The Unlucky Investor's Guide to Options Trading. Julia Spina
Чтение книги онлайн.

Читать онлайн книгу The Unlucky Investor's Guide to Options Trading - Julia Spina страница 9

Название: The Unlucky Investor's Guide to Options Trading

Автор: Julia Spina

Издательство: John Wiley & Sons Limited

Жанр: Ценные бумаги, инвестиции

Серия:

isbn: 9781119882664

isbn:

СКАЧАТЬ 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.

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 (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 СКАЧАТЬ

<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.