The Volatility Smile. Park Curry David

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 principle states that markets abhor an arbitrage opportunity. It is equivalent to the law of one price in that, if two securities were to have identical future payoffs but different current prices, a suitably weighted long position in the cheaper security and a short position in the more expensive one would create an arbitrage opportunity.

      Given enough time and enough information, market participants will end up enforcing the law of one price and the principle of no riskless arbitrage as they seek to quickly profit by buying securities that are too cheap and selling securities that are too expensive, thereby eliminating arbitrage opportunities. In the long run, in liquid markets, the law of one price usually holds. But the law of one price is not a law of nature. It is a statement about what prices should be, not what they must be. In practice, in the short run, in illiquid markets or during financial crises and panics, and in some other instances too, the law of one price may not hold.

      The law of one price requires that payoffs be identical under all possible future scenarios. Trying to imagine all future scenarios is an impossible task. Even if markets are not strictly random, their vagaries are too rich to capture in a few thoughts, sentences, or equations. In practice, extreme and often unimaginable scenarios (September 11, 2001, for example) are considered possible only after they have happened. Before they happen, these events are not just considered unlikely, but are entirely excluded from the distribution.

       Valuation by Replication

      How do you use the law of one price to determine value? If you want to estimate the unknown value of a target security, you must find some replicating portfolio, a set of more liquid securities with known prices, that has the same payoffs as the target, no matter how the future turns out. The target's value is then simply the known price of the replicating portfolio.

      Where do models enter? It takes a model to demonstrate that the target and the replicating portfolio have identical future payoffs under all circumstances. To demonstrate similarity, you must (1) specify what you mean by “under all circumstances” for each security, and (2) find a strategy for creating a replicating portfolio that, in each future scenario or circumstance, will have payoffs identical to those of the target.

      The first step is reductive and involves science. We need to take some very complicated things – the economy and financial markets – and reduce them to mathematical equations that describe their potential range of future behavior. The second step