FE 522 Part 4 Monte Carlo Simulation: Basics

Intro

Monte Carlo Simulation is a mathematical technique that allows you to account for risk in quantitative analysis and decision making. It provides a range of possible outcomes and the probabilities they will occur for any choice of action. In finance, it’s often used to determine the value at risk for a specific portfolio of assets.

Here’s a brief overview of the basics:

Random Sampling: The core idea behind Monte Carlo simulations is to generate a large number of random samples from a given process or distribution.
Statistical Modeling: Each random sample represents a possible outcome. By running many such samples, you can build a statistical model of possible outcomes.
Probability Distributions: Input variables in a Monte Carlo model are usually modeled using probability distributions rather than single-point estimates.
Analysis of Results: After running the simulation many times, the results can be analyzed to get statistics like the mean, median, variance, and to generate confidence intervals for predictions.
Applications in Finance: Monte Carlo simulation can be used in options pricing, risk management, capital budgeting, and many other areas of financial decision-making.

Motivation

EUR/USD Rate: $S(t)$ represents the exchange rate between the Euro and US Dollar at a given time ( t ).
Geometric Brownian Motion (GBM) Model: The slide states that the exchange rate $S(t)$ follows a GBM model. This means its logarithm follows a Brownian motion with drift, making $S(t)$ log-normally distributed. In simpler terms, the GBM model is often used to describe stock prices because it assumes that percentage changes in stock prices are normally distributed, which often fits real-world data reasonably well.
Put Option on Average FX Rate: The slide introduces a specific type of put option. This option’s payoff is based on the average exchange rate over 60 days, rather than the rate at a specific expiration date. This is an example of an Asian option, which takes the average price of the underlying asset over a certain period of time as its strike price.
Dates Considered: The ( t_1, t_2, … t_{60} ) are the specific days over which the averaging takes place.
Average FX Rate Formula:

A = \frac{1}{60} \sum_{n=1}^{60} S(t_n)

Strike Rate and Payoff: ( K ) is the predetermined strike rate. The payoff for this put option is ( \max(0, K – A) ). This means if the average rate ( A ) is below the strike rate ( K ), the holder of the option can make a profit.

For the formula of ( S(t) ) under the Geometric Brownian Motion (GBM) model:

S(t) = S(0) e^{(\mu - \frac{\sigma^2}{2})t + \sigma W(t)}

Where:

$S(0)$ is the initial stock price or, in this case, the initial EUR/USD rate.
$\mu$ is the expected return or drift.
$\sigma$ is the volatility of the returns.
$W(t)$ is a standard Brownian motion (or Wiener process) up to time $t$ .

This formula describes how the exchange rate or stock price $S(t)$ evolves over time under the GBM model.

How can we determine the premium of such an option?
According to the risk-neutral evaluation (FE 520), the computation can be expressed as:

p = e^{-r t_{60}} \mathbb{E}\left[ (K - A)^+ \right]

Where:

$p$ denotes the premium.
$e$ represents the base of natural logarithm.
$r$ is the risk-free rate.
$t_{60}$ is the time at the end of the 60th day.
$\mathbb{E}$ represents the expected value.
$K$ is the strike price of the option.
$A$ is the average rate.
The notation $(K - A)^+$ represents the positive part of $(K - A)$ , which means it’s equivalent to $\max(0, K-A)$ .
How do we compute the expectation above?
To determine this, we require the distribution of $A$ .
It’s known that each $S(t_i)$ value follows a lognormal distribution. However, the distribution of the sum:

[ S(t_1) + S(t_2) + \dots + S(t_{60}) ]

is not immediately apparent.

While the sum of normally distributed random variables results in another normal distribution, it is important to note that the distribution of the sum of lognormally distributed variables isn’t straightforwardly known.