### A Sequential Monte Carlo Approach for the pricing of barrier option in a Stochastic Volatility Model

#### Abstract

In this paper we propose a numerical scheme to estimate

the price of a barrier option in a general framework.

More precisely, we extend a classical Sequential

Monte Carlo approach, developed under the hypothesis

of deterministic volatility, to Stochastic Volatility models,

in order to improve the efficiency of Standard Monte Carlo

techniques in the case of barrier options whose underlying

approaches the barriers. The paper concludes with the

application of our procedure to two case studies in

`a SABR model. `

#### References

Back, K. (2006). A course in derivative securities: Introduction to theory and computa- tion. Springer Science & Business Media.

Baldi, P., Caramellino, L., and Iovino, M. G. (1999). Pricing general barrier options: a numerical approach using sharp large deviations. Mathematical Finance, 9(4):293–321.

Broadie, M., Glasserman, P., and Kou, S. (1997). A continuity correction for discrete barrier options. Mathematical Finance, 7(4):325–349.

Carmona, R., Fouque, J.-P., and Vestal, D. (2009). Interacting particle systems for the computation of rare credit portfolio losses. Finance and Stochastics, 13(4):613–633.

Carpenter, J., Clifford, P., and Fearnhead, P. (1999). Improved particle filter for non-

Electronic Journal of Applied Statistical Analysis 13 linear problems. IEE Proceedings-Radar, Sonar and Navigation, 146(1):2–7.

Cuomo, S., Campagna, R., Di Somma, V., and Severino, G. (2016). Numerical remarks on the estimation of the option price. In 2016 12th International Conference on Signal-Image Technology & Internet-Based Systems (SITIS), pages 746–749. IEEE.

Deborshee, S., Ajay, J., and Yan, Z. (2017). Some contributions to sequential monte carlo methods for option pricing. Journal of Statistical Computation and Simulation.

Del Moral, P. and Patras, F. (2011). Interacting path systems for credit risk. Credit Risk Frontiers: Subprime Crisis, Pricing and Hedging, CVA, MBS, Ratings, and Liquidity, pages 649–673.

Dewynne, J. and Wilmott, P. (1993). Partial to the exotic. Risk, 6(3):38–46.

Dey, S. and Maiti, S. S. (2012). Bayesian estimation of the parameter of rayleigh dis- tribution under the extended jeffrey’s prior. Electronic journal of applied statistical analysis, 5(1):44–59.

Douc, R. and Capp ́e, O. (2005). Comparison of resampling schemes for particle filtering. In Image and Signal Processing and Analysis, 2005. ISPA 2005. Proceedings of the 4th International Symposium on, pages 64–69. IEEE.

Efron, B. (1992). Bootstrap methods: another look at the jackknife. In Breakthroughs in statistics, pages 569–593. Springer.

Efron, B. and Tibshirani, R. J. (1994). An introduction to the bootstrap. CRC press. Giles, M. B. (2008). Multilevel monte carlo path simulation. Operations Research,

(3):607–617.

Glasserman, P. and Staum, J. (2001). Conditioning on one-step survival for barrier

option simulations. Operations Research, 49(6):923–937.

Glassermann, P. (2004). Monte Carlo Methods in Financial Engineering, volume 53 of

Applications of Mathematics. Springer-Verlag.

Gobet, E. (2009). Advanced monte carlo methods for barrier and related exotic options.

volume 15, pages 497 – 528.

Gobet, E. and Menozzi, S. (2010). Stopped diffusion processes: boundary corrections

and overshoot. Stochastic Processes and Their Applications, 120(2):130–162.

Hagan, P., Lesniewski, A., and Woodward, D. (2015). Probability distribution in the sabr model of stochastic volatility. In Large deviations and asymptotic methods in finance, pages 1–35. Springer.

Hagan, P. S., Kumar, D., Lesniewski, A. S., and Woodward, D. E. (2002). Managing smile risk. The Best of Wilmott, 1:249–296.

Hammersley, J. (2013). Monte carlo methods. Springer Science & Business Media.

Hasan, T., Ali, S., and Khan, M. F. (2013). A comparative study of loss functions for bayesian control in mixture models. Electronic Journal of Applied Statistical Analysis, 6(2):175–185.

Heynen, R. and Kat, H. (1994b). Partial barrier options. Partial barrier options. The Journal of Financial Engineering, 3:253–274.

Hull, J. and White, A. (1993). Efficient procedures for valuing european and american

path-dependent options. The Journal of Derivatives, 1(1):21–31.

Hull, J. C. (2003). Options futures and other derivatives. Pearson Education India.

Iorio, C., Frasso, G., D’Ambrosio, A., and Siciliano, R. (2016). Parsimonious time series clustering using p-splines. Expert Systems with Applications, 52:26–38.

Iorio, C., Frasso, G., D’Ambrosio, A., and Siciliano, R. (2018). A p-spline based clus- tering approach for portfolio selection. Expert Systems with Applications, 95:88–103.

Jackel, P. (2002). Monte Carlo methods in finance. J. Wiley.

Jasra, A. and Del Moral, P. (2011). Sequential monte carlo methods for option pricing.

Stochastic analysis and applications, 29(2):292–316.

Kat, H. and Verdonk, L. (1995). Tree surgery.

Kitagawa, G. (1996). Monte carlo filter and smoother for non-gaussian nonlinear state space models. Journal of computational and graphical statistics, 5(1):1–25.

Kunitomo, N. and Ikeda, M. (1992). Pricing options with curved bound- aries1. Math- ematical finance, 4:275–298.

Liu, J. S. and Chen, R. (1998). Sequential monte carlo methods for dynamic systems. Journal of the American statistical association, 93(443):1032–1044.

McCallum, A., Nigam, K., et al. (1998). A comparison of event models for naive bayes text classification. In AAAI-98 workshop on learning for text categorization, volume 752, pages 41–48. Citeseer.

Merton, R. (1973). Theory of rational option pricing. The Bell Journal of Economics and Management Science, 4:141–183.

Moral, P. (2004). Feynman-kac formulae: Genealogical and interacting particle systems with applications, Probability and its applications. Springer, New York.

Pandolfo, G., D’Ambrosio, A., and Porzio, G. C. (2018). A note on depth-based classifica- tion of circular data. Electronic Journal of Applied Statistical Analysis, 11(2):447–462.

Prakash, G. (2013). Bayes estimation in the inverse rayleigh model. Electronic Journal of Applied Statistical Analysis, 6(1):67–83.

Rish, I. et al. (2001). An empirical study of the naive bayes classifier. In IJCAI 2001 workshop on empirical methods in artificial intelligence, volume 3, pages 41–46.

Rubinstein, M. and Reiner, E. (1991). Breaking down the barriers. Risk, 4:28–35. Shevchenko, P. and Del Moral, P. (2016). Valuation of barrier options using sequential

monte carlo. Journal of Computational Finance.

Targino, R. S., Peters, G. W., and Shevchenko, P. V. (2015). Sequential monte carlo samplers for capital allocation under copula-dependent risk models. Insurance: Math- ematics and Economics, 61:206–226.

Whitley, D. (1994). A genetic algorithm tutorial. Statistics and computing, 4(2):65–85.

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