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

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