Analytics on conditional moment generating functions of stochastic volatility models

Moment generating functions (MGFs) of the terminal log-asset price and integrated variance of stochastic volatility models conditional on the terminal variance value are required in exact simulation algorithms and Fourier based algorithms for pricing European path dependent options. Broadie and Kaya (Exact simulation of stochastic volatility and other affine jump diffusion processes. Oper. Res., 2006, 54(2), 217–231) initiate the analytic derivation of conditional MGF of integrated variance of the Heston model based on related analytic results for the Bessel bridge. Kang et al. (Exact simulation of the Wishart multidimensional stochastic volatility model. Oper. Res., 2017, 65(5), 1190–1206) and Zeng et al. (Analytical solvability and exact simulation in models with affine stochastic volatility and Lévy jumps. Math. Finance, 2023, 33, 842–890) employ different techniques of measure changes to obtain the conditional MGFs of the Heston model, multidimensional Wishart stochastic volatility model, 4/2-model and Ornstein–Uhlenbeck-driven stochastic volatility model. In this paper, we develop systematic and comprehensive measure change techniques that provide effective derivation procedures for the associated conditional MGFs. We establish an interesting linkage between joint conditional MGFs and their unconditional counterparts. Interestingly, the conditional MGFs under the 4/2-model can be deduced from those under the Heston model via an appropriate measure change. Besides, we employ the partial transform method to derive conditional MGFs that go beyond the Heston-type stochastic volatility models.