Sunday, November 17, 2013

How do you model the dynamics of asset prices? A self-reflection (1 of 3)

Having been working on/studying mathematical finance for a while now, I feel like it would be useful to step back and look at one of the most important topics in this area: how does one model asset price dynamics?

Gauß
The canonical account would inevitably begin with Gauss. The Gaussian distribution (or the normal distribution, depending on your academic upbringing) is undeniable the most well-investigated probability distribution in human history. We don't need to go into all the details and properties of it, but it's good to be reminded of a number of facts regarding how the normal distribution is related to some other entities:
  • It is closely related to the phenomenon of diffusion;
  • The Central Limit Theorem says that (under some technical conditions) the sum of many i.i.d. random variables would converge to a normal distribution;
  • Brownian motion is mathematically described by Wiener process, which follows a Gaussian distribution.
I don't want to delve into historical investigation in whether Louis Bachelier is the first person to model financial time series with (arithmetic) Brownian motion, but it's fair to say that he is at least among the first ones who formalize it and apply it in derivative pricing. Later down the road, some would argue that arithmetic Brownian motion is prone to producing negative values and would propose other alternative processes (e.g. geometric Brownian motion for stock prices, O-U process for rates), but those would all be in the same spirit of arithmetic Brownian motion, save some transformations (e.g. taking the natural log as in geometric Brownian motion)

Our  Checklist
Before proceeding further, let's compile a checklist that we will re-visit multiple times in this series:

       Candidate: Brownian motion
  • Markovian
  • "Tractable"(-ish, depending on how you transform it)
  • Excess kurtosis = 0
  • Market completeness (if number of hedging instruments >= number of sources of randomness)
Since this is the first time the checklist is presented, we should go through each point:
  • A Markov process has no memory. It doesn't matter if the market went up 20%, up 3%, down 3%, or down 20% yesterday. Today's probability of movement is not affected in any way.
  • "Tractable" refers to a closed-form expression for the asset price process itself, or the price of derivatives with the asset as underlying.
  • Excess kurtosis is a measure of "tail effect"
  • Market completeness means a contingent claim can be fully hedged
The paradigm of thinking of asset price as following a Brownian motion is commonly known as the random walk hypothesis. It is well-known and widely used: it's the canonical/default view in the majority of quant finance textbooks; it is mentioned and discussed in "pop-finance" books such as A Random Walk Down Wall Street. But how realistic is it and what are the caveats?

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