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

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(which equals 1 in this case). This figure indicates that horizontal step sizes between –1 and 1 are most common, and step sizes with a larger magnitude than 1 are less common.

Schematic illustration of the (a) daily returns, (b) price, and (c) daily returns histogram for SPY from 2010–2015.

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

(1.10)italic d upper S left-parenthesis t right-parenthesis equals upper S left-parenthesis t right-parenthesis left-parenthesis mu italic d t plus sigma italic d upper W left-parenthesis t right-parenthesis right-parenthesis

      where upper S left-parenthesis t right-parenthesis is the price of the stock at time t, upper W left-parenthesis t right-parenthesis is the Wiener process at time t , μ is a drift rate, and σ is the volatility of the stock. The drift rate and volatility of the stock are assumed to be constant, and it's important to reiterate that neither of these variables are directly observable. These constants can be approximated using the average return of a stock and the standard deviation of historical returns, but they can never be precisely known.

      The equation states that each stock price increment left-parenthesis italic d upper S left-parenthesis t right-parenthesis right-parenthesis is driven by a predictable amount of drift (with expected return mu italic d t) and some amount of random noise left-parenthesis sigma italic d upper W left-parenthesis t right-parenthesis right-parenthesis. In other words, this equation has two components: one that models deterministic price trends left-parenthesis upper S left-parenthesis t right-parenthesis mu italic d t right-parenthesis and one that models probabilistic price fluctuations left-parenthesis upper S left-parenthesis t right-parenthesis sigma italic d upper W left-parenthesis t right-parenthesis right-parenthesis. The important takeaway from this observation is that inherent uncertainty is in the price of stock, represented with the contributions from the Wiener process. Because the increments of a Wiener process are independent of one another, it also is common to assume that the weak EMH holds at minimum, in addition to the normality of log returns.

      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)StartFraction partial-differential upper C Over partial-differential t EndFraction plus italic r upper S left-parenthesis StartFraction partial-differential upper C Over partial-differential upper S EndFraction right-parenthesis plus one half sigma squared upper S squared left-parenthesis StartFraction partial-differential squared upper C Over partial-differential upper S squared EndFraction right-parenthesis equals italic r upper C

      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)upper C left-parenthesis upper S comma t right-parenthesis equals upper N left-parenthesis d 1 right-parenthesis upper S left-parenthesis t right-parenthesis minus upper N left-parenthesis d 2 right-parenthesis upper K e Superscript minus r left-parenthesis upper T minus t right-parenthesis

      where upper N left-parenthesis d 1 right-parenthesis is the value of the standard normal cumulative distribution function at d 1 and similarly for upper N left-parenthesis d 2 right-parenthesis, T is the time that the option will expire (upper T minus t is the duration of the contract), upper S left-parenthesis t right-parenthesis is the price of the stock at time t, K is the strike price of the option, and d 1 and d 2 are given by the following:

      (1.13)StartLayout 1st Row 1st Column d 1 2nd Column equals StartFraction 1 Over sigma StartRoot upper T minus t EndRoot EndFraction left-bracket ln left-parenthesis StartFraction upper S left-parenthesis t right-parenthesis Over upper K EndFraction right-parenthesis plus left-parenthesis r plus one half sigma squared right-parenthesis left-parenthesis upper T minus t right-parenthesis right-bracket EndLayout

(1.14)StartLayout 1st Row 1st Column d 2 2nd Column equals d 1 minus sigma StartRoot upper T minus t EndRoot EndLayout

      where σ is the volatility of the stock. If the equations seem gross, it's because they are.

Again, the purpose of this section is not to describe the underlying mechanics of the Black‐Scholes model in detail. Rather, Equations (1.10) through (1.14) are included to emphasize three important points.

      1. There is inherent uncertainty in the price of stock. Stock price movements are also assumed to be independent of one another and log‐normally distributed.12

      2. An estimate for the fair price of an option can be calculated according to the price of the stock, the volatility of the stock, the risk‐free rate, the duration of the contract, and the strike price.

      3. The volatility of a stock, which plays an important role in estimating the risk of an asset and the valuation of an option, cannot be directly observed. This suggests that the “true risk” of an instrument can never be exactly known. Risk can only be approximated using a metric, such as historical volatility or the standard deviation of the historical returns over some timescale, typically matching the duration of the contract. Other than using a past‐looking metric, such as historical volatility to estimate the risk of an asset, one can also infer the risk of an asset from the price of its options.

      As stated previously, the Black‐Scholes model only gives a theoretical estimate for the fair price of an option. Once the

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<p>11</p>

d is a symbol used in calculus to represent a mathematical derivative. It equivalently represents an infinitesimal change in the variable it's applied to. dS(t) is merely a very small, incremental movement of the stock price at time t. ∂ is the partial derivative, which also represents a very small change in one variable with respect to variations in another.