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

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.