A subdiffusive stochastic volatility jump model

Abstract

Subdiffusions appear as good candidates for modeling illiquidity in financial markets. Existing subdiffusive models of asset prices are indeed able to capture the motionless periods in the quotes of thinly-traded assets. However, they fail at reproducing simultaneously the jumps and the time-varying random volatility observed in the price of these assets. The aim of this work is hence to propose a new model of subdiffusive asset prices reproducing the main characteristics exhibited in illiquid markets. This is done by considering a stochastic volatility jump model, time changed by an inverse subordinator. We derive the forward fractional partial differential equations (PDE) governing the probability density function of the introduced model and we prove that it leads to an arbitrage-free and incomplete market. By proposing a new procedure for estimating the model parameters and using a series expansion for solving numerically the obtained fractional PDE, we are able to price various European-type derivatives on illiquid assets and to depart from the common Markovian valuation setup. This way, we show that the introduced subdiffusive stochastic volatility jump model yields consistent and reliable results in illiquid markets.

Keywords:

• Illiquidity modeling
• Subdiffusion
• Fractional Fokker–Planck equations
• Stochastic volatility jump model
• Option pricing

Acknowledgments

Thanks are due to the referees whose remarks and suggestions have significantly improved the initial draft of the paper.

Data Availability Statement

The datasets and code generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 A similar argument can be used for $V_{S_t}$. We refer to Kobayashi (2011) for a more thorough discussion on stochastic calculus for integrals driven by a time-changed semimartingale (note that since S is a.s. continuous in our context, any process is synchronized with S so that lemma 2.3 of the above paper applies).

2 The approximation (67(67) ${\varphi }_S(t,z_1,z|s,w,u,x_s,v_s)\approx \sum _{k=0}^{\mathrm{\infty }}\sum _{h=0}^{\mathrm{\infty }}{w}_{\alpha }(k,h)\times z^h(t-u{)}^{\alpha k},$(67) ) is exact if ${\varphi }_S(t,z_1,z|\cdot )$ is the product of one function of t and one function of z.

Additional information

Funding

This work was supported by the FNRS, Belgium under Grant no. 33658713.