The Unlucky Investor's Guide to Options Trading - Julia Spina

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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.

A detailed plot of the normal distribution and the corresponding probabilities at each standard deviation mark.

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.

      ● upper W left-parenthesis 0 right-parenthesis equals 0 (i.e., the process initially begins at location 0).

      ● upper W left-parenthesis t right-parenthesis is almost surely continuous.

      ● The increments of upper W left-parenthesis t right-parenthesis, defined as upper W left-parenthesis t right-parenthesis minus upper W left-parenthesis s right-parenthesis where 0 less-than s less-than t, are normally distributed with mean 0 and variance t minus s (i.e., the steps of the Wiener process are normally distributed with constant mean of 0 and variance of Δt ).

      ● Disjoint increments of upper W left-parenthesis t right-parenthesis are independent of one another (i.e., the current step of the process is not influenced by the previous steps, nor does it influence the subsequent steps).

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 the dynamics of SPY from 2010–20158 for the purposes of comparison.

      The price trends of SPY in Figure 1.8(b) appear fairly similar to the Brownian motion cumulative horizontal displacements shown in Figure 1.6(c). The daily returns for SPY are more prone to outlier moves compared to the horizontal displacements of Brownian motion but share some characteristics. The symmetric geometry of the SPY returns histogram bears resemblance to the fairly normal distribution of horizontal displacements, with the tails of the distribution being more prominent as a result of the history of large price moves.

Schematic illustration of (a) The 2D position of a particle in a fluid, moving with Brownian motion. The particle begins at a coordinate of (X = 0, Y = 0) and drifts to a new location over 1,000 steps. (b) The horizontal displacements8 of the particle. (c) The cumulative horizontal displacement of the particle over 1,000 steps.

Figure 1.6 (a) The 2D position of a particle in a fluid, moving with Brownian motion. The particle begins at a coordinate of left-parenthesis upper X equals 0 comma upper Y equals 0 right-parenthesis and drifts to a new location over 1,000 steps. (b) The horizontal displacements9 of the particle (i.e., the movements of the particle along the X‐axis over 1,000 steps). (c) The cumulative horizontal displacement of the particle over 1,000 steps.

      Similarities are clear between price dynamics and Brownian motion, but this remains a highly simplified model of price dynamics. In reality, stock log returns are not normal and are typically skewed to the upside or downside, depending on the specific underlying. Additionally, the drift and volatility of a stock are not directly observable, and it cannot be experimentally confirmed whether or not these variables are constant. Stock volatility approximated with historical return data is rarely constant with time (a phenomenon known as heteroscedasticity). Stock returns are also not typically independent of one another across time (a phenomenon known as autocorrelation), which is a requirement for this model.

Schematic illustration of the distribution of the horizontal displacements of the particle over 1,000 steps. As characteristic of a Wiener process, the increments are normally distributed, have a mean of zero and variance t – s.

Figure 1.7 The distribution of the horizontal displacements of the particle over 1,000 steps. As characteristic of a Wiener process, the increments are normally distributed, have a mean of zero and variance t minus s Скачать книгу

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The skew of the returns distribution is also used to estimate the directional risk of an asset. The fourth moment (kurtosis) quantifies how heavy the tails of a returns distribution are and is commonly used to estimate the outlier risk of an asset.

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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 plus-or-minus 1 steps with probability 1 slash 2. The scaling limit is reached by shrinking the size of the steps while speeding up their rate in such a way that the process neither sits at its initial location nor runs off to infinity immediately.