połoczański, and Sundar (2017), Piryatinska, Saichev, and Woyczynski (2005), in which the parent process VðτÞ is a geometric Brownian motion and T α ðtÞ is the inverse α-stable subordinator defined as follows: Here, Q α ðtÞ is a strictly increasing α-stable subordinator with Laplace transform: E e ÀηQαðτÞ À Á ¼ e Àτη α , 0 < α < 1, where E denotes the mathematical expectation. Magdziarz (2009a) demonstrated that the considered model is free-arbitrage but is incomplete and proposed the corresponding subdiffusive BS formula for the fair prices of European options.
Subdiffusion is a well-known and established phenomenon in statistical physics. The usual model of subdiffusion in physics is developed in terms of FFPE (fractional Fokker-Planck equations). This equation was first derived from the continuous-time random walk scheme with heavy-tailed waiting times (Metzler & Klafter, 2000). It provides a useful way for the description of transport dynamics in complex systems (Magdziarz, Weron, & Weron, 2007). Another description of subdiffusion is in terms of subordination, where the standard diffusion process is time-changed by the so-called inverse subordinator (Gu, Liang, & Zhang, 2012;Guo, 2017;Janczura, Orzeł, & Wyłomańska, 2011;Magdziarz, 2009b, Magdziarz et al., 2007Scalas, Gorenflo, & Mainardi, 2000Yang, 2017).
The objective of this paper is to study the European call currency option by a mean self financing delta hedging argument. The main contribution of this paper is to derive an analytical formula for European call currency option without using the arbitrage argument in discrete time setting when the exchange rate follows a subdiffusive FBS (1:4) S 0 ¼Vð0Þ > 0: We then apply the result to value European put currency option. We also provide representative numerical results.
Making the change of variable, B H ðtÞ ¼ (1:5) This formula is similar to the Black-Scholes option pricing formula, but with the volatility being different.
We denote the subordinated process W α;H ðtÞ ¼ B H ðT α ðtÞÞ, here the parent process B H ðτÞ is a FBM and T α ðtÞ is assumed to be independent of B H ðτÞ. The process W α;H ðtÞ is called a subdiffusion process. Particularly, when H ¼ 1 2 , it is a subdiffusion process presented in Karipova and Magdziarz (2017), Kumar et al. (2017), and Magdziarz (2010). Figure 1 shows typically the differences and relationships between the sample paths of the spot exchange rate in the FBS model and the subdiffusive FBS model. The rest of the paper proceeds as follows: In Section 2, we provide an analytic pricing formula for the European currency option in the subdiffusive FBS environment and some Greeks of our pricing model are also obtained. Section 3 is devoted to analyze the impact of scaling and long-range dependence on currency option pricing. Moreover, the comparison of our subdiffusive FBS model and traditional models is undertaken in this section. Finally, Section 4 draws the concluding remarks. The proof of Theorems are provided in Appendix.

Pricing model for the European call currency option
In this section, we derive a pricing formula for the European call currency option of the subdiffusive FBS model under the following assumptions: (i) We consider two possible investments: (1) a stock whose price satisfies the equation: 1Þ, α þ αH > 1, and r d ; and r f are the domestic and the foreign interest rates, respectively. (2) A money market account: ( 2:2) where r d shows the domestic interest rate.
(ii) The stock pays no dividends or other distributions, and all securities are perfectly divisible. There are no penalties to short selling. It is possible to borrow any fraction of the price of a security to buy it or to hold it, at the short-term interest rate. These are the same valuation policy as in the BS model. (iii) There are transaction costs that are proportional to the value of the transaction in the underlying stock. Let k denote the round trip transaction cost per unit dollar of transaction. Suppose U shares of the underlying stock are bought ðU > 0Þ or sold ðU < 0Þ at the price S t , then the transaction cost is given by k 2 U j jS t in either buying or selling. Moreover, trading takes place only at discrete intervals. (iv) The option value is replicated by a replicating portfolio Å with UðtÞ units of stock and riskless bonds with value FðtÞ. The value of the option must equal the value of the replicating portfolio to reduce (but not to avoid) arbitrage opportunities and be consistent with economic equilibrium. (v) The expected return for a hedged portfolio is equal to that from an option. The portfolio is revised every Δt and hedging takes place at equidistant time points with rebalancing intervals of (equal) length Δt, where Δt is a finite and fixed, small time-step.
Let C ¼ Cðt; S t Þ be the price of a European currency option at time t with a strike price K that matures at time T. Then, the pricing formula for currency call option is given by the following theorem.
Theorem 2.1. C ¼ Cðt; S t Þ is the value of the European currency call option on the stock S t satisfied (1.5) and the trading takes place discretely with rebalancing intervals of length Δt. Then, C satisfies the partial differential equation with boundary condition CðT; S T Þ ¼ max S T À K; 0 f g . The value of the currency call option is Cðt; S t Þ ¼ S t e Àr f ðTÀtÞ Φðd 1 Þ À Ke Àr d ðTÀtÞ Φðd 2 Þ; (2:6) and the value of the put currency option is Pðt; S t Þ ¼ Ke Àr d ðTÀtÞ ΦðÀd 2 Þ À S t e Àr f ðTÀtÞ ΦðÀd 1 Þ; (2:7) where (2:8) (2:9) where Φð:Þ is the cumulative normal distribution function.
In what follows, the properties of the subdiffusive FBS model are discussed, such as Greeks, which summarize how option prices change with respect to underlying variables and are critically important to asset pricing and risk management. The model can be used to rebalance a portfolio to achieve the desired exposure to certain risk. More importantly, by knowing the Greeks, particular exposure can be hedged from adverse changes in the market by using appropriate amounts of other related financial instruments. In contrast to option prices that can be observed in the market, Greeks cannot be observed and must be calculated given a model assumption. The Greeks are typically computed using a partial differentiation of the price formula.
Letting α " 1, from Equation (2.9), we obtain Remark 2.3. The modified volatility under transaction costs is given bŷ ( 2:19) that is in line with the findings in Wang (2010).

Empirical studies
The objective of this section is to obtain the minimal price of an option with transaction costs and to show the impact of time scaling Δt, transaction costs k, and subordinator parameter α on the subdiffusive FBS model. Moreover, in the last part, we compute the currency option prices using our model and make comparisons with the results of the G À K and FBS models.
Moreover, the option rehedging time interval for traders to take is Δt ¼ t αÀ1 ΓðαÞ À1 2 π À Á 1 2H k σ À Á1 H . The minimal price C min ðt; S t Þ can be used as the actual price of an option.

ΓðαÞ
shows that fractal scaling Δt has not any impact on option pricing if a mean selffinancing delta-hedging strategy is applied in a discrete time setting, while subordinator parameter β has remarkable impact on option pricing in this case. In particular, from Equations (3.4) and ( ! , which displays that the fractal scaling Δt and sabordinator parameter α have a significant impact on option pricing. Furthermore, for kÞ0, from Equation (2.8), we know that option pricing is scaling dependent in general. Now, we present the values of currency call option using subdiffusive FBS model for different parameters. For the sake of simplicity, we will just consider the out-of-the-money case. Indeed, using the same method, one can also discuss the remaining cases: in-the-money and at-the-money. First, the prices of our subdiffusive FBS model are investigated for some Δt and prices for different exponent parameters. The prices of the call currency option versus its parameters H; Δt; α and k are revealed in Figure 2. The selected parameters are S t ¼ 1:4; K ¼ 1:5; σ ¼ 0:1; r d ¼ 0:03; r f ¼ 0:02; T ¼ 1; t ¼ 0:1; Δt ¼ 0:01; k ¼ 0:01; H ¼ 0:8; α ¼ 0:9: Figure 2 indicates that the option price is an increasing function of k and Δt, while it is a decreasing function of H and α.
For a detailed analysis of our model, the prices calculated by the G À K, FBS and subdiffusive FBS models are compared for both out-of-the-money and in-the-money cases. The following parameters are chosen: S t ¼ 1:2; σ ¼ 0:5; r d ¼ 0:05; r f ¼ 0:01; t ¼ 0:1; Δt ¼ 0:01; k ¼ 0:001, and H ¼ 0:8, along with time maturity T 2 ½0:1; 2, strike price K 2 ½0:8; 1:19 for the in-the-money case and K 2 ½1:21; 1:4 for the out-of-the-money case. Figures 3 and 4 show the theoretical values difference by the G À K, FBS, and our subdiffusive FBS models for the in-the-money and out-of-the-money, respectively. As indicated in these figures, the values computed by our subdiffusive FBS model are better fitted to the G À K values than the FBS model for both in-the-money and out-of-the money cases. Hence, when compared to these figures, our subdiffusive FBS model seems reasonable.

Conclusion
Without using the arbitrage argument, in this paper, we derive a European currency option pricing model with transaction costs to capture the behavior of the spot exchange rate price, where the spot exchange rate follows a subdiffusive FBS with transaction costs. In discrete time case, we show that the time scaling Δt and the Hurst exponent H play an important role in option pricing with or without transaction costs and option pricing is scaling dependent. In particular, the minimal price of an option under transaction costs is obtained.  Figure 3. Relative difference between the G À K, FBS, and subdiffusive FBS models for the in-the-money case.