Subdiffusive models with local volatility for illiquid markets

Анотація

The paper focuses on subdiffusive models with local volatility that extend classical approaches to capture illiquidity effects and trapping events through stochastic time changes. The illiquidity effects mean lower trading volume, wider bid-ask spreads, greater price volatility, and long periods without any trading for risky assets. These effects occur during crisis periods that negatively affect financial activity, or in emerging markets where the number of participants and, thus, the number of transactions is relatively low. All models based on Brownian motion are perpetually moving. They are not adapted for modeling periods with motionless stock returns. Nevertheless, in statistical physics, there are subdiffusion processes that are well suited to describe the dynamics of underlying returns with trapping events. However, for modeling the dynamics of the financial market with illiquid effects, we need not only to describe stock prices, but also to propose a method for option evaluation. The main option pricing tool in this paper is a fractional extension of the Dupire equation, which gives the fair price for call and put options for subdiffusive underlying assets and takes into account local volatility. It is a forward partial differential equation in which the derivative with respect to time is replaced by a Dzerbayshan – Caputo (D – C) derivative. The form of the D – C derivative and fractal Dupire equation depends upon the chosen Lévy subordinator. We apply the equation for subdiffusive in a Brownian setting in the case of the inverted tempered stable subordinator and demonstrate numerical results for real financial data. Due to the proposed approaches and numerical algorithm, the investor gets tools that allow him to take into account both the market’s illiquidity and local volatility.

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