# American Institute of Mathematical Sciences

September 2018, 8(3&4): 625-635. doi: 10.3934/mcrf.2018026

## Recovery of local volatility for financial assets with mean-reverting price processes

 Institute of Scientific Computation and Financial Data Analysis, Shanghai University of Finance and Economics, Shanghai 200433, China

Received  October 2017 Revised  May 2018 Published  September 2018

Fund Project: This research is supported in part by Natural Science Foundation of China under Grant 71771142, 71271127

This article is concerned with the model calibration for financial assets with mean-reverting price processes, which is an important topic in mathematical finance.

The discussion focuses on the recovery of local volatility from market data for Schwartz(1997) model. It is formulated as an inverse parabolic problem, and the necessary condition for determining the local volatility is derived under the optimal control framework. An iterative algorithm is provided to solve the optimality system and a synthetic numerical example is provided to illustrate the effectiveness.

Citation: Qihong Chen. Recovery of local volatility for financial assets with mean-reverting price processes. Mathematical Control & Related Fields, 2018, 8 (3&4) : 625-635. doi: 10.3934/mcrf.2018026
##### References:
 [1] M. Avellaneda, C. Friedman, R. Holmes and D. Samperi, Calibrating volatility surfaces via relative entropy minimization, Appl. Math. Finance, 4 (1997), 37-64. [2] H. Berestycki1, J. Busca and I. Florent, Asymptotics and calibration of local volatility models, Quantitative Finance, 2 (2002), 61-69. doi: 10.1088/1469-7688/2/1/305. [3] F. Black and M. S. Scholes, The pricing of options and corporate liabilities, J. Political Economy, 81 (1973), 637-654. doi: 10.1086/260062. [4] I. Bouchouev and V. Isakov, The inverse problem of option pricing, Inverse Problems, 13 (1997), L11-L17. doi: 10.1088/0266-5611/13/5/001. [5] I. Bouchouev and V. Isakov, Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets, Inverse Problems, 15 (1999), R95-R116. doi: 10.1088/0266-5611/15/3/201. [6] L. Clewlow and C. Strickland, Valuing energy options in a one factor model fitted to forward prices, Lacima Research Papers & Articles, Lacima Group, 1999. [7] L. Clewlow and C. Strickland, Energy Derivatives: Pricing and Risk Management, Lacima Publications, London, England, 2000. [8] T. F. Coleman, Y. Li and A. Verma, Reconstructing the unknown local volatility function, World Scientific Book Chapters, 2 (2015), 77-102. [9] R. W. Cottle, J. S. Pang and R. B. Stone, The Linear Complementarity Problem, Academic Press, Inc., Boston, MA, 1992. [10] S. Crépey, Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM J. Math. Anal., 34 (2003), 1183-1206. doi: 10.1137/S0036141001400202. [11] E. Derman and I. Kani, Riding on a smile, Risk, 7 (1994), 32-39. [12] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. [13] H. Egger and H. W. Engl, Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates, Inverse Problems, 21 (2005), 1027-1045. doi: 10.1088/0266-5611/21/3/014. [14] J. Gatheral, E. P. Hsu, P. Laurence, C. Ouyang and T. H. Wang, Asymptotics of implied volatility in local volatility models, Mathematical Finance, 22 (2012), 591-620. doi: 10.1111/j.1467-9965.2010.00472.x. [15] S. Heston, A closed form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), 327-343. [16] N. Jackson, E. Süli and S. Howison, Computation of deterministic volatility surfaces, J. Computational Finance, 2 (1999), 5-32. [17] L. Jiang, Q. Chen, L. Huang and J. Zhang, A new well-posed algorithm to recover implied local volatility, Quantitative Finance, 3 (2003), 451-457. doi: 10.1088/1469-7688/3/6/304. [18] R. Lagnado and S. Osher, Reconciling differences, Risk, 10 (1997), 79-83. [19] T. Leung and X. Li, Optimal Mean-Reversion Trading: Mathematical Analysis and Practical Applications, World Scientific Publishing Co., 2016. doi: 10.1142/9839. [20] A. Lipton, A. Gal and A. Lasis, Pricing of vanilla and first generation exotic options in the local stochastic volatility framework: Survey and new results, Quantitative Finance, 14 (2014), 1899-1922. doi: 10.1080/14697688.2014.930965. [21] R. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144. [22] M. Rubinstein, Implied binomial trees, The Journal of Finance, 49 (1994), 771-818. [23] E. S. Schwartz, The stochastic behavior of commodity prices: Implications for valuation and hedging, The Journal of Finance, 52 (1997), 923-973. [24] S. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Springer, 2004. [25] E. Stein and J. Stein, Stock price distributions with stochastic volatility: An analytic approach, Review of Financial Studies, 4 (1991), 727-752. [26] M. Tikhonov, Regularization of incorrectly posed problems, Sov. Math., 4 (1963), 1624-1627. [27] G. E. Uhlenbeck and L. S. Ornstein, On the theory of Brownian Motion, Phys. Rev., 36 (1930), 823-841.

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##### References:
 [1] M. Avellaneda, C. Friedman, R. Holmes and D. Samperi, Calibrating volatility surfaces via relative entropy minimization, Appl. Math. Finance, 4 (1997), 37-64. [2] H. Berestycki1, J. Busca and I. Florent, Asymptotics and calibration of local volatility models, Quantitative Finance, 2 (2002), 61-69. doi: 10.1088/1469-7688/2/1/305. [3] F. Black and M. S. Scholes, The pricing of options and corporate liabilities, J. Political Economy, 81 (1973), 637-654. doi: 10.1086/260062. [4] I. Bouchouev and V. Isakov, The inverse problem of option pricing, Inverse Problems, 13 (1997), L11-L17. doi: 10.1088/0266-5611/13/5/001. [5] I. Bouchouev and V. Isakov, Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets, Inverse Problems, 15 (1999), R95-R116. doi: 10.1088/0266-5611/15/3/201. [6] L. Clewlow and C. Strickland, Valuing energy options in a one factor model fitted to forward prices, Lacima Research Papers & Articles, Lacima Group, 1999. [7] L. Clewlow and C. Strickland, Energy Derivatives: Pricing and Risk Management, Lacima Publications, London, England, 2000. [8] T. F. Coleman, Y. Li and A. Verma, Reconstructing the unknown local volatility function, World Scientific Book Chapters, 2 (2015), 77-102. [9] R. W. Cottle, J. S. Pang and R. B. Stone, The Linear Complementarity Problem, Academic Press, Inc., Boston, MA, 1992. [10] S. Crépey, Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM J. Math. Anal., 34 (2003), 1183-1206. doi: 10.1137/S0036141001400202. [11] E. Derman and I. Kani, Riding on a smile, Risk, 7 (1994), 32-39. [12] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. [13] H. Egger and H. W. Engl, Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates, Inverse Problems, 21 (2005), 1027-1045. doi: 10.1088/0266-5611/21/3/014. [14] J. Gatheral, E. P. Hsu, P. Laurence, C. Ouyang and T. H. Wang, Asymptotics of implied volatility in local volatility models, Mathematical Finance, 22 (2012), 591-620. doi: 10.1111/j.1467-9965.2010.00472.x. [15] S. Heston, A closed form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), 327-343. [16] N. Jackson, E. Süli and S. Howison, Computation of deterministic volatility surfaces, J. Computational Finance, 2 (1999), 5-32. [17] L. Jiang, Q. Chen, L. Huang and J. Zhang, A new well-posed algorithm to recover implied local volatility, Quantitative Finance, 3 (2003), 451-457. doi: 10.1088/1469-7688/3/6/304. [18] R. Lagnado and S. Osher, Reconciling differences, Risk, 10 (1997), 79-83. [19] T. Leung and X. Li, Optimal Mean-Reversion Trading: Mathematical Analysis and Practical Applications, World Scientific Publishing Co., 2016. doi: 10.1142/9839. [20] A. Lipton, A. Gal and A. Lasis, Pricing of vanilla and first generation exotic options in the local stochastic volatility framework: Survey and new results, Quantitative Finance, 14 (2014), 1899-1922. doi: 10.1080/14697688.2014.930965. [21] R. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144. [22] M. Rubinstein, Implied binomial trees, The Journal of Finance, 49 (1994), 771-818. [23] E. S. Schwartz, The stochastic behavior of commodity prices: Implications for valuation and hedging, The Journal of Finance, 52 (1997), 923-973. [24] S. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Springer, 2004. [25] E. Stein and J. Stein, Stock price distributions with stochastic volatility: An analytic approach, Review of Financial Studies, 4 (1991), 727-752. [26] M. Tikhonov, Regularization of incorrectly posed problems, Sov. Math., 4 (1963), 1624-1627. [27] G. E. Uhlenbeck and L. S. Ornstein, On the theory of Brownian Motion, Phys. Rev., 36 (1930), 823-841.
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