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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[26]

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

[27]

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

[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.  Google Scholar

[29]

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

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[26]

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

[27]

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

[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.  Google Scholar

[29]

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

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

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
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