Computing the implied volatility in stochastic volatility models

Berestycki, Henri; Busca, Jérôme; Florent, Igor

doi:10.1002/cpa.20039

Cited by 197 publications

(220 citation statements)

References 21 publications

(29 reference statements)

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“…Due to their importance for model calibration and testing, small-time asymptotics of option prices have received considerable attention in recent years; see [2,5,6,7,10,11,12,13,14,15,16,17,22,24]. A survey of recent literature is given in the introduction of Forde et al [16].…”

Section: Introductionmentioning

confidence: 99%

Small-Time Asymptotics of Option Prices and First Absolute Moments

Muhle‐Karbe

Nutz

2011

J. Appl. Probab.

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We study the leading term in the small-time asymptotics of atthe-money call option prices when the stock price process follows a general martingale. This is equivalent to studying the rst centered absolute moment of . We show that if has a continuous part, the leading term is of order √ in time and depends only on the initial value of the volatility. Furthermore, the term is linear in if and only if is of nite variation. The leading terms for pure-jump processes with in nite variation are between these two cases; we obtain their exact form for stable-like small jumps. To derive these results, we use a natural approximation of so that calculations are necessary only for the class of Lévy processes.

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Section: Introductionmentioning

confidence: 99%

Small-Time Asymptotics of Option Prices and First Absolute Moments

Muhle‐Karbe

Nutz

2011

J. Appl. Probab.

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“…This paper deals with the relationship between these two equations. It is clear that, once the process σ is known, option prices are characterized by (2). The reverse direction (i.e., from C to σ) is less obvious.…”

Section: Results and Outline Of The Papermentioning

confidence: 99%

“…This has been known for some time in the financial literature, see for example, [21], [6], and [3]. Rigorous proofs were given in [1] and [2] in a different context. Theorem 1.…”

Section: Recovering the Spot Volatilitymentioning

confidence: 94%

“…, [15] and [2] provide with methods to get approximate formulas for the implied volatility at any order. In Section 7, in a spirit close to [2], we obtained expansions in θ about 0 by solving partial differential equations (PDE for short) and/or ordinary differential equations (ODE for short) in a given model.…”

Section: Computing Expansions For Implied Volatilitiesmentioning

confidence: 99%

“…In Section 7, in a spirit close to [2], we obtained expansions in θ about 0 by solving partial differential equations (PDE for short) and/or ordinary differential equations (ODE for short) in a given model. In spirit closer to Section 5, we can also obtain expansions in θ and x about (0, 0).…”

Section: Computing Expansions For Implied Volatilitiesmentioning

confidence: 99%

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From implied to spot volatilities

Durrleman

2009

Finance Stoch

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This paper is concerned with the link between spot and implied volatilities. The main result is the derivation of the stochastic differential equation driving the spot volatility based on the shape of the implied volatility surface. This equation is a consequence of no-arbitrage constraints on the implied volatility surface right before expiry. We investigate the regularity of this surface at maturity in the case of the Constant Elasticity of Variance and Heston models. We also show that a simple way to link spot and implied volatilities is to relate the coefficients of the implied volatility surface Taylor expansion to the coefficients of a certain chaos expansion of the spot volatility process. As a byproduct, we give expansions for the implied volatility surface for a general stochastic volatility model.

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Model Calibration

Cont

2010

Encyclopedia of Quantitative Finance

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A derivative pricing model is said to be calibrated to a set of benchmark instruments if the value of these instruments, computed in the model, correspond to their market prices. Model calibration is the procedure of selecting model parameters in order to verify the calibration condition. Model calibration can be viewed as the inverse problem associated with the pricing of derivatives. In the theoretical situation where prices of call options are available for all strikes and maturities, the calibration problem can be explicitly solved using an inversion formula. In real situations, given a finite (and often sparse) set of derivative prices, model calibration is an ill‐posed problem whose solution often requires a regularization method. We discuss various—deterministic and stochastic—algorithms for solving them.

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Computing the implied volatility in stochastic volatility models

Cited by 197 publications

References 21 publications

Small-Time Asymptotics of Option Prices and First Absolute Moments

Small-Time Asymptotics of Option Prices and First Absolute Moments

From implied to spot volatilities

Model Calibration

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