Simulation Approaches to Delta Hedging in the Black-Scholes Model.

A delta neutral portfolio in the Black-Scholes model is analysed using both simulated and historical data. An in-depth analysis is conducted in order to investigate what happens when the hedging interval is discrete and when the market maker does not know the true volatility. Based on simulated data, the risk measured by Expected Shortfall decreases the higher the volatility is for a sold call option. This is not the case in reality, where the risk is minimized at about 60-80% of the implied volatility depending on the time to maturity.The so called volatility skew observed in the implied volatilities of the market can either remain steady or move as the spot price of the underlying asset changes. This skew is in contradiction with the Black- Scholes model assumptions. Therefore, what will theoretically happen if it remains frozen in place or moves cannot be analysed in that model. Instead, it is empirically tested how this skew should behave as the market moves to yield the lowest risk. It turns out that it differs depending on the time to maturity of the option. An option which has a long time to maturity seems to generate the lowest risk with a skew that moves in alignment with the spot price. In the case of decreasing time to maturity, a skew should remain stable and not move to yield the lowest possible risk.While most assumptions of the Black-Scholes model does not conform with reality, it still gives a reasonable result in this study. The uncertainty of the input parameters seems to have a greater effect on the returns than the shortcomings of the model does.