The Volatility Smile. Park Curry David

mathematics. The problems of financial modeling are less mathematical than they are conceptual. In this book, we want to first concentrate on understanding concepts and their implementation, and then use mathematics as a tool. We're less interested here in great numerical accuracy or computational efficiency than in making the ideas we're using clear.

      We know so little that is absolutely right about the fundamental behavior of assets. Are there really strict laws they satisfy? Are those laws stationary? It's best to assume as little as possible and rely on models as little as possible. And when we do rely on models, simpler is better. With that in mind, we proceed to a brief overview of the principles of financial modeling.

      The Purpose of Models

      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