Читать книгу The Volatility Smile - Park Curry David - Страница 17

We never know what the future holds. An extremely important question in life as well as in finance is how to act in the face of risk or uncertainty. In finance, as outlined in the previous section, we think about securities in terms of their anticipated risk and return. The key question of investing can therefore be stated as follows:

What anticipated possible future reward justifies a particular anticipated risk?

The law of one price states that securities with identical payoffs under all possible circumstances should have identical prices. For the binomial model described earlier, the payoffs for a security are entirely characterized by its volatility σ and its expected return μ. Within the binomial framework, on which we will focus for now, the key question of finance then becomes:

What is the relation between μ and σ?

To answer this question, we must think more deeply about risk and return.

Some Investment Risks Can Be Avoided

The law of one price states that securities with identical payoffs under all possible circumstances should have identical prices, and therefore identical expected returns. It is tempting to reformulate the law of one price to say that securities with identical risks should have identical expected returns. It's not quite that simple, though. Not all risks are the same. The risk of a security depends on its relation to other securities. Two securities with the same numerical volatility σ might, for example, have different correlations with the Standard & Poor's (S&P) 500 index, and, therefore, when one hedges their exposure to the S&P 500, they would have different risks. In other words, when more than one stock exists, σ alone is not an adequate characterization of risk.

In life, there are certain risks that we can avoid, alter, or voluntarily expose ourselves to, while there are other risks that cannot be avoided. The same is true in financial markets. By combining assets in various ways via financial engineering, we can alter, avoid, or eliminate many forms of financial risk. It's only unavoidable investment risk that is truly fundamental. We must therefore consider whether risk is avoidable or unavoidable.

In general, as we will illustrate in the following sections, there are three ways to alter or avoid risk: by dilution, by diversification, and by hedging away common risk factors. We propose that you should expect to earn a return in excess of the riskless rate on an investment only if that investment's risk is unavoidable or irreducible. An irreducible or unavoidable risk is the risk of an asset that is uncorrelated with all other assets. We therefore reformulate our law of one price to state:

Identical unavoidable risks should have identical expected returns.

To examine the relation between a security's μ and σ, we will consider a stock with volatility σ and return μ. We will then evaluate its risk in a sequence of imaginary, but increasingly realistic, model worlds that involve ensembles of securities, to determine how much of the security's risk is avoidable by dilution, diversification, or hedging. Whatever is left over has only unavoidable risk, and we will then assume (1) that it has the same return as other unavoidable risks of the same size, and (2) that the principle of replication applies to it and all other securities. In particular, we will use the principle of replication to show that a portfolio with zero risk should earn the riskless return. This will allow us to derive a relation between the risk and return of any stock.

The three model worlds we now consider are:

■ World #1: a simple world with a finite number of uncorrelated stocks and a riskless bond.

■ World #2: a world with an infinite number of uncorrelated stocks and a riskless bond.

■ World #3: a world with an infinite number of stocks all simultaneously correlated with the market M, and a riskless bond.

We will now use the simple Worlds #1 and #2 as warm-up exercises to deduce a relation between μ and σ from the law of one price. The results we deduce in those worlds will be logically consistent, but will not resemble the relation between μ and σ in actual markets. We are using those worlds to illustrate an argument so that when we apply it to World #3, which is more complicated, the logic will be clearer. World #3 is the one that most closely resembles the world we live in. By applying the reformulated law of one price to it, we will show how it leads, in that world, to a renowned relation between risk and expected return, the capital asset pricing model5 or the arbitrage pricing theory (APT) (Ross 1976). In all cases, we restrict ourselves to a world in which securities evolve according to the binomial model, so that every security is entirely characterized by its volatility σ and its expected return μ.6

World #1: Only a Few Uncorrelated Stocks and a Riskless Bond

In this simple world, there are a finite number of stocks and a riskless bond. Each stock is uncorrelated with all of the other stocks (and any combination of the other stocks). In other words, in this world, stocks have only unavoidable risk. Suppose we are interested in investing in a risky stock S with volatility σ and expected return μ. Since there are only a finite number of uncorrelated stocks in this world, we cannot entirely avoid its risk by hedging or by diversification. We can, however, reduce our overall investment risk by combining it in a portfolio with a riskless bond. For example, given $100, instead of investing all $100 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.”