Option pricing with illiquidity during a high volatile period

• Published in: Mathematical methods in the applied sciences 2022-03, Vol.45 (5), p.3213-3224
• Main Authors: El‐Khatib, Youssef, Hatemi‐J, Abdulnasser
• Format: Article
• Language: English
• Subjects: Brownian motion; derivatives securities; financial crisis; Financial management; illiquid markets; Mathematical models; numerical methods; PDEs; Risk management; Securities markets; stochastic processes; Volatility
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.7612

This paper deals with the valuation of options in markets without liquidity and under stress. More precisely, a European option is considered when the dynamic of the underlying asset is governed by a Brownian motion. Following Liu and Young (2005), a term related to the number of invested stocks is embedded into the model. Moreover, the volatility of the asset is augmented by a separate function that models the abnormal increase of the volatility. Under these settings, we deal with the evaluation of European options. Dealing with illiquidity during a financial crisis is an important issue in terms of financial risk management, which has not been tackled with in the literature to the best of our knowledge. It is during these non‐normal periods that volatility in the financial markets is very high and tools for reducing the underlying risk are urgently needed for hedging purposes. In this work, the PDE of the option price within this context is derived, which is a generalization of the PDE of Liu and Young (2005). Moreover, the hedging strategy to replicate the price of the option is obtained. Numerical simulations for the underlying asset price are conducted. The illustrations support several stylized facts of the model that seem to reflect what is operational in the real financial markets.