Heston option pricing with trapezoidal fuzzy parameters

Abstract

We propose a deterministic and conservative method for pricing European options under parameter ambiguity in the Heston model. The primary inputs $(\kappa ,\theta ,\sigma ,\rho ,v_0,r)$ are represented as trapezoidal fuzzy numbers, and uncertainty is propagated via $\alpha$-cuts through the characteristic-function representation. Our main contribution is a backward-recursive enclosure pipeline: we first construct conservative $\alpha$-cut bands for the volatility and stock-price processes at a prescribed normal-quantile level $p$, then derive principal-branch complex-interval (rectangle) enclosures for the Heston discriminant and auxiliary terms $(d_j,g_j)$ and the affine coefficients $(A_j,B_j)$. These bounds yield conservative enclosures for the risk-neutral probabilities $(Q_1,Q_2)$ through Fourier quadrature with interval-valued integrands, and finally monotone $\alpha$-cut price intervals for call and put options. We further report expected intervals and expected values of the resulting fuzzy option prices by integrating $\alpha$-cut endpoints, providing concise summaries alongside the full price bands. Numerical experiments show that the exact (generally nonlinear) $\alpha$-cut bounds vary smoothly across $\alpha$, and that a simple trapezoidal surrogate obtained by linear interpolation between $\alpha =0$ and $\alpha =1$ approximates the exact bounds with very small relative errors under moderate uncertainty. A core/support sensitivity study highlights that the support width dominates the effect on $E\left({\tilde{C}}_t\right)$ and $E\left({\tilde{P}}_t\right)$, consistent with the support governing extreme $\alpha$-cut tails. Overall, the proposed complex-interval/$\alpha$-cut methodology delivers deterministic enclosures and expected values without sampling error and offers a transparent alternative to outer–inner Monte Carlo under parameter uncertainty.