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Dynamic-programming-based heuristic for multi-objective operating theater planning
An alternative tree method for calibration of the local volatility
1. | College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China |
2. | School of Mathematics, Renmin University of China, Beijing, 100872, China |
In this paper, we combine the traditional binomial tree and trinomial tree to construct a new alternative tree pricing model, where the local volatility is a deterministic function of time. We then prove the convergence rates of the alternative tree method. The proposed model can price a wide range of derivatives efficiently and accurately. In addition, we research the optimization approach for the calibration of local volatility. The calibration problem can be transformed into a nonlinear unconstrained optimization problem by exterior penalty method. For the optimization problem, we use the quasi-Newton algorithm. Finally, we test our model by numerical examples and options data on the S & P 500 index. Numerical results confirm the excellent performance of the alternative tree pricing model.
References:
[1] |
J. Ahn and M. Song,
Convergence of the trinomial tree method for pricing European/American options, Appl. Math. Comput., 189 (2007), 575-582.
doi: 10.1016/j.amc.2006.11.132. |
[2] |
K. Amin,
On the computation of continuous time option prices using discrete approximations, Journal of Financial and Quantitative Analysis, 26 (1991), 477-495.
doi: 10.2307/2331407. |
[3] |
L. Andersen and J. Andreasen,
Jump-Diffusion processes: Volatility smile fitting and numerical methods for option pricing, Rev. Derivatives Res., 4 (2000), 231-262.
doi: 10.2139/ssrn.171438. |
[4] |
K. Atkinson, An Introduction to Numerical Analysis, 2$^{nd}$ edition, John Wiley & Sons, New York, 1989. |
[5] |
S. Barle and N. Cakici,
How to grow a smiling tree, J. Financ. Eng., 7 (1999), 127-146.
|
[6] |
F. Black and M. Scholes,
The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654.
doi: 10.1086/260062. |
[7] |
P. P. Boyle,
Option valuation using a three-jump process, Int. Options J., 3 (1986), 7-12.
|
[8] |
D. M. Chance,
A synthesis of binomial option pricing models for lognormally distributed asset, J. Appl. Finance, 18 (2008), 38-56.
doi: 10.2139/ssrn.969834. |
[9] |
L. B. Chang and K. Palmer,
Smooth convergence in the binomial model, Finance and Stochastics, 11 (2007), 91-105.
doi: 10.1007/s00780-006-0020-6. |
[10] |
C. Charalambous, N. Christofides, E. Constantinide and S. Martzoukos,
Implied non-recombining trees and calibration for the volatility smile, Quant. Finance, 7 (2007), 459-472.
doi: 10.1080/14697680701488692. |
[11] |
J. C. Cox, S. A. Ross and M. Rubinstein,
Option pricing: A simplified approach, J. Financ. Econ., 7 (1979), 229-263.
doi: 10.1016/0304-405X(79)90015-1. |
[12] |
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. |
[13] |
T. S. Dai and Y. D. Lyuu, The Bino-Trinomial tree: A simple model for efficient and accurate option pricing, J. Deriv., (2010), 7–24. |
[14] |
E. Derman, I. Kani and N. Chriss,
Implied trinomial trees of the volatility smile, J. Deriv., 3 (1996), 7-22.
|
[15] |
F. Diener and M. Diener,
Asymptotics of the price oscillations of a European call option in a tree model, Math. Finance, 14 (2004), 271-293.
doi: 10.1111/j.0960-1627.2004.00192.x. |
[16] |
B. Dupire,
Pricing with a smile, Risk, 7 (1994), 18-20.
|
[17] |
W. X. Gong and Z. L. Xu,
Non-recombining trinomial tree pricing model and calibration for the volatility smile, J. Inverse Ill-Posed Probl., 27 (2019), 353-366.
doi: 10.1515/jiip-2018-0005. |
[18] |
D. P. J. Leisen and M. Reimer,
Binomial models for option value-examining and improving convergence, Appl. Math. Finance, 3 (1996), 319-346.
|
[19] |
Y. Li,
A new algorithm for constructing implied binomial trees: Does the implied model fit any volatility smile?, J. Comput. Finance, 4 (2001), 69-98.
|
[20] |
U. H. Lok and Y. D. Lyuu,
The waterline tree for separable local-volatility models, Comput. Math. Appl., 73 (2017), 537-559.
doi: 10.1016/j.camwa.2016.12.008. |
[21] |
J. T. Ma and T. F. Zhu,
Convergence rates of trinomial tree methods for option pricing under regime-switching models, Appl. Math. Lett., 39 (2015), 13-18.
doi: 10.1016/j.aml.2014.07.020. |
[22] |
J. Rendleman, J. Richard and B. J. Bartter,
Two-state option pricing, J. Finance, 34 (1979), 1093-1110.
doi: 10.1111/j.1540-6261.1979.tb00058.x. |
[23] |
K. Talias, Implied Binomial Trees and Genetic Algorithms, Ph.D thesis, Imperial College, 2005. |
[24] |
J. B. Walsh,
The rate of convergence of the binomial tree scheme, Finance Stoch., 7 (2003), 337-361.
doi: 10.1007/s007800200094. |
show all references
References:
[1] |
J. Ahn and M. Song,
Convergence of the trinomial tree method for pricing European/American options, Appl. Math. Comput., 189 (2007), 575-582.
doi: 10.1016/j.amc.2006.11.132. |
[2] |
K. Amin,
On the computation of continuous time option prices using discrete approximations, Journal of Financial and Quantitative Analysis, 26 (1991), 477-495.
doi: 10.2307/2331407. |
[3] |
L. Andersen and J. Andreasen,
Jump-Diffusion processes: Volatility smile fitting and numerical methods for option pricing, Rev. Derivatives Res., 4 (2000), 231-262.
doi: 10.2139/ssrn.171438. |
[4] |
K. Atkinson, An Introduction to Numerical Analysis, 2$^{nd}$ edition, John Wiley & Sons, New York, 1989. |
[5] |
S. Barle and N. Cakici,
How to grow a smiling tree, J. Financ. Eng., 7 (1999), 127-146.
|
[6] |
F. Black and M. Scholes,
The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654.
doi: 10.1086/260062. |
[7] |
P. P. Boyle,
Option valuation using a three-jump process, Int. Options J., 3 (1986), 7-12.
|
[8] |
D. M. Chance,
A synthesis of binomial option pricing models for lognormally distributed asset, J. Appl. Finance, 18 (2008), 38-56.
doi: 10.2139/ssrn.969834. |
[9] |
L. B. Chang and K. Palmer,
Smooth convergence in the binomial model, Finance and Stochastics, 11 (2007), 91-105.
doi: 10.1007/s00780-006-0020-6. |
[10] |
C. Charalambous, N. Christofides, E. Constantinide and S. Martzoukos,
Implied non-recombining trees and calibration for the volatility smile, Quant. Finance, 7 (2007), 459-472.
doi: 10.1080/14697680701488692. |
[11] |
J. C. Cox, S. A. Ross and M. Rubinstein,
Option pricing: A simplified approach, J. Financ. Econ., 7 (1979), 229-263.
doi: 10.1016/0304-405X(79)90015-1. |
[12] |
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. |
[13] |
T. S. Dai and Y. D. Lyuu, The Bino-Trinomial tree: A simple model for efficient and accurate option pricing, J. Deriv., (2010), 7–24. |
[14] |
E. Derman, I. Kani and N. Chriss,
Implied trinomial trees of the volatility smile, J. Deriv., 3 (1996), 7-22.
|
[15] |
F. Diener and M. Diener,
Asymptotics of the price oscillations of a European call option in a tree model, Math. Finance, 14 (2004), 271-293.
doi: 10.1111/j.0960-1627.2004.00192.x. |
[16] |
B. Dupire,
Pricing with a smile, Risk, 7 (1994), 18-20.
|
[17] |
W. X. Gong and Z. L. Xu,
Non-recombining trinomial tree pricing model and calibration for the volatility smile, J. Inverse Ill-Posed Probl., 27 (2019), 353-366.
doi: 10.1515/jiip-2018-0005. |
[18] |
D. P. J. Leisen and M. Reimer,
Binomial models for option value-examining and improving convergence, Appl. Math. Finance, 3 (1996), 319-346.
|
[19] |
Y. Li,
A new algorithm for constructing implied binomial trees: Does the implied model fit any volatility smile?, J. Comput. Finance, 4 (2001), 69-98.
|
[20] |
U. H. Lok and Y. D. Lyuu,
The waterline tree for separable local-volatility models, Comput. Math. Appl., 73 (2017), 537-559.
doi: 10.1016/j.camwa.2016.12.008. |
[21] |
J. T. Ma and T. F. Zhu,
Convergence rates of trinomial tree methods for option pricing under regime-switching models, Appl. Math. Lett., 39 (2015), 13-18.
doi: 10.1016/j.aml.2014.07.020. |
[22] |
J. Rendleman, J. Richard and B. J. Bartter,
Two-state option pricing, J. Finance, 34 (1979), 1093-1110.
doi: 10.1111/j.1540-6261.1979.tb00058.x. |
[23] |
K. Talias, Implied Binomial Trees and Genetic Algorithms, Ph.D thesis, Imperial College, 2005. |
[24] |
J. B. Walsh,
The rate of convergence of the binomial tree scheme, Finance Stoch., 7 (2003), 337-361.
doi: 10.1007/s007800200094. |







Auther | Tree method | volatility function |
Derman(1996), Barle(1999) | Recombining TTM | |
Li(2001) | Recombining BTM | |
Crépey (2003) | TTM with regularization | |
Charalambous et al. (2007) | Nonrecombining BTM | |
Lok and Lyuu (2017) | Recombining waterline tree | |
Gong and Xu (2019) | Nonrecombining TTM |
Auther | Tree method | volatility function |
Derman(1996), Barle(1999) | Recombining TTM | |
Li(2001) | Recombining BTM | |
Crépey (2003) | TTM with regularization | |
Charalambous et al. (2007) | Nonrecombining BTM | |
Lok and Lyuu (2017) | Recombining waterline tree | |
Gong and Xu (2019) | Nonrecombining TTM |
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