زمان پیوسته معادله سیاه شولز با هزینه معاملات در رژیم subdiffusive با حرکت براونی جزء به جزء

In this paper, we study the problem of continuous time option pricing with transaction costs by using the homogeneous subdiffusive fractional Brownian motion (HFBM) Z(t)=X(Sα(t))Z(t)=X(Sα(t)), 0<α<10<α<1, here dX(τ)=μX(τ)(dτ)2H+σX(τ)dBH(τ)dX(τ)=μX(τ)(dτ)2H+σX(τ)dBH(τ), as a model of asset prices, which captures the subdiffusive characteristic of financial markets. We find the corresponding subdiffusive Black–Scholes equation and the Black–Scholes formula for the fair prices of European option, the turnover and transaction costs of replicating strategies. We also give the total transaction costs.

where μ,σμ,σ are constants, and B(τ)B(τ) is the Brownian motion. In the presence of transaction costs (TC), Leland [3] first examined option replication in a discrete time setting, and pose a modified replicating strategy, which depends upon the level of transaction costs and upon the revision interval, as well as upon the option to be replicated and the environment. Since then, a lot of authors study this problem, but all in a discrete time setting [4], [5], [6], [7], [8], [9], [10] and [11]. The option pricing theory as developed by Black–Scholes [1] and [2] rests on an arbitrage argument: by continuously adjusting a portfolio consisting of a stock and a risk-free bond, an investor can exactly replicate the returns to any option on the stock. It leads us naturally to pose the following question. In the presence of transaction costs, is there an alternative replicating strategy depending upon the level of transaction costs and a technique leading to the Black–Scholes equation in a continuous time setting? Does the perfect replication incur an infinite amount of transaction costs? The Black–Scholes (BS) model is based on the diffusion process called geometric Brownian motion (GBM). However, the empirical studies show that many characteristic properties of markets cannot be captured by the BS model, such as: long-range correlations, heavy-tailed and skewed marginal distributions, lack of scale invariance, periods of constant values, etc. Therefore, in recent years one observes many generalizations of the BS model based on the ideas and methods known from statistical and quantum physics [12].

Using the strategy of Leland [3], this paper has developed a technique for replicating option returns in a continuous time setting in the presence of transaction costs and obtain the corresponding Black–Scholes equations, Black–Scholes formulas and total transaction costs with transaction costs both in subdiffusive regime and in real time.