Option pricing under mixed hedging strategy in time-changed mixed fractional Brownian model
Introduction
Over the last years, studies have shown that many financial market time series display scaling laws and long-range dependence. Although the works of Black and Scholes [1] and Merton [2] have contributed distinctly in option pricing, their approaches are based on the assumption of continuous-time trading, so the price of any option does not depend on the time scaling and investors’ risk preferences (scaling-free pricing and preference-free pricing). However, in actual markets, continuous trading is physically impossible to realize and the investors amend the option risk from time to time. Therefore, it has been emphasized that the scaling and the risk preference play an important role in portfolio hedging and option pricing in a discrete time case [3], [4], thus one should consider the scaling laws and risk preference. In recent years with respect to these topics, a lot of research has been progressed [5], [6], [7], [8], [9], [10], [11], [12].
Wang [13] proposed a mixed hedging strategy under the classical B–S model in a discrete time case and obtained a pricing formula for European call option through the mixed hedging strategy, and showed that the mixed hedging is better than the delta hedging in some cases. In addition, he generalized his results to mixed hedging and option pricing with the proportional transaction costs in the same model [14].
However, some empirical studies have shown that the B–S model cannot capture many of the abnormal behaviors of prices, such as heavy tailed, long-range dependence and periods of constant values, etc. Therefore, one should replace the classical B–S model by other models.
Since a fractional Brownian motion with (Hereafter FBM) has two important properties such as self-similarity and long-range dependence, but unfortunately, it is neither a Markov process nor a semi-martingale, so one cannot apply the Ito stochastic analysis to the FBM model [15], [16], [17]. Actually, it admits arbitrages in a complete and frictionless market [18], [19]. To overcome this difficulty, Cheridito [20] introduced a mixed fractional Brownian motion (MFBM hereafter). The MFBM is a family of Gaussian processes that is a linear combination of a BM and a FBM, it is equivalent to one with a BM, and hence it does not only imply arbitrage-free but exhibit long-memory property [21]. In this environment, pricing formulae for a European call option were derived [22], [23].
As another generalization of B–S model, Magdziarz [24] applied the sub-diffusive mechanism to describe properly financial data exhibiting periods of constant values and introduced the sub-diffusive geometric Brownian motion as a model of underlying asset processes, which has been referred to time-changed Brownian motion model. He pointed out that this model is arbitrage-free but incomplete and also he derived the corresponding sub-diffusive B–S formula for a European option in this model [25]. As above mentioned, as well as extension of B–S model to the MFBM model, time-changed Brownian model has been also extended to time-changed mixed fractional Brownian models of two forms. They are as follows, the first is the case that underlying asset processes are represented by time-changed stochastic differential equation [26] and the second case has exponential representation of time-changed mixed fractional Brownian motion [27]. In these models, pricing formulae for the European call option have been derived respectively.
The aim of this paper is to extend a mixed hedging strategy and a pricing formula for the European call option in the B–S model [13] to the time-changed mixed fractional Brownian model [27].
The rest of the paper is as follows. In Section 2, we introduce the time-changed mixed fractional Brownian model. In Section 3, a mixed hedging strategy and a pricing formula for European call option was derived. In Section 4, we consider some numerical experiments and discussion; on different trading frequencies compare performances of mixed hedging and delta hedging in our model and investigate the influence of parameters. In Section 5, we apply delta hedging and mixed hedging for empirical data by our model and investigate the performance.
Section snippets
Time-changed mixed fractional Brownian model
Let us consider a financial market model, which consists of a stock and a bond with price dynamics given by and where and are risk preference parameter, volatility parameter and risk-free interest rates respectively. The time transformation process is the inverse -stable subordinator defined as below where is a strictly increasing -stable Lévy process with Laplace transform: . The
The mixed hedging strategy and option pricing under the time-changed mixed fractional Brownian model
In the section a mixed hedging strategy is given, and based on the strategy, we obtain a discrete-time pricing formula for the European call option under the time-changed mixed fractional Brownian model.
Let be the price at time of a European call option with expiration date and exercise price . In addition, we assume that partial derivatives of are continuous on their respective domains.
Theorem 3.1 The mixed hedging strategy under the time-changed mixed fractional Brownian model
Comparison of delta hedging and mixed hedging in time-changed mixed fractional Brownian model
It turned out that the scaling law and the risk preference parameter play an important role in option pricing. In this section, with respect to the hedging error ratio, we examine the effects of the delta hedging and our mixed hedging strategy in the time-changed mixed fractional Brownian model. We note that the delta hedging strategy is .
Performance of delta hedging and mixed hedging in time-changed mixed fractional Brownian motion model for exchange rates
Fig. 2 shows the graph of EUR/USD exchange rates from 01/02/2014 to 28/02/2014. At a glance in Fig. 2, they seem to exhibit some constant periods and to be smoother than usual behavior, moreover, to have the long-range dependence. Therefore, we can assume that the EUR/USD exchange rates are generated by time-changed mixed fractional Brownian motion model. The beginning times of constant periods are , , , , , . We analyze EUR/USD exchange rates from
Conclusions
This paper deals with a mixed hedging strategy and a pricing formula for the European call option in the discrete time under the time-changed mixed fractional Brownian model. It has been shown that in the discrete time case, a stability parameter and a Hurst index as well as a risk preference parameter and a fractal scaling play important roles in option pricing under the mixed hedging strategy. In particular, through some numerical studies and empirical analysis for stock prices
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