Analytical Solvability and Exact Simulation of Stochastic Volatility Models with Jumps

Willow Algorithm for Consumption-Investment under Stochastic Volatility Model with Jump Diffusion

“…Zeng et al [19] Proposition 2.4 gave an analytical expression of the characteristic function of the log-price process of assets at time t under the 4/2 model. It should be noted here that since the expression for the characteristic function of the Heston model in our derivation of Theorem 2.2 is more concise than that of Zeng et al [19]. We use Theorem 2.2 below for the Heston model problem and Zeng et al [19] for the 3/2 model problem.…”

“…• In this paper, the characteristic function of the underlying asset process is derived based on affine structure theory and Zeng et al [19]. The precise sampling of the underlying asset process is realized by moment matching method and Johnson curve transformation theory, which not only overcomes the approximate treatment of the difference integral in [18] but improves the accuracy and efficiency of sampling.…”

“…This study opens up a new numerical method for solving the multi-cycle optimal investment consumption decision problem in the discrete case and provides solutions for other optimal investment consumption problems, such as the dividendcapital injection optimal investment consumption problem. The paper is structured as follows: In Section II, using the 4/2 model as an example, we derive the characteristic function of the logarithmic asset price based on affine structure theory and Zeng et al [19]. Section III shows the construction of the two-dimensional willow frame.…”

Simulation schemes for the Heston model with Poisson conditioning

“…However, the Broadie-Kaya exact simulation algorithm is not competitive in accuracy-speed comparison because it requires extensive computational time to sample the conditional integrated variance via the numerical inversion of the Laplace transform in each simulation path. To improve computational efficiency, one may use a caching technique to sample the terminal variance and conditional integrated variance via precomputation and interpolation of the appropriate inverse distribution functions [24,26]. Despite its limitations, Broadie and Kaya [5]'s pioneering work triggers the construction of exact simulation schemes for other stochastic volatility models, such as the stochastic-alpha-beta-rho (SABR) [6,8], 3/2 [2,26], Wishart [13], and Ornstein-Uhlenbeck-driven stochastic volatility model [17,7].…”

“…To improve computational efficiency, one may use a caching technique to sample the terminal variance and conditional integrated variance via precomputation and interpolation of the appropriate inverse distribution functions [24,26]. Despite its limitations, Broadie and Kaya [5]'s pioneering work triggers the construction of exact simulation schemes for other stochastic volatility models, such as the stochastic-alpha-beta-rho (SABR) [6,8], 3/2 [2,26], Wishart [13], and Ornstein-Uhlenbeck-driven stochastic volatility model [17,7].…”