Multifactor Heston's stochastic volatility model for European option pricing

Abstract

In this study, we extend the multiscale stochastic volatility model of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254] by incorporating a slow varying factor of volatility. The resulting model can be viewed as a multifactor extension of the Heston model with two additional factors driving the volatility levels. An asymptotic analysis consisting of singular and regular perturbation expansions is developed to obtain an approximation to European option prices. We also find explicit expressions for some essential functions that are available only in integral formulas in the work of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254]. This finding basically leads to considerable reduction in computational time for numerical calculation as well as calibration problems. An accuracy result of the asymptotic approximation is also provided. For numerical illustration, the multifactor Heston model is calibrated to index options on the market, and we find that the resulting implied volatility surfaces fit the market data better than those produced by the multiscale stochastic volatility model of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254], particularly for long‐maturity call options.

The complete Gaussian kernel in the multi-factor Heston model: Option pricing and implied volatility applications

“…Some descriptive statistics for the estimated model parameters, initial variance, Feller ratio, objective function and observed implied volatility are given for the two sets in Table 6. The values of the objective function compare favourably with those in Table 1 of Veng et al (2019). The two sets of parameters are almost identical, with the exception of the estimate of the long-term mean parameter.…”

“…Alós et al (2012) use Malliavin calculus to study the short-term behaviour of implied volatility for jump-diffusion models with stochastic volatility. Veng et al (2019) derive an asymptotic expansion for put prices, extending the results of Benhamou et al (2010) to the general multi-factor Heston model with time-dependent parameters.…”