This thesis deals with derivatives, the pricing of them and the eect they can have on nancial markets. In the SABR model we develop a new expansion for call prices and a fast arbitrage free pricing scheme that uses a one step implicit nite dierence scheme. Next we consider perpetual claims. We suggest a simple nite dierence method to price them and prove convergence for the method. We then apply the method to dierent examples: We calculate the expected exit time of a diusion from an interval, price perpetual CDOs on households with exploding credit risk, calculate the value of cash when the interest rate can go negative and value perpetual range accruals in the presence of transaction costs. Thirdly we apply modications of the Ninomiya-Victoir Monte Carlo scheme to the double-mean-reverting DMR) model (a three factor stochastic volatility model). Thereby we demonstrate on the one hand that fast calibration of the DMR model is practical, and on the other that suitably modied Ninomiya-Victoir schemes are applicable to the simulation of much more complicated time-homogeneous models than may have been thought previously. Last we construct a simple model to capture feedback eects in nancial markets. Using the model we can under certain assumptions back out the option position of delta hedgers in a market.