Emanual Derman Answers Edge.org's Annual Question With: "No more time decay"
Today we visit Emanual Derman whom I once described, back when he was blogging at Reuters, thusly:
Mr. Derman is a blogger based in New York City.From Inference Review, volume 4, issue 4, July 2019:
He also teaches at Columbia.
Before that he was head of the quantitative strategies group in the equities division, and then head of quantitative risk strategies at Goldman Sachs
And before that he was a theoretical physicist.
I hate him....
Here is his personal homepage.
...Psst...I don't really hate him.
But do visit his homepage to see why one could.
Emanuel Derman is a Professor of Professional Practice and Director of the Financial Engineering Program at Columbia University.
The Black–Scholes model describes the value of a stock option as a function of the underlying stock price and its volatility. The model suggests an analogy between the random fluctuations of a stock price and a concept from physics, the Brownian motion of randomly diffusing particles. The analogy is not entirely new. More than 100 years ago, the French mathematician Louis Bachelier developed the theory of Brownian stock-price fluctuations to analyze stock options.1 The model has become the foundation for valuing options of all kinds. The Black–Scholes equation allows traders to treat volatility as an asset and trade it by buying or selling portfolios of options. An impalpable property of a financial system—volatility—has become concrete enough to be bought and sold. The result has been a transformation in mathematical finance and an explosion of new financial products. Although the Black–Scholes model is genuinely useful, it fails to capture the reality of the market’s behavior. This failure has triggered further ingenious extensions of the model.
These extensions, too, eventually fail. Markets and prices are social phenomena, not physical ones, and it is unlikely that there is an accurate predictive theory of human behavior.
Stocks and Options
A call option on a stock is a contract between a buyer A and a seller, or counterparty, B, that gives A the right, but not the obligation, to purchase the stock for a specified price on some future date. If one share of a stock, S, is selling for $200 today, a call option with a strike of $250 and an expiration of one year, gives A the right to buy S one year from today for $250. A is long the option, and B short. If S trades in the market at $300, A can exercise the option, purchasing one share at $250 from B and selling it for $300, obtaining a payoff of $50. If S trades below $250, A need do nothing. A put option is the corresponding right to sell the stock on a future date for a specified price.
What is the rational price of a one-year call option on S? Before 1973, it seemed obvious that the current value of a call option should reflect the expected future price of its stock. Different people would have different expectations about that price. What was obvious turned out to be false. In 1973, Fischer Black, Myron Scholes, and Robert Merton demonstrated that if stock prices fluctuated randomly as described by the theory of Brownian motion, the future payoff of a stock option could be replicated by a portfolio comprising borrowed money and a fractional share of the stock.2 Suppose the solution to the Black–Scholes equation yields a value today of one dollar for the call option on S described above. The equation then also shows that this one-dollar call option can be replicated by investing one dollar, borrowing 15 dollars, and then using their sum to purchase 16 dollars’ worth of S. If the current stock price of S is $200, and if it fluctuates randomly, this portfolio will behave precisely like an option in the future, with one year to expiration and a strike of $250. The amount of money borrowed, and the amount of stock in the portfolio, must be continuously adjusted as the stock price changes. At expiration, the portfolio and the call option have the same payoff. The adjustments represent a manufacturing cost. The greater the volatility of the stock, the more rapidly the stock price changes, and the greater the manufacturing cost.....MUCH MORE
An option’s value does not depend on the expected future price of its stock....
We are fans.
Previous Dermanettes:
Emanuel Derman Reviews "The Age of Cryptocurrency"
Emanuel Derman, Tyler Cowen et al On "Why is Thomas Piketty's 700-page book a bestseller?"
"Derman, Rodrik and the nature of statistical models"
Emanuel Derman: "Money Changes Everything?" or Spinoza on Pain, Pleasure and Desire
Book Review: "Models.Behaving.Badly: Why Confusing Illusion with Reality Can Lead to Disaster, on Wall Street and in Life "
"Emanuel Derman on Twitter, on volatility products"
The Financial Modelers' Manifesto
And on option pricing:
The Guy Who Discovered Black-Scholes Before Mssrs. Black and Scholes
Heckuva job, Brownian.
See also:"Pricing the Future: Finance, Physics, and the 300-Year Journey to the Black-Scholes Equation "
"Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen"
at Austria's Zentralbibliothek für Physik
And:
Marian Smoluchowski
(he missed it by that much)
Variables: σ = volatility of returns of the underlying asset/commodity; S = its spot (current) price; δ = rate of change; V = price of financial derivative; r = risk-free interest rate; t = time.
Climateer Quote of the Day: I May be a Wonk Edition
“In [Bachelier’s] paper we find the Chapman-Kolmogorov-Smoluchowski equation for continuous stochastic processes, the derivation of the Einstein-Wiener Brownian motion process and the recognition that this process is a solution of the partial differential equation for heat diffusion. The Einstein-Wiener process is the analogue, for continuous time and continuous random variables, of the discrete random walk process. … Most of this theory was later to be developed by the mathematicians who were transforming probability theory into a rigorous discipline, Levy, Kolmogorov, Borel, Khinchine, and Feller. Compared to these standards of rigor, Bachelier’s work was heuristic, and scorn for the heuristics led to an underestimation by contemporaries of the significance of the contributions.” (p. 3)That's via a post at Brenda Jubin's blog on the literature of investing, Reading the Markets:
-Cootner, The Random Character of Stock Market Prices, II
