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

TL;DR

This paper derives a closed-form partial transform for the 3/2 stochastic volatility model, enabling efficient pricing of variance derivatives and exotic options with discrete monitoring, which better captures market volatility features.

Contribution

It provides the first closed-form formula for the partial transform of the 3/2 model's joint density, facilitating advanced option pricing methods.

Findings

Closed-form partial transform derived for the 3/2 model

Enables pricing of discretely monitored variance derivatives

Numerical examples demonstrate practical applicability

Abstract

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