says, roughly, that if a random variable is made by adding together many independently random pieces, then, regardless of what those pieces are, the result will be normally distributed. For example, the distribution in the two‐dice example is fairly non‐normal, being relatively triangular and lacking tails. If one considered the sum of more and more dice, each of which is an independent random variable, the distribution would gradually take on a bell shape. This is shown in Figure 1.4.
The normal distribution is a symmetric, bell‐shaped distribution, meaning that equidistant events on either side of the center are equally likely and the skew is zero. The distribution is centered around the mean, and outcomes further away from the mean are less likely. The normal distribution has the intriguing property that 68% of occurrences fall within
These probabilities can be used to roughly contextualize distributions with similar geometry. For example, in the fair dice pair model, the expected value of the fair dice experiment was 7.0, and the standard deviation was 2.4. With the assumption of normality, one would infer there is roughly a 68% chance that future outcomes will fall between five and nine. The true probability is 66.67% for this random variable, indicating that the normality assumption is not exactly correct but can be used for the purposes of approximation. As more dice are added to the example, this approximation becomes increasingly accurate.
Figure 1.4 A histogram for 100,000 simulated rolls with a group of fair, six‐sided dice numbering (a) 2, (b) 4, or (c) 6.
Understanding distribution statistics and the properties of the normal distribution is incredibly useful in quantitative finance. The expected return of a stock is usually estimated by the mean return, and the historic risk is estimated with the standard deviation of returns (historical volatility). Stock log returns are also widely assumed to be normally distributed. Although, this is only approximately true because the overwhelming majority of stocks and ETFs have skewed returns distributions.5 Regardless, this normality estimation provides a quantitative framework for expectations around future price moments. This approximation also simplifies mathematical models of price dynamics and options pricing, the most notable of which is the Black‐Scholes model.
Figure 1.5 A detailed plot of the normal distribution and the corresponding probabilities at each standard deviation mark.
The Black‐Scholes Model
The Black‐Scholes options pricing formalism revolutionized options markets when it was published in 1973. It provided the first popular quantitative framework for estimating the fair price of an option according to the contract parameters and the characteristics of the underlying. The Black‐Scholes equation models the price evolution of a European‐style option (an option that can only be exercised at expiration) within the context of the broader financial market. The corresponding Black‐Scholes formula uses this equation to estimate the theoretical price of that option according to its parameters.
It's important to note that the purpose of this Black‐Scholes section is not to elucidate the underlying mathematics of the model, which can be quite complicated. The output of the model is merely a theoretical value for the fair price of an option. In practice, an option's price typically deviates from this value because of market speculation and supply and demand, which this model does not take into account. Rather, it is essential to have at least a superficial grasp of the Black‐Scholes model to understand (1) the foundational assumptions of financial markets and (2) where implied volatility (a gauge for the market's perception of risk) comes from.
The Black‐Scholes model is based on a set of assumptions related to the dynamics of financial assets and the market as a whole. The assumptions are as follows:
● The market is frictionless (i.e., there are no transaction fees).
● Cash can be borrowed and lent in any amount, even fractional, at the risk‐free rate (the theoretical rate of return of an investment with no risk, a macroeconomic variable assumed to be constant).
● There is no arbitrage opportunity (i.e., profits in excess of the risk‐free rate cannot be made without risk).
● Stocks can be bought and sold in any amount, even fractional amounts.
● Stocks do not pay dividends.6
● Stock log returns follow Brownian motion with constant drift and volatility (the theoretical mean and standard deviation of annual log returns).
A Brownian motion, or a Wiener process, is a type of stochastic process or a system that experiences random fluctuations as it evolves with time. Traditionally used to describe the positional fluctuations of a particle suspended in fluid at thermal equilibrium,7 a standard Wiener process (denoted W(t)) is mathematically defined by the conditions in the grey box. The mathematical definition can be overlooked if preferred, as the intuition behind the mathematics is more crucial for understanding the theoretical foundation of options pricing and follows after.
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● The increments of
● Disjoint increments of
Simplified, a Wiener process is a process that follows a random path. Each step in this path is probabilistic and independent of one another. When disjoint steps of equal duration are plotted in a histogram, that distribution is normal with a constant mean and variance. Brownian motion dynamics are driven by this underlying process. These conditions can be best understood visually, which will also demonstrate why this assumption appears in the development of the Black‐Scholes model as an approximation for price dynamics. Figures 1.6 and 1.7 illustrate the characteristics of Brownian motion, and Figure 1.8 illustrates
This application of Wiener processes as well as their use in financial mathematics are due to them arising as the scaling limit of simple random walk. A simple random walk is a discrete process that takes independent
