Asymptotic Implied Volatility at the Second Order with Application to the SABR Model

Paulot, Louis

doi:10.1007/978-3-319-11605-1_2

Cited by 32 publications

(5 citation statements)

References 11 publications

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“…For instance,

c_{BS} (k, t; σ) = exp (- \frac{Λ_{BS} (k)}{t} (() 1 + o (1))), k > 0 fixed, t ↓ 0,

with

{normalΛ}_{BS} false(k false) = \frac{1}{2} k^{2} / σ^{2}

in the Black–Scholes model. Similar results appear in the literature, with different levels of mathematical rigor, for other and/or generic diffusion models; see Berestycki, Busca, and Florent (), Carr and Wu (), Forde and Jacquier (), and Paulot (). Table summarizes first‐order call price asymptotics in various models and regimes.…”

Section: Introductionsupporting

confidence: 82%

Option pricing in the moderate deviations regime

Friz

Gerhold

Pinter

2017

Mathematical Finance

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We consider call option prices close to expiry in diffusion models, in an asymptotic regime (“moderately out of the money”) that interpolates between the well‐studied cases of at‐the‐money and out‐of‐the‐money regimes. First and higher order small‐time moderate deviation estimates of call prices and implied volatilities are obtained. The expansions involve only simple expressions of the model parameters, and we show how to calculate them for generic local and stochastic volatility models. Some numerical computations for the Heston model illustrate the accuracy of our results.

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“…For instance,

c_{BS} (k, t; σ) = exp (- \frac{Λ_{BS} (k)}{t} (() 1 + o (1))), k > 0 fixed, t ↓ 0,

with

{normalΛ}_{BS} false(k false) = \frac{1}{2} k^{2} / σ^{2}

Section: Introductionsupporting

confidence: 82%

Option pricing in the moderate deviations regime

Friz

Gerhold

Pinter

2017

Mathematical Finance

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“…This is a different feature from other current approaches in the literature that rely on the assumption of very small total volatility and usually degrades for longer than 10 years maturity, e.g. Hagan et al (2002), Hagan, Lesniewski, and Woodward (2005), Wu (2012), Henry-Labordere (2005) and Paulot (2009). The works that we mentioned earlier, i.e.…”

Section: Introductionmentioning

confidence: 60%

On the Approximation of the SABR with Mean Reversion Model: A Probabilistic Approach

Kennedy

Pham

2014

Applied Mathematical Finance

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In this paper, we study the stochastic alpha beta rho with mean reversion model (SABR-MR). We first compare the SABR model with the SABR-MR model in terms of future volatility to point out the fundamental difference in the models' dynamics. We then derive an efficient probabilistic approximation for the SABR-MR model to price European options. Similar to the method derived in Kennedy, J. E., Mitra, S., & Pham, D. (2012). On the approximation of the SABR model: A probabilistic approach. Applied Mathematical Finance, 19(6), 553-586., we focus on capturing the terminal distribution of the underlying asset (conditional on the terminal volatility) to arrive at the implied volatilities of the corresponding European options for all strikes and maturities. Our resulting method allows us to work with a wide range of parameters that cover the long-dated option and different market conditions.

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“…We then review how this equation can be related to a geometric heat equation, which will motivate the heat kernel expansion ansatz; alternative expositions of these topics can be found in Labordère (2008) and Paulot (2007). Finally, we write the ansantz out explicitly in the one dimensional setting.…”

Section: Background and The Heat Kernel Ansatzmentioning

confidence: 98%

Explicit Density Approximations for Local Volatility Models Using Heat Kernel Expansions

Taylor¹,

Glasgow

Taylor

et al. 2015

Methodol Comput Appl Probab

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Heat kernel perturbation theory is a tool for constructing explicit approximation formulas for the solutions of linear parabolic equations. We review the crux of this perturbative formalism and then apply it to differential equations which govern the transition densities of several local volatility processes. In particular, we compute all the heat kernel coefficients for the CEV and quadratic local volatility models; in the later case, we are able to use these to construct an exact explicit formula for the processes' transition density. We then derive low order approximation formulas for the cubic local volatility model, an affine-affine short rate model, and a generalized mean reverting CEV model. We finally demonstrate that the approximation formulas are accurate in certain model parameter regimes via comparison to Monte Carlo simulations.

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Asymptotic Implied Volatility at the Second Order with Application to the SABR Model

Cited by 32 publications

References 11 publications

Option pricing in the moderate deviations regime

Option pricing in the moderate deviations regime

On the Approximation of the SABR with Mean Reversion Model: A Probabilistic Approach

Explicit Density Approximations for Local Volatility Models Using Heat Kernel Expansions

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