Monte Carlo simulations are a class of computational algorithms that use repeated random sampling to solve any problems that have a probabilistic interpretation. In finance, one of the reasons they gained popularity is that they can be used to accurately estimate integrals. The main idea of Monte Carlo simulations is to produce a multitude of sample paths—possible scenarios/outcomes, often over a given period of time. The horizon is then split into a specified number of time steps and the process of doing so is called discretization. Its goal is to approximate continuous time, since the pricing of financial instruments happens in continuous time.
The results from all these simulated sample paths can be used to calculate metrics such as the percentage of times an event occurred, the average value of an instrument at the last step, and so on. Historically, the main problem with the Monte Carlo approach was that it required heavy computational power to calculate all possible scenarios. Nowadays, it is becoming less of a problem as we can run fairly advanced simulations on a desktop computer or a laptop.
By the end of this chapter, we will have seen how we can use Monte Carlo methods in various scenarios and tasks. In some of them, we will create the simulations from scratch, while in others, we will use modern Python libraries to make the process even easier. Due to the method's flexibility, Monte Carlo is one of the most important techniques in computational finance. It can be adapted to various problems, such as pricing derivatives with no closed-form solution (American/Exotic options), valuating bonds (for example, a zero-coupon bond), estimating the uncertainty of a portfolio (for example, by calculating Value-at-Risk and Expected Shortfall), or carrying out stress-tests in risk management. We show how to solve some of these problems in this chapter.
In this chapter, we cover the following recipes:
- Simulating stock price dynamics using Geometric Brownian Motion
- Pricing European options using simulations
- Pricing American options with Least Squares Monte Carlo
- Pricing American options using Quantlib
- Estimating value-at-risk using Monte Carlo