Forma, Vol. 31 (Special Issue), pp. S29S40, 2016

European Option Pricing under Fractional Brownian Motion with an Application to Realized Volatility

Takayuki Morimoto

Department of Mathematical Sciences, Kwansei Gakuin University, Japan
E-mail address:

(Received February 13, 2015; Accepted October 30, 2015)

Abstract. This study investigates European option pricing under fractional Brownian motion (fBm) and applies it to realized volatility (RV). The RV measure is selected because it uniquely exhibits simultaneous stationarity and long-range dependency properties in financial time series, as shown in our empirical study. Meanwhile, the Black-Scholes differential equation is not well defined when the underlying assets follow fBm with the Hurst exponent H ≠ ½ because fBm is not a semimartingale. Thus, we compute the European option prices using a previously proposed fractional Black-Scholes formula. Our empirical study is conducted on Tokyo Stock Price Index data from January 06, 1997 to December 30, 2013 with a sample size of 4177.

Keywords: Fractional Brownian Motion, Realized Volatility, Long-range Dependency, Fractional Black-Scholes Formula, Tokyo Stock Price Index Data

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