One way you might do this is by taking a big step forward, then looking back and seeing how much higher or lower you are compared to where you started. The difference in height over the
distance of your step gives you an idea of the slope at that point.
Now, let's say you want a more accurate idea of the steepness right where you're standing. Instead of taking a big step, you take a very tiny step. The tinier the step, the more accurate
your measurement of the slope at that exact point.
Finite difference methods are a lot like this hiking analogy. They measure how much a function (like the mountain trail) changes as you make tiny "steps" in the variable of interest (like
your position on the trail). And just like with hiking, the smaller the step, the more accurate your measurement—but also, the more steps you might need to take to cover the whole
In math and many other fields, we use this method to approximate how things change, whether it's the steepness of a curve, the temperature in a material, or the price of a financial
When we talk about options, one of the key components we're interested in is how the price of an option changes as the price of the underlying asset (like a stock) changes. This change
rate is known as the "delta" of the option.
Now, imagine the mountain trail from our analogy represents the option price, and every step you take represents a small change in the stock price. As you move along this trail (i.e., as
the stock price changes), the steepness of the trail represents the delta. So, when you're taking tiny steps on the mountain trail to figure out its steepness, you're essentially using
the finite difference method to approximate the delta of the option.
But here's the kicker: just like in our mountain trail, the landscape of options isn't flat. The steepness (or delta) itself can change! This "change of the change" or the rate at which
delta changes is known as "gamma."
Now, if you think about taking another set of steps and seeing how the steepness changes between those steps, you're approximating gamma using finite differences.
In real-world options pricing:
1. Delta is like asking: "If the stock price goes up by $1, how much will my option price change?"
2. Gamma is then asking: "If the stock price goes up by $1 again, how will the change rate (delta) itself change?"
Finite difference methods allow us to answer these questions without having to know the full, exact formula for the option price.
Just like how taking small steps helps us understand the steepness of a mountain trail without having to view the entire mountain from afar.