Monte Carlo Simulations in Finance

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...

Simulating stock price dynamics using Geometric Brownian Motion

Thanks to the unpredictability of financial markets, simulating stock prices plays an important role in the valuation of many derivatives, such as options. Due to the aforementioned randomness in price movement, these simulations rely on stochastic differential equations (SDE).

A stochastic process is said to follow the Geometric Brownian Motion (GBM) when it satisfies the following SDE:

Here, we have the following:

• S: Stock price
• μ: The drift coefficient, that is, the average return over a given period or the instantaneous expected return
• σ: The diffusion coefficient, that is, how much volatility is in the drift
• W_t: The Brownian Motion

We will not investigate the properties of the Brownian Motion in too much depth, as it is outside the scope of this book. Suffice to say, Brownian increments are calculated...

Pricing European options using simulations

Options are a type of derivative instrument because their price is linked to the price of the underlying security, such as stock. Buying an options contract grants the right, but not the obligation, to buy or sell an underlying asset at a set price (known as a strike) on/before a certain date. The main reason for the popularity of options is because they hedge away exposure to an asset's price moving in an undesirable way.

A European call/put option gives us the right (but again, no obligation) to buy/sell a certain asset on a certain expiry date (commonly denoted as T).

Some popular methods of options' valuation:

• Using analytic formulas
• Binomial tree approach
• Finite differences
• Monte Carlo simulations

European options are an exception in the sense that there exist an analytical formula for their valuation, which is not the...

Pricing American options with Least Squares Monte Carlo

In this recipe, we learn how to valuate American options. The key difference between European and American options is that the latter can be exercised at any time before and including the maturity date – basically, whenever the underlying asset's price moves favorably for the option holder.

This behavior introduces additional complexity to the valuation and there is no closed-form solution to this problem. When using Monte-Carlo simulations, we cannot only look at the terminal value on each sample path, as the option's exercise can happen anywhere along the path. That is why we need to employ a more sophisticated approach called Least Squares Monte Carlo (LSMC), which was introduced by Longstaff and Schwartz (2001).

First of all, the time axis spanning [0, T] is discretized into a finite number of equally...

Pricing American options using Quantlib

In the previous recipe, we showed how to manually code the Longstaff-Schwartz algorithm. However, we can also use already existing frameworks for valuation of derivatives. One of the most popular ones is QuantLib. It is an open source C++ library that provides tools for the valuation of financial instruments. By using Simplified Wrapper and Interface Generator (SWIG), it is possible to use QuantLib from Python (and some other programming languages, such as R or Julia). In this recipe, we show how to price the same American put option that we priced in the Pricing American options with Least squares Monte Carlo recipe, but the library itself has many more interesting features to explore.

How to do it...

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Estimating value-at-risk using Monte Carlo

Value-at-risk is a very important financial metric that measures the risk associated with a position, portfolio, and so on. It is commonly abbreviated to VaR, not to be confused with Vector Autoregression. VaR reports the worst expected loss – at a given level of confidence – over a certain horizon under normal market conditions. The easiest way to understand it is by looking at an example. Let's say that the 1-day 95% VaR of our portfolio is $100. This means that 95% of the time (under normal market conditions), we will not lose more than $100 by holding our portfolio over one day.

It is common to present the loss given by VaR as a positive (absolute) value. That is why in this example, a VaR of $100 means losing no more than $100.

There are several ways to calculate VaR, some of which are:

• Parametric Approach (Variance...