Local Variance Gamma and Explicit Calibration to Option Prices

Abstract

In some options markets (e.g., commodities), options are listed with only a single maturity for each underlying. In others (e.g., equities, currencies), options are listed with multiple maturities. In this paper, we analyze a special class of pure jump Markov martingale models and provide an algorithm for calibrating such models to match the market prices of European options with multiple strikes and maturities. This algorithm matches option prices exactly and only requires solving several one‐dimensional root‐search problems and applying elementary functions. We show how to construct a time‐homogeneous process which meets a single smile, and a piecewise time‐homogeneous process which can meet multiple smiles.

Robust Trading of Implied Skew

“…The models presented herein are closely related to the Local Variance Gamma (LVG) models, introduced in [18]. Let the process D, taking values in [0, ∞), be defined for t ≥ 0 as the unique weak solution to…”

“…The name is motivated by the similarity of the above construction and the LVG models introduced in [18]. Indeed, the main difference between the two is the different choice of a time change: the latter is assumed to be a Gamma process in [18]. However, it is also mentioned in [18] that any other independent time change will produce models with similar features, as long as the marginal distributions of the time change are exponential.…”

“…Proof of Lemma 1. It is shown in [18] that χ = C σ (U, ·, τ ) is continuously differentiable, with an absolutely continuous derivative, and it satisfies…”

“…Proof of Proposition 2. Denote by X s the PCLVG process with the same σ and with the initial condition s. As discussed in [18], {X s } is a Markov family. Using the Markov property, we obtain…”

“…Finally, in order to achieve our goal, we introduce and analyze a new family of models, called Piecewise Constant Local Variance Gamma models (see [18] for related results). These turn out to be rather remarkable.…”