target="_blank" rel="nofollow" href="#n8" type="note">8 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.
Figure 1.6 (a) The 2D position of a particle in a fluid, moving with Brownian motion. The particle begins at a coordinate of
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.
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
Figure 1.8 The (a) daily returns, (b) price, and (c) daily returns histogram for SPY from 2010–2015.
Although the normality assumption is not entirely accurate, making this simplification allows the development of the rest of this theoretical framework shown in the gray box. The formalism in the gray box is supplemental material for the mathematically inclined. The interpretation of the math, which is more significant, follows after. It should be noted that the Black‐Scholes model technically assumes that stock prices follow geometric Brownian motion, which is more accurate because price movements cannot be negative. Geometric Brownian motion is a slight modification of Brownian motion and requires that the logarithm of the signal follow Brownian motion rather than the signal itself. As it relates to price dynamics, this suggests that the log returns are normally distributed with constant drift (return rate) and volatility.10
For the price of a stock that follows a geometric Brownian motion, the dynamics of the asset price can be represented with the following stochastic differential equation:11
where
The equation states that each stock price increment
Using this equation as a basis for the derivation, assuming a riskless options portfolio must earn the risk‐free rate, and rearranging terms, the Black‐Scholes equation follows:
(1.11)
where C is the price of a European call (with a dependence on S and t ), S is the price of the stock (with a dependence on t ), r is the risk‐free rate, and σ is the volatility of the stock. The Black‐Scholes formula can be calculated by solving the Black‐Scholes equation according to boundary conditions given by the payoff at expiration of European options. The formula, which provides the value of a European call option for a non‐dividend‐paying stock, is given by the following equation:
(1.12)
where
