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Review of Quantitative Finance and Accounting

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01.10.2015 | Original Research

R-2GAM stochastic volatility model: flexibility and calibration

verfasst von: Cheng-Few Lee, Oleg Sokolinskiy

Erschienen in: Review of Quantitative Finance and Accounting | Ausgabe 3/2015

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Abstract

This paper investigates the potential of the 2GAM stochastic volatility model for capturing varying properties of option prices represented by the implied volatility surface. The 2GAM model is shown to be a generalization of the Heston model. Then, taking the original Heston model as the benchmark, the paper explores the flexibility allowed by the 2GAM model. More precisely, the focus is on the restricted 2GAM (R-2GAM) model which builds upon the Heston model reproducing a given short-term implied volatility skew. Going from theory to practice, the paper suggests a numerically-feasible calibration procedure for the R-2GAM model. In an application to the valuation of the S&P 500 option contracts this paper addresses the challenges of calibrating the R-2GAM model to market prices and raises concerns of possible over-parameterization.

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Jones (2003) does not explain the nomenclature, but the name of the model likely reflects the presence of two parameters denoted by the Greek letter gamma (γ ₁ and γ ₂).

However, in some markets the Black–Scholes model is still effective due to its sheer robustness (see Harikumar et al. 2004).

Binder and Merges (2001) study the economic factors contributing to the dynamics of volatility.

The latter statement applies to the restricted 2GAM (R-2GAM) model.

Relative to Heston (1993), Gatheral (2006) provides a numerically more stable, but otherwise equivalent, solution for plain vanilla European option prices in the framework of the Heston (1993) model.

The moneyness characteristic of the option is captured by the ratio of the strike price to the spot or forward price of the underlying asset, or some meaningful function of this ratio (like the Black–Scholes delta).

ρ and η in the Heston (1993) model versus σ ₁, σ ₂, and γ ₂ in the R-2GAM model. Once ρη is fixed to re-produce the short-term implied volatility skew, the Heston (1993) model has one degree of freedom, while the R-2GAM model (with σ ₁ = ρη) has two: σ ₂, γ ₂.

This direct fix (akin to the method described in Glasserman 2003 on p. 121) addresses the consequence of the discretization error. Given a sufficiently fine discretization of the time axis (500 time steps are used in the current application of the procedure), the applied heuristic method is highly unlikely to significantly influence the results. Whenever the discretization error leads to a simulated negative variance, the true variance (while strictly positive) is highly likely to be very small.

This increase in the level of the implied volatility surface is not tied to raising the value of v _LR.

Not reported here, but available on demand.

Using the parameters grid \(\sigma_2 \times \gamma_2 = \left\{0.01,0.02,\ldots,0.25\right\} \times \left\{0.250, 0.275,\ldots,0.550\right\}\).

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Titel: R-2GAM stochastic volatility model: flexibility and calibration
verfasst von: Cheng-Few Lee
Oleg Sokolinskiy
Publikationsdatum: 01.10.2015
Verlag: Springer US
Erschienen in: Review of Quantitative Finance and Accounting / Ausgabe 3/2015
Print ISSN: 0924-865X
Elektronische ISSN: 1573-7179
DOI: https://doi.org/10.1007/s11156-014-0443-7

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