Option Pricing with Stochastic Volatility, Equity Premium, and Interest Rates

TL;DR

This paper introduces a comprehensive options pricing model that incorporates stochastic volatility, interest rates, and equity premiums, extending classical models to better reflect real market dynamics and using numerical methods for option valuation.

Contribution

It develops a new model combining stochastic elements of volatility, interest rates, and equity premiums, and derives PDEs for pricing complex options with numerical solutions.

Findings

Derived a new PDE for options with stochastic parameters.

Extended the model to Asian options, providing a framework for their pricing.

Implemented finite difference methods for numerical approximation of option prices.

Abstract

This paper presents a new model for options pricing. The Black-Scholes-Merton (BSM) model plays an important role in financial options pricing. However, the BSM model assumes that the risk-free interest rate, volatility, and equity premium are constant, which is unrealistic in the real market. To address this, our paper considers the time-varying characteristics of those parameters. Our model integrates elements of the BSM model, the Heston (1993) model for stochastic variance, the Vasicek model (1977) for stochastic interest rates, and the Campbell and Viceira model (1999, 2001) for stochastic equity premium. We derive a linear second-order parabolic PDE and extend our model to encompass fixed-strike Asian options, yielding a new PDE. In the absence of closed-form solutions for any options from our new model, we utilize finite difference methods to approximate prices for European call and up-and-out barrier options, and outline the numerical implementation for fixed-strike Asian call options.