The Volatility Smile. Park Curry David
This assumption, a very strong one, will be used later, in combination with the law of one price, to derive some famous results of neoclassical finance, in particular the capital asset pricing model (CAPM), and later, the famous Black-Scholes-Merton option pricing formula.
The symmetric distribution of our simple model is at odds with the observed return distributions of almost all securities, which are characterized by negatively skewed distributions and fat tails. Nevertheless, the binomial model is a reasonable starting point for modeling risk. Though the actual behavior of securities is more complex and unpredictable, the binomial model provides an easily accessible intuitive and mathematical treatment of risk. Actual risk is wilder than the model and the normal distribution can accommodate. This should never be forgotten. We will investigate some more ambitious models, which go beyond these assumptions, later in this book.
Riskless Bonds
In the binomial model in the limit when σ is zero, the up-move and the down-move are identical, and risk vanishes. We refer to the rate earned by a riskless security as the riskless rate, often denoted by r. The riskless rate is ubiquitous throughout economics and finance and is central to the replication and valuation of options.
Figure 2.4 shows the binomial tree for a riskless security. The two branches of our tree, though we've kept them separate in the drawing, are identical. No matter which branch we take, the end value is the same.
Figure 2.4 Binomial Tree for a Riskless Security
For any risky security, the riskless rate must lie in the zone between the up-return and the down-return. If this were not the case – if, for example, both the up- and down-returns were greater than the riskless return – you could create a portfolio that is long $100 of stock and short $100 of a riskless bond with zero net cost and a paradoxically positive payoff under all future scenarios in the binomial model. Any model with such possibilities is in trouble before it leaves the ground, because it immediately provides an opportunity for a riskless profit, an arbitrage opportunity that violates the principle of no riskless arbitrage.
How do we determine the riskless rate in practice? One possibility is to use the yield of a bond with no risk of default, such as a U.S. Treasury bill, commonly considered to be entirely safe. Rather than talking about borrowing or lending at the riskless rate, in fact, we often talk about buying or selling a riskless bond. The problem of determining the riskless rate is then a problem of defining and then finding a riskless bond. While this may sound simple, in practice agreeing on what number to use for the riskless rate can become complicated, especially in crisis-ridden markets. Here we will simply assume the riskless rate is known.
The Key Question of Investing
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
In this section and in what follows, we have been assuming that all that matters for valuing a security is its volatility