*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.

**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)

- 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

*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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