The Black-Scholes formula provides us a way to calculate the price of European options. \(C\) is the value of a call option. \(S\) is the stock price (spot price), and \(K\) is the exercise price (strike price). \(\sigma\) is a common measure of volatility (standard deviation of log returns). \(N\) is a function that provides the CDF of the normal distribution at its argument. \(r\) is the risk-free interest rate (a hypothetical rate of return on an investment with zero risk). Finally, \(T\) is the amount of time left to maturity of this option.

\[ C = S N(d_1) - Ke^{-rT}N(d_2) \]

\[ d_1 = \frac{\ln \frac{S}{K} + \left( r + \frac{\sigma^2}{2}T \right)}{\sigma \sqrt{T}} \]

\[ d_2 = \frac{\ln \frac{S}{K} + \left( r - \frac{\sigma^2}{2}T \right)}{\sigma \sqrt{T}} \]

This gives us the price for an option based on a perfect hedge (a way to reduce risk) via buying and selling its underlying asset. This formula is based on the Black-Scholes equation, which tells us how the price of an option changes over time. As above, \(\sigma\) is volatility and \(r\) is risk-free interest rate. \(t\) is time. \(V\) is the price of the option (based on the underlying stock price \(S\)).

\[ \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - r V = 0 \]

This is a very useful model! It’s relatively easy to work with, and we have simple ways to calculate or observe most of the properties it depends on (time, stock price, even risk-free interest rate has a decent method for approximation). As we all know, models in the realm of finance always reflect the underlying reality appropriately, so we should be able to….

Hang on a second. Every significant financial model is an approximation of a wildly complicated underlying system. What are the ways in which Black-Scholes could lead us astray?

Per *Options, Futures, and Other Derivatives*^{1},
“the one parameter in the Black-Scholes pricing formulas that cannot be
directly observed is the volatility of the stock price”. In that book,
§12.11 describes “implied volatilities”, which are volatilities based on
what we see happening to option prices already in the market.
Interestingly, the procedure described to discover these volatilities
isn’t some amazing mathematical formula, but instead just a binary
search.

…an iterative search procedure can be used to find the implied \(\sigma\)…the range for \(\sigma\) can be halved at each iteration and the correct value of \(\sigma\) can be calculated to any required accuracy.

This doesn’t exactly bestow supreme confidence on the uninitiated reader (such as myself). If we’re just basing our volatility on what’s already out there (and what’s more, finding our numbers by searching for them in some space!), it seems like we could quickly run into problems of circular definition.

This doesn’t quite happen though, because we bottom out immediately.
Since we already know the true prices of the options we’re looking for
the volatility of, this search is just an easy way to calculate an
implied \(\sigma\). It’s guess and
check, not just guess. We can then use this implied volatility to
calculate the price of a *different* option. It should be noted
that certain option prices are somewhat insensitive to volatility, and
so shouldn’t be used for this procedure.

I stated above that “risk-free interest rate has a decent method for approximation”. A highly operative word here is “decent”. We don’t know the true risk-free rate, and it also is free to change over time, counter the assumptions of Black-Scholes. This is based on another set of approximations. Often, the way to calculate the risk-free rate is by taking a look at the lowest-risk things you can get your hands on (e.g. government bonds). However, who can guarantee that the government will be around forever, or won’t change its policies regarding these bonds? There is still some amount of risk that Black-Scholes disregards.

The formula makes a few more assumptions that may or may not hold during any given time period. First, it assumes that trading incurs no cost and is instantaneous. The first may or may not be true, and the second is true only by approximation. It also assumes that trading is continuous and the market is open continuously. Hours that the market is closed violate the second assumption.

The first is true enough of many exchanges now, but proposals have
been made to implement more discrete-time markets. These would be
intended to cut down on the “race to the bottom” for trading speed,
which has caused high frequency traders to take extreme measures such as
building an extremely-straight fiber optic cable between Chicago and New
York, or to move their server racks a few feet closer to cut down on
nanoseconds of data exchange time.^{2}

More importantly, Black-Scholes misjudges the severity of extreme situations. It assumes that everything is going swimmingly, markets are acting normal, etc. Like many of the entry-level financial models I’ve run into, normalcy is a prerequisite. This isn’t too dangerous, of course, unless everything is falling apart anyways, so maybe it’s a valid-enough assumption to depend on for everyday purposes.

If you couldn’t tell already, I’m pretty new to learning about finance, but the ways in which financial models are adjusted to guard against risk, and the kinds of decisions that have to be made about which models to trust (since models are all approximations in the end) fascinate me. My conclusion is fairly obvious at this point: use models to help you make sense of the world, but remember that they could be making some invalid assumptions about the underlying reality.