Option Pricing under the Subordinated Market Models

Longjin Lv,Changjuan Zheng,Luna Wang,Benjamin Miranda Tabak

doi:10.1155/2022/6213803

Longjin Lv, Changjuan Zheng + Show 2 more

Open Access

https://doi.org/10.1155/2022/6213803

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Abstract

This paper aims to study option pricing problem under the subordinated Brownian motion. Firstly, we prove that the subordinated Brownian motion controlled by the fractional diffusion equation has many financial properties, such as self-similarity, leptokurtic, and long memory, which indicate that the fractional calculus can describe the financial data well. Then, we investigate the option pricing under the assumption that the stock price is driven by the subordinated Brownian motion. The closed-form pricing formula for European options is derived. In the comparison with the classic Black–Sholes model, we find the option prices become higher, and the “volatility smiles” phenomenon happens in the proposed model. Finally, an empirical analysis is performed to show the validity of these results.

Highlights

Brownian motion and normal distribution have been widely used in the Black–Scholes option pricing framework to the return of assets. e classical Black–Scholes model is based on the diffusion process called geometric Brownian motion (GBM) [1, 2]
Leland [5] first examined option replication in the presence of transaction costs (TC) in a discrete time setting and posed a modified replicating strategy, which depends upon the level of transactions costs and upon the revision interval, as well as upon the option to be replicated and the environment
In the paper [8], Magdziarz applied the subdiffusive mechanism of trapping events to describe properly financial data exhibiting periods of constant values and introduced the subdiffusive geometric Brownian motion (SGBM) St X(Sα(t)) as the model of asset prices exhibiting subdiffusive dynamics

Summary

Introduction

Brownian motion and normal distribution have been widely used in the Black–Scholes option pricing framework to the return of assets. e classical Black–Scholes model is based on the diffusion process called geometric Brownian motion (GBM) [1, 2]. Brownian motion and normal distribution have been widely used in the Black–Scholes option pricing framework to the return of assets. In the paper [8], Magdziarz applied the subdiffusive mechanism of trapping events to describe properly financial data exhibiting periods of constant values and introduced the subdiffusive geometric Brownian motion (SGBM) St X(Sα(t)) as the model of asset prices exhibiting subdiffusive dynamics. In this paper, we suppose that the underlying of the option contract is driven by a subordinated Brownian motion, i.e., the price of underlying St follows the stochastic differential equation: dSt μStdt + σStdB Sα(t)􏼁.

The Subordinated Brownian Motion

Option Pricing under the Subordinated Models

Conclusions