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. AvellanedaC. FriedmanR. Holmes and D. Samperi, Calibrating volatility surfaces via relative entropy minimization, Appl. Math. Finance, 4 (1997), 37-64.   Google Scholar

[2]

H. Berestycki1J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

L. Clewlow and C. Strickland, Energy Derivatives: Pricing and Risk Management, Lacima Publications, London, England, 2000. Google Scholar

[8]

T. F. ColemanY. Li and A. Verma, Reconstructing the unknown local volatility function, World Scientific Book Chapters, 2 (2015), 77-102.   Google Scholar

[9]

R. W. Cottle, J. S. Pang and R. B. Stone, The Linear Complementarity Problem, Academic Press, Inc., Boston, MA, 1992.  Google Scholar

[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.  Google Scholar

[11]

E. Derman and I. Kani, Riding on a smile, Risk, 7 (1994), 32-39.   Google Scholar

[12]

B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20.   Google Scholar

[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.  Google Scholar

[14]

J. GatheralE. P. HsuP. LaurenceC. 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.  Google Scholar

[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.   Google Scholar

[16]

N. JacksonE. Süli and S. Howison, Computation of deterministic volatility surfaces, J. Computational Finance, 2 (1999), 5-32.   Google Scholar

[17]

L. JiangQ. ChenL. 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.  Google Scholar

[18]

R. Lagnado and S. Osher, Reconciling differences, Risk, 10 (1997), 79-83.   Google Scholar

[19]

T. Leung and X. Li, Optimal Mean-Reversion Trading: Mathematical Analysis and Practical Applications, World Scientific Publishing Co., 2016. doi: 10.1142/9839.  Google Scholar

[20]

A. LiptonA. 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.  Google Scholar

[21]

R. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.   Google Scholar

[22]

M. Rubinstein, Implied binomial trees, The Journal of Finance, 49 (1994), 771-818.   Google Scholar

[23]

E. S. Schwartz, The stochastic behavior of commodity prices: Implications for valuation and hedging, The Journal of Finance, 52 (1997), 923-973.   Google Scholar

[24]

S. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Springer, 2004.  Google Scholar

[25]

E. Stein and J. Stein, Stock price distributions with stochastic volatility: An analytic approach, Review of Financial Studies, 4 (1991), 727-752.   Google Scholar

[26]

M. Tikhonov, Regularization of incorrectly posed problems, Sov. Math., 4 (1963), 1624-1627.   Google Scholar

[27]

G. E. Uhlenbeck and L. S. Ornstein, On the theory of Brownian Motion, Phys. Rev., 36 (1930), 823-841.   Google Scholar

show all references

References:
[1]

M. AvellanedaC. FriedmanR. Holmes and D. Samperi, Calibrating volatility surfaces via relative entropy minimization, Appl. Math. Finance, 4 (1997), 37-64.   Google Scholar

[2]

H. Berestycki1J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

L. Clewlow and C. Strickland, Energy Derivatives: Pricing and Risk Management, Lacima Publications, London, England, 2000. Google Scholar

[8]

T. F. ColemanY. Li and A. Verma, Reconstructing the unknown local volatility function, World Scientific Book Chapters, 2 (2015), 77-102.   Google Scholar

[9]

R. W. Cottle, J. S. Pang and R. B. Stone, The Linear Complementarity Problem, Academic Press, Inc., Boston, MA, 1992.  Google Scholar

[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.  Google Scholar

[11]

E. Derman and I. Kani, Riding on a smile, Risk, 7 (1994), 32-39.   Google Scholar

[12]

B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20.   Google Scholar

[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.  Google Scholar

[14]

J. GatheralE. P. HsuP. LaurenceC. 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.  Google Scholar

[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.   Google Scholar

[16]

N. JacksonE. Süli and S. Howison, Computation of deterministic volatility surfaces, J. Computational Finance, 2 (1999), 5-32.   Google Scholar

[17]

L. JiangQ. ChenL. 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.  Google Scholar

[18]

R. Lagnado and S. Osher, Reconciling differences, Risk, 10 (1997), 79-83.   Google Scholar

[19]

T. Leung and X. Li, Optimal Mean-Reversion Trading: Mathematical Analysis and Practical Applications, World Scientific Publishing Co., 2016. doi: 10.1142/9839.  Google Scholar

[20]

A. LiptonA. 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.  Google Scholar

[21]

R. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.   Google Scholar

[22]

M. Rubinstein, Implied binomial trees, The Journal of Finance, 49 (1994), 771-818.   Google Scholar

[23]

E. S. Schwartz, The stochastic behavior of commodity prices: Implications for valuation and hedging, The Journal of Finance, 52 (1997), 923-973.   Google Scholar

[24]

S. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Springer, 2004.  Google Scholar

[25]

E. Stein and J. Stein, Stock price distributions with stochastic volatility: An analytic approach, Review of Financial Studies, 4 (1991), 727-752.   Google Scholar

[26]

M. Tikhonov, Regularization of incorrectly posed problems, Sov. Math., 4 (1963), 1624-1627.   Google Scholar

[27]

G. E. Uhlenbeck and L. S. Ornstein, On the theory of Brownian Motion, Phys. Rev., 36 (1930), 823-841.   Google Scholar

Figure 1.  Local volatility fitting result
[1]

Yanqing Hu, Zaiming Liu, Jinbiao Wu. Optimal impulse control of a mean-reverting inventory with quadratic costs. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1685-1700. doi: 10.3934/jimo.2018027

[2]

Hoi Tin Kong, Qing Zhang. An optimal trading rule of a mean-reverting asset. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1403-1417. doi: 10.3934/dcdsb.2010.14.1403

[3]

Edward Allen. Environmental variability and mean-reverting processes. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2073-2089. doi: 10.3934/dcdsb.2016037

[4]

Weiwei Wang, Ping Chen. A mean-reverting currency model with floating interest rates in uncertain environment. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1921-1936. doi: 10.3934/jimo.2018129

[5]

Vinicius Albani, Uri M. Ascher, Xu Yang, Jorge P. Zubelli. Data driven recovery of local volatility surfaces. Inverse Problems & Imaging, 2017, 11 (5) : 799-823. doi: 10.3934/ipi.2017038

[6]

Jingzhen Liu, Ka Fai Cedric Yiu, Alain Bensoussan. Ergodic control for a mean reverting inventory model. Journal of Industrial & Management Optimization, 2018, 14 (3) : 857-876. doi: 10.3934/jimo.2017079

[7]

Qinghua Ma, Zuoliang Xu, Liping Wang. Recovery of the local volatility function using regularization and a gradient projection method. Journal of Industrial & Management Optimization, 2015, 11 (2) : 421-437. doi: 10.3934/jimo.2015.11.421

[8]

Volker Rehbock, Iztok Livk. Optimal control of a batch crystallization process. Journal of Industrial & Management Optimization, 2007, 3 (3) : 585-596. doi: 10.3934/jimo.2007.3.585

[9]

Lili Chang, Wei Gong, Guiquan Sun, Ningning Yan. PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem. Inverse Problems & Imaging, 2015, 9 (3) : 791-814. doi: 10.3934/ipi.2015.9.791

[10]

Jie Chen, Maarten de Hoop. The inverse problem for electroseismic conversion: Stable recovery of the conductivity and the electrokinetic mobility parameter. Inverse Problems & Imaging, 2016, 10 (3) : 641-658. doi: 10.3934/ipi.2016015

[11]

Jingzhen Liu, Ka Fai Cedric Yiu, Alain Bensoussan. The optimal mean variance problem with inflation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 185-203. doi: 10.3934/dcdsb.2016.21.185

[12]

Francesco Cordoni, Luca Di Persio. Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable. Evolution Equations & Control Theory, 2018, 7 (4) : 571-585. doi: 10.3934/eect.2018027

[13]

Victor Isakov. Recovery of time dependent volatility coefficient by linearization. Evolution Equations & Control Theory, 2014, 3 (1) : 119-134. doi: 10.3934/eect.2014.3.119

[14]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97

[15]

Shuang Li, Chuong Luong, Francisca Angkola, Yonghong Wu. Optimal asset portfolio with stochastic volatility under the mean-variance utility with state-dependent risk aversion. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1521-1533. doi: 10.3934/jimo.2016.12.1521

[16]

Pedro Caro. On an inverse problem in electromagnetism with local data: stability and uniqueness. Inverse Problems & Imaging, 2011, 5 (2) : 297-322. doi: 10.3934/ipi.2011.5.297

[17]

Victor Isakov. On uniqueness in the inverse conductivity problem with local data. Inverse Problems & Imaging, 2007, 1 (1) : 95-105. doi: 10.3934/ipi.2007.1.95

[18]

Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967

[19]

Tan H. Cao, Boris S. Mordukhovich. Optimal control of a perturbed sweeping process via discrete approximations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3331-3358. doi: 10.3934/dcdsb.2016100

[20]

Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709

2018 Impact Factor: 1.292

Metrics

  • PDF downloads (44)
  • HTML views (300)
  • Cited by (0)

Other articles
by authors

[Back to Top]