European Power Option Pricing with Extended Vasic\v{e}k Interest Rate and Exponential Ornstein-Uhlenbeck Asset Process under Different Market Assumptions

TL;DR

This paper develops a comprehensive framework for pricing European power options considering extended Vasicek interest rates and exponential Ornstein-Uhlenbeck asset processes, under different market assumptions and correlations.

Contribution

It introduces a unified approach to derive closed-form pricing formulas for European power options under two market assumptions with correlated stochastic processes.

Findings

Derived explicit pricing formulas under both market assumptions.

Established the use of Girsanov transform for real-world to risk-neutral measure mapping.

Unified theoretical framework for different market conditions.

Abstract

We propose a general framework of European power option pricing under two different market assumptions about extended Vasic\v{e}k interest rate process and exponential Ornstein-Uhlenbeck asset process with continuous dividend as underlying, in which the Brownian motions involved in Vasic\v{e}k interest rate and exponential Ornstein-Uhlenbeck process are time-dependent correlated in equivalent martingale measure probability space or real-world probability space respectively. We first develop European power option pricing in two types of payoffs with martingale method under the market assumption that Vasic\v{e}k interest rate and exponential Ornstein-Uhlenbeck process are correlated in equivalent martingale measure probability space. Then, we solve the European power option pricing under the market assumption that Vasic\v{e}k interest rate and exponential Ornstein-Uhlenbeck process are correlated in real-world probability by constructing a Girsannov transform to map real-world probability to risk-neutral equivalent martingale measure. Finally, the European power option pricing formulae are derived with numeraire change and T-forward measure under the above two market assumptions in a uniform theoretical framework and close formulae expression.