تقریب بسته شکل برای مدل سیاه شولز کسری با هزینه های معاملاتی

In this paper, we investigate option valuation problems under the fractional Black–Scholes model. The aim is to propose a pricing formula for the European option with transaction costs, where the costs structure contains fixed costs, a cost propositional to the volume traded, and a cost proportional to the value traded. Precisely, we provide an approximate solution of the nonlinear Hoggard–Whalley–Wilmott equation. The comparison results reveal that our approximate solutions are close to the numerical computations. Moreover, the comparison results demonstrate that the price of the European option decreases as the Hurst exponent increases.

The fractional Brownian motion (fBm) was first introduced by Kolmogorov in 1940 in [1]. Mandelbrot and Van Ness provided a stochastic integral representation of this process in terms of a standard Brownian motion in 1968 in [2]. The name fBm is due to the stochastic integral representation in terms of a standard Brownian motion. To capture the property of long-range dependence in financial market series [3], the fractional Black–Scholes model replaces the Brownian motion in the Black–Scholes (BS) model [4] with the fBm. Since the fBm is not a semimartingale, the arbitrage opportunities exist in the fractional Black–Scholes model under a complete and frictionless setting. A considerable number of arbitrage strategies for fBm models is provided by Rogers [5], Shiryaev [6], Salopek [7] and Cheridito [8]. Besides the financial problems, the fractional calculus is also applied to other real problems [9] and [10]. In the complete and frictionless market, Black and Scholes [4] constructed a self-finance trading strategy that replicates the option’s value continuously. However, the replicating portfolio requires continuous trading and, in the market with transaction costs, the continuous trading policy will incur an infinite amount of trading cost. Therefore, the BS replicating strategy is no longer valid. Barles and Soner [11] pointed out that in such a market, there is no portfolio that replicates the final payoff of a European option.

In this paper, we obtain the following three conclusions. (1) By using VIM, we have proposed an approximation for the extended Leland model in Theorem 3.1. When the transaction cost is only the fixed cost, the exact solution has been given in Corollary 3.2. These solutions provide a useful formula for the valuation of the European option with the transaction costs under the fractional Black–Scholes models. (2) In the comparison results, we have found that our approximate solution is close to the numerical computations. The results revealed that the VIM is very effective and efficient for finding an approximate price of a European option in the fractional Black–Scholes model. (3) An advantage of the VIM is that there is no need to make the assumption of small parameters in the transaction costs terms.