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2015 | OriginalPaper | Buchkapitel

Asymptotics for \(d\)-Dimensional Lévy-Type Processes

verfasst von : Matthew Lorig, Stefano Pagliarani, Andrea Pascucci

Erschienen in: Large Deviations and Asymptotic Methods in Finance

Verlag: Springer International Publishing

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Abstract

We consider a general \(d\)-dimensional Lévy-type process with killing. Combining the classical Dyson series approach with a novel polynomial expansion of the generator \(\mathcal {A}(t)\) of the Lévy-type process, we derive a family of asymptotic approximations for transition densities and European-style options prices. Examples of stochastic volatility models with jumps are provided in order to illustrate the numerical accuracy of our approach. The methods described in this paper extend the results from Corielli et al. (SIAM J Financ Math 1:833–867, 2010, [4]), Pagliarani and Pascucci (Int. J. Theor. Appl. Financ. 16(8):1–35, 2013, [20]) to Lorig et al. (Analytical expansions for parabolic equations, 2013, [13]) for Markov diffusions to Markov processes with jumps.

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Fußnoten
1
The operator \(\hat{\mathcal {L}}^{\xi }_{n}\) is not a function as in the classical theory of pseudo-differential calculus. However \({\mathrm {e}}^{-\mathtt {i}\langle \xi ,x\rangle } \hat{\mathcal {L}}^{\xi }_{n} {\mathrm {e}}^{\mathtt {i}\langle \xi ,x\rangle }\) is the symbol of \(\mathcal {L}_n^x(t,T)\).
 
2
This was one of the main points of the adjoint expansion method proposed by Pagliarani et al. [21].
 
Literatur
1.
Zurück zum Zitat Benhamou, E., Gobet, E., Miri, M.: Smart expansion and fast calibration for jump diffusions. Financ. Stoch. 13(4), 563–589 (2009)MATHMathSciNetCrossRef Benhamou, E., Gobet, E., Miri, M.: Smart expansion and fast calibration for jump diffusions. Financ. Stoch. 13(4), 563–589 (2009)MATHMathSciNetCrossRef
2.
Zurück zum Zitat Bompis, R., Gobet, E.: Asymptotic and non asymptotic approximations for option valuation. Recent Developments in Computational Finance. Foundations, Algorithms and Applications, pp. 159–241. World Scientific, Hackensack (2013)CrossRef Bompis, R., Gobet, E.: Asymptotic and non asymptotic approximations for option valuation. Recent Developments in Computational Finance. Foundations, Algorithms and Applications, pp. 159–241. World Scientific, Hackensack (2013)CrossRef
3.
Zurück zum Zitat Carr, P., Wu, L.: Time-changed Lévy processes and option pricing. J. Financ. Econ. 71(1), 113–141 (2004)CrossRef Carr, P., Wu, L.: Time-changed Lévy processes and option pricing. J. Financ. Econ. 71(1), 113–141 (2004)CrossRef
4.
Zurück zum Zitat Corielli, F., Foschi, P., Pascucci, A.: Parametrix approximation of diffusion transition densities. SIAM J. Financ. Math. 1, 833–867 (2010)MathSciNetCrossRef Corielli, F., Foschi, P., Pascucci, A.: Parametrix approximation of diffusion transition densities. SIAM J. Financ. Math. 1, 833–867 (2010)MathSciNetCrossRef
5.
Zurück zum Zitat Deuschel, J.-D., Friz, P., Jacquier, A., Violante, S.: Marginal density expansions for diffusions and stochastic volatility, part i: theoretical foundations. Commun. Pure Appl. Math. 67(1), 40–82 (2014)MATHMathSciNetCrossRef Deuschel, J.-D., Friz, P., Jacquier, A., Violante, S.: Marginal density expansions for diffusions and stochastic volatility, part i: theoretical foundations. Commun. Pure Appl. Math. 67(1), 40–82 (2014)MATHMathSciNetCrossRef
6.
Zurück zum Zitat Fouque, J.-P., Papanicolaou, G., Sircar, R., Solna, K.: Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives. Cambridge University Press, Cambridge (2011)MATHCrossRef Fouque, J.-P., Papanicolaou, G., Sircar, R., Solna, K.: Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives. Cambridge University Press, Cambridge (2011)MATHCrossRef
7.
Zurück zum Zitat Friz, P. K., Gerhold, S., Yor, M.: How to make Dupire’s local volatility work with jumps. Quant. Financ. 14(8), 1327–1331 (2014) Friz, P. K., Gerhold, S., Yor, M.: How to make Dupire’s local volatility work with jumps. Quant. Financ. 14(8), 1327–1331 (2014)
8.
Zurück zum Zitat Heston, S.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6(2), 327–343 (1993)CrossRef Heston, S.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6(2), 327–343 (1993)CrossRef
9.
Zurück zum Zitat Jeanblanc, M., Yor, M., Chesney, M.: Mathematical Methods for Financial Markets. Springer, London (2009)MATHCrossRef Jeanblanc, M., Yor, M., Chesney, M.: Mathematical Methods for Financial Markets. Springer, London (2009)MATHCrossRef
10.
Zurück zum Zitat Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995). Reprint of the 1980 editionMATH Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995). Reprint of the 1980 editionMATH
11.
Zurück zum Zitat Lorig, M.: Pricing derivatives on multiscale diffusions: an eigenfunction expansion approach. Math. Finance 24(2), 331–363 (2014) Lorig, M.: Pricing derivatives on multiscale diffusions: an eigenfunction expansion approach. Math. Finance 24(2), 331–363 (2014)
12.
Zurück zum Zitat Lorig, M., Lozano-Carbassé O.: Multiscale Exponential Lévy models. Quant. Finance 15(1), 91–100 (2015) Lorig, M., Lozano-Carbassé O.: Multiscale Exponential Lévy models. Quant. Finance 15(1), 91–100 (2015)
13.
Zurück zum Zitat Lorig, M., Pagliarani, S., Pascucci, A.: Analytical expansions for parabolic equations. SIAM J. Appl. Math. 75(2), 468–491 (2015) Lorig, M., Pagliarani, S., Pascucci, A.: Analytical expansions for parabolic equations. SIAM J. Appl. Math. 75(2), 468–491 (2015)
14.
Zurück zum Zitat Lorig, M., Pagliarani, S., Pascucci, A.: Explicit implied volatilities for multifactor local-stochastic volatility models. Math. Finance (to appear) (2015). ArXiv preprint arXiv:1306.5447 Lorig, M., Pagliarani, S., Pascucci, A.: Explicit implied volatilities for multifactor local-stochastic volatility models. Math. Finance (to appear) (2015). ArXiv preprint arXiv:​1306.​5447
15.
Zurück zum Zitat Lorig, M., Pagliarani, S., Pascucci, A.: A family of density expansions for Lévy-type processes with default. Ann. Appl. Probab. 25(1), 235–267 (2015) Lorig, M., Pagliarani, S., Pascucci, A.: A family of density expansions for Lévy-type processes with default. Ann. Appl. Probab. 25(1), 235–267 (2015)
16.
Zurück zum Zitat Lorig, M., Pagliarani, S., Pascucci, A.: Pricing approximations and error estimates for local Lévy-type models with default. Comp. Math. App. 69(10), 1189–1219 (2015) Lorig, M., Pagliarani, S., Pascucci, A.: Pricing approximations and error estimates for local Lévy-type models with default. Comp. Math. App. 69(10), 1189–1219 (2015)
17.
Zurück zum Zitat Lorig, M., Pagliarani, S., Pascucci, A.: A Taylor series approach to pricing and implied vol for LSV models. J. Risk 17(2), 1–17 (2014) Lorig, M., Pagliarani, S., Pascucci, A.: A Taylor series approach to pricing and implied vol for LSV models. J. Risk 17(2), 1–17 (2014)
18.
Zurück zum Zitat Øksendal, B., Sulem, A.: Applied Stochastic Control of Jump Diffusions. Springer, Berlin (2005) Øksendal, B., Sulem, A.: Applied Stochastic Control of Jump Diffusions. Springer, Berlin (2005)
19.
Zurück zum Zitat Pagliarani, S., Pascucci, A.: Analytical approximation of the transition density in a local volatility model. Cent. Eur. J. Math. 10(1), 250–270 (2012)MATHMathSciNetCrossRef Pagliarani, S., Pascucci, A.: Analytical approximation of the transition density in a local volatility model. Cent. Eur. J. Math. 10(1), 250–270 (2012)MATHMathSciNetCrossRef
20.
Zurück zum Zitat Pagliarani, S., Pascucci, A.: Local stochastic volatility with jumps: analytical approximations. Int. J. Theor. Appl. Financ. 16(8), 1–35 (2013)MathSciNetCrossRef Pagliarani, S., Pascucci, A.: Local stochastic volatility with jumps: analytical approximations. Int. J. Theor. Appl. Financ. 16(8), 1–35 (2013)MathSciNetCrossRef
21.
22.
Zurück zum Zitat Pascucci, A.: PDE and Martingale Methods in Option Pricing. Bocconi & Springer Series, vol. 2. Springer, Milan (2011)MATHCrossRef Pascucci, A.: PDE and Martingale Methods in Option Pricing. Bocconi & Springer Series, vol. 2. Springer, Milan (2011)MATHCrossRef
23.
Zurück zum Zitat Sakurai, J.J., Tuan, S.F.: Modern Quantum Mechanics, vol. 104. Addison-Wesley, Reading (Mass.) (1994) Sakurai, J.J., Tuan, S.F.: Modern Quantum Mechanics, vol. 104. Addison-Wesley, Reading (Mass.) (1994)
Metadaten
Titel
Asymptotics for -Dimensional Lévy-Type Processes
verfasst von
Matthew Lorig
Stefano Pagliarani
Andrea Pascucci
Copyright-Jahr
2015
DOI
https://doi.org/10.1007/978-3-319-11605-1_12