An orthogonal expansion approach to joint SPX and VIX calibration in affine stochastic volatility models with jumps

Thomas K. Kloster and Elisa Nicolato

Quantitative Finance, 2025, vol. 25, issue 1, 63-89

Abstract: We discuss the joint calibration to SPX and VIX options of an affine Stochastic Volatility model with Jumps in price and Jumps in volatility (the SVJJ model). Conventionally, the SVJJ model assumes exponential jumps in the variance process, leaving the potential benefits of more flexible jump distributions unexplored. The purpose of our study is twofold. First, we show that choosing the gamma distributions for the jumps in variance significantly improves the performance of the joint calibration. However, this improvement comes at the cost of increased computational time. Second, we mitigate this loss of tractability by constructing novel approximations to option prices based on orthogonal polynomial expansions. Unlike the classical method of selecting an explicit reference density, our approach generalizes to all densities with explicit Laplace transform. We apply this methodology to the SVJJ model with gamma jumps and we find that the proposed price expansions achieve the same accuracy as exact transform inversion formulas while requiring only a fraction of the computational time.

Date: 2025
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DOI: 10.1080/14697688.2024.2433142

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