# Stochastic Differential Equations & Geometric Brownian Motion

## Table of Contents

## Attention Conservation Notice

The 2nd post in a series of 5 (first post here) in which the Black-Scholes-Merton Model for pricing European put and call options is derived.

This follows Stephen Blyth's Excellent Book An Introduction to Quantitative Finance closely, with embellishment using python and some additional notes. This is probably:

- not very helpful if you pragmatically want to price an option
- overwhelming if you don't like math
- may miss some of the contexts you'd want if you have a lot of mathematical maturity

## Preamble

In the last post, we derived Brownian motion by considering a symmetric random walk (one with an equal likelihood of each change) and taking the limit as the time difference between steps goes to 0. Now that we know what Brownian motion is, we can use that to intuit stochastic differential equations.

## Difference Equations

For \(\Delta t > 0\), we can write a difference equation such as

\(S_{t + \Delta t} - S_t = \mu \Delta t + \sigma (W_{t + \Delta t} - W_t) = \mu \Delta t + \sigma \sqrt{\Delta t} W\), \(W \sim \mathcal{N}(0, 1)\)

That is, we say that the difference between a stock at time \(t + \Delta t\) and time \(t\) is the weighted sum of two functions \(\mu\) and \(\sigma\).

If we fix \(\mu\) and/or \(\sigma\) we can view the effect that it has on the outcome process:

Intuitively, we see that \(\mu\) dictates our slope. It adds (or removes) a certain amount at each timestep and is only dictated by the length of the timestep.

\(\sigma\) dictates the strength of the Brownian motion effect on the difference. Higher \(\sigma\) can lead us to some pretty rocky territory which can be intuitively seen when we look at the middle row of the plot which isolates the effect of \(\sigma\) by fixing \(\mu\) at 0. We can also see that consistent steady returns from \(\mu\) can 'tame' the variability introduced by \(\sigma\) if we look at the upper or lower right plots.

## Stochastic Differential Equations

We write the limits as \(\Delta t \to 0\):

\(dS_t = \mu dt + \sigma d W_t\)

or more generally

\(dS_t = \mu(S_t, t)dt + \sigma(S_t, t) d W_t\)

This is a **stochastic differential equation**.

The meaning of \(dW_t\) is non-trivial, although the discrete analog \(\sqrt{\Delta t}W\) where \(W \sim \mathcal{N}(0, 1)\) is helpful for conceptual understanding.

### Motivation

Stochastic differential equations have a rich general theory. The idea is that we have a differential equation (i.e. an equation with a derivative as a variable) and we add some random variable to it. In this case, we are using a normal random variable, but we can also add different random variables to get other effects.

There are many different ways that folks in finance use SDEs. One
example is the Cox-Ingersoll-Ross model for creating short-rate
models. This uses square-root diffusion and has some nice properties
(namely **mean reversion**). A short-rate is the continuously compounded, annualized
interest rate in which you can borrow money for a very small period of
time.

The plot above shows simulated paths from the Cox-Ingersoll-Ross Square-Root Diffusion model. We can see the mean-reverting characteristic

Another interesting example of a stochastic differential equation is the Merton Jump Diffusion Model. Our SDE above generates a 'continuous' sample path. By changing our normal R.V. for a Poisson and changing the structure a bit, we can tackle the more general case where the underlying values are generated by a mixture of both continuous and jump processes. We can think of this being useful by considering events that cause jumps in a stock price, like an earnings report or a major news event.

The plot above shows 2 jump diffusion processes, one low and one high volatility. In this case, the difference is made by tweaking the rate parameter \(\lambda\).

Low Volatility \(\sim \mathrm{Poisson}(\lambda = 0.2)\)

High Volatility \(\sim \mathrm{Poisson}(\lambda = 2.0)\)

In the plot, you can see that the points where a jump was made, and they are quite drastic for the high volatility case.

All that said, We're getting ahead of ourselves. Let's return to \(dS_t = \mu(S_t, t)dt + \sigma(S_t, t) d W_t\)

## Choice of \(\mu(S_t, t)\), \(\sigma(S_t, t)\) to model the behavior of a stock price

A reasonable choice of functions \(\mu(S_t, t), \sigma(S_t, t)\) to model the behavior of a stock price is to:

- assume that the drift \(\mu\) of a stock is proportional to its price, \(\mu(S_t, t) = \mu S_t\)
- with no random term, the ODE would be \(dS_t = \mu S_tdt \implies S_t = S_0e^{\mu t}\)

- For \(\sigma(S_t, t)\), we might assume that the uncertainty about the stock's percentage return is constant
- i.e. \(p(S_t) = 100 \to \not \in (99, 101) \equiv p(S_t) = 50 \to \not \in (49.5, 50.5)\)
- Then over small \(\Delta t\), \(\mathrm{Var}(\frac{\Delta S_t}{S_t}) \approx \sigma^2 \Delta t \implies \mathrm{Var}(\Delta S_t) \approx \sigma^2 S_t^2 \Delta t\)

## Geometric Brownian Motion

Therefore, one possible process for the evolution of the stock is

\(dS_t = \mu S_t dt + \sigma S_t d W_t\)

which is the definition that the stock follows **Geometric Brownian
Motion**. **[1]**

The plot above looks like the jump processes we saw earlier without any jumps.

Geometric Brownian motion is used widely in finance. It has a bunch of characteristics that make it a natural fit for modeling stocks:

- it looks like a stock
- it quacks like a stock
- it waddles like a stock

It is not a duck though, because:

- It does not account for volatility changing over time
- If we assume that volatility \(\propto t, S_t\), then we have a local volatility model
- If we assume that the volatility is it's own random variable
driven by a different equation (usually another Brownian motion),
then we have a stochastic volatility model
**[2]**

- It does not account for jumps caused by unpredictable events or news
- Attempts to model these discontinuities are given by using jump processes as briefly alluded to above

## Next Time

The next post in this series will take a look at Ito's lemma. We will need it for some of the steps that will follow in our quest to derive the Black-Scholes formula.

## Extra Notes

**Note [1]**
Most plots look pretty cool in 3D. Here is a Brownian surface:

**Note [2]**
Stochastic volatility models are important. They resolve a shortcoming of the
Black-Scholes model that we are deriving, namely the assumption that
the underlying volatility is constant over the life of the derivative
and unaffected by changes in the price of the underlying security. By
modeling this change in volatility as its own stochastic process, the
model gets more accurate.