# The Black Scholes Equation

Since I don’t have a finance background, I’ve always wondered what the big deal was with the Black-Scholes equation – like, what’s it supposed to do? Why is it the holy grail of finance equations? I found this great resource the other day, explaining the equation at a very high level: A Beginner’s Guide To The Black-Scholes Option Pricing Formula.

# What’s an option?

Basically, the equation is used to price options, which are contracts that allow a person to buy an asset at a certain price after a period of time. I liked the example from Rich’s blog:

Consider a European call option on a Microsoft share (the ‘asset’), with a strike of 30 (the ‘predetermined price’) and maturity of one year from today (the ‘predetermined date’). If I pay to enter into this contract I have the right but not the obligation to buy one share at 30 in a year’s time. Whether I actually exercise my right clearly depends on the share price in the market at that date:

– If the share price is above 30, say at 35, I can buy the share in the contract at 30 and sell it immediately at 35, making a profit of 5. Similarly if the share price is 40 I make a profit of 10.
– If the share price is below 30, say at 25, the fact that I have the right to buy at 30 is worthless: I can buy more cheaply in the open market.

Alternatively, you can think of insurance as an option – you pay a fee to your insurance company each month so that in the event of an accident, they will cover your medical expenses. If you don’t have an accident, you’re still out the money from the fee.

You should know there are lots of different types of options, like European options, American options, exotic options, etc. And for each type of option, you can either get a “call option” or a “put option.” A call option is when you want to buy an asset at a strike price, and a put option is when you want to sell an asset at a strike price. I found this Forbes article particularly well-written: Options Basics.

# The Black Scholes Equation

Now, how to price such a squirrely thing? For starters, you would need to know things like how volatile your asset is, what the risk free interest rate is (so that you can discount the future value back to present), an appropriate strike price, and the date you’re willing to allow the option to expire. So Scholes, Black, and Merton came up with this equation to describe the value of the option:

$\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$

Initial Condition for a European call option:
$V(S,0) = \max{(S-E,0)}$

Boundary Conditions for a European call option:
$V(0,t) = 0$
$\lim_{S \to\infty}\frac{V(S,t)}{S} \rightarrow 1$

where:

• $V$ is the value of the option
• $S$ is the value of the asset
• $\sigma$ is the volatility of the asset
• $r$ is the risk free interest rate
• $t$ is time
• $E$ is the strike price

In mathematical terms, all of these option types (puts, calls, European, American, etc.) translate to different boundary conditions and initial conditions for the partial differential equation (PDE). It turns out that there is a closed form, analytical solution to the Black Scholes equation for just the European option case, which is the kind of option that only allows you to buy/sell after the expiration date. Other kinds of options, like American options, allow you to buy/sell at any time. Analytical solutions for the other types of options don’t exist or are too hard to come up with, so most people use numerical solutions.

# Analytical Solution

Remember the heat equation? (A refresher on the closely related wave equation is here: Why do drums and guitars sound differently?) The strategy for solving the BS equation analytically is to transform the equation to the form of the heat equation because we know how to solve equations in that form (we don’t know how to solve them when they’re in different forms). Yes, it’s very sad; of all the PDEs in the world, we can solve only a small subset of them analytically. The rest of them rely on numerical approximations. After a lot of work, the solution uses the cumulative distribution function for the Normal distribution (from probability & statistics).

# Numerical Solution

A numerical solution is different than the analytical solution because where the analytical solution gives us an equation, a numerical solution gives us only the equation evaluated at certain points. So the output of an analytical solution is an equation, but we get a set of points from the numerical solution. Numerical solutions may be evaluated at any point, or at as many points as you want, so it’s basically just as good as the analytical solution. When plotted with the analytical solution, really good numerical solutions are essentially identical.

The basic strategy in a numerical approximation is to discretize the space into a mesh, and then solve the PDE for each time step. As we continue solving, the mesh evolves in time until we tell it to stop (the expiration date for the option). To do this, you would use a Taylor series expansion to approximate the derivatives in the PDE. At a high level, the Taylor series is an infinite sum of the function and its derivatives (and some coefficients).

$f(x) = \sum^{\infty}_{k = 0} \frac{f^{(k)}(a) (x-a)^k}{k!}$

We expand around the nodes in the grid (see below), and then truncate the series at an acceptable error term. The farther you expand the series, the smaller your error becomes, and the better your approximation is. However, in practice, people usually expand only to the third or fourth term of the series. From the Taylor series, we get an approximation to the first and second derivatives, and those get plugged into the BS PDE.

Forward approximation to the first time derivative:
$\frac{\partial V}{\partial t} = \frac{w_{i,j+1}-w_{i,i}}{\Delta t}$

Centered approximation to the first asset value derivative:
$\frac{\partial V}{\partial S} = \frac{w_{i+1,j}-w_{i-1,j}}{2\Delta S}$

Centered approximation to the second asset value derivative:
$\frac{\partial^2 V}{\partial S^2} = \frac{w_{i+1,j}+w_{i-1,j}-2w_{i,j}}{\Delta S^2}$

where each $w_{i,j}$ corresponds to a point in this grid:

At the end, you get a numerical approximation (set of points) to the solution of the PDE, $V(S,t=T)$, where $T$ is the expiration date.

# Problems with assumptions

Like in any model, there are some assumptions we made in solving this equation that are unrealistic. For instance, the risk-free interest rate $r$ and the volatility of the stock $\sigma$ are assumed to be constant. In reality, this isn’t the case. But the bigger flaw is with the underlying logic in the PDE itself. Benoit Mandelbrot wrote in The (Mis)Behavior of Markets that the main issue in Black Scholes equation is that it assumes that stock prices behave according to the laws of Brownian Motion (this is why the Normal Distribution shows up in the analytical solution), which basically states that, like a vibrating particle in a molecule, stock prices deviate from their true price randomly, but within a certain radius.

In reality, stock prices are known to skyrocket and plummet well beyond this radius, so it seems that the underlying logic in the Black Scholes PDE doesn’t totally capture the true behavior of the market.