In the financial industry, a derivative is a contract whose value is derived from the value of the underlying asset price. The model of reference for the pricing of derivatives is the Black-Scholes model. This thesis examines the limitations of the Black-Scholes model and focuses on plausible alternative models in the equity markets. First, I investigate the effects of hedging strategies on the so-called pinning effect, which is the tendency of stock prices to close near the strike prices of heavily traded options as the expiration date nears. Then I concentrate on alternative stock price models that are more flexible and maintain a degree of analytical tractability for the pricing and hedging of barrier options. Based on the Weiner-Hopf factorisation, I develop some transform algorithms to derive semi-analytical expressions, in the case of barrier options, for first-passage-time probabilities, prices, and Greeks. The outcomes are compared to Monte-Carlo simulations, and I find that the proposed algorithms lead to faster, more accurate results.