Small noise expansion and importance sampling

Fournié, Éric; Lebuchoux, Jérôme; Touzi, Nizar

doi:10.3233/asy-1997-14404

Cited by 25 publications

(7 citation statements)

References 6 publications

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“…Note also that the parameter reduction outlined in Section 2.6.1 can be applied to this implied volatility expansion as well (σ replaced by σ * and V 2 -terms removed), and this will be used in the calibration in the next section. We also remark that the formal second order expansion for the case of a single slow volatility factor had previously been considered in [9], [19] and [23], for instance.…”

Section: Implied Volatility Expansionmentioning

confidence: 96%

Second Order Multiscale Stochastic Volatility Asymptotics: Stochastic Terminal Layer Analysis & Calibration

2012

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Multiscale stochastic volatility models have been developed as an efficient way to capture the principle effects on derivative pricing and portfolio optimization of randomly varying volatility. The recent book Fouque, Papanicolaou, Sircar and Sølna (2011, CUP) analyzes models in which the volatility of the underlying is driven by two diffusions -one fast mean-reverting and one slow-varying, and provides a first order approximation for European option prices and for the implied volatility surface, which is calibrated to market data. Here, we present the full second order asymptotics, which are considerably more complicated due to a terminal layer near the option expiration time. We find that, to second order, the implied volatility approximation depends quadratically on log-moneyness, capturing the convexity of the implied volatility curve seen in data. We introduce a new probabilistic approach to the terminal layer analysis needed for the derivation of the second order singular perturbation term, and calibrate to S&P 500 options data.

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Section: Implied Volatility Expansionmentioning

confidence: 96%

Second Order Multiscale Stochastic Volatility Asymptotics: Stochastic Terminal Layer Analysis & Calibration

2012

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“…The classical "noise addition in decibels" order zero lognormal approximation was studied by Huynh (1994), Musiela & Rutkowski (1997) and Brace et al (1999) when the underlying instruments follow a Black & Scholes (1973) like lognormal diffusion. Here, we approximate the price of a basket using stochastic expansion techniques similar to those used by Fournié, Lebuchoux & Touzi (1997) or Fouque, Papanicolaou & Sircar (2000) on other stochastic volatility problems. This provides a theoretical justification for the classical price approximation and allows us to compute additional terms, better accounting for the stochastic nature of the basket volatility.…”

Section: Basket Price Approximationmentioning

confidence: 99%

“…and develop around small values of ε > 0. As in Fournié et al (1997), we want to evaluate the price and develop its series expansion in ε around 0.…”

Section: Diffusion Approximationmentioning

confidence: 99%

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Interest rate model calibration using semidefinite Programming

d’Aspremont¹

2003

Applied Mathematical Finance

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It is shown that, for the purpose of pricing swaptions, the swap rate and the corresponding forward rates can be considered lognormal under a single martingale measure. Swaptions can then be priced as options on a basket of lognormal assets and an approximation formula is derived for such options. This formula is centred around a Black-Scholes price with an appropriate volatility, plus a correction term that can be interpreted as the expected tracking error. The calibration problem can then be solved very efficiently using semidefinite programming.semidefinite programming, Libor market model, calibration, basket options,

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“…Stratified sampling is another technique that has been widely used in Monte Carlo simulation: see, for instance, Glynn and Iglehart (1989), Ross (1991) and Fournie, Lebuchoux and Touzi (1997). A simple version of it can be described by dividing the whole sample space into M sets of disjoint events A\,..., AM with P(A{) = -^ for all i G { 1 , .…”

Section: Variance Reduction Techniquesmentioning

confidence: 99%

An introduction to numerical methods for stochastic differential equations

1999

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This paper aims to give an overview and summary of numerical methods for the solution of stochastic differential equations. It covers discrete time strong and weak approximation methods that are suitable for different applications. A range of approaches and results is discussed within a unified framework. On the one hand, these methods can be interpreted as generalizing the well-developed theory on numerical analysis for deterministic ordinary differential equations. On the other hand they highlight the specific stochastic nature of the equations. In some cases these methods lead to completely new and challenging problems.

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Small noise expansion and importance sampling

Cited by 25 publications

References 6 publications

Second Order Multiscale Stochastic Volatility Asymptotics: Stochastic Terminal Layer Analysis & Calibration

Second Order Multiscale Stochastic Volatility Asymptotics: Stochastic Terminal Layer Analysis & Calibration

Interest rate model calibration using semidefinite Programming

An introduction to numerical methods for stochastic differential equations

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