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Chapter 1
Introduction
This is a book about model risk in the equity derivatives market. From the crash of 1987, until the crash of 2008, the outstanding notional in interest rate, credit and equity derivatives grew from just under USD 1 trillion to USD 451 trillion [ISDA (2009)], equivalent to over seven times global GDP [IMF (2009)]. So-called ‘exotic’ derivatives made a significant contribution to this growth, so much so that the ‘vanilla’ market failed to keep step and provide a sufficient basis for hedging and price discovery. This resulted in model choice, based on one view of unobservable dynamics or another, becoming a major source of uncertainty. This book focuses on the significance of such model choices to valuation. Of the many lessons we may learn from this analysis, principal among them is how much there is still to learn. Instruments we might expect to be exquisitely sensitive to certain model choices in fact turn out not to be, whilst conversely, instruments deemed to be essentially vanilla, and thereby insensitive to model choice, turn out to be even more sensitive than their more exotic looking counterparts. In the field of derivatives, possibly the most untrustworthy of all our faculties is our intuition. It is hoped, nonetheless, that this book may go some way to improving that faculty, and at least provide some tools for testing it.
1.1 Equity, fairness and arbitrage
Equity, by definition, is supposed to be fair. It entitles the holder to a percentage of the income distributed by the associated company equal to the percentage of stock held. Shareholders do not have equal voting rights in the decisions of the company; that would clearly be unfair. Why would someone who'd invested twice as much as someone else only have an equal say? On the contrary, shareholders have equitable rights in the company, their vote carries proportionally to the amount invested. Such a scheme is clearly fair from a democratic point of view, but is this the same as financially fair? Let us define the latter as follows:
Definition 1.1. In the context of a financial transaction, a fair transaction is one in which neither the buyer nor seller is able to make a riskless profit.
Such a definition is equivalent to the statement that the transaction is free from arbitrage. Note that the transaction must yield a profit greater than or equal to zero with certainty to be classified as an arbitrage. A simple example would be a coin tossing game between two people, call them A and B, where A receives two dollars if the coin falls heads, and B receives 1 dollar if the coin falls tails. As the game is obviously more attractive to A than B, B should clearly demand payment to enter into it. Suppose that the coin is fair, and that interest rates are zero, then the transaction would be fair provided neither A nor B would be guaranteed a profit. Clearly, if A pays B 2 dollars or more, the game is an arbitrage for B. Conversely, if B somehow ends up paying A 1 dollar or more, the game is an arbitrage for A. Defining the amount paid by A to B as P, the game would be fair provided −1 < P < 2. Note the distinction with expectation. We might think that the fair price for this game should simply be 1/2, as for this price E[P] = 0. If we played this game a large number of times, and paid the average profit to the profit taker, then as the number of games tended to infinity, this would indeed be the fair price, as the variance of the profit would tend to zero, making the game riskless. For a finite number of games, however, the fair price must necessarily exist over a range. Whilst the example is obviously somewhat contrived, it does serve to illustrate an important aspect of valuation theory, namely that the fair price for a transaction is, in general, unique only under idealised circumstances, such as infinite portfolio diversification, or infinitesimal transaction time. We will examine this in more detail in the rest of this chapter.
For a stable financial system to exist, parties within it must be able to transact freely and agree on a price. The more they transact, the faster any opportunities for immediate wealth creation will be eroded, so that indeed, in a fully transparent, liquid, continuously traded system, the market should always be fair. Such a theory is an example of an efficient market hypothesis, of which there are three forms. The strong efficient market hypothesis [Fama (1965)], maintains that the prices of traded assets reflect all possible information, including ‘insider’ information (such as how much dividend a stock is going to pay in a year's time). The semi-strong efficient market hypothesis, maintains that the prices of traded assets reflect all publicly available information (such as the next announced dividend on a stock) is more reasonable. The weak efficient market hypothesis maintains that the prices of traded assets reflect only all past available information (for example, the financial health of a company and how its performance has compared to its peers). Until Black and Scholes published their seminal paper [Black and Scholes (1973)], one of the most popular methods for determining the fair price of a security was based on the Capital Asset Pricing Model (CAPM) introduced originally by Jack Treynor in 1961 [Treynor (1961)], and lat...
