American Institute of Mathematical Sciences

Advanced
January  2022, 18(1): 137-156. doi: 10.3934/jimo.2020146

An alternative tree method for calibration of the local volatility

Wenxiu Gong 1, and  Zuoliang Xu 2,,

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

* Corresponding author: Zuoliang Xu

Received  March 2019 Revised  July 2020 Published  January 2022 Early access  September 2020

Fund Project: This work is supported by National Natural Science Foundation of China(11571365) and the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China(18XNH107)

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.

Keywords: Volatility, Alternative tree, Option pricing, Calibration, Nonlinear constrained optimization.
Mathematics Subject Classification: Primary: 91G20; Secondary: 91G60.
Citation: Wenxiu Gong, Zuoliang Xu. An alternative tree method for calibration of the local volatility. Journal of Industrial and Management Optimization, 2022, 18 (1) : 137-156. doi: 10.3934/jimo.2020146
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.

Figure 1.  The left figure presents CRR method and steps while the right figure presents TTM and steps. The blue line denotes the BS price. The red line denotes the CRR and TTM price. The green line denotes CRR price with odd steps while the black line denotes CRR price with even steps
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The alternative tree
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  CRR and TTM price with different time steps
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Volatility function $ \sigma_{ex}(t) $ and volatility estimation for $ n = 7 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Stability analysis of the algorithm
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Comparison of the exact value and the optimal with alternative tree, TTM and CRR tree
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Volatility calibrated by linear and quadratic penalty method
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Local volatility and calibrated volatility with $ \frac{K}{S_0} = 100\%, 110\% $
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Some tree methods for calibration of the local volatility
Auther Tree method volatility function
Derman(1996), Barle(1999) Recombining TTM $ \sigma=\sigma(S, t) $
Li(2001) Recombining BTM $ \sigma=\sigma(S, t) $
Crépey (2003) TTM with regularization $ \sigma=\sigma(S, t) $
Charalambous et al. (2007) Nonrecombining BTM $ \sigma=\sigma(t) $
Lok and Lyuu (2017) Recombining waterline tree $ \sigma=\sigma(S)\sigma(t) $
Gong and Xu (2019) Nonrecombining TTM $ \sigma=\sigma(t) $
Table Options

Download as excel

Auther Tree method volatility function
Derman(1996), Barle(1999) Recombining TTM $ \sigma=\sigma(S, t) $
Li(2001) Recombining BTM $ \sigma=\sigma(S, t) $
Crépey (2003) TTM with regularization $ \sigma=\sigma(S, t) $
Charalambous et al. (2007) Nonrecombining BTM $ \sigma=\sigma(t) $
Lok and Lyuu (2017) Recombining waterline tree $ \sigma=\sigma(S)\sigma(t) $
Gong and Xu (2019) Nonrecombining TTM $ \sigma=\sigma(t) $
[1]

Michael C. Fu, Bingqing Li, Rongwen Wu, Tianqi Zhang. Option pricing under a discrete-time Markov switching stochastic volatility with co-jump model. Frontiers of Mathematical Finance, 2022, 1 (1) : 137-160. doi: 10.3934/fmf.2021005
[2]

Han Zhao, Bangdong Sun, Hui Wang, Shiji Song, Yuli Zhang, Liejun Wang. Optimization and coordination in a service-constrained supply chain with the bidirectional option contract under conditional value-at-risk. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022021
[3]

Laurent Devineau, Pierre-Edouard Arrouy, Paul Bonnefoy, Alexandre Boumezoued. Fast calibration of the Libor market model with stochastic volatility and displaced diffusion. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1699-1729. doi: 10.3934/jimo.2019025
[4]

Mikhail Dokuchaev, Guanglu Zhou, Song Wang. A modification of Galerkin's method for option pricing. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021077
[5]

Cuilian You, Le Bo. Option pricing formulas for generalized fuzzy stock model. Journal of Industrial and Management Optimization, 2020, 16 (1) : 387-396. doi: 10.3934/jimo.2018158
[6]

Jia Yue, Nan-Jing Huang. Neutral and indifference pricing with stochastic correlation and volatility. Journal of Industrial and Management Optimization, 2018, 14 (1) : 199-229. doi: 10.3934/jimo.2017043
[7]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial and Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119
[8]

Lishang Jiang, Baojun Bian. The regularized implied local volatility equations -A new model to recover the volatility of underlying asset from observed market option price. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2017-2046. doi: 10.3934/dcdsb.2012.17.2017
[9]

Kai Zhang, Song Wang. Convergence property of an interior penalty approach to pricing American option. Journal of Industrial and Management Optimization, 2011, 7 (2) : 435-447. doi: 10.3934/jimo.2011.7.435
[10]

Kun Fan, Yang Shen, Tak Kuen Siu, Rongming Wang. On a Markov chain approximation method for option pricing with regime switching. Journal of Industrial and Management Optimization, 2016, 12 (2) : 529-541. doi: 10.3934/jimo.2016.12.529
[11]

Tak Kuen Siu, Howell Tong, Hailiang Yang. Option pricing under threshold autoregressive models by threshold Esscher transform. Journal of Industrial and Management Optimization, 2006, 2 (2) : 177-197. doi: 10.3934/jimo.2006.2.177
[12]

Kai Zhang, Xiaoqi Yang, Kok Lay Teo. A power penalty approach to american option pricing with jump diffusion processes. Journal of Industrial and Management Optimization, 2008, 4 (4) : 783-799. doi: 10.3934/jimo.2008.4.783
[13]

Zhuo Jin, Linyi Qian. Lookback option pricing for regime-switching jump diffusion models. Mathematical Control and Related Fields, 2015, 5 (2) : 237-258. doi: 10.3934/mcrf.2015.5.237
[14]

Cuilian You, Le Bo. Pricing of European call option under fuzzy interest rate. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022033
[15]

Antonio Attalienti, Michele Bufalo. Expected vs. real transaction costs in European option pricing. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022063
[16]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks and Heterogeneous Media, 2021, 16 (1) : 1-29. doi: 10.3934/nhm.2020031
[17]

Boshi Tian, Xiaoqi Yang, Kaiwen Meng. An interior-point $l_{\frac{1}{2}}$-penalty method for inequality constrained nonlinear optimization. Journal of Industrial and Management Optimization, 2016, 12 (3) : 949-973. doi: 10.3934/jimo.2016.12.949
[18]

Xiaodi Bai, Xiaojin Zheng, Xiaoling Sun. A survey on probabilistically constrained optimization problems. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 767-778. doi: 10.3934/naco.2012.2.767
[19]

Miao Tian, Xiangfeng Yang, Yi Zhang. Lookback option pricing problem of mean-reverting stock model in uncertain environment. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2703-2714. doi: 10.3934/jimo.2020090
[20]

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

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (418)
  • HTML views (593)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy