The Volatility Smile. Park Curry David

in the risky stock, we could invest only $40 in the stock and the remaining $60 in a riskless bond. This can be thought of as diluting the risk of the stock.

      More generally, assume that we dilute the risk of stock S by investing a percentage of our portfolio, w, in a risky stock and (1 – w) in riskless bonds. If w is 1, our portfolio is entirely invested in risky securities. If w is 0, our portfolio is entirely invested in riskless bonds. If 0 < w < 1 then our portfolio is a mix of risky and riskless securities. If w is greater than 1, then (1 – w) is negative and we are borrowing at the riskless rate in order to leverage our investment in the risky security.

Figure 2.5 shows the binomial tree of returns for a mixture of a risky security and riskless bonds. The expected return of this portfolio, μ_P, is simply the weighted average of the risky security and the riskless bonds:

      (2.1)

      Because the riskless bonds have no volatility, the volatility σ_P of the portfolio is simply wσ. By decreasing volatility from σ to wσ, we decrease the expected excess return to w(μ − r), the excess return being the return of a security or portfolio minus the riskless rate.

Figure 2.5 Binomial Tree for a Mixture of a Risky Stock S and a Riskless Bond

      Define a new variable λ, the ratio of a security's excess return to its volatility, so that

      (2.2)

      The variable λ is the well-known Sharpe ratio. Now, for the portfolio of a risky security and riskless bonds in Equation 2.1, the Sharpe ratio is

      (2.3)

      The Sharpe ratio of the portfolio is equal to the Sharpe ratio of the risky security. Diluting a portfolio by investing part of the portfolio in riskless bonds has no effect on the Sharpe ratio.7

      Now consider another uncorrelated stock S′ that has the same volatility wσ as the portfolio P. It has the same numerical risk as portfolio P consisting of S and a riskless bond, but, since it is a separate source of risk, uncorrelated with the behavior of S,both risks are unavoidable. The reformulated law of one price tells us that any security with unavoidable risk wσ must have expected excess return w(μ − r). Therefore, S′ must have the same return as P. Thus,

      (2.4)

      Equation 2.4 shows that the Sharpe ratio is the same both for the security S′ and for the security S. Therefore, in World #1, the Sharpe ratio must be the same for all stocks. By varying w in Figure 2.5, we can create portfolios P of any risk σ_P. Equation 2.3 shows that the excess return of any uncorrelated security will be proportional to its volatility. It confirms the popular maxim “More risk, more return,” which strictly speaking should read “More unavoidable risk, more expected return.”

      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 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