Abstract
ABSTRACTMost of the empirical studies on stochastic volatility dynamics favour 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 (SVM) 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 SVM by proposing a closed-form formula for the partial transform of the triple joint transition density 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 moment swaps and timer options with discrete monitoring feature are given to demonstrate the versatility of the partial transform under the 3/2 model.
Highlights
Stochastic volatility models (SVMs) were introduced to option pricing theory to resolve the incapability of the Black–Scholes framework in capturing the volatility smile/skew
We demonstrate how the partial transform of the triple and its induced marginal characteristic functions and transforms can be used to price various exotic derivative products that may be embedded with path-dependent features, and have a complicated terminal payoff structure that depends on the asset price and quadratic variation
We explore the analytic tractability of the 3/2 stochastic volatility model (SVM) by investigating the joint distribution of the triple consisting of the log asset price, quadratic variation and instantaneous variance
Summary
Stochastic volatility models (SVMs) were introduced to option pricing theory to resolve the incapability of the Black–Scholes framework in capturing the volatility smile/skew. Lewis (2009) derives the joint transition density of the log asset price and the instantaneous variance for the 3/2 model with constant parameters. As we are informed, the full characterization of the joint distribution of the triple ðX; I; VÞ, which stand for the log asset price, quadratic variation and the instantaneous variance, has not yet been done in the literature Our paper fills this gap by providing a complete description of the joint distribution through the closed-form partial transform of the triple transition density. PDE by converting the equation into a Riccati system of ODEs. Our probabilistic approach explores the relationship between the partial transform and the conditional characteristic function of the integrated variance and works out the closed-form expression for the latter with the change of measure technique. Notice that under normal market condition, κ > 0 and ρ automatically satisfied
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