Power Option Pricing Based on Time-Fractional Model and Triangular Interval Type-2 Fuzzy Numbers

The problem of generalizing the power option-pricing model to incorporate more empirical features becomes an urgent and necessary event. A new power option pricing method is designed for the financial market uncertainty that simultaneously involves randomness and fuzziness. The randomness in market uncertainty is modeled by a time-fractional diffusion model, which describes trend memory in underlying asset prices. The fuzziness in market uncertainty is characterized by a triangular interval type-2 fuzzy numbers, which better captures the fuzziness of underlying asset prices. Considering the decision-maker’s subjective judgment, we show the price mean value with the possibility-necessity weight and pessimistic-optimistic index under the type-2 fuzzy environment. We develop the power option pricing model with the time-fractional diffusion model and the triangular interval type-2 fuzzy numbers. Furthermore, the analytic solutions of pricing call power option and put power option are obtained and verified by the variational iterative reconstruction method. Our study shows that power option pricing, which adopts the time-fractional model and the triangular interval type-2 fuzzy numbers, can better capture the trend memory and double fuzziness of the real market. In addition, a numerical example is provided to illustrate that the power option means the value is decreasing with respect to the pessimistic-optimistic index and is fluctuating with respect to the possibility-necessity weight index.