5 Things You Need to Know About Geometric Brownian Motion
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Geometric Brownian Motion, also known as “GBM”, is the first and a fundamental model one learns in quantitative finance. The same is used in the famous Black-Scholes Option Pricing formula, meaning questions around GBM may pop up during the interview.
If the job role you’re applying is primarily in modelling area in a bank, this topic is likely to surface in some form so it’s important to understand this intuitively.
Here are 5 things you need to know about Geometric Brownian Motion.
#1 – The “Brownian Motion” in GBM is a Physics concept
Brownian Motion is in truth a physics concept that was ported over to finance; check out the wiki definition below.
Brownian Motion: the erratic random movement of microscopic particles in a fluid, as a result of continuous bombardment from molecules of the surrounding medium.
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The key phrase above is “erratic random movement of particles” and in our context of quantitative finance, these “particles” refer to financial variables such as stock prices, interest rates, foreign exchange rates, and so on.
#2 – GBM is type of dynamics
Hang on, what does “dynamics” mean anyway?
In quantitative finance, “dynamics” describes how the value of a financial quantity evolves over time. A financial quantity refers to the value of an asset e.g. stock price or a rate e.g. foreign exchange rate, interest rate.
However, without further specification, “dynamics” is still very general and does not sufficiently describe the how. Sure, we all know stock prices fluctuate over time, but how do you actually describe that movement precisely, and in a way that is consistent to everyone?
This is where the math comes in – it enables us to precisely specify this “dynamics” so everyone’s understanding is .
#3 – GBM is closely related to Brownian Motion
Geometric Brownian Motion is a modified version of Brownian Motion and it’s also commonly known as the “Log-Normal model”. This is because the distribution at any time $t$ of the modelled quantity is Log-Normal. Equivalently, the log of the quantity is Normal.
For example, let $S_t$ represent the price of a stock at time $t$.
Suppose $S_t$ follows Brownian motion (Normal) i.e. $S_t$ ~ BM:
$S_t = S_0 + \mu t + \sigma W_t$
Now, if $S_t$ follows Geometric Brownian Motion (Log-Normal) instead i.e. $S_t$ ~ GBM:
$S_t = S_0 e^{(\mu – \frac{1}{2} \sigma^2 )t +\sigma W_t}$
Where:
– $S_t$: Stock price at time $t$ (in years)
– $\mu$: Annual drift (%)
– $\sigma $: Annual volatility (%)
– $W_t$: Wiener process
The above means: the future stock price at time $t$ is the current stock price impacted by some deterministic (known) drift that scales with time + some random shock that’s scaled up with volatility $ \sigma $.
Finally, another way to think of it is: if the modelled quantity follows GBM, the log of the quantity follows BM.
#4 – The terminal distribution of GBM is Log-Normal
The terminal distribution of a variable following GBM is Log-Normal. That is, at the end of a certain time period e.g. 1Y, the distribution of a variable that follows GBM is a Log-Normal one.
The above is how a typical Log-normal distribution looks like. Key things to note are the variable (x) cannot take negative values and it has a long tail to the right i.e. right-skewed or positively-skewed.
The limitation of positive-only values of the Log-Normal distribution has forced some changes in interest rate modelling, in order to cater to negative interest rates.
#5 – GBM is most often used to model stock prices
In the classic Black-Scholes option pricing model, one of the fundamental assumptions is the stock price follows GBM.
It is not immediately clear what the significance of stock prices following GBM is. When we take log, however, the intuition becomes clearer. Saying that a stock price follows GBM i.e. log-normally distributed is equivalent to saying that the stock returns follows BM i.e. normally distributed.
Returns following a Normal distribution is much more intuitive. Most of the time, returns are around the average return value. Sometimes, it’s a lot more and sometimes, it’s a lot less – in a symmetrical fashion.
Although the Normal distribution is not the most accurate description of real-life returns – market crashes are generally larger than rallies – it’s still very popular due to it’s tractability and analytical properties.
What are some of the things you believe is a must-know about GBM?
Share your thoughts with me below!
Risk Manager by Profession, Mentor and Coach by Passion.
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