Actuarial Finance

19 Simulation methods

As we have seen in Chapter 18, it can be a real challenge to obtain explicit expressions for the price of complex derivatives. In practice, contracts are even more complex than what we have studied so far and the market models used are usually more sophisticated than the Black-Scholes-Merton model. Therefore, deriving analytical expressions for the price of a financial contract is often cumbersome or even impossible. Then, we need to resort to numerical methods.

This chapter focuses on a popular approach to price complex derivatives, namely simulation. Simulation in general refers to a set of techniques meant to artificially imitate a complex phenomenon. For example, pilots commonly use flight simulators to train in situations that would be too costly or dangerous to perform otherwise. An oft-cited example in mathematics is Monte Carlo integration in which random numbers are used to approximate the area under a curve when a closed-form expression is not available.

In statistics and actuarial science, simulation is used to generate artificial random scenarios from a specified model, e.g. a random sample from a given distribution. Once the sample is generated, it can be used to approximate quantities that could not be computed otherwise. Typical applications of simulation in actuarial finance include the pricing of advanced derivatives such as ELIAs and path-dependent exotic options, and the analysis of sensitivities (of profits and solvency) with respect ...