Mixed Fractional Merton Model to Evaluate European Options with Transaction Costs

This paper deals with the problem of discrete-time option pricing by the mixed fractional version of Merton model with transaction costs. By a mean-self-financing delta hedging argument in a discrete-time setting, a European call option pricing formula is obtained. We also investigate the effect of the time-step δt and the Hurst parameter H on our pricing option model, which reveals that these parameters have high impact on option pricing. The properties of this model are also explained.


Introduction
Over the last few years, the financial markets have been regarded as complex and nonlinear dynamic systems. A series of studies has found that many financial market time series display scaling laws and long-range dependence. Therefore, it has been proposed that the Brownian motion in the classical Black-Scholes (BS) model [1] should be replaced by a process with long-range dependence.
Nowadays, the BS model is the one most commonly used for analyzing financial data, and some scholars have presented modified forms of the BS model which have influential and significant outcomes on option pricing. However, they are still theoretical adaptations and not necessarily consistent with the empirical features of financial return series, such as nonnormality, long-range dependence, etc. For example, some scholars [2] [3] [4] [5] [6] have showed that returns are of long-range (or short-range) dependence, which suggests strong time-correlations between different events at different time How to cite this paper: Shokrollahi [20] [21] [22].
The above researches have an important implication for option pricing. Merton [23] created a revolution in option pricing when the underlying asset was governed by a diffusion process. Based on this theory, Kou [24], Cont and Tankov [25] also considered the problems of pricing options under a jump diffusion environment in a larger setting. In this paper, to capture jumps or discontinuities, fluctuations and to take into account the long memory property of financial markets, a mixed fractional version of the Merton model is introduced, which is based on a combination of Poisson jumps and MFBM. The mixed fractional Merton (MFM) model is based on the assumption that the underlying asset price is generated by a two-part stochastic process: 1) small, continuous price movements are generated by an MFBM, 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 impressive market in which major information arrives infrequently and randomly. This process may provide a description for empirically observed distributions of exchange rate changes that are skewed, leptokurtic, have long memory and fatter tails than comparable normal distributions and apparent nonstationary variance. Further, we will show the impact of the time-step and long-range dependence in return series exactly on option pricing, regardless of whether proportional transaction costs are considered or not in a discrete time setting.
Leland [26] is a pioneer scholar, who investigated option replication where transaction costs exist in a discrete time setting. In this view, the arbitrage-free arguments presented by Black and Scholes [1] are not applicable in a model where transaction costs occur at all moments of trading of the stock or bond. The problem is that perfect replication incurs an infinite number of transaction costs because of the infinite variation which exists in the geometric Brownian motion. In this regard, a delta hedge strategy is constructed in accordance with revision conducted a discrete number of times. Transaction costs lead to the failure of the no arbitrage principle and the continuous time trade in general: instead of no arbitrage, the principle of hedge pricing, according to which the price of an option is defined as the minimum level of initial wealth needed to hedge the option, comes into force.
According to the empirical findings obtained before and the views of behavioral finance and econophysics, we are motivated to examine the problem that exists in option pricing, while the dynamics of price t S follows a mixed fractional jump-diffusion process under the transaction costs. We assume that t S satisfies

Pricing Option by Mixed Fractional Version of Merton Model with Transaction Costs
The groundwork of modeling the effects of transaction costs was done by Leland [26]. He adopted the hedging strategy of rehedging at every time-step That is, with every t δ the portfolio is rebalanced, whether or not this is optimal in any sense. In the following proportional transaction cost option pricing model, we follow the other usual assumptions in the Black-Scholes model, but with the following exceptions: 1) The price t S of the underlying stock at time t satisfies Equation (2.2).
2) The portfolio is revised every t δ where t δ is a finite and fixed, small time-step.
3) Transaction costs are proportional to the value of the transaction in the underlying. Let k denote the round trip transaction cost per unit dollar of 4) The hedge portfolio has an expected return equal to that from an option.
This is exactly the same valuation policy as earlier on discrete hedging with no transaction costs.

5) Traditional economics assumes that traders are rational and maximize their
utility. However, if their behaviour is assumed to be boundedly rational, the traders' decisions can be explained both by their reaction to the past stock price, according to a standard speculative behaviour, and by imitation of other traders' past decisions, according to common evidence in social psychology. It is well known that the delta-hedging strategy plays a central role in the theory of option pricing and that it is popularly used on the trading floor. Therefore, based on the availability heuristic, suggested by Tversky and Kahneman [27], traders are assumed to follow, anchor, and imitate the Black-Scholes delta-hedging strategy to price an option. In this case, delta-hedging argument is a partial and imperfect hedging strategy, which does not eliminate all of the risk. However, as mentioned in the paper [28], in most models of stock fluctuations, except for very special cases, risk in option trading cannot be eliminated and strict arbitrage opportunities do not exist, whatever be the price of the option. Then,

( )
, t C t S is derived by the following theorem.
sign Γ is the signum function of φ is the cumulative normal distribution.
Moreover, using the put call parity, we can easily obtain the valuation model for a put currency option, which is provided by the following corollary.
Corollary 2.1. The value of European put option with transaction costs is given by

Properties of Pricing Formula
In this section, we present the properties of MFM's log-return density. The effects of Hurst parameter and time-step on our modified volatility ( )

Log-Return Density
In the case of MFM the log return jump size is assumed to be ; , The term ( ) ( )  (Table 1).
Secondly, larger value of intensity λ (which means that jumps are expected to occur more frequently) makes the density fatter-tailed as illustrated in Figure   2. Note that the excess kurtosis in the case 20 λ = is much smaller than in the case 1 λ = or 10 λ = . This is because excess kurtosis is a standardized measure (by standard deviation) ( Table 2).

The Impact of Parameters
Mantegna and Stanley [29] as pioneer scholars proposed the scaling invariance method from the complex science of economic systems which led to numerous investigations into scaling laws in finance. The major question in economics is whether the price impact of scaling law and long-range dependence is significant in option pricing. The answer to this question is assured. For instance, one of the significant issues in finance concerning the modeling of high-frequency data is related to analyzing the volatility in different time scales.
which shows that the Hurst parameter H and time-step t δ have no effect on option pricing model in a continuous time setting ( ) 0 t δ = .
which demonstrates that the delta hedging strategy in a discrete time case is fundamentally different in comparison with a continuous time case. It also indicates that the scaling exponent 2 1 H − and time-step t δ play a significant role in option pricing theory. Figure 3 illustrates the impacts of the time-step, Hurst parameter, mean jump, and jump intensity on our option pricing model.
as k is small enough.

Conclusion
To capture the long memory and discontinuous property, this article focuses on the problem of pricing European option in a mixed fractional Merton environment without using the arbitrage argument. We obtain a mixed fractional version of Merton model for pricing European option with transaction costs. Some properties of mixed fractional Merton's log-return density are discussed. Moreover, we derive that the Hurst parameter H and time-step t δ play a significant role in pricing option in a discrete time setting for cases both with and without transaction costs. But these parameters have no impact on option pricing in a continuous time setting.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.
Now, the movement in t S and t P is considered under discrete time interval t δ . In view of this, we suppose that trading takes place at the specific time points of t and t t δ + . It can be said that the number of shares through the use of delta-hedging strategy and the present stock price t S are constantly held during the rebalancing interval [ ) . Then, the movement in the value of the portfolio after time interval t δ is defined as follows: