EXPLICIT CALIBRATION OF PURE JUMP PROCESSES: THE BIST30 EUROPEAN OPTION CASE

Year 2023, Volume: 16 Issue: 1, 200 - 202, 01.02.2023

Bilgi Yılmaz A. Alper Hekimoglu

https://doi.org/10.17261/Pressacademia.2023.1689

Abstract

Purpose- The study has two main purposes. First, it aims to show the efficiency of pure jump processes, more specifically Variance Gamma (VG) and Normal Inverse Gaussian models (NIG), in option pricing by comparing with the Black Scholes (BS) option pricing mode. Second, it aims to calibrate the European options written on BIST30 index.
Methodology- We introduce an alternative derivation of option pricing formulas under the VG and NIG model assumption. We analyze the VG and NIG models' pricing performance by comparing their pricing result with the classical BS model for the BIST30 index. Our data includes the BIST30 index daily price and European options written on it from 05 May 2018 to 05 May 2020 with a maturity of 90 days. In the given period, the European call options' strike prices range from 1200 to 1650, and the European put options' strike prices range from 1000 to 1400. To compare their efficiency, first, the models are calibrated by minimizing the sum of squared deviations between the historically recorded and theoretical European option prices. Second, the theoretical option prices are computed, and the results are compared with the historically recorded option prices.
Findings- Our significant contribution is the calibration of pure jump processes, VG and NIG processes, with the help of characteristic functions, the BS model prices for the BIST30 index, and the computation of European options prices traded in BIST. The study showed that the NIG process outperforms both VG and BS models, and the BS model is the worst model in option pricing for BIST.
Conclusion- The VG and NIG processes can be calibrated by the help of their characteristic functions, and European option price estimations with these models superior to BS model option prices for BIST. Therefore, these processes are more efficient in BIST than the classical BS model.

Keywords

Variance Gamma Process, Normal Inverse Gaussian Process, Black-Scholes Model, Options Pricing; Calibration

References

• Akyapı, B. (2014). An analysis of BIST30 index options market. Master's thesis, Middle East Technical University.
• Alan, N. S., Karagozoglu, A. K., & Korkmaz, S. (2016). Growing pains: The evolution of new stock index futures in emerging markets. Research in International Business and Finance, 37, 1-16.
• Alp, Ö. S. (2016). The Performance of Skewness and Kurtosis Adjusted Option Pricing Model in Emerging Markets: A case of Turkish Derivatives Market. International Journal of Finance & Banking Studies, 5(3), 70-79.
• Barndorff-Nielsen, O. (1977). Exponentially decreasing distributions for the logarithm of particle size. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 353(1674), 401-419.
• Barndorff‐Nielsen, O. E. (1997). Normal inverse Gaussian distributions and stochastic volatility modelling. Scandinavian Journal of statistics, 24(1), 1-13.
• Barndorff-Nielsen, O. E. (1997). Processes of normal inverse Gaussian type. Finance and stochastics, 2(1), 41-68.
• Bastı, E., Kuzey, C., & Delen, D. (2015). Analyzing initial public offerings' short-term performance using decision trees and SVMs. Decision Support Systems, 73, 15-27.
• Behr, A., & Pötter, U. (2009). Alternatives to the normal model of stock returns: Gaussian mixture, generalised logF and generalised hyperbolic models. Annals of Finance, 5(1), 49-68.
• Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637-654.
• Carr, P., Geman, H., Madan, D. B., & Yor, M. (2002). The fine structure of asset returns: An empirical investigation. The Journal of Business, 75(2), 305-332.
• Cont, R., & Da Fonseca, J. (2002). Dynamics of implied volatility surfaces. Quantitative finance, 2(1), 45-51.
• Coşkun, Y., Selcuk-Kestel, A. S., & Yilmaz, B. (2017). Diversification benefit and return performance of REITs using CAPM and Fama-French: Evidence from Turkey. Borsa Istanbul Review, 17(4), 199-215.
• Daal, E. A., & Madan, D. B. (2005). An empirical examination of the variance‐gamma model for foreign currency options. The Journal of Business, 78(6), 2121-2152.
• Demir, S., & Tutek, H. (2004). Pricing of options in emerging financial markets using martingale simulation: An example from Turkey. WIT Transactions on Modelling and Simulation, 38, 141-52.
• Eberlein, E. (2001). Application of generalized hyperbolic Lévy motions to finance. In Lévy processes (pp. 319-336). Birkhäuser, Boston, MA.
• Eberlein, E. (2014). Fourier-based valuation methods in mathematical finance. In Quantitative energy finance (pp. 85-114). Springer, New York, NY.
• Eberlein, E., & Keller, U. (1995). Hyperbolic distributions in finance. Bernoulli, 281-299.
• Eriksson, A., Ghysels, E., & Wang, F. (2009). The normal inverse Gaussian distribution and the pricing of derivatives. The Journal of Derivatives, 16(3), 23-37.
• Ersoy, E., & Bayrakdaroğlu, A. (2013). The lead-lag relationship between ISE 30 index and the TURKDEX-ISE 30 index futures contracts. İstanbul Üniversitesi İşletme Fakültesi Dergisi, 42(1), 26-40.
• Geman, H., Madan, D. B., & Yor, M. (2001). Asset prices are Brownian motion: only in business time. In Quantitative Analysis in Financial Markets: Collected Papers of the New York University Mathematical Finance Seminar, V.2, 103-146.
• Gokgoz, F., & Sezgin-Alp, O. (2014). Estimating the Turkish sectoral market returns via arbitrage pricing model under neural network approach. International Journal of Economics and Finance, 7(1), 154-166.
• Harrison, J. M., & Pliska, S. R. (1983). A stochastic calculus model of continuous trading: complete markets. Stochastic Processes and Their Applications, 15(3), 313-316.
• Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6(2), 327-343.
• Hull, J., & White, A. (1987). The pricing of options on assets with stochastic volatilities. The Journal of Finance, 42(2), 281-300.
• Ivanov, R. V. (2013). Closed form pricing of European options for a family of normal-inverse Gaussian processes. Stochastic Models, 29(4), 435-450.
• Kayalidere, K., Araci, H., & Aktaş, H. (2012). Türev ve spot piyasalar arasındaki etkileşim: VOB üzerine bir inceleme. Muhasebe ve Finansman Dergisi, (56), 137-154.
• Konikov, M., & Madan, D. B. (2002). Option pricing using variance gamma Markov chains. Review of Derivatives Research, 5(1), 81-115.
• Leicht, J. J., & Rathgeber, A. W. (2014). Guaranteed stop orders as portfolio insurance–An analysis for the German stock market. Journal of Derivatives & Hedge Funds, 20(4), 257-278.
• Loregian, A., Mercuri, L., & Rroji, E. (2012). Approximation of the variance gamma model with a finite mixture of normals. Statistics & Probability Letters, 82(2), 217-224.
• Luciano, E., Marena, M., & Semeraro, P. (2016). Dependence calibration and portfolio fit with factor-based subordinators. Quantitative Finance, 16(7), 1037-1052.
• Luciano, E., & Schoutens, W. (2006). A multivariate jump-driven financial asset model. Quantitative Finance, 6(5), 385-402.
• Luciano, E., & Semeraro, P. (2010). Multivariate Variance Gamma and Gaussian dependence: a study with copulas. In Mathematical and Statistical Methods for Actuarial Sciences and Finance (pp. 193-203). Springer, Milano.
• Luciano, E., Marena, M., & Semeraro, P. (2013). Dependence calibration and portfolio fit with factor-based time changes. Carlo Alberto Notebooks, (307).
• Madan, D. B., & Seneta, E. (1987). Simulation of estimates using the empirical characteristic function. International Statistical Review, 153-161.
• Madan, D. B., & Seneta, E. (1990). The variance gamma (VG) model for share market returns. Journal of Business, 511-524.
• Madan, D. B., Carr, P. P., & Chang, E. C. (1998). The variance gamma process and option pricing. Review of Finance, 2(1), 79-105.
• Mandelbrot, B. (1963). New methods in statistical economics. Journal of Political Economy, 71(5), 421-440.
• Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3(1-2), 125-144.
• Praetz, P. D. (1972). The distribution of share price changes. Journal of Business, 49-55.
• Rathgeber, A. W., Stadler, J., & Stöckl, S. (2016). Modeling share returns-an empirical study on the Variance Gamma model. Journal of Economics and Finance, 40(4), 653-682.
• Semeraro, P. (2008). A multivariate variance gamma model for financial applications. International Journal of Theoretical and Applied Finance, 11(1), 1-18.
• Stein, E. M., & Stein, J. C. (1991). Stock price distributions with stochastic volatility: an analytic approach. The Review of Financial Studies, 4(4), 727-752.
• Tankov, P. (2003). Financial modelling with jump processes. Chapman and Hall/CRC.
• Tokat, H. A. (2009). Re-examination of volatility dynamics in Istanbul Stock Exchange. Investment Management and Financial Innovations, 6(1), 192-198.
• Viens, F. G., Mariani, M. C., & Florescu, I. (2011). Handbook of modeling high-frequency data in finance (Vol. 4). John Wiley & Sons.
• Zwillinger, D., & Jeffrey, A. (Eds.). (2007). Table of integrals, series, and products. Elsevier.
• Gomes, M, and Chaibi, A., (2014). Volatility Spillovers Between Oil Prices and Stock Returns: A Focus on Frontier Markets. Post-Print Hall-02314397, HAL.
• Hatty, H., (2002). Airline strategies against crises. Presentation at 5th Hamburg Aviation Conference. 14 February, Hamburg.
• Jones, C., M. and Kaul, G., (1996). Oil and the stock markets. Journal of Finance, 51(2), 463-491.
• Malik, F. and Hammoudeh, S., (2007). Shock and volatility transmission in the oil, US, and Gulf equity markets. International Review of Economics and Finance, 16(3), 357-368.
• Miller, J.I. and Ratti, R.A., (2009). Crude oil and stock markets: Stability, instability, and bubbles. Energy Economics, 31(4), 559-568.
• Mohanty, S, K. and Nandha, M., (2011). Oil risk exposure: the case of the U.S. oil and gas sector. Financial Review, 46(1), 165-19.
• Narayan, P, K., Sharma, S, S., (2011). New evidence on oil price and firm returns. Journal of Banking & Finance, 35(12), 3253-3262.
• Liu, Z, Ding, Z, Li, R, Jiang, X, Wu, Lv, T., (2017). Research on differences of spillover effects between international crude oil price and stock markets in China and America. Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards. 88(1), 575-590.
• Ling, S., & McAleer, M., (2003). Asymptotic theory for a vector ARMA-GARCH model. Econometric Theory, 19(2), 280-310.
• Phan, D, H, B., Sharma,S, S,. and Narayan, P, K, (2015). Oil price and stock returns of consumers and producers of crude oil. Journal of International Financial Markets, Institutions and Money, 34(C), 245-262.
• Reboredo, J.C., (2015). Is there dependence and systemic risk between oil and renewable energy stock prices? Energy Economics, 48(1), 32–45.