# American Institute of Mathematical Sciences

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

## 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

* 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.

Citation: Wenxiu Gong, Zuoliang Xu. An alternative tree method for calibration of the local volatility. Journal of Industrial & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] K. Atkinson, An Introduction to Numerical Analysis, 2$^{nd}$ edition, John Wiley & Sons, New York, 1989.  Google Scholar [5] S. Barle and N. Cakici, How to grow a smiling tree, J. Financ. Eng., 7 (1999), 127-146.   Google Scholar [6] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar [7] P. P. Boyle, Option valuation using a three-jump process, Int. Options J., 3 (1986), 7-12.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [14] E. Derman, I. Kani and N. Chriss, Implied trinomial trees of the volatility smile, J. Deriv., 3 (1996), 7-22.   Google Scholar [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.  Google Scholar [16] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20.   Google Scholar [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.  Google Scholar [18] D. P. J. Leisen and M. Reimer, Binomial models for option value-examining and improving convergence, Appl. Math. Finance, 3 (1996), 319-346.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] K. Talias, Implied Binomial Trees and Genetic Algorithms, Ph.D thesis, Imperial College, 2005. Google Scholar [24] J. B. Walsh, The rate of convergence of the binomial tree scheme, Finance Stoch., 7 (2003), 337-361.  doi: 10.1007/s007800200094.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] K. Atkinson, An Introduction to Numerical Analysis, 2$^{nd}$ edition, John Wiley & Sons, New York, 1989.  Google Scholar [5] S. Barle and N. Cakici, How to grow a smiling tree, J. Financ. Eng., 7 (1999), 127-146.   Google Scholar [6] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar [7] P. P. Boyle, Option valuation using a three-jump process, Int. Options J., 3 (1986), 7-12.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [14] E. Derman, I. Kani and N. Chriss, Implied trinomial trees of the volatility smile, J. Deriv., 3 (1996), 7-22.   Google Scholar [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.  Google Scholar [16] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20.   Google Scholar [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.  Google Scholar [18] D. P. J. Leisen and M. Reimer, Binomial models for option value-examining and improving convergence, Appl. Math. Finance, 3 (1996), 319-346.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] K. Talias, Implied Binomial Trees and Genetic Algorithms, Ph.D thesis, Imperial College, 2005. Google Scholar [24] J. B. Walsh, The rate of convergence of the binomial tree scheme, Finance Stoch., 7 (2003), 337-361.  doi: 10.1007/s007800200094.  Google Scholar
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
The alternative tree
CRR and TTM price with different time steps
Volatility function $\sigma_{ex}(t)$ and volatility estimation for $n = 7$
Stability analysis of the algorithm
Comparison of the exact value and the optimal with alternative tree, TTM and CRR tree
Volatility calibrated by linear and quadratic penalty method
Local volatility and calibrated volatility with $\frac{K}{S_0} = 100\%, 110\%$
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 $\sigma=\sigma(t)$
 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 $\sigma=\sigma(t)$
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