Exact and approximate solutions for options with time-dependent stochastic volatility

Abstract

In this paper it is shown how symmetry methods can be used to find exact solutions for European option pricing under a time-dependent 3/2-stochastic volatility model $\mathit{dv}=\mathit{kv}(A(t)-v)\mathit{dt}+{\mathit{bv}}^{\frac{3}{2}}\mathit{dZ}$. This model with $A(t)$ constant has been proven by many authors to outperform the Heston model in its ability to capture the behaviour of volatility and fit option prices. Further, singular perturbation techniques are used to derive a simple analytic approximation suitable for pricing options with short tenor, a common feature of most options traded in the market.