Cumulant Formulas for Implied Volatility

R. Lee

Chapter 10 in Options — 45 Years since the Publication of the Black–Scholes–Merton Model:The Gershon Fintech Center Conference, 2023, pp 185-193 from World Scientific Publishing Co. Pte. Ltd.

Abstract: The following sections are included:Implied Volatility and its ImportanceRelate Implied Volatility Skew to Underlying DistributionNotationsExact Relationship: Distribution ↔ Implied VolatilitiesMomentsCumulantsSkewness and KurtosisHow do Skewness/Kurtosis Relate to Implied Volatilty?BFW ApproachOur ApproachCumulant Expansion for Implied VolatilityFive MomentsThe Book of Five MomentsMoment FormulaImplied Volatility: BFW vs Refined vs Exact (Figure 1)Intuition: Jump–Diffusion DynamicsOption Price ApproximationImplied Volatility ApproximationConclusions

Keywords: Options; Call; Put; Stock; Equity; Bond; Debt; Dividend; Investment; Diversification; Volatility; Black–Scholes; Merton Model; Stochastic; Swap; Commodity; Index; Contingent Claims; Exotic Option (search for similar items in EconPapers)
JEL-codes: C C02 G1 G11 G12 G15 (search for similar items in EconPapers)
Date: 2023
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