The Volatility Smile. Park Curry David

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      To be clear, the total value of a firm, what financial analysts refer to as a company's enterprise value, includes the value of both the company's stock and its debt, and Apple, like most large firms, does issue debt. In fact, in 2013 Apple issued what was, at the time, the largest corporate debt issue in history. The value of a company's deb

1

See, for example, Mandelbrot (2004) and Gabaix et al. (2003).

2

To be clear, the total value of a firm, what financial analysts refer to as a company's enterprise value, includes the value of both the company's stock and its debt, and Apple, like most large firms, does issue debt. In fact, in 2013 Apple issued what was, at the time, the largest corporate debt issue in history. The value of a company's debt is generally fixed and largely predictable, except perhaps when it enters a credit crisis. The interesting part of determining the value of a company is, in most cases, almost entirely concerned with determining the value of its stock. This is what we focus on here, though more advanced models do treat the enterprise value as the fundamental underlier.

3

Ole Bjerg, a philosopher working in the framework of Slavoj Žižek, sees the corporation as “the real” and the stock price as its “symbol,” and this seems right. What interests Bjerg is the way fantasy and ideology fill the gap between reality and symbol, as discussed in his book Making Money: The Philosophy of Crisis Capitalism (Verso Press, 2014).

4

Throughout the book, whenever we specify the return or volatility of a security without specifying a time period, you can assume these values are being expressed per year. In our current example, when we said “with.. expected return μ,” this was shorthand for “with an expected return of μ per year.”

5

A more complete version of the following presentation is contained in E. Derman, “The Perception of Time, Risk and Return during Periods of Speculation,” Quantitative Finance 2 (2002): 282–296.

6

In this section and in what follows, we have been assuming that all that matters for valuing a security is its volatility σ and its expected return μ. In actual markets, security returns can have higher-order moments and cross moments. In the real world, two securities could both be uncorrelated with all other securities and have equal standard deviations, but have different skewness and/or kurtosis. Securities can also differ in their liquidity, in their tax treatment, and in a whole host of other ways that investors care about. These factors could, in turn, cause expected returns to be higher or lower. In the derivations in this chapter, when we say equal unavoidable risk, we are basically assuming that all of these other risk factors do not matter. That is an implicit assumption of this model that assumes everything of interest to valuation is captured by the first two moments.

7

We're assuming that w > 0. If we allow w to be negative, effectively shorting the risky asset, then σ_P = |w|σ = −wσ but μ_P is still w(μ – r) and the Sharpe ratio

. The magnitude of the Sharpe ratio for a short position in the risky asset is still the same, but with opposite sign.