MARKOVIAN STOCHASTIC VOLATILITY WITH STOCHASTIC CORRELATION — JOINT CALIBRATION AND CONSISTENCY OF SPX/VIX SHORT-MATURITY SMILES

Martin Forde and Benjamin Smith ()
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Martin Forde: Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, UK
Benjamin Smith: Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, UK

International Journal of Theoretical and Applied Finance (IJTAF), 2023, vol. 26, issue 02n03, 1-42

Abstract: In this paper, we show how to calibrate a general Markovian stochastic volatility model with stochastic correlation to the VIX implied volatility smile and the overall level, slope and curvature of the SPX smile in the T→0 limit. Explicit formulae are obtained for the asymptotic VIX smile for Heston and SABR-type models with mean reversion, and the Lewis CEV-p-model. We also discuss how the Bass martingale can be used to give an exact fit to a single VIX smile for T>0. In the second half of this paper, we derive a more involved integral equation for the correlation function Ï (y) to be perfectly consistent with the short-maturity SPX and VIX smiles at all strikes (or all strikes in an interval) as T→0, and discuss consistency conditions between the wings of the two asymptotic smiles and how to avoid |Ï (y)|>1 for the calibrated Ï (y) in practice.

Keywords: Markov models; stochastic volatility; small-time smile asymptotics; Freidlin-Wentzell theory; large deviations; geodesics (search for similar items in EconPapers)
Date: 2023
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Citations: View citations in EconPapers (2)

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DOI: 10.1142/S0219024923500073

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