December  2020, 13(12): 3473-3489. doi: 10.3934/dcdss.2020235

The inverse volatility problem for American options

Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA

* Corresponding author

Received  September 2018 Published  December 2020 Early access  January 2020

The problem of determining equity volatility from a knowledge of American option prices for a range of exercise (strike) prices and expirations is solved by minimization of a convex functional.

Citation: Ian Knowles, Ajay Mahato. The inverse volatility problem for American options. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3473-3489. doi: 10.3934/dcdss.2020235
References:
[1]

Y. Achdou, G. Indragoby and O. Pironneau, Volatility calibration with American options, Methods Appl. Anal., 11 (2004), 533–556, https://projecteuclid.org/euclid.maa/1144939946. doi: 10.4310/MAA.2004.v11.n4.a6.

[2]

A. Alfonsi and B. Jourdain, Exact volatility calibration based on a Dupire-type Call-Put duality for perpetual American options, Nonlinear Differ. Equ. Appl., 16 (2009), 523-554.  doi: 10.1007/s00030-009-0027-8.

[3]

F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.

[4]

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.

[5]

P. Carr and A. Hirsa, Why be backward?, Risk, 16 (2003), 103-107. 

[6]

P. Carr and A. Hirsa, Forward evolution equations for knock-out options, in Advances in Mathematical Finance, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2007,195–217. doi: 10.1007/978-0-8176-4545-8_11.

[7]

S. Crépey, Calibration of the local volatility in a trinomial tree using Tikhonov regularization, Inverse Problems, 19 (2003), 91-127.  doi: 10.1088/0266-5611/19/1/306.

[8]

B. Dupire, Pricing with a smile, RISK, 7 (1994), 18-20. 

[9]

E. Fama, The behavior of stock-market prices, Journal of Business, 38 (1965), 34-105.  doi: 10.1086/294743.

[10]

P. Hartman, Ordinary Differential Equations, S. M. Hartman, Baltimore, Md., 1973.

[11]

J. Huang and J.-S. Pang, A mathematical programming with equilibrium constraints approach to the implied volatility surface of American options, Journal of Computational Finance, 4 (2000), 21-56.  doi: 10.21314/JCF.2000.054.

[12]

V. A. Kholodnyi, A nonlinear partial differential equation for American options in the entire domain of the state variable, in Proceedings of the Second World Congress of Nonlinear Analysts, Part 8 (Athens, 1996), 30 (1997), 5059–5070. doi: 10.1016/S0362-546X(97)00207-1.

[13]

I. Knowles, Uniqueness for an elliptic inverse problem, SIAM J. Appl. Math., 59 (1999), 1356-1370.  doi: 10.1137/S0036139997327782.

[14]

I. Knowles and M. A. LaRussa, Conditional well-posedness for an elliptic inverse problem, SIAM J. Appl. Math., 71 (2011), 952-971.  doi: 10.1137/09077566X.

[15]

I. Knowles and M. A. LaRussa, Lavrentiev's theorem and error estimation in elliptic inverse problems, in Spectral Theory, Function Spaces and Inequalities, vol. 219 of Oper. Theory Adv. Appl., Birkhäuser/Springer Basel AG, Basel, 2012, 91–103. doi: 10.1007/978-3-0348-0263-5_6.

[16]

I. KnowlesT. Le and A. Yan, On the recovery of multiple flow parameters from transient head data, J. Comp. Appl. Math., 169 (2004), 1-15.  doi: 10.1016/j.cam.2003.10.013.

[17]

I. KnowlesM. TeubnerA. YanP. Rasser and J. Lee, Inverse groundwater modelling in the Willunga Basin, South Australia, Hydrogeology Journal, 15 (2007), 1107-1118.  doi: 10.1007/s10040-007-0189-6.

[18]

I. Knowles and R. Wallace, A variational method for numerical differentiation, Numerische Mathematik, 70 (1995), 91-110.  doi: 10.1007/s002110050111.

[19]

S. Kon, Models of stock returns - a comparison, Journal of Finance, 39 (1984), 147-165. 

[20]

G. W. Kutner, Determining the implied volatility for American options using the QAM, The Financial Review, 33 (1998), 119-130.  doi: 10.1111/j.1540-6288.1998.tb01611.x.

[21]

O. A. Ladyzhenskaya and N. N. Uraltzeva, Linear and Quasilinear Equations of Parabolic Type, Second edition, revised. Izdat. "Nauka", Moscow, 1973.

[22]

D. Madan and E. Seneta, The variance gamma model for share market returns, Journal of Business, 63 (1990), 511-524.  doi: 10.1086/296519.

[23]

B. Mandelbrot, The variation of certain speculative prices, Journal of Business, 36 (1963), 394-41. 

[24]

J. W. Neuberger, Sobolev Gradients and Differential Equations, vol. 1670 of Lecture Notes in Mathematics, 2nd edition, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04041-2.

[25]

R. Officer, The distribution of stock returns, Journal of the American Statistical Association, 67 (1972), 807-812.  doi: 10.1080/01621459.1972.10481297.

[26]

P. Praetz, The distribution of share price changes, Journal of Business, 45 (1972), 49-55.  doi: 10.1086/295425.

[27]

S. Press, A compound events model for security prices, Journal of Business, 40 (1967), 317-335.  doi: 10.1086/294980.

[28]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in C, 2nd edition, Cambridge University Press, Cambridge, 1992, The art of scientific computing.

[29]

A. Sen, An optimization approach to computing the implied volatility of American options, Optimization Online Digest.

[30] P. WilmottS. Howison and J. Dewynne, Option Pricing, Oxford Financial Press, Oxford, 1993. 
[31]

E. Zeidler, Nonlinear Functional Analysis and its Applications. III, Springer-Verlag, New York, 1985, Variational Methods and Optimization, Translated from the German by Leo F. Boron. doi: 10.1007/978-1-4612-5020-3.

show all references

References:
[1]

Y. Achdou, G. Indragoby and O. Pironneau, Volatility calibration with American options, Methods Appl. Anal., 11 (2004), 533–556, https://projecteuclid.org/euclid.maa/1144939946. doi: 10.4310/MAA.2004.v11.n4.a6.

[2]

A. Alfonsi and B. Jourdain, Exact volatility calibration based on a Dupire-type Call-Put duality for perpetual American options, Nonlinear Differ. Equ. Appl., 16 (2009), 523-554.  doi: 10.1007/s00030-009-0027-8.

[3]

F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.

[4]

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.

[5]

P. Carr and A. Hirsa, Why be backward?, Risk, 16 (2003), 103-107. 

[6]

P. Carr and A. Hirsa, Forward evolution equations for knock-out options, in Advances in Mathematical Finance, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2007,195–217. doi: 10.1007/978-0-8176-4545-8_11.

[7]

S. Crépey, Calibration of the local volatility in a trinomial tree using Tikhonov regularization, Inverse Problems, 19 (2003), 91-127.  doi: 10.1088/0266-5611/19/1/306.

[8]

B. Dupire, Pricing with a smile, RISK, 7 (1994), 18-20. 

[9]

E. Fama, The behavior of stock-market prices, Journal of Business, 38 (1965), 34-105.  doi: 10.1086/294743.

[10]

P. Hartman, Ordinary Differential Equations, S. M. Hartman, Baltimore, Md., 1973.

[11]

J. Huang and J.-S. Pang, A mathematical programming with equilibrium constraints approach to the implied volatility surface of American options, Journal of Computational Finance, 4 (2000), 21-56.  doi: 10.21314/JCF.2000.054.

[12]

V. A. Kholodnyi, A nonlinear partial differential equation for American options in the entire domain of the state variable, in Proceedings of the Second World Congress of Nonlinear Analysts, Part 8 (Athens, 1996), 30 (1997), 5059–5070. doi: 10.1016/S0362-546X(97)00207-1.

[13]

I. Knowles, Uniqueness for an elliptic inverse problem, SIAM J. Appl. Math., 59 (1999), 1356-1370.  doi: 10.1137/S0036139997327782.

[14]

I. Knowles and M. A. LaRussa, Conditional well-posedness for an elliptic inverse problem, SIAM J. Appl. Math., 71 (2011), 952-971.  doi: 10.1137/09077566X.

[15]

I. Knowles and M. A. LaRussa, Lavrentiev's theorem and error estimation in elliptic inverse problems, in Spectral Theory, Function Spaces and Inequalities, vol. 219 of Oper. Theory Adv. Appl., Birkhäuser/Springer Basel AG, Basel, 2012, 91–103. doi: 10.1007/978-3-0348-0263-5_6.

[16]

I. KnowlesT. Le and A. Yan, On the recovery of multiple flow parameters from transient head data, J. Comp. Appl. Math., 169 (2004), 1-15.  doi: 10.1016/j.cam.2003.10.013.

[17]

I. KnowlesM. TeubnerA. YanP. Rasser and J. Lee, Inverse groundwater modelling in the Willunga Basin, South Australia, Hydrogeology Journal, 15 (2007), 1107-1118.  doi: 10.1007/s10040-007-0189-6.

[18]

I. Knowles and R. Wallace, A variational method for numerical differentiation, Numerische Mathematik, 70 (1995), 91-110.  doi: 10.1007/s002110050111.

[19]

S. Kon, Models of stock returns - a comparison, Journal of Finance, 39 (1984), 147-165. 

[20]

G. W. Kutner, Determining the implied volatility for American options using the QAM, The Financial Review, 33 (1998), 119-130.  doi: 10.1111/j.1540-6288.1998.tb01611.x.

[21]

O. A. Ladyzhenskaya and N. N. Uraltzeva, Linear and Quasilinear Equations of Parabolic Type, Second edition, revised. Izdat. "Nauka", Moscow, 1973.

[22]

D. Madan and E. Seneta, The variance gamma model for share market returns, Journal of Business, 63 (1990), 511-524.  doi: 10.1086/296519.

[23]

B. Mandelbrot, The variation of certain speculative prices, Journal of Business, 36 (1963), 394-41. 

[24]

J. W. Neuberger, Sobolev Gradients and Differential Equations, vol. 1670 of Lecture Notes in Mathematics, 2nd edition, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04041-2.

[25]

R. Officer, The distribution of stock returns, Journal of the American Statistical Association, 67 (1972), 807-812.  doi: 10.1080/01621459.1972.10481297.

[26]

P. Praetz, The distribution of share price changes, Journal of Business, 45 (1972), 49-55.  doi: 10.1086/295425.

[27]

S. Press, A compound events model for security prices, Journal of Business, 40 (1967), 317-335.  doi: 10.1086/294980.

[28]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in C, 2nd edition, Cambridge University Press, Cambridge, 1992, The art of scientific computing.

[29]

A. Sen, An optimization approach to computing the implied volatility of American options, Optimization Online Digest.

[30] P. WilmottS. Howison and J. Dewynne, Option Pricing, Oxford Financial Press, Oxford, 1993. 
[31]

E. Zeidler, Nonlinear Functional Analysis and its Applications. III, Springer-Verlag, New York, 1985, Variational Methods and Optimization, Translated from the German by Leo F. Boron. doi: 10.1007/978-1-4612-5020-3.

Figure 1.  Flowchart
Figure 2.  GOOGLE put option data, 19th April, 2013
Figure 4.  Maturity interval $ 14\le T\le 21 $: iterations of $ g(S_0, t_0, K, T) $
Figure 5.  The functional $ G $ for maturity $ 14\le T\le 21 $
Figure 3.  Recovered Google Volatility
Figure 6.  Comparing option and recovered prices from the Dupire Equation
[1]

Kai Zhang, Kok Lay Teo. A penalty-based method from reconstructing smooth local volatility surface from American options. Journal of Industrial and Management Optimization, 2015, 11 (2) : 631-644. doi: 10.3934/jimo.2015.11.631

[2]

Xiaoyu Xing, Hailiang Yang. American type geometric step options. Journal of Industrial and Management Optimization, 2013, 9 (3) : 549-560. doi: 10.3934/jimo.2013.9.549

[3]

Walter Allegretto, Yanping Lin, Ningning Yan. A posteriori error analysis for FEM of American options. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 957-978. doi: 10.3934/dcdsb.2006.6.957

[4]

Xingchun Wang, Yongjin Wang. Variance-optimal hedging for target volatility options. Journal of Industrial and Management Optimization, 2014, 10 (1) : 207-218. doi: 10.3934/jimo.2014.10.207

[5]

Kais Hamza, Fima C. Klebaner, Olivia Mah. Volatility in options formulae for general stochastic dynamics. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 435-446. doi: 10.3934/dcdsb.2014.19.435

[6]

Jacinto Marabel Romo. A closed-form solution for outperformance options with stochastic correlation and stochastic volatility. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1185-1209. doi: 10.3934/jimo.2015.11.1185

[7]

Yu Xing, Wei Wang, Xiaonan Su, Huawei Niu. Equilibrium valuation of currency options with stochastic volatility and systemic co-jumps. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022022

[8]

Lorella Fatone, Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli. Pricing realized variance options using integrated stochastic variance options in the Heston stochastic volatility model. Conference Publications, 2007, 2007 (Special) : 354-363. doi: 10.3934/proc.2007.2007.354

[9]

Wen Li, Song Wang. Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme. Journal of Industrial and Management Optimization, 2013, 9 (2) : 365-389. doi: 10.3934/jimo.2013.9.365

[10]

María Teresa V. Martínez-Palacios, Adrián Hernández-Del-Valle, Ambrosio Ortiz-Ramírez. On the pricing of Asian options with geometric average of American type with stochastic interest rate: A stochastic optimal control approach. Journal of Dynamics and Games, 2019, 6 (1) : 53-64. doi: 10.3934/jdg.2019004

[11]

Donny Citra Lesmana, Song Wang. A numerical scheme for pricing American options with transaction costs under a jump diffusion process. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1793-1813. doi: 10.3934/jimo.2017019

[12]

Baojun Bian, Shuntai Hu, Quan Yuan, Harry Zheng. Constrained viscosity solution to the HJB equation arising in perpetual American employee stock options pricing. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5413-5433. doi: 10.3934/dcds.2015.35.5413

[13]

Wen Chen, Song Wang. A finite difference method for pricing European and American options under a geometric Lévy process. Journal of Industrial and Management Optimization, 2015, 11 (1) : 241-264. doi: 10.3934/jimo.2015.11.241

[14]

Hwa-Sung Kim, Bara Kim, Jerim Kim. Catastrophe equity put options under stochastic volatility and catastrophe-dependent jumps. Journal of Industrial and Management Optimization, 2014, 10 (1) : 41-55. doi: 10.3934/jimo.2014.10.41

[15]

Wan-Hua He, Chufang Wu, Jia-Wen Gu, Wai-Ki Ching, Chi-Wing Wong. Pricing vulnerable options under a jump-diffusion model with fast mean-reverting stochastic volatility. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2077-2094. doi: 10.3934/jimo.2021057

[16]

Ingrid Daubechies, Gerd Teschke, Luminita Vese. Iteratively solving linear inverse problems under general convex constraints. Inverse Problems and Imaging, 2007, 1 (1) : 29-46. doi: 10.3934/ipi.2007.1.29

[17]

Yaoping Chen, Jianqing Chen. Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1531-1552. doi: 10.3934/cpaa.2017073

[18]

Xinfu Chen, Huibin Cheng. Regularity of the free boundary for the American put option. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1751-1759. doi: 10.3934/dcdsb.2012.17.1751

[19]

Junkee Jeon, Jehan Oh. Valuation of American strangle option: Variational inequality approach. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 755-781. doi: 10.3934/dcdsb.2018206

[20]

Mourad Bellassoued, Raymond Brummelhuis, Michel Cristofol, Éric Soccorsi. Stable reconstruction of the volatility in a regime-switching local-volatility model. Mathematical Control and Related Fields, 2020, 10 (1) : 189-215. doi: 10.3934/mcrf.2019036

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (620)
  • HTML views (371)
  • Cited by (0)

Other articles
by authors

[Back to Top]