A Hybrid Importance Sampling Algorithm for Estimating VaR under the Jump Diffusion Model
ABSTRACT
Value at Risk (VaR) is an important tool for estimating the risk of a financial portfolio under significant loss. Although Monte Carlo simulation is a powerful tool for estimating VaR, it is quite inefficient since the event of significant loss is usually rare. Previous studies suggest that the performance of the Monte Carlo simulation can be improved by impor-tance sampling if the market returns follow the normality or the distributions. The first contribution of our paper is to extend the importance sampling method for dealing with jump-diffusion market returns, which can more precisely model the phenomenon of high peaks, heavy tails, and jumps of market returns mentioned in numerous empirical study papers. This paper also points out that for portfolios of which the huge loss is triggered by significantly distinct events, naively applying importance sampling method can result in poor performance. The second contribution of our paper is to develop the hybrid importance sampling method for the aforementioned problem. Our method decomposes a Monte Carlo simulation into sub simulations, and each sub simulation focuses only on one huge loss event. Thus the perform-ance for each sub simulation is improved by importance sampling method, and overall performance is optimized by determining the allotment of samples to each sub simulation by Lagrange’s multiplier. Numerical experiments are given to verify the superiority of our method.
Value at Risk (VaR) is an important tool for estimating the risk of a financial portfolio under significant loss. Although Monte Carlo simulation is a powerful tool for estimating VaR, it is quite inefficient since the event of significant loss is usually rare. Previous studies suggest that the performance of the Monte Carlo simulation can be improved by impor-tance sampling if the market returns follow the normality or the distributions. The first contribution of our paper is to extend the importance sampling method for dealing with jump-diffusion market returns, which can more precisely model the phenomenon of high peaks, heavy tails, and jumps of market returns mentioned in numerous empirical study papers. This paper also points out that for portfolios of which the huge loss is triggered by significantly distinct events, naively applying importance sampling method can result in poor performance. The second contribution of our paper is to develop the hybrid importance sampling method for the aforementioned problem. Our method decomposes a Monte Carlo simulation into sub simulations, and each sub simulation focuses only on one huge loss event. Thus the perform-ance for each sub simulation is improved by importance sampling method, and overall performance is optimized by determining the allotment of samples to each sub simulation by Lagrange’s multiplier. Numerical experiments are given to verify the superiority of our method.
Cite this paper
nullT. Dai and L. Liu, "A Hybrid Importance Sampling Algorithm for Estimating VaR under the Jump Diffusion Model," Journal of Software Engineering and Applications, Vol. 2 No. 4, 2009, pp. 301-307. doi: 10.4236/jsea.2009.24039.
nullT. Dai and L. Liu, "A Hybrid Importance Sampling Algorithm for Estimating VaR under the Jump Diffusion Model," Journal of Software Engineering and Applications, Vol. 2 No. 4, 2009, pp. 301-307. doi: 10.4236/jsea.2009.24039.
References
[1] S. K. Lin, C. D. Fuh, and T. J. Ko, “A bootstrap method with importance resampling to evaluate value-at-risk,” J. Financial Studies, Vol. 12, pp. 81–116, 2004. [2] H. G. Fong and K. C. Lin, “A new analytical approach to value at risk,” J. Portfolio Management, Vol. 25, pp. 88–97, 1999. [3] P. Glasserman, P. Heidelberger, and P. Shahabuddin, Vaniance Reduction Technique for Estimating Value-at- Risk, Management Sci., Vol. 46, pp. 1349–1364, 2000. [4] E. Eberlein, U. Keller, and K. Prause, “New insights into smile, mispricing, and value-at-risk: The hyperbolic model, J. Business, Vol. 71, pp. 371–406, 1998. [5] J. R. M. Hosking, G. Bonti, and D. Siegel, “Beyond the Lognormal,” Risk, Vol. 13, pp. 59–62, 2000. [6] K. Koedijk, R. Huisman, and R. Pownall, “VaR-x: Fat tails in financial risk management,” J. Risk 1, pp. 47–62, 1998. [7] P. Glasserman, P. Heidelberger, and P. Shahabuddin, “Portfolio value-at-risk with heavy-tailed risk factors,” Math. Finance, Vol. 12, pp. 239–269, 2002. [8] R. C. Merton, “Option pricing when underlying stock returns are discontinuous,” J. Financial Econ., Vol. 3, pp. 125–144, 1976. [9] J.-C. Duan, “The GARCH option pricing model,” Math. Finance, Vol. 5, pp. 13–32, 1995. [10] S. Heston, “A closed-form folution for options with stochas-tic volatility with applications to bond and currency op-tions,” Rev. Financial Studies, Vol. 6, pp. 327–343, 1993. [11] P. Jorion, Value at risk, McGraw-Hill, New York, 1997. [12] C. Rouvnez, Going Greek with VaR, Risk 10 (1997) 57–65. [13] F. Black and M. Scholes, “The pricing of options and corporate liabilities,” J. Political Econ., Vol. 81, pp. 637–659, 1973.
[1] S. K. Lin, C. D. Fuh, and T. J. Ko, “A bootstrap method with importance resampling to evaluate value-at-risk,” J. Financial Studies, Vol. 12, pp. 81–116, 2004. [2] H. G. Fong and K. C. Lin, “A new analytical approach to value at risk,” J. Portfolio Management, Vol. 25, pp. 88–97, 1999. [3] P. Glasserman, P. Heidelberger, and P. Shahabuddin, Vaniance Reduction Technique for Estimating Value-at- Risk, Management Sci., Vol. 46, pp. 1349–1364, 2000. [4] E. Eberlein, U. Keller, and K. Prause, “New insights into smile, mispricing, and value-at-risk: The hyperbolic model, J. Business, Vol. 71, pp. 371–406, 1998. [5] J. R. M. Hosking, G. Bonti, and D. Siegel, “Beyond the Lognormal,” Risk, Vol. 13, pp. 59–62, 2000. [6] K. Koedijk, R. Huisman, and R. Pownall, “VaR-x: Fat tails in financial risk management,” J. Risk 1, pp. 47–62, 1998. [7] P. Glasserman, P. Heidelberger, and P. Shahabuddin, “Portfolio value-at-risk with heavy-tailed risk factors,” Math. Finance, Vol. 12, pp. 239–269, 2002. [8] R. C. Merton, “Option pricing when underlying stock returns are discontinuous,” J. Financial Econ., Vol. 3, pp. 125–144, 1976. [9] J.-C. Duan, “The GARCH option pricing model,” Math. Finance, Vol. 5, pp. 13–32, 1995. [10] S. Heston, “A closed-form folution for options with stochas-tic volatility with applications to bond and currency op-tions,” Rev. Financial Studies, Vol. 6, pp. 327–343, 1993. [11] P. Jorion, Value at risk, McGraw-Hill, New York, 1997. [12] C. Rouvnez, Going Greek with VaR, Risk 10 (1997) 57–65. [13] F. Black and M. Scholes, “The pricing of options and corporate liabilities,” J. Political Econ., Vol. 81, pp. 637–659, 1973.