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

Area of Science:

• Quantitative Finance
• Mathematical Finance
• Stochastic Processes

Background:

• Empirical studies often favor the 3/2 stochastic volatility model (SVM) over the square-root (CIR) process in the Heston model.
• The 3/2 SVM demonstrates superior ability in capturing volatility skew evolution for option pricing compared to the Heston model.

Purpose of the Study:

• To investigate the analytic tractability of the 3/2 SVM.
• To derive a closed-form formula for the partial transform of the triple joint transition density for log asset price, quadratic variation, and instantaneous variance.

Main Methods:

• Derivation of a closed-form partial transform formula using two distinct formulations.
• Analysis of the analytic properties of the 3/2 SVM.

Main Results:

• A closed-form formula for the partial transform of the triple joint transition density is obtained.
• The derived formula facilitates the deduction of marginal partial transforms and characteristic functions.
• The partial transform is shown to be crucial for pricing discretely sampled variance derivatives and exotic options.

Conclusions:

• The 3/2 SVM exhibits enhanced analytic tractability through the derived closed-form partial transform.
• The methodology provides a versatile tool for pricing various financial derivatives, including moment swaps and timer options with discrete monitoring.