How to make Dupire’s local volatility work with jumps

Friz, Peter K.; Gerhold, Stefan; Yor, Marc

doi:10.1080/14697688.2013.874622

Cited by 19 publications

(14 citation statements)

References 15 publications

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“…Example Hermite expansions can be useful when the diffusion coefficients are not smooth. A remarkable example in financial mathematics is given by the Dupire's LV formula for models with jumps (see Friz, Gerhold, and Yor ). In some cases, e.g., the well‐known variance‐gamma model, the fundamental solution (i.e., the transition density of the underlying stochastic model) has singularities.…”

Section: Asymptotic Pricing For a General Class Of Lsv Modelsmentioning

confidence: 99%

Explicit Implied Volatilities for Multifactor Local‐stochastic Volatility Models

Lorig

Pagliarani

Pascucci

2015

Mathematical Finance

123

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We consider an asset whose risk-neutral dynamics are described by a general class of local-stochastic volatility models and derive a family of asymptotic expansions for European-style option prices and implied volatilities. We also establish rigorous error estimates for these quantities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under four different model dynamics: constant elasticity of variance local volatility, Heston stochastic volatility, three-halves stochastic volatility, and SABR local-stochastic volatility.

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Section: Asymptotic Pricing For a General Class Of Lsv Modelsmentioning

confidence: 99%

Explicit Implied Volatilities for Multifactor Local‐stochastic Volatility Models

Lorig

Pagliarani

Pascucci

2015

Mathematical Finance

123

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“…See theorem 1 and proposition 2 inMijatović and Tankov (2016).5 We write CLT for central limit theorem and LDP for large deviation principle. For readers unfamiliar with moderate deviations, we recall some of the basics toward the end of the Introduction.6 The situation is very different with jumps; seeFriz, Gerhold, and Yor (2014).7 Recall that the MD rate function for a centered i.i.d. sequence ( ) ≥1 is given by  → 2 ∕(2Var( 1 )).…”

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confidence: 99%

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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“…Note that in the proposed algorithm we do not compute the implied volatility and do not use Dupire's formula [12,[19][20][21]. Instead, we directly compute the time-dependent volatility function from option prices and the BS equation.…”

Section: Algorithm Of Volatilitymentioning

confidence: 99%

Reconstruction of the Time-Dependent Volatility Function Using the Black–Scholes Model

Jin

Wang

Kim

et al. 2018

Discrete Dynamics in Nature and Society

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We propose a simple and robust numerical algorithm to estimate a time-dependent volatility function from a set of market observations, using the Black–Scholes (BS) model. We employ a fully implicit finite difference method to solve the BS equation numerically. To define the time-dependent volatility function, we define a cost function that is the sum of the squared errors between the market values and the theoretical values obtained by the BS model using the time-dependent volatility function. To minimize the cost function, we employ the steepest descent method. However, in general, volatility functions for minimizing the cost function are nonunique. To resolve this problem, we propose a predictor-corrector technique. As the first step, we construct the volatility function as a constant. Then, in the next step, our algorithm follows the prediction step and correction step at half-backward time level. The constructed volatility function is continuous and piecewise linear with respect to the time variable. We demonstrate the ability of the proposed algorithm to reconstruct time-dependent volatility functions using manufactured volatility functions. We also present some numerical results for real market data using the proposed volatility function reconstruction algorithm.

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How to make Dupire’s local volatility work with jumps

Cited by 19 publications

References 15 publications

Explicit Implied Volatilities for Multifactor Local‐stochastic Volatility Models

Explicit Implied Volatilities for Multifactor Local‐stochastic Volatility Models

Option pricing in the moderate deviations regime

Reconstruction of the Time-Dependent Volatility Function Using the Black–Scholes Model

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