Читать книгу The Volatility Smile - Park Curry David - Страница 13

Before examining the notion of modeling, we must distinguish between price and value. Price is simply what you have to pay to acquire a security, or what you get when you sell it; value is what a security is worth (or, more accurately, what you believe it is worth). Not everyone will agree on value. A price is considered fair when it is equal to the value.

But what is the fair value? How do you estimate it? Judging value, in even the simplest way, involves the construction of a model or theory.

A Simple but Prototypical Financial Model

Suppose a financial crisis has just occurred. Wall Street is laying off people, apartments innearby Battery Park are changing hands daily, but large luxurious apartments are still illiquid. How would you estimate the value of a seven-room apartment on Park Avenue, whose price is unknown, if someone tells you the price of a two-room apartment in Battery Park? This would be a reasonable model: First, figure out the price per square foot of the Battery Park apartment; second, multiply by the square footage of the Park Avenue apartment; third, make some adjustments for location, views, light, staff, facilities, and so forth.

For example, suppose the two-room Battery Park apartment cost $1.5 million and was 1,000 square feet in size. That comes to $1,500 per square foot. Now suppose the seven-room Park Avenue apartment occupies 5,000 square feet. According to our model, the price of the Park Avenue apartment should be roughly $7.5 million. But Park Avenue is a very desirable location, and so we understand that there is about a 33 % premium over Battery Park, which raises our estimate to $10 million. Furthermore, large apartments are scarce and carry their own premium, raising our estimate further to $13 million. Suppose further that the Park Avenue apartment is on a high floor with great views and its own elevator, so we bump up our estimate to $15 million. On the other hand, say the same Park Avenue apartment is being sold by the family of a recently deceased parent who hasn't renovated it for 40 years. It will need a lot of work, which causes us to lower our estimate to $12 million.

Our model's one initial parameter is the implied price per square foot. You calibrate the model to Battery Park and then use it to estimate the value of the Park Avenue apartment. The price per square foot is truly implied from the price; $1,500 is not the price of one square foot of the apartment, because there are other variables – views, quality of construction, neighborhood – that are subsumed into that one number.

With financial securities, too, as in the apartment example, models are used to interpolate or extrapolate from prices you know to values you don't – in our example, from Battery Park prices to Park Avenue prices. Models are mostly used to value relatively illiquid securities based on the known prices of more liquid securities. This is true both for structural option models and purely statistical arbitrage models. In that sense, and unlike models in physics, models in finance don't really predict the future. Whereas Newton's laws tell you where a rocket will go in the future given its initial position and velocity, a financial model tells you how to compare different prices in the present. The BSM model tells you how to go from the current price of a stock and a riskless bond to the current value of an option, which it views as a mixture of the stock and the bond, by means of a very sophisticated and rational kind of interpolation. Once you calibrate the model to a stock's implied volatility for one option whose price you know, it tells you how to interpolate to the value of options with different strikes. The volatility in the BSM model, like the price per square foot in the apartment pricing model, is implied, because all sorts of other variables – trading costs, hedging errors, and the cost of doing business, for example – are subsumed into that one number. The way property markets use implied price per square foot illustrates the general way in which most financial models operate.

Additional Advantages of Using a Model

Models do more than just extrapolate from liquid prices to illiquid values.

Ranking Securities

A security's price doesn't tell you whether it's worth buying. If its value is more than its price, it may be. But sometimes, faced with an array of similar securities, you want to know which security is the best deal. Models are often used by investors or salespeople to rank securities in attractiveness. Implied price per square foot, for example, can be used to rank and compare similar, but not identical, apartments. Suppose, to return to our apartment example, that we are interested in purchasing a new apartment in the Financial District. The apartment lists at $3 million, but is 1,500 square feet, or $2,000 per square foot, appreciably higher than the $1,500 per square foot for the Battery Park apartment. What justifies the difference? Perhaps the Financial District apartment has better features. We might even go one level deeper and start to build a comparative model for the features themselves, or for both the features and the square footage, to see if the features are fairly priced.

Implied price per square foot provides a simple, one-dimensional scale on which to begin ranking apartments by value. The single number given by implied price per square foot does not truly reflect the value of the apartment; it provides a starting point, after which other factors must be taken into account. Similarly, yield to maturity for bonds allows us to compare the values of many similar but not identical bonds, each with a different coupon, maturity, and/or probability of default, by mapping their yields onto a linear scale from high (attractive) to low (less so). We can do the same thing with price-earnings (P/E) ratio for stocks or with option-adjusted spread (OAS) for mortgages or callable bonds. All these metrics project a multidimensional universe of securities onto a one-dimensional ruler. The implied volatility associated with options obtained by filtering prices through the BSM model provides a similar way to collapse instruments with many qualities (strike, expiration, underlier, etc.) onto a single value scale.

Quantifying Intuition

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.