The valuation of currency options by fractional Brownian motion

This research aims to investigate a model for pricing of currency options in which value governed by the fractional Brownian motion model (FBM). The fractional partial differential equation and some Greeks are also obtained. In addition, some properties of our pricing formula and simulation studies are presented, which demonstrate that the FBM model is easy to use.

In view of this, the independency of logarithmic returns of the exchange rate was pointed out in all these studies along with the distribution of normal random variables. In addition, the empirical studies reveal that the logarithmic returns disseminations in the asset markets widely manifest excess kurtosis with high possibility of mass around the origin and in the tails, and indicate low possibility in the flanks in comparison with normal distribution of data. It means that financial return series include the properties, which are not normal, independent, linear and are self-similar, with heavy tails. Both autocorrelations and cross-correlations and also volatility clustering are considered to these properties.
In this regard, two fundamental features are considered in FBM namely self-similarity and long-range dependence. Then, employing this process is more feasible in terms of capturing the behavior from financial asset (Carbone et al. 2004;Wang et al. 2010). Although, FBM is neither a semi-martingale nor a Markov process then, we are not able to employ the conventional stochastic calculus for analyzing 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-Ito-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 FBM 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 selffinancing. We noted that this arbitrage example in discrete-time does not, however, rule out the use of FBM 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 FBM to capture the behavior of underlying asset and to obtain fractional Black-Scholes formulas for pricing options, including Necula (2002) and Bayraktar et al. (2004).
In this paper, the pricing formula is investigated for pricing currency options by using the FBM model. Furthermore, we obtain risk neutral valuation model and fractional Black-Scholes equation. Some properties and numerical studies of our pricing formula are also analyzed. "Preparations" section deals with the definition and features of the FBM process, and some results regarding quasi-conditional expectation are also investigated. In "Pricing model" section, option pricing formula for the European currency options is derived by the FBM model. "Properties of pricing formula" section describe the fractional differential equation and also investigates some Greeks of our model. We show empirical studies and simulation in "Numerical studies" section in order to indicate the efficiency of the FBM model and final section of the paper is "Conclusion".
Let f (x) = 1 A thus, the following corollary is obtained.
Corollary 5 Assume A ∈ B(R). Therefore Assume θ , w ∈ R. Then, this process considered According to the Girsanov formula, there is a measure P * such that Z * t is a new FBM. We will denote E * t [.] is a quasi-conditional expectation under P * . Consider where r shows the fixed rate of riskless interest.

Pricing model
Since, the system in finance is considered as an intricate system in investments in which investors avoid to make instant decisions after obtaining financial information in a fractional system. It means that achieving information to its threshold limit value is the major criteria for making decisions of investors rather than financial information with high flexibility. The asymmetric leptokurtic and long memory properties result from this behavior. In this regard, the beneficial model seems to be FBM model.
To derive the new currency option pricing formula in a fractional market. The following hypothesis will be provided: 1. there are no transaction costs or taxes; 2. security trading is continuous; 3. The rate of domestic interest r d and the rate of foreign interest r f are known and fixed throughout time; 4. There are no riskfree arbitrage opportunities. Now, we consider a fractional Black-Scholes currency market that has two investments: (a) a money market account where r d show the rate of domestic interest. (b) a stock whose price satisfies the following equation:

Properties of pricing formula
Assume that V is the value of currency options which depends just on t and S t . Thus, the value of whole portfolio satisfies in the partial differential equation that present in this theorem.
Theorem 10 The value of a currency options V (t, S t ) satisfies in the following PDE Now, we discuss the properties of the FBM model such as Greeks, which summarize how option prices change with respect to underlying variables that are critically important in asset pricing and risk management. In addition, it can be used to rebalance the portfolio to achieve desired exposure to a certain risk. It is significant to note that, knowing the Greek, a particular exposure can be hedged from adverse changes in the market by employing the appropriate amount of other related financial instruments. Contrary to option prices, observed in the market, Greeks can not be found and have to be calculated by a model assumption. Typically, the Greeks are computed using a partial differentiation of the price formula Shokrollahi et al. ( , 2016.

Theorem 11 The Greeks can be written as
The Hurst parameter H play a significant role in the FBM model. Then, we represents the influence of this parameter in the following theorm.

Theorem 12
The impact of the Hurst parameter as follows Fig. 1 shows the impact of parameters on our pricing formula.

The following theorem presents the estimation of volatility by R / S method.
Theorem 13 Assume 0 ≤ T 1 < T 2 be given, and let a partition of this interval is chosen,

Suppose S t i show the time series of observed price. Thus, the volatility of interval [T 1 , T 2 ] is
Remark 14 The relationship of call-put parity is given by

Remark 15 The relationship of put-call parity satisfies
Remark 16 The delta of spot exercise price has a space-homogeneity feature, such that for every b > 0, and Furthermore, differenting both sides with under b and thus by b = 1 we have and In fact, these equation is other model of the pricing currency option, when the value of stock is measured in a various unit. Moreover, C ′ S t (t, S t ), C ′ K (t, S t ), P ′ S t (t, S t ) and P ′ K (t, S t ) can be obtained by comparing this model with Eqs. (15), (17). These methods gives a new model for calculate delta.

Numerical studies
This section deals with how implement the FBM model and shows the impact of Hurst parameter H. In the present study, we consider the real call currency options values from Philadelphia Stock exchange (PHLX) in order to investigate some information concerning our pricing formula. By applying the R/S method, we estimate the exponent parameter for EUR/USD and then we obtain H = 0.6102. Furthermore, the volatility estimation is obtained by utilizing the historical volatility as follows; where q i show the daily value of exchange rate.
These data are extracted from 01/06/2010 to 01/12/2010 (six months) with the following parameters: K = 1.35, σ = 0.1201, r d = 0.0231, r f = 0.0352, T = 0.5, and t = 0.1. We use the MATLAB software for obtaining results by different models such as G-K, BS and FBM models. The values calculated by these models are represented in Table 1, where P Actual indicates the price of call currency options from PHLX, and the P BS is the values computed by the BS model. In addition, the P FBM points to the values calculated by FBM model. According to Table 1 our findings are more consistent with the actual price rather than the results of the other models. These properties reveal that our FBM model is able to get the behavior from financial market, which leads to creation of a satisfactory currency pricing model. To further understand the preference of the FBM model, we calculated the theoretical prices of the our pricing formula and then we compare it with derived results from the G-K model and the BS model. For our propose, these parameter valuation are selected: r d = 0.0210, r f = 0.0320, σ = 0.1050, t = 0.1, H = 0.78, S t = 49 for out-of-the-money case, S t = 61 for in-the-money case with different exercise price K ∈ [50, 60] and expiration date, T ∈ [0.11, 20]. Figures 2 and 3 show the theoretical value discrepancy by the G-K model, FBM model and BS model, for in-the-money case and out-of-the-money case, respectively. These figures reveal that our pricing model are better matched with the G-K model. Then, from Table 1 and Figs. 2 and 3, we can conclude that our FBM model seems reasonable.

Conclusion
This study provided a new framework for pricing currency options in accordance with the FBM model to capture long-memory property of the spot exchange rate. In addition, a obtained a new formula for pricing European call currency options and the volatility estimation were presented. Some certain features and Greeks of currency options model are also obtained. Finally, we reported the empirical results for several models, which demonstrate that the FBM model would be reasonable.
and the last equality follows since σ B H t = ln K S − (r d − r f )t + σ 2 2 t 2H . Now, we consider E t [S T 1 S T >K ]; setting Let Then we have X t = e −rt S t . According to the Lemma 6, we obtain But By setting d * 1 = ln K S − (r d − r f )T − 1 2 σ 2 T 2H , we obtain The last equality follows since (42) E t S T 1 S T >K = e rt E t X T 1 x>d * 2 σ B H T = e rt X t E * t 1 x>d * 2 σ B H T = e rt X t E * t 1 S T >K . (44)