This thesis describes the modelling of stock price volatility as a function of the stock price and an exponentially weighted moving average. The data used is the Dow Jones Industrial Average Index for the years 1901 until 1995. The random walk of asset prices and the Black-Scholes model are modified to include a moving average as a third variable, the other two being stock price and time. The ideas behind technical analysis are introduced while no attempt is made to justify its use. Particular mention is made to the widely used technical indicator, the moving average. Numerous tests are performed on the Dow Jones data in order to determine whether or not the volatility, 𝜎, can be a function of the ratio of the stock price and the corresponding exponentially weighted moving average, S/I. Explicit finite differencing is carried out on the p.d.e. associated with the modified Black- Scholes model and various checks are made to study the agreement between the numerical scheme and the exact results. A 2-D option, which has a payoff dependent on the performance of an exponentially weighted moving average, is introduced and solved numerically. Finally, the results detailed in the thesis are discussed along with possible future extensions. Also mentioned is an Asian option type payoff used by traders to trade the cross-overs of two different (time scale) moving averages.