Boundary Conditions for Computing Densities in Hybrid Models via PDE Methods

Lucic, Vladimir

doi:10.2139/ssrn.1191962

Cited by 14 publications

(13 citation statements)

References 12 publications

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“…Moreover, the choice of boundary conditions for the numerical method is also very important. We have chosen the zero flux condition, see for instance [Luc12]. Note that, by using an ADI method to solve for the Fokker-Planck equation, p(t n , ·, ·) would depend on L(t n , ·) and L(t n−1 , ·).…”

Section: Numerical Methods For the Fokker-planck Pdementioning

confidence: 99%

“…(1 − θA 1 )W = [1 + A 0 + (1 − θ)A 1 + A 2 ]p (n) (1 − θA 2 )p (n+1) = W − θA 2 p (n) Note that, here, for notational reason, we assume the rate r −d is constant. In our experiment, with a slight modification of A 1 and A 2 , we developed the method for the case of r − d being time-dependent and also the zero flux condition [Luc12] .…”

Section: A Specification Of the Adi Methodsmentioning

confidence: 99%

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The calibration of stochastic local-volatility models: An inverse problem perspective

Saporito¹,

Yang²,

Zubelli³

2019

Computers & Mathematics with Applications

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We tackle the calibration of the so-called Stochastic-Local Volatility (SLV) model. This is the class of financial models that combines the local and stochastic volatility features and has been subject of the attention by many researchers recently. More precisely, given a local volatility surface and a choice of stochastic volatility parameters, we calibrate the corresponding leverage function. Our approach makes use of regularization techniques from the inverse-problem theory, respecting the integrity of the data and thus avoiding data interpolation. The result is a stable and robust algorithm which is resilient to instabilities in the regions of low probability density of the spot price and of the instantaneous variance. We substantiate our claims with numerical experiments using simulated as well as real data.

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Section: Numerical Methods For the Fokker-planck Pdementioning

confidence: 99%

Section: A Specification Of the Adi Methodsmentioning

confidence: 99%

The calibration of stochastic local-volatility models: An inverse problem perspective

Saporito¹,

Yang²,

Zubelli³

2019

Computers & Mathematics with Applications

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“…A seven-point stencil for discretization of the mixed-derivative operator that preserves the positivity of the solution was proposed in Chiarella et al (2008), Toivanen (2010) for correlations ρ < 0, and in Ikonen & Toivanen (2007, 2008 for positive correlations. However, in their schemes the mixed derivative term was treated implicitly (that is the reason they needed a discretized matrix to be an Mmatrix).…”

Section: Positivity Of the Solutionmentioning

confidence: 99%

High-Order Splitting Methods for Forward PDEs and PIDEs

Itkin¹

2017

Pseudo-Differential Operators

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This paper is dedicated to the construction of high-order (in both space and time) finite-difference schemes for both forward and backward PDEs and PIDEs, such that option prices obtained by solving both the forward and backward equations are consistent. This approach is partly inspired by Andreasen & Huge (2011) who reported a pair of consistent finite-difference schemes of first-order approximation in time for an uncorrelated local stochastic volatility model. We extend their approach by constructing schemes that are second-order in both space and time and that apply to models with jumps and discrete dividends. Taking correlation into account in our approach is also not an issue.

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“…Also, setting the boundary conditions in the v-domain should be done with a care. See Lucic (2008) for a discussion on this matter.…”

Section: Lsv Model With Jumpsmentioning

confidence: 99%