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The following is an interview question from Mark Joshi et al. Quant Job Interview.

Question: Why is Brownian motion useful in finance?

I am from a Pure Maths PhD background (functional analysis, particularly Banach Space Theory). I would like to venture into quant finance industry after my PhD graduation.

Thus, I have no idea on how to answer question above as it seems that most stochastic calculus books would involve talking about Brownian motion but never give motivations.

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    $\begingroup$ Hi: One reason is that it's a martingale and some are okay with viewing log prices as a martingale. Therefore, BW can be a reasonable process to use for modelling changes in log prices. In fact, the whole black scholes framework is based on that assumption. $\endgroup$ – mark leeds Dec 8 '19 at 13:45
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    $\begingroup$ standard Brownian motion, or geometric Brownian motion? $\endgroup$ – develarist Dec 9 '19 at 4:39
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    $\begingroup$ I guess can answer both? $\endgroup$ – Idonknow Dec 9 '19 at 6:14
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Brownian motion is simply the limit of a scaled (discrete-time) random walk and thus a natural candidate to use. It is very intuitive and arguably one of the simplest and best understood time-continuous stochastic processes. Also, don't forget that you obtain many more stochastic processes as functions of a (time-changed) Brownian motion. In many books on stochastic calculus, you first define the Ito integral with respect to a Brownian motion before you extend it to general semimartingales. Assuming that log-returns follow a Brownian motion (with drift), you can easily derive closed-form solutions for option prices. Brownian motion is furthermore Markovian and a martingale which represent key properties in finance.

Brownian motion was first introduced by Bachelier in 1900. Samuelson then used the exponential of a Brownian motion (geometric Brownian motion) to avoid negativity for a stock price model. Based on this work, Black and Scholes found their famous formula in 1973.

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    $\begingroup$ This looks like the answer they want you to give in an interview. A word of warning as you have a pure math background. All these modells make assumptions that various quantities are Gaussian normally distributed. Real life data is not. Whether the models are still useful or not is exactly the question they should want to hire a pure math PhD for. $\endgroup$ – quarague Dec 9 '19 at 8:20
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    $\begingroup$ But why is a random walk a natural candidate for modeling assets? The answer is an economic question rather than a mathematical question (if returns could be "predicted", then trading would occur such that returns would no longer be "predictable") $\endgroup$ – user9403 Dec 9 '19 at 21:14
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Physical objects move according to simple smooth curves that can be represented by low order polynomials: a straight line, a parabola, an ellipse, etc.

Financial market prices move in a completely different way, as can be seen by looking at any graph of stock prices, interest rates etc. in a newspaper: there are constant, erratic fluctuations, sometimes in one direction, sometimes in the other, sometimes small and sometimes big, that give the curve a rough, random appearance. The Brownian Motion is a suitable model for this kind of curve.

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