Models with Uncertain Volatility

Models in which volatility is one of the possible trajectories are considered in the paper. As an example of a model with a certain volatility, the Black-Scholes model is considered. As an example of models with uncertain volatility three models are considered: the Heston model with random trajectories and two models with deterministic trajectories from a confidence set of possible trajectories. Three computational methods are proposed for finding the range of fair prices for a European option. The first method is based on solving viscosity equations using difference schemes. The second is the Monte-Carlo method, which is based on the simulation of the initial stock price process. The third is the tree method, which is based on approximating the original continuous model with a discrete model and obtaining recurrent formulas on a binary tree to calculate the upper and lower prices. The results of calculations using the listed methods are presented. It is shown that the ranges of fair prices obtained using the three numerical methods practically coincide.

Keywords

Black-Scholes model; Heston model; uncertain volatility; viscosity equation; option; fair price.

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