Document Zbl 1465.91128

Hambly, Ben; Mariapragassam, Matthieu; Reisinger, Christoph

A forward equation for barrier options under the Brunick & Shreve Markovian projection. (English) Zbl 1465.91128

Quant. Finance 16, No. 6, 827-838 (2016).

From the text: In this paper, we provided a new and numerically effective way to work with observed barrier option prices to determine the Brunick-Shreve Markovian projection of a stochastic variance of a semi-martingale \(S\) with running maximum \(M\) onto \((S_t, M_t)\). The present Dupire-type formula, which provides similar advantages (and disadvantages) as the standard Dupire approach for vanillas, was then re-arranged to obtain a forward PIDE, which is convenient to control the numerical stability of best fit algorithms [S. Crépey, “Tikhonov regularization”, in: Encyclopedia of quantitative finance. Chichester: Wiley. 1807–1812 (2010; doi:10.1002/9780470061602.eqf12016)]. There is a well-known literature on the calibration of vanilla options through Markovian projection for LSV models [J. Guyon and P. Henry-Labordere, “The smile calibration problem solved”, SSRN Preprint, 17 p. (2011; doi:10.2139/ssrn.1885032); Y. Ren et al., “Calibrating and pricing with embedded local volatility models”, Risk Mag. 20, No. 9, 138–143 (2007)]. We believe that extending these methods combined with the forward PIDE we presented in this paper will lead to novel calibration algorithms for barrier options.

Cited in 9 Documents

MSC:

91G60	Numerical methods (including Monte Carlo methods)
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65R99	Numerical methods for integral equations, integral transforms
91G20	Derivative securities (option pricing, hedging, etc.)

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References:

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