Abstract
In order to gauge foreign exchange market expectations prior to and after the Brexit vote in June, 2016, this paper examines European options written on the GBP/USD and GBP/EUR exchange rates in 2016. First, the parameter estimates from a non-parametric option pricing model with a homogeneity hint show that the Brexit announcement was to a certain extent expected because the implicit probability density functions were negatively skewed in January–February, 2016 and April–June, 2016. This effect was more pronounced for the GBP/USD exchange rates, indicating an increased pessimism of the U.S. currency traders relative to their European counterparts. Entropic risk measures based on skewness premia of deepest out-of-the-money options confirm the findings from implicit distributions. Moreover, these new risk measures are found to statistically significantly predict foreign exchange market volatility at daily to monthly time horizons.
Similar content being viewed by others
Notes
Options are financial derivatives whose value (i.e., price) depends on the value of the underlying security. For example, a European call (put) option based on an exchange rate provides its buyer with a right to purchase (sell) a predetermined amount of currency at a contracted price (“strike exchange rate”) on a specific future date (“maturity”). For this right, the buyer of an option pays a price called the premium.
RSI is a technical indicator that measures the momentum behind price movements. An oversold condition (RSI<30) suggests the price has moved too low too quickly. A break back above 30 should indicate a correction (Bloomberg: Technical Analysis Handbook).
Out-of-the-money options are those that are not profitable to exercise (e.g., for a buyer/holder of a put option, it is the situation when the spot exchange rate is greater than the strike exchange rate).
The SPD is the second derivative of an option-pricing formula with respect to the strike price.
In an additional exercise performed by the author, it is found that the coefficient of correlation between the daily smirk measure and the daily changes in entropic measures ranges from − 0.10 to − 0.15. The implied volatility smirk measure for exchange rate i on day t is calculated as the difference between the implied volatilities of out-of-the-money (OTM) puts and at-the-money (ATM) calls: \(SKEW_{i,t}=VOL^{OTMP}_{i,t}-VOL^{ATMC}_{i,t},~~t=1,\ldots ,T\) and \(i\in \{GBP/USD,GBP/EUR\}\). Therefore, the negative sign of the correlation coefficient is as expected because larger smirk measures correspond to the prevalence of pessimistic FX market’s aggregate expectations and lower entropy.
The author is grateful to the Guest Editors and the anonymous Referees for this and other insightful comments and suggestions.
This variable will be explained in data description.
This choice is based on the forecast horizon of interest that is roughly two months. The main results of the paper are not sensitive to moving window sizes from 30 to 70 days. Moving windows that are shorter than 30 days do not provide sufficient information for successful predictive performance. Moving windows that are longer than 70 days do not leave a sufficient number of data points for any meaningful analysis.
More information on Euronext currency derivatives can be found at: https://www.euronext.com/en/market-data/products/euronext-currency-derivatives.
One such example of the misspecifications of the Black–Scholes model is its substantial inaccuracy related to the pricing of the deep out-of-the-money options (Gençay and Altay-Salih 2003; Gradojevic et al. 2009). For these options, it was found that the Black–Scholes prices overestimate market prices while feedforward NN models provide a superior pricing performance.
The results for the GBP/EUR exchange rate are similar to those for the GBP/USD exchange rate, but the fluctuations in the implied skewness and kurtosis are more moderate. The additional estimates and figures can be available from the author by request.
The number in parentheses is the bootstrap standard error. One leave-out bootstrap with replacement for a window size of \(K=\) 50 observations is applied.
References
Adesina, T. (2017). Estimating volatility persistence under a Brexit-vote structural break. Finance Research Letters, 23, 65–68.
Aït-Sahalia, Y., & Lo, A. (1998). Nonparametric estimation of state-price densities implicit in financial asset prices. Journal of Finance, 53, 499–547.
Asmussen, S., Blanchet, J., Juneja, S., & Rojas-Nandayapa, L. (2011). Efficient simulation of tail probabilities of sums of correlated lognormals. Annals of Operations Research, 189(1), 5–23.
Bates, D. S. (1991). The crash of ’87: Was it expected? The evidence from options markets. Journal of Finance, 46, 1009–1044.
Bekiros, S. (2014). Timescale analysis with an entropy-based shift-invariant discrete wavelet transform. Computational Economics, 44(2), 231–251.
Bekiros, S., & Marcellino, M. (2013). The multiscale causal dynamics of foreign exchange markets. Journal of International Money and Finance, 38, 282–305.
Ben Sita, B. (2017). Volatility patterns of the constituents of FTSE100 in the aftermath of the U.K. Brexit referendum. Finance Research Letters, 23, 137–146.
Bentes, S., & Menezes, R. (2012). Entropy: A new measure of stock market volatility? Journal of Physics: Conference Series, 394(1), 012033.
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659.
Borland, L. (2002). A theory of non-Gaussian option pricing. Quantitative Finance, 2, 415–431.
Borland, L. (2004). The pricing of stock options. In M. Gell-Mann & C. Tsallis (Eds.), Nonextensive entropy: Interdisciplinary applications (pp. 305–320). Oxford: Oxford University Press.
Bowden, R. J. (2011). Directional entropy and tail uncertainty, with applications to financial hazard. Quantitative Finance, 11(3), 437–446.
Breitung, J., & Candelon, B. (2006). Testing for short- and long-run causality: A frequency-domain approach. Journal of Econometrics, 132, 363–378.
Castro, D., & de Carvalho, M. (2017). Spectral density regression for bivariate extremes. Stochastic Environmental Research and Risk Assessment, 31(7), 1603–1613.
de Boer, P.-T., Kroese, D. P., Mannor, S., & Rubinstein, R. Y. (2011). A tutorial on the cross-entropy method. Annals of Operations Research, 134(1), 19–67.
de Carvalho, M., & Martos, G. (2018). Brexit: Tracking and disentangling the sentiment towards leaving the EU. International Journal of Forecasting. https://doi.org/10.1016/j.ijforecast.2018.07.002.
Diks, C., & Panchenko, V. (2006). A new statistic and practical guidelines for nonparametric Granger causality testing. Journal of Economic Dynamics and Control, 30(9–10), 1647–1669.
Garcia, R., & Gençay, R. (2000). Pricing and hedging derivative securities with neural networks and a homogeneity hint. Journal of Econometrics, 94(1/2), 93–115.
Gell-Mann, M., & Tsallis, C. (2004). Nonextensive entropy: Interdisciplinary applications. Oxford: Oxford University Press.
Gençay, R., & Altay-Salih, A. (2003). Degree of mispricing with the Black–Scholes model and nonparametric cures. Annals of Economics and Finance, 4, 73–101.
Gençay, R., & Gibson, R. (2009). Model risk for European-style stock index options. IEEE Transactions on Neural Networks, 18(1), 193–202.
Gençay, R., & Gradojevic, N. (2010). Crash of ’87: Was it expected? Aggregate market fears and long range dependence. Journal of Empirical Finance, 17(2), 270–282.
Gençay, R., & Gradojevic, N. (2017). The tale of two financial crises: An entropic perspective. Entropy, 19(6), 244.
Gradojevic, N. (2016). Multi-criteria classification for pricing European options. Studies in Nonlinear Dynamics & Econometrics, 20(2), 123–139.
Gradojevic, N., & Caric, M. (2017). Predicting systemic risk with entropic indicators. Journal of Forecasting, 36(1), 16–25.
Gradojevic, N., & Gençay, R. (2011). Financial applications of nonextensive entropy. IEEE Signal Processing Magazine, 28(5), 116–121.
Gradojevic, N., Gençay, R., & Kukolj, D. (2009). Option pricing with modular neural networks. IEEE Transactions on Neural Networks, 20(4), 626–637.
Granger, C. W. J. (1969). Investigating causal relations by econometric models and cross-spectral methods. Econometrica, 37, 424–438.
Hammer, P. L., Kogan, A., & Lejeune, M. A. (2011). Reverse-engineering country risk ratings: A combinatorial non-recursive model. Annals of Operations Research, 188(1), 185–213.
Hutchinson, J. M., Lo, A. W., & Poggio, T. (1994). A nonparametric approach to pricing and hedging derivative securities via learning networks. Journal of Finance, 49(3), 851–889.
Ishizaki, R., & Inoue, M. (2013). Time-series analysis of foreign exchange rates using time-dependent pattern entropy. Physica A: Statistical Mechanics and Its Applications, 392(16), 3344–3350.
Martin, M. T., Plastino, A. R., & Plastino, A. (2000). Tsallis-like information measures and the analysis of complex signals. Physica A, 275(1), 262–271.
Namaki, A., Lai, Z. K., Jafari, G., Raei, R., & Tehrani, R. (2013). Comparing emerging and mature markets during times of crises: A non-extensive statistical approach. Physica A: Statistical Mechanics and Its Applications, 392(14), 3039–3044.
Pincus, S. M. (1991). Approximate entropy as a measure of system complexity. Proceedings of the National Academy of Sciences, 88, 2297–2301.
Schiereck, D., Kiesel, F., & Kolaric, S. (2016). Brexit: (Not) another Lehman moment for banks? Finance Research Letters, 19, 291–297.
Shaw, D., Smith, C. M., & Scully, J. (2017). Why did Brexit happen? Using causal mapping to analyse secondary, longitudinal data. European Journal of Operational Research, 263(3), 1019–1032.
Stutzer, M. J. (2000). Simple entropic derivation of a generalized Black–Scholes option pricing model. Entropy, 2, 70–77.
Stutzer, M. J., & Kitamura, Y. (2002). Connections between entropic and linear projections in asset pricing estimation. Journal of Econometrics, 107, 159–174.
Thakor, N. V., & Tong, S. (2004). Advances in quantitative electroencephalogram analysis methods. Annual Review of Biomedical Engineering, 6, 453–495.
Tsallis, C. (1988). Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics, 52, 479–487.
Tsallis, C. (2009). Introduction to nonextensive statistical mechanics: Approaching a complex world. New York: Springer.
Tsallis, C. (2011). The nonadditive entropy Sq and its applications in physics and elsewhere: Some remarks. Entropy, 13, 1765–1804.
Xing, Y., Zhang, X., & Zhao, R. (2010). What does the individual option volatility smirk tell us about future equity returns? Journal of Financial and Quantitative Analysis, 45(3), 641–662.
Zhu, Y., Yang, F., & Ye, W. (2018). Financial contagion behavior analysis based on complex network approach. Annals of Operations Research, 268(1), 93–111.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gradojevic, N. Brexit and foreign exchange market expectations: Could it have been predicted?. Ann Oper Res 297, 167–189 (2021). https://doi.org/10.1007/s10479-020-03582-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-020-03582-z