Pricing and calibration with stochastic local volatility models in a Monte Carlo setting

A general purpose of mathematical models is to accurately mimic some observed phenomena in the real world. In financial engineering, for example, one aim is to reproduce market prices of financial contracts with the help of applied mathematics. In the Foreign Exchange (FX) market, the so-called implied volatility smile plays a key role in the pricing and hedging of financial derivative contracts. This volatility smile is a phenomenon that reflects the prices of European-type options for different strike prices; the implied volatility tends to be higher for options that are deeper In The Money and Out of The Money than options that are approximately At The Money. In order for a pricing model to be accepted in the financial industry, it should at least be able to accurately price back the most simple financial derivative contracts, namely European call and put options. In other words, the model should calibrate well to the implied volatility smile observed in the financial market. The calibration should not only be accurate, but also reasonably fast. Another feature we wish the financial asset model to possess, is an accurate pricing of so-called exotic financial products. Exotic products are not traded on regular exchanges, but over-the-counter, i.e. directly between two parties without the supervision of an exchange. An example is a barrier option, which is a financial contract of which its payoff depends on the possible event that the underlying asset price hits a certain pre-determined level. The model prices of these path-dependent contracts are determined by the transition densities of the relevant underlying asset(s) between future time-points. These transition densities are reflected by the forward volatility smile the model implies; in order for the model to accurately price exotic products, it should yield realistic forward volatilities

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