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How well can each simulate the real-life behavior of stock prices, and what considerations or (dis-)advantages must we be aware of when deciding to use each:

  • Random walk with drift
  • Random walk without drift
  • standard Brownian motion
  • geometric Brownian motion
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    $\begingroup$ When you simulate a Brownian motion (with drift), you really only simulate a Gaussian random walk (with very small step sizes): Brownian motion is just the limit of a scaled random walk and the maths/notation is often easier in continuous time - but the intuition is the same. Geometric Brownian motion is then the result of exponentiating a Brownian motion, ensuring positivity. All these processes are Markovian, which agrees with informationally efficient markets. Asset prices may also include fat tails, jumps, stochastic volatility (+ leverage effect), seasonal patterns etc. $\endgroup$ – Kevin Oct 14 '20 at 18:39
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    $\begingroup$ Yes, random walks (with drift) are just discrete analogues of Brownian motion (with drift). These processes can be negative, so they clearly don't model asset prices. Even ensuring positivity doesn't really give you a good fit. You need fatter tails, heteroscedasticity etc. What a good model is depends on your intended application though. $\endgroup$ – Kevin Oct 14 '20 at 19:01
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    $\begingroup$ @develarist: James Hamilton's "Time Series Analysis" has a nice explanation of how the random walk becomes brownian motion as the step size gets smaller and smaller. $\endgroup$ – mark leeds Oct 15 '20 at 0:22
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    $\begingroup$ @develarist it really depends on what you’re doing and what your intended application is. But yeah, none of these processes would be used for derivatives pricing. Time varying volatility, jumps in asset prices (both relate to skewness and kurtosis) etc are important features which we know as stylised facts but are not captured by simple random walks and their time continuous analogues (Brownian motion). $\endgroup$ – Kevin Oct 20 '20 at 6:57
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    $\begingroup$ @develarist - Markovian processes essentially believe in different "states" with different parameters. So "it's raining" is different from "it's sunny" or "it's overcast". Typically with stockmarkets, such models call for a high-vol,neg-return "chaos" state and a low-vol, pos-return "stability" state. Irrespective of how you measure these states, let alone the dynamics of moving from one to the other, the parameters of any RW or Gaussian will be time-varying. $\endgroup$ – demully Oct 23 '20 at 20:58
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Echoing some of the comments to the OP above, the only real difference between random walks and Brownian motions is a question of time frequency. IE a Brownian motion is just an aggregation of a (binary) random walk with higher frequency. Given both will always be at best an approximation of reality, asking for which is "better" becomes a bit of a superfluous question. How pixelated do you want your thumbnail of the Mona Lisa? ;-)

The real question is the degree of drift you want to assume. A simple perusal of stock price charts will tell you that there is clearly drift, at least in headline nominal terms. As such, stock prices, as quoted, are non-stationary. Maybe you could argue that stock prices are "real-stationary" (with respect to say money supply) or "output-stationary" (with respect to earnings growth being cointegrated with respect to GDP, investment, consumption, etc.).

But then you'll probably end up arguing more about the correct economic deflator to correct for this drift than about useful conclusions from the model ;-( [Been there; done that; no T-shirts]. So the drift exists; but almost becomes a bigger problem handling it than the problem of stock returns... crazy, but sadly all too common.

The "standard" versus "geometric" Brownian motion distinction boils down to whether you believe that prices are normal versus lognormal in nature. Which ceases to matter if you allow drift, because a "variance drag" (of half sigma squared) will make the two equivalent. At least over the timeframe you've chosen to measure this over, reference comments above about binary random walks versus normal Brownians.

The short - and I'm sorry - answer is that there really isn't that much of a distinction between the choices above. Another way of saying this is that the errors of ALL of these models compared to reality are so correlated, it maybe doesn't matter which one you choose.

I know probably not what you were hoping for here...

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Echoing the comments earlier in the thread, I suggest we work backwards by first picking the key aspect of the market (e.g. flash crashes, liquidity shocks, long-memory) we seek to replicate. Then we select the stochastic model that best approximates our target behavior. We'll need to decide how to calibrate our models to the market and picking a salient feature can inform our choice of calibration method.

Inevitably, we'll discover several suitable candidates that offer good enough approximations of the target market behavior. I would advise against picking a winner. Instead, we can surface model risk and better understand exposure to modeling assumptions, when we evaluate the results of competing models and keep a skeptical eye towards results.

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