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or synthetic, and involves mostly engineering. We must create a replicating portfolio of liquid securities whose payoffs match the payoffs of the target in all future scenarios.

       Styles of Replication

      There are two kinds of replication, static and dynamic. Static replication reproduces the payoffs of the target security over its entire lifetime with an initial portfolio of securities whose weights will never need to be changed. Once the static replicating portfolio is created, by buying and selling the necessary securities, no additional trading is required for the lifetime of the target security. Assuming that the replicating portfolio can be set up, the only thing that can go wrong is a failure of credit: Counterparties may not pay what they owe you when the securities you purchased from them require that they make payments to you. Static replication is the simplest and most straightforward method of valuation, but is feasible only in the rare cases when the target security closely resembles the available liquid securities. Even when the resemblance isn't perfect, the attraction of a static portfolio is so great that traders often try to create static portfolios that only approximately replicate the target. We will illustrate this for barrier options in Chapter 12.

      With dynamic replication, the components and weights of the replicating portfolio must change over time. We need to continually buy and sell securities as time passes and the price of the underlier changes in order to achieve theoretically accurate replication. As practitioners who work with trading desks know, dynamic replication can be very complex, both in theory and in practice. Part of the trouble is the mismatch between the model of the markets (the science) and the actual behavior of markets. When it does work, though, dynamic replication allows us to value a wide range of securities, many of which would be difficult or impossible to value otherwise. In 1973, Fischer Black and Myron Scholes, and separately Robert Merton, published papers explaining how to replicate a stock option by constructing a dynamic portfolio containing shares of the underlying stock and a riskless bond. This allowed traders to determine the value of an option based on the price of the underlying stock, the prevailing level of interest rates, and an estimate of future stock price volatility. That this replicating portfolio could be constructed was unsuspected until it was achieved, and its discovery dramatically changed the financial world. This insight would eventually earn Scholes and Merton the Nobel Prize in Economics. Black unfortunately died before the award was given, and Nobel Prizes are not granted posthumously.

      Dynamic replication is very elegant, and almost all of the advances in the field of derivatives over the past 40 years have been connected with extending the fundamental insight that you can sometimes replicate a complex security by dynamically adjusting the weights of a portfolio of the security's underliers.

       The Limits of Replication

      As noted in Chapter 1, all financial models are based on assumptions. Models are toy-like descriptions of an idealized world. They don't accurately describe the world we operate in, though they may resemble it. At best, therefore, financial models are only approximations to reality. Understanding the assumptions of our models is the key to understanding the limits of replication.

      The first step in replication involves science: specifying as accurately as possible the future scenarios for underliers, interest rates, and so forth. Much of the mathematical complexity in finance originates in our attempt to define and describe possible future scenarios. Complete accuracy is virtually impossible in finance. We would like our financial model to be as simple as possible while still capturing the essential characteristics of the underlier's behavior. Choosing a financial model, then, often comes down to selecting the model that is just complicated enough.

      The second step, constructing a replicating portfolio, is mostly engineering. In theory, given the necessary securities, constructing the replicating portfolio is simply a matter of determining a set of portfolio weights at any instant. The efficacy of dynamic hedging rests on the correctness of the assumed evolution for the price of the underliers, and on the assumption that the person executing the replication strategy can react instantly to any price change by adjusting the associated portfolio weights. In practice, adjusting the weights by trading in the market can be problematic. Bid-ask spreads, illiquidity, and market impact can all affect the replication strategy. If we try to buy too much of a security we may push the price up, and when we need to sell we may find it difficult to sell at the market price. If we need to short a security, we must consider borrowing costs, which rise when the security is hard to borrow. Financing costs, transaction costs, and operational risks may vary from firm to firm. These problems are all much worse for dynamic hedging than for static hedging, because dynamic hedging requires continuous trading. Finally, dynamic hedging often requires us to estimate the future values of certain parameters that are difficult or impossible to observe in the market. The most important of these parameters, the future volatility of an option's underlier, is the main topic of this book.

      Wherever we can, we will first try to use static replication for valuing securities. If we cannot, then we will use dynamic replication. In actual markets, one cannot always find a replicating strategy. In that case, one must resort to using economic models. This last approach often requires assumptions about how market participants feel about risk and return – that is, about their utility function. Utility functions are the hidden variables of economic theory, quantities never directly observed, and our policy in this book will be to avoid them. Much of the charm of option theory lies in its seeming ability to ignore these personal preferences.

      Modeling the Risk of Underliers

      As described earlier, replication begins with the science, the descriptive model of underlier behavior. Modern portfolio theory rests on the efficient market hypothesis (EMH), a framework that has come under renewed and very severe attack since the onset of the great financial crisis of 2007–2008. Let's try to understand what it proposes.

       The Efficient Market Hypothesis

      Empirically, no one is very good at stock price prediction, whether using magical thinking or deep fundamental analysis. To be sure, there have been a few investors who have significantly outperformed the market in the past. Whether you believe their performance was due to luck or to skill, to significantly outperform the market you do not need to be very good at stock price prediction. Being right just 55 % to 60 % of the time, consistently, over many trades, is remarkable and can lead to great profit.

      In the 1960s, faced with this failure at price prediction, a group of academics associated with Eugene Fama at the University of Chicago developed what has become known as the efficient market hypothesis. Over the years, many formulations of the theory have evolved, some more mathematical and rigorous, and some less so. Economists have defined strong, weak, and other kinds of “efficiency.” No matter how we define it, though, at its core the EMH acknowledges the following more or less true fact of life:

      It is difficult or well-nigh impossible to successfully and consistently predict what is going to happen to a stock's price tomorrow based on all the information you have today.

      The EMH formalizes this concept by stating that it is impossible to beat the market in the long run, because current prices reflect all current economic and market information.

      Converting the experience of failed attempts at systematic stock price prediction into a hypothesis was a fiendishly clever jiu-jitsu response on the part of economists. It was an attempt to turn weakness into strength: “I can't figure out how things work, so I'll make the inability to do that a principle.”

       Uncertainty, Risk, and Return

      It might seem as though the efficient market hypothesis claims that the stock's price and value are identical, and that nothing more can be said. That's not the case. Let's proceed to understand how the assumption of efficient markets can lead to a model for valuing securities. The elephant in the room of finance, as in the realm of all things human, is the unknown future. Uncertainty implies risk; risk means danger; danger means the possibility of loss.

      In economics, thoughtful people have come to distinguish between quantifiable and unquantifiable uncertainty. Examples of unquantifiable uncertainty include the likelihood of a revolution in Russia within two years, the probability of a terrorist attack in midtown Manhattan this year, or the chance of finding intelligent

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