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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 is constructive 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.