We deal with the stability problem of the fractional order Black-Scholes model driven by fractional Brownian motion (fBm). First, necessary and sufficient conditions are established for almost sure asymptotic stability and pth moment asymptotic stability by means of the largest Lyapunov exponent and the pth moment Lyapunov exponent, respectively. Moreover, we are able to present large deviations results for this fractional process. In particular, for the first time it is found that the Hurst parameter affects both stability conclusions and large deviations. Interestingly, large deviations always happen for the considered system when 1/2 < H < 1. This fact is due to the long-range dependence (LRD) property of the fBm. Numerical simulation results are presented to illustrate the above findings.