style="font-size:15px;"> Models provide an entry point for intuition, which the model then quantifies. A model transforms linear quantities, which you can have intuition about, into nonlinear dollar values. Our apartment model transforms price per square foot into the estimated dollar value of the apartment. It is easier to develop intuition about variation of price per square foot than it is about an apartment's dollar value.
In physics, as we stressed, a theory predicts the future. In finance, a model translates intuition into current dollar values. As a further example, equity analysts have an intuitive sense, based on experience, about what constitutes a reasonable P/E ratio. Developing intuition about yield to maturity, option-adjusted spread, default probability, or return volatility may be harder than thinking about price per square foot. Nevertheless, all of these parameters are directly related to value and easier to judge than dollar value itself. They are intuitively graspable, and the more experienced you become, the richer your intuition will be. Models advance by leapfrogging from a simple, intuitive mental concept (e.g., volatility) to the mathematics that describes it (geometric Brownian motion and the BSM model), to a richer concept (the volatility smile), to experience-based intuition (the variation in the shape of the smile), and, finally, to a model (a stochastic volatility model, for example) that incorporates an extension of the concept.
Styles of Modeling: What Works and What Doesn't
The apartment model is an example of relative valuation. With relative valuation, given one set of prices, one can use the model to determine the value of some other security. One could also hope to develop models that value securities absolutely rather than relatively. In physics, Newton's laws are absolute laws. They specify a law of motion, F = ma, and a particular force law, the gravitational inverse-square law of attraction, which allow one to calculate any planetary trajectory. Geometric Brownian motion and other more elaborate hypotheses for the movement of primitive assets (stocks, commodities, etc.) look like models of absolute valuation, but in fact they are based on analogy between asset prices and physical diffusion phenomena. They aren't nearly as accurate as physics theories or models. Whereas physics theories often describe the actual world – so much so that one is tempted to ignore the gap between the equations and the phenomena – financial models describe an imaginary world whose distance from the world we live in is significant.
Because absolute valuation doesn't work too well in finance, in this book we're going to concentrate predominantly on methods of relative valuation. Relative valuation is less ambitious, and that's good. Relative valuation is especially well suited to valuing derivative securities.
Why do practitioners concentrate on relative valuation for derivatives valuation? Because derivatives are a lot like molecules made out of simpler atoms, and so we're dealing with their behavior relative to their constituents. The great insight of the BSM model is that derivatives can be manufactured out of stocks and bonds. Options trading desks can then regard themselves as manufacturers. They acquire simple ingredients – stocks and Treasury bonds, for example – and manufacture options out of them. The more sophisticated trading desks acquire relatively simple options and construct exotic ones out of them. Some even do the reverse: acquiring exotic options and deconstructing them into simpler parts to be sold. In all cases, relative value is important, because the desks aim to make a profit based on the difference in price of inputs and outputs – the difference in what it costs you to buy the ingredients and the price at which you can sell the finished product.
Relative value modeling is nothing but a more sophisticated version of the fruit salad problem: Given the price of apples, oranges, and pears, what should you charge for fruit salad? Or the inverse problem: Given the price of fruit salad, apples, and oranges, what is the implied price of pears? You can think of most option valuation models as trying to answer the options' analogue of this question.
In this book we'll mostly take the viewpoint of a trading desk or a market maker who buys what others want to sell and sells what others want to buy, willing to go either way, always seeking to make a fairly safe profit by creating what its clients want out of the raw materials it acquires, or decomposing what its clients sell into raw materials it can itself sell or reuse. For trading desks that think like that, valuation is always a relative concept.
Chapter 2
The Principle of Replication
■ The law of one price: Similar things must have similar prices.
■ Replication: the only reliable way to value a security.
■ A simple up-down model for the risk of stocks, in which expected return μ and volatility σ are all that matter.
■ The law of one price leads to CAPM for stocks.
■ Replicating derivatives via the law of one price.
Replication
Replication is the strategy of creating a portfolio of securities that closely mimics the behavior of another security. In this section we will see how replication can be used to value a security of interest. We define different styles of replication, and discuss the power and limits of this method of valuation.
The One Law of Quantitative Finance
Hillel, a famous Jewish sage, when asked to recite the essence of God's laws while standing on one leg, replied:
Do not do unto others as you would not have them do unto you. All the rest is commentary. Go and learn.
Andrew Lo, a professor at MIT, has quipped that while physics has three laws that explain 99 % of the phenomena, finance has 99 laws that explain only 3 %. It's a funny joke at finance's expense, but finance actually has one more or less reliable law that forms the basis of almost all of quantitative finance.
Though it is often stated in different ways, you can summarize the essence of quantitative finance somewhat like Hillel, on one leg:
If you want to know the value of a security, use the price of another security or set of securities that's as similar to it as possible. All the rest is modeling. Go and build.
This is the law of analogy: If you want to value something, do it by comparing it to something else whose price you already know.
Financial economists like a different statement of this principle, which they call the law of one price:
If two securities have identical payoffs under all possible future scenarios, then the two securities should have identical current prices.
If two securities (or portfolios of securities) with identical payoffs were to have different prices, you could buy the cheaper one and short the more expensive one, immediately pocket the difference, and experience no positive or negative cash flows in the future, since the payoffs of the long and short positions would always exactly cancel.
In practice, we will rarely be able to construct a replicating portfolio that is exactly the same in all scenarios. We may have to settle for a replicating portfolio that is approximately the same in most scenarios.
What both of the aforementioned formulations hint at is the impossibility of arbitrage, the ability to trade in such a way that will guarantee a profit without any risk. Another version of the law of one price is therefore the principle of no riskless arbitrage, which can be stated as follows:
It should be impossible to obtain for zero cost a security that has nonnegative payoffs in all future scenarios, with at least one scenario having a positive payoff.
This
