ABSTRACT

If a financial derivative can be traded consecutively and its terminal payoffs can be adjusted as the sum of a bounded process and a stationary process, then we can use the moving average of the historical payoffs to forecast and the corresponding errors form a generalised mean reversion process. Thus we can price the financial derivatives by its moving average. One can even possibly get statistical arbitrage from certain derivative pricing. We particularly discuss the example of European call options. We show that there is a possibility to get statistical arbitrage from Black–Scholes's option price.

References

1. Ansley, C. F., Spivey, W. A., & Wrobleski, W. J. (1977). On the structure of moving average processes. Journal of Econometrics, 6, 121–134. doi: 10.1016/0304-4076(77)90058-6 [Crossref], [Google Scholar]
2. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654. doi: 10.1086/260062 [Crossref], [Web of Science ®], [Google Scholar]
3. Canina, L., & Figlewski, S. (1993). The informational content of implied volatility. The Review of Financial Studies, 6(3), 659–681. doi: 10.1093/rfs/5.3.659 [Crossref], [Web of Science ®], [Google Scholar]
4. Chrisensen, B. J., & Prabhala, N. R. (1998). The relation between implied and realized volatility. Journal of Financial Economics, 50(2), 125–150. doi: 10.1016/S0304-405X(98)00034-8 [Crossref], [Web of Science ®], [Google Scholar]
5. Dumas, B., Fleming, J., & Whaley, R. E. (1998). Implied volatility functions: Empirical tests. The Journal of Finance, 53(6), 2059–2106. doi: 10.1111/0022-1082.00083 [Crossref], [Web of Science ®], [Google Scholar]
6. Fama, E. F, & French, K. R. (1988). Dividend yields and expected stock returns. Journal of Financial Economics, 22, 3–25. doi: 10.1016/0304-405X(88)90020-7 [Crossref], [Web of Science ®], [Google Scholar]
7. Heston, S. L. (1993). A closed form solution for options with stochastic volatility with applications to bond and currency option. The Review of Financial Studies, 6(2), 327–343. doi: 10.1093/rfs/6.2.327 [Crossref], [Web of Science ®], [Google Scholar]
8. Hogan, S., Jarrow, R., & Warachka, M. (2002). Statistical arbitrage and tests of market efficiency. Singapore: Singapore Management University Pre-Prints. [Google Scholar]
9. Karatzas, I., & Shreve, S. E. (1987). Brownian motion and stochastic calculus. Berlin, Germany: Springer-Verlag. [Google Scholar]
10. Lo, A. (2010). Hedge funds: An analytic perspective (Revised and Expanded ed, pp. 260). New Jersey, USA: Priceton University Press. [Crossref], [Google Scholar]
11. Loeve, M. (1977). Probability theory II. Berlin, Germany: Springer-Verlag. [Google Scholar]
12. Merton, R. C. (1973). A rational theory of option pricing. The Bell Journal of Economics and Management Science, 4(1), 141–183. doi: 10.2307/3003143 [Crossref], [Google Scholar]
13. Mukherji, S. (2011). Are stock returns still mean-reverting? Review of Financial Economics, 20, 22–27. doi: 10.1016/j.rfe.2010.08.001 [Crossref], [Google Scholar]
14. Pole, A. (2007). Statistical arbitrage. Hoboken, New Jersey, USA: John Wiley & Sons Inc. [Google Scholar]
15. Poon, S. H., & Granger, C. W. J. (2003). Forecasting volatility in financial markets: a review. Journal of Economic Literature, XLI, 478–539. doi: 10.1257/jel.41.2.478 [Crossref], [Google Scholar]
16. Poterba, J. M, & Summers, L. H. (1988). Mean reversion in stock prices. Journal of Financial Economics, 22, 27–59. doi: 10.1016/0304-405X(88)90021-9 [Crossref], [Web of Science ®], [Google Scholar]
17. Wang, Z. D., & Zheng, W. A. (2014). High frequency trading and probability theory. Singapore: World Scientific. [Crossref], [Google Scholar]