Models

Emanuel Derman, author of the popular My Life as a Quant, is always worth reading. His piece in Haslett’s Risk Management (Wiley, 2010), “Models” (pp. 681-88), contrasts the models of hobbyists, scientific models, and financial models. Hobbyists are satisfied, sometimes delighted, with resemblance: a model airplane resembles the real thing. Scientists with their models aim to foretell the future and control it. These models can be either fundamental (laws of the universe) or phenomenological. Phenomenological models “make pragmatic analogies between things one would like to understand and things one already understands from fundamental models.” They are approximations and “often have a toylike quality.”

Financial models “are used less for divination than for interpolation or extrapolation from the known dollar prices of liquid securities to the unknown dollar values of illiquid securities.” For example, the Black-Scholes model “proceeds from a known stock price and a riskless bond price to the unknown price of a hybrid security—an option—much in the same way one estimates the value of fruit salad from its constituent fruits or, inversely, the way one estimates the price of one fruit from the prices of the other fruits in the salad. None of these metrics is strictly accurate, but they all provide immensely helpful ways to begin to estimate value.”

Financial models transform intuitive linear quantities into nonlinear dollar values. We can transform price per square foot into the dollar value of an apartment; this is intuitively easy because price per square foot “captures much of the variability of apartment prices. Similarly, P/E describes much of the variability of share prices. Developing intuition about yield to maturity, option-adjusted spread, default probability, or return volatility is harder than thinking about price per square foot. Nevertheless, all of these parameters are clearly related to value and easier to think about than dollar value itself. They are intuitively graspable, and the more sophisticated one becomes, the richer one’s intuition becomes. Models are developed by leapfrogging from a simple, intuitive mental concept (e.g., volatility) to the mathematics that describes it (e.g., geometric Brownian motion, the Black-Scholes model), to a richer mental concept (e.g., the volatility smile), to experienced-based intuition about it, and, finally, to a model (e.g., a stochastic volatility model) that incorporates the new concept.”

Alas, “the gap between a successful financial model and the correct value is nearly indefinable because fair value is finance’s fata morgana, undefined by prices, which themselves are not stationary. So, model success is temporary at best. If fair value were precisely calculable, markets would not exist.”

The essence of financial modeling is to use the known price of a security that is as similar as possible to the security whose value you want to know. “The law of one price [that any two securities with identical estimated future payoffs, no matter how the future turns out, should have identical current prices]—this valuation by analogy—is the only genuine law in quantitative finance, and it is not a law of nature. It is a general reflection on the practices of human beings—who, when they have enough time and enough information, will grab a bargain when they see one.” The modeler’s job is to show that “the target and the replicating portfolio have identical future payoffs under all circumstances.” That’s tricky, of course. The Black-Scholes model, for example, sees a future that is not real since stock returns are not normally distributed nor do stock prices move continuously.

Financial models change over time to reflect changing economic conditions and increasing financial sophistication, but their correctness is always uncertain and this uncertainty is much vaguer than probabilistic risk. In the final analysis “models are best regarded as a collection of parallel, inanimate ‘thought universes’ to explore. Each universe should be internally consistent, but the financial/human world, unlike the world of matter, is vastly more complex and vivacious than any model we could ever make of it.”

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A footnote to this summary of Derman’s paper. In The Business of Options (Wiley, 2001) Martin O’Connell recalls a 1985 seminar on interest rate options where he was one of four speakers. The best known was Myron Scholes. “One of the participants was quite persistent in hassling Dr. Scholes about perceived imperfections in his model. Finally, things came to a head when the guy said: ‘Your model is just wrong.’ Dr. Scholes, who so far had not said anything funny or ironic, came back with: ‘Of course, it’s wrong. That’s why we call it a model.’” (p. 37)