Brownian Motion Options

[Reggie Lamborn]

Iam a mathematician who knows nothing about finance. I heard from a popular source that a something called the Black-Scholes equation is used to model the prices of options. Out of curiosity, I turned to Wikipedia to learn about the model. I was shocked to learn that it assumes that the log of the price of an asset follows a Brownian motion with drift (and then the asset price itself is said to follow a "geometric" Brownian motion). Why, I wondered, should that be a good model? I can understand that asset prices have to be unpredictable or else smart traders would be able to beat the market by predicting them, but there would seem to be many unpredictable alternatives to geometric Brownian motion.

I have found one source that addresses my question, the following book chapter: and an argument it alludes to in chapter 11 of the same book. The analysis here looks very interesting, and I am curious if it is generally accepted in the finance community. I have not studied it enough to understand how realistic its assumptions are, however. It apparently depends on a "continuous time" assumption that seems like it might not be very realistic given that real markets move in response to discrete news events such as earnings announcements.

To provide a straight forward answer: It is not a good model. It never was, it never will be. Until we all do not come up with a better model that provides better modeling accuracy while it is equally intuitive and makes similarly simplifying assumptions the BS model with its geometric brownian motion component is here to stay.

It actually does not matter what model the market agrees on to use for the purpose of translating between an option price and its implied volatility. BS is merely a translation tool, nothing more, nothing less. What is really priced by the market is implied volatility. What is traded, however, is the option price. Hence, as long as the market agrees on one standardized model it does not matter what exact model we are talking about. Example: When a broker dealer and buy side trader agree on the particulars of a standard European or American option they obviously have to agree on price, however, if the price diverts by a significant amount between both desks then both traders interact which implied vols they are taking into account. Hence the model and underlying driving brownian motion plays a very insignificant role in this particular context.

Models and their underlying assumptions become much more important when forecasting asset prices as well as when pricing non-standard (aka non vanilla) derivatives products where the model is sufficiently complex and there are a number of input variables that the choice of models make a significant difference. Example: A more complex interest rate structured note, such as a PRDC. It is very easy to arrive at a 50bps-1% pricing differential when making slight adjustments to modeling assumptions, the way correlations are computed, or what have you.

Models are not created or chosen on the basis of whether "smart traders" are able to "beat" the market. The market imho is maybe the second most complex construct after the human brain, more complex than any concept in Physics, Mathematics, or other science. Nothing in the market is stable, we are exposed to ever changing complexities, correlations, and micro market dynamics. Models are chosen to somewhat approximate the statistical properties of market behavior but much for their simplicity and intuition. It might be shocking news to some academicians that most seasoned trading practitioners attach much more importance, time, and efforts to improving risk managing and risk limiting models as well as approaches than to pricing models in the full knowledge that pricing models will always be imperfect and never capture all the dynamics in the market. It might also be in complete disagreement with many pure quants when I say that it is utterly unimportant whether I price an option using a geometric brownian motion or an arithmetic one. Sure we will end up with a different option price, but so what? Which one is more accurate? It does not matter. Why? Example: If my model constantly overprices option prices then I pay each time above fair market when I buy and I am not able to sell to other market practitioners at the prices I believe are fair. What will I do? I will tweak my model until I get to where most other practitioners price. Result? Most traders align their models like ducks in a row. More or less all models on the street are identical. And if a new model pops up or someone makes slight improvements that are worthy of studying then you can trust me that such model makes it (legally or illegally) across most firms' desk in no time. The result is the same in that most practitioners price with very similar models, you can call it Black Scholes or anything else you want.

My sobering assessment of how to "engineer" statistically significant profits in the market is that there are very few who truly understand the relationships between the real economy and market dynamics. Very few consistently outperform the broad market. Guess what most of them have in common? An uncanny willingness and ability to buy risk off those whose risk limits are breached (emotionally or through hard-coded limits)and an incredible attention to detail when it comes to managing risk exposure. A small handful of such minority are applying quantitative or mathematical concepts to how they invest and trade.

If at first you don't have a model at all, then geometric Brownian motion is not bad.As others before me said: log-returns are normally distributed in this model. This is debatable and there are times and markets where this is not true. There is more than enough research about this.

But why is a model based on Brownian motion not that bad?The reason is that if you have a continuous process with independent increments then it simply is Brownian motion (it is some kind of application of the central limit theorem). This is the only process possible.

If you assume independent increments and other distributions then the processes are so called Levy processes which have jumps. E.g. if you assume that returns are t-distributed then you are in the world of Levy processes and the trajectories are not continuous anymore.

addon: geometric Brownian motion can be enriched: apply time-dependent or stochastic parameters (especially volatility) or apply a time-changed Brownian motion (which will lead to a Levy process) and so on.

It has a closed form solution, which is rare in option pricing models. It is also relative to simple to understand. Otherwise, you usually need to rely on Monte Carlo simulation or some other way. And honestly the added level of sophistication is not that desirable.

Is BS wrong? Sure. Many of its assumptions can be damned for being unrealistic (constant volatility assumption for example). But these assumptions are required to get that simple formula with some approximation to reality.

Is GBM wrong? Sure. Many studies show that log-return behavior is leptokurtic (high tails, high head), asymmetric and prone to jumps. See volatility smile. But it is adequate for most of the time. But the difference is made in extraordinary times (2008 crisis, 1987 crash, etc.)

In academia, BS is a slapstick for new methods and benchmark studies. As in any pioneer in any field new methods are always eager to show they are "better than the benchmark". Otherwise it means your method is so trash that it cannot beat one from almost 40 years ago.

Number 2, when dealing with finance you need a model that seems plausible. An arithmetic Brownian motion could go negative, but stock prices can't. On the other hand, it seems quite plausible that returns, in percent, could be normally distributed - and, indeed, they do within the ability to test that hypothesis with data. This is the same as geometric Brownian motion.

Number three, there aren't obvious and plausible alternatives. There could be, but like solutions to an engineering ODE about voltages and currents in a circuit with solely capacitors and resistors, even when there are actual ODE solutions that make sense mathematically they don't make physical sense unless the solutions are decaying exponentials or sinusoids.

Number four, geometric Brownian motion corresponds with logical discrete models that are internally consistent mathematically from a financial perspective. For example, if a security has a return of 21% in two years it is consistent to have a return of 10% for each of the one-year sub-intervals. There are other ways this could happen, but all that make sense, when taken to a continuous model via a limiting approach, hold together if the continuous model is geometric Brownian motion.

By the way, this is true of stocks. If you want to do currency or interest rates there are other solutions that better fit those situations - just as the engineering solution would give you different answers if there were energy being put into the circuit. In the end it comes down to PDEs that make sense and boundary conditions. The same is true in engineering and physics: why do you believe the wave equation makes sense?

I think Matt Wolf had the best answer by far, but the only point I would add is that the normal distribution can actually be a bit of a dangerous assumption at times, I actually believe this is the reason that more emphasis has been placed on risk management (especially recently) as opposed to pricing models. The main reason for GBM is that it creates effective and simple closed form solutions, modelling asset prices with jumps creates tremendous mathematical difficulties that some have frankly calculated to be more trouble then they're worth. Truly successful market practitioners are rarely simply mathematicians, they are some combination of a mathematician, economist and philosopher. While it is important to understand and be able to handle elements of quant finance (especially if you work in a more exotic area), you need a feel for market dynamics before you can really succeed. I think people continue to use more simplistic models because there has been a bit of a psychological risk/benefit analysis. The benefit to using an extremely complex model, that may or may not represent the market a bit more efficiently, is limited at best and impossible to quantify at worst. Instead, moving forward with an understanding that the model is flawed, while simultaneously creating strong risk management to counter these failures, appears to be the most popular route.

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