Modeling Implied Volatility Surfaces Using Two-dimensional Cubic Spline with Estimated Grid Points

- Journal title : Industrial Engineering and Management Systems
- Volume 9, Issue 4, 2010, pp.323-338
- Publisher : Korean Institute of Industrial Engineers
- DOI : 10.7232/iems.2010.9.4.323

Title & Authors

Modeling Implied Volatility Surfaces Using Two-dimensional Cubic Spline with Estimated Grid Points

Yang, Seung-Ho; Lee, Jae-wook; Han, Gyu-Sik;

Yang, Seung-Ho; Lee, Jae-wook; Han, Gyu-Sik;

Abstract

In this paper, we introduce the implied volatility from Black-Scholes model and suggest a model for constructing implied volatility surfaces by using the two-dimensional cubic (bi-cubic) spline. In order to utilize a spline method, we acquire grid (knot) points. To this end, we first extract implied volatility curves weighted by trading contracts from market option data and calculate grid points from the extracted curves. At this time, we consider several conditions to avoid arbitrage opportunity. Then, we establish an implied volatility surface, making use of the two-dimensional cubic spline method with previously estimated grid points. The method is shown to satisfy several properties of the implied volatility surface (smile, skew, and flattening) as well as avoid the arbitrage opportunity caused by simple match with market data. To show the merits of our proposed method, we conduct simulations on market data of S&P500 index European options with reasonable and acceptable results.

Keywords

Implied Volatility Surface;Arbitrage;Two-dimensional Cubic Spline;S&P500 Index European option;

Language

English

References

1.

Andersen, L. B. G. and Brotherton-Ratcliffe, R. (1997), The equity option volatility smile: An implicit finite- difference approach, Journal of Computational Finance, 1(2), 5-37.

2.

Bakshi, G., Cao, C., and Chen, Z. (1997), Empirical Performance of Alternative Option Pricing Models, Journal of Finance, 52, 2003-2049.

3.

Black, F. and Scholes, M. (1973), The Pricing of Options and Corporate Liabilities, Journal of Politics and Economics, 81, 637-654.

4.

Breeden, D. and Litzenberger, R. (1978), Price of statecontingent claims implicit in options prices, Journal of Business, 51, 621-651.

5.

Byrd, R. H., Gilbert, J. C., and Nocedal, J. (2000), A Trust Region Method Based on Interior Point Techniques for Nonlinear Programming, Mathematical Programming, 89, 149-185.

6.

Byrd, R. H., Hribar, M. E., and Nocedal, J. (1999), An Interior Point Algorithm for Large-Scale Nonlinear Programming, SIAM Journal on Optimization, 9, 877-900.

7.

Carr, P. and Madan, D. B. (2005), A Note on Sufficient Conditions for No Arbitrage, Finance Research Letters, 2, 125-130.

8.

Cont, R. and da Fonseca, J. (2002), Dynamics of Implied Volatility Surfaces, Quantitative Finance, 2, 45-60.

9.

Cont, R. and Tankov, P. (2004), Financial Modeling with Jump Processes, Chapman and Hall, Florida.

10.

de Boor, C. (1962), Bicubic Spline Interpolation, Journal of Mathematical Physics, 41, 212-218.

11.

Dempster, M. A. H. and Richards, D. G. (2000), Pricing American options fitting the smile, Mathematical Finance, 10(2), 157-177.

12.

Derman, E. and Kani, I. (1994), Riding on a smile, RISK, 7(2), 32-39.

13.

Dumas, B., Fleming, J., and Whaley, R. E. (1998), Implied Volatility Functions: Empirical Tests, Journal of Finance, 53, 2059-2106.

14.

Dupire, B. (1994), Pricing with a Smile, RISK, 7(1), 18- 20.

15.

Konstantinidi, E., Skiadopoulos, G., and Tzagkaraki, E. (2008), Can the Evolution of Implied Volatility be Forecasted? Evidence from European and US Implied Volatility Indices, Journal of Banking and Finance, 32, 2401-2411.

16.

Fengler, M. R. (2005), Arbitrage-free Smoothing of the Implied Volatility Surface, Quantitative Finance, 9, 417-428.

17.

Han, G.-S. and Lee, J. (2008), Prediction of pricing and hedging errors for equity linked warrants with Gaussian process models, Expert Systems with Applications, 35, 515-523.

18.

Han, G.-S., Kim, B.-H., and Lee, J. (2009), Kernelbased Monte Carlo simulation for American option pricing, Expert Systems with Applications, 36, 4431- 4436.

19.

Hull, J. H. (2009), Option, Futures, and Other Derivatives $7^{th}$ edition, Prentice Hall, New Jersey.

20.

Kim, N.-H., Lee, J., and Han, G.-S. (2009), Model Averaging Methods for Estimating Implied Volatility and Local Volatility Surfaces, Industrial Engineering and Management Science, 8(2), 93-100.

21.

Lee, J. and Lee, D.-W. (2005), An Improved Cluster Labeling Method for Support Vector Clustering, IEEE Trans. on Pattern Analysis and Machine Intelligence, 27, 461-464.

22.

Lee, J. and Lee, D.-W. (2006), Dynamic Characterization of Cluster Structures for Robust and Inductive Support Vector Clustering, IEEE Trans. on Pattern Analysis and Machine Intelligence, 28, 1869-1874.

23.

Lee, D.-W. and Lee, J. (2007), Domain Described Support Vector Classifier for Multi-Classification Problems, Pattern Recognition, 40, 41-51.

24.

Lee, D.-W. and Lee, J. (2007), Equilibrium-Based Support Vector Machine for Semi-Supervised Classification, IEEE Trans. on Neural Networks, 18, 578- 583.

25.

Lee, D.-W., Jung, K.-H., and Lee, J. (2009), Constructing Sparse Kernel Machines Using Attractors, IEEE Trans. on Neural Networks, 20, 721-729.

26.

Lindstrom, M. J. (1999), Penalized Estimation of Free- Knot Splines, Journal of Computational and Graphical Statistics, 8, 333-352.

27.

Parkinson, M. (1980), The Extreme Value Method for Estimating the Variance of the Rate of Return, Journal of Business, 53, 61-68.

28.

Rogers, L. and Satchell. S. (1991), Estimating Variance from High, Low and Closing Prices, Annals of Applied Probability, 1, 504-512.

29.

Rogers, L., Satchell, S., and Yoon, Y. (1994), Estimating the Volatility of Stock Prices: A Comparison of Methods that Use High and Low Prices, Applied Financial Economics, 4, 241-247.

31.

Späth, H. (1995), Two Dimensional Spline Interpolation Algorithms, A K Peters, Boston.

32.

Skiadopoulos G., Hodges, S., and Clewlow, L. (1999), The Dynamics of the S&P 500 Implied Volatility Surface, Review of Derivatives Research, 3, 263- 282.