Closed-Form Formula for the Conditional Moments of Log Prices under the Inhomogeneous Heston Model

Abstract

Several financial instruments have been thoroughly calculated via the price of an underlying asset, which can be regarded as a solution of a stochastic differential equation (SDE), for example the moment swap and its exotic types that encourage investors in markets to trade volatility on payoff and are especially beneficial for hedging on volatility risk. In the past few decades, numerous studies about conditional moments from various SDEs have been conducted. However, some existing results are not in closed forms, which are more difficult to apply than simply using Monte Carlo (MC) simulations. To overcome this issue, this paper presents an efficient closed-form formula to price generalized swaps for discrete sampling times under the inhomogeneous Heston model, which is the Heston model with time-parameter functions. The obtained formulas are based on the infinitesimal generator and solving a recurrence relation. These formulas are expressed in an explicit and general form. An investigation of the essential properties was carried out for the inhomogeneous Heston model, including conditional moments, central moments, variance, and skewness. Moreover, the closed-form formula obtained was numerically validated through MC simulations. Under this approach, the computational burden was significantly reduced.