Uniform asymptotic expansions for pricing European options (Q816972)

A fast varying stochastic volatility \({\sigma}(t,{\omega})=f(y^{\varepsilon}(t))\) which plays the role of the constant volatility in the Black-Scholes' SDE is considered to describe the risky asset price, where \(y^{\varepsilon}(t)\) satisfies the following SDE: \[ dy^{\varepsilon}(t)=\frac{1}{\varepsilon}m(y^{\varepsilon}(t))dt +\frac{1}{\sqrt{\varepsilon}}{\beta}(y^{\varepsilon}(t))d \tilde{z^{\varepsilon}} (t) \] driven by \(\tilde{z^{\varepsilon}}(t)\), a ``Brownian motion on a circle'' of circumference \textit{a}. Let \(P^{\varepsilon}(t,x,y)\) be the option price at \(t<T\) of a payoff at maturity time \textit{T} of the stock with the present value \textit{x} and a current value of the volatility \textit{y}. Under conditions of smoothness of the coefficient and a compact support restriction of the payoff function, it is proven that \(P^{\varepsilon}(t,x,y)\) has an asymptotic expansion of \(P_0(t,x,y)+ {\varepsilon}^{1/2}P_1(t,x,y)+{\varepsilon}P_2(t,x,y) +{\varepsilon}Q_2((T-t)/\varepsilon,x,y)\) unifomly in \(t<T,x,y\) with an error of \(O({\varepsilon}^{3/2})\).