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Most of the empirical studies on stochastic volatility dynamics favor the 3/2specification over the square-root (CIR) process in the Heston model. In thecontext of option pricing, the 3/2 stochastic volatility model is reported tobe able to capture the volatility skew evolution better than the Heston model.In this article, we make a thorough investigation on the analytic tractabilityof the 3/2 stochastic volatility model by proposing a closed-form formula forthe partial transform of the triple joint transition density $(X,I,V)$ whichstand for the log asset price, the quadratic variation (continuous realizedvariance) and the instantaneous variance, respectively. Two distinctformulations are provided for deriving the main result. The closed-form partialtransform enables us to deduce a variety of marginal partial transforms andcharacteristic functions and plays a crucial role in pricing discretely sampledvariance derivatives and exotic options that depend on both the asset price andquadratic variation. Various applications and numerical examples on pricingexotic derivatives with discrete monitoring feature are given to demonstratethe versatility of the partial transform under the 3/2 model.

Pricing timer options and variance derivatives with closed-form partial transform under the 3/2 model