Brownian motion - because it is simple, and results in intuitive closed form solutions, and it's not a terrible description of asset prices, especially when employed in high-frequency event time.
Geometric - because the returns compound, and equities cannot go below zero due to the fact that they are limited liability corporations. There are many, many other models, but sometimes what you gain in power you lose in calibration stability of the parameters, which is important for cost-effective hedging.
Basically, Black-Scholes is an "industry standard" formula. It is widely used by practitioners and usually augmented with extra specifications or intuition. 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.
The parts of the formula are used for hedging. See Greeks. Many traders use this kind of information. Is BS wrong?
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? 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 crisis, 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. There is no magic bullet in finance world and BS is something accepted that you can justify yourself in your moves in the market.
Number one, the central limit theorem means a lot of things that may not be normal end up looking normal when lots of little 'experiments' or impacts are added up. 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. 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.
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.
Every practitioner knows there are serious flaws with relying on the basic Black-Scholes model. If you look at the distribution as a part of the solution then you have to throw the baby out with the bath water and all of that Black-Scholes stuff is worthless.
If you look at the distribution of asset prices as one of the assumptions then you have a lot of mathematical literature that can help you put the pieces together for your pricing model.
Consider how wide some of the spreads are in options with relatively liquid underlying assets. If you don't know then I'll tell you that the difference in option bids and asks can get theoretically very wide on some relatively liquid underlyings. Even with liquidity lots of money changing hands in the underlying asset, the option spreads tend to stay wide until you also have a very liquid options market. If practitioners had more confidence in their models that used underlying asset price distribution as inputs then there would be a gold rush to make tighter markets and take more of the order flow in the options.
That competition happens to an extent, but the people using theoretical models require a wide disparity from the theoretical prices to set spreads. And those models may use numerical methods like the Black-Scholes framework, but they're never priced assuming normal distributions of asset prices.
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Learn more. Why should we expect geometric Brownian motion to model asset prices? Ask Question. Asked 7 years, 1 month ago. Active 6 years, 8 months ago. Viewed 13k times. Improve this question. Mary S Mary S 1 1 gold badge 1 1 silver badge 3 3 bronze badges. Add a comment. Active Oldest Votes. Improve this answer. Matt Matt Isn't it similar to the EMH part 'new information is quickly absorbed by the market'?
Ric Ric 13k 4 4 gold badges 31 31 silver badges 83 83 bronze badges. Then this is a constant process - of course I don't mean constant processes.
Here is a chart of the lognormal distribution superimposed on our illustrated assumptions e. A Monte Carlo simulation applies a selected model that specifies the behavior of an instrument to a large set of random trials in an attempt to produce a plausible set of possible future outcomes.
In regard to simulating stock prices, the most common model is geometric Brownian motion GBM. GBM assumes that a constant drift is accompanied by random shocks. While the period returns under GBM are normally distributed, the consequent multi-period for example, ten days price levels are lognormally distributed. Portfolio Management. Risk Management. Tools for Fundamental Analysis. Your Privacy Rights. To change or withdraw your consent choices for Investopedia.
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Partner Links. Monte Carlo simulations are used to model the probability of different outcomes in a process that cannot easily be predicted. What Is the Black-Scholes Model? The Black-Scholes model is a mathematical equation used for pricing options contracts and other derivatives, using time and other variables. What Is the Heston Model? The Heston Model, named after Steve Heston, is a type of stochastic volatility model used by financial professionals to price European options.
Vomma Vomma is the rate at which the vega of an option will react to volatility in the market. What Is Nonlinearity? Options have a high degree of nonlinearity, which may make them seem unpredictable. Learn about nonlinearity and how to manage your options trading risk.
Downside Risk Definition Downside risk is an estimation of a security's potential loss in value if market conditions precipitate a decline in that security's price.
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