What is the value of a callable bond (a bond that the issuer can choose to pay off early)? A foreign currency option? A security backed by mortgage cash flows?
In today's financial system, companies that trade bonds, options, and other financial instruments put a lot of resources into financial engineering. This rigorous discipline combines statistics, mathematics, economics, and computer science.
Several factors combine to make financial engineering difficult. They include:
At any given time, there are bonds trading that have maturities of one year, ten years, thirty years, and everything in between. They have to satisfy certain equilibrium relationships to one another.
If you simulate the performance of a security, its value will depend not only on the initial conditions and the final conditions. If the random variables (interest rates, foreign exchange rates, and so forth) take different paths between the same two end points, the value will be different.
The curvature of the function that relates value to a random variable will determine how the mean and the variance of the random variable affect value.
The first two topics are outside of our scope for now. But we have learned enough to talk about convexity.
Suppose that we have a function of a random variable. We can write this as f(X). We can talk about the expected value of f(X), which we can write as E(f(X)).
For example, suppose that X is the number of heads we get when we flip two coins. X could be 0, 1, or 2. P(X) is 0.25, 0.5, and 0.25, respectively.
Now, let f(X) = X2. What is E(f(X))?
The possible values for f(X) are 02, 12, and 22. That is, they are 0, 1, and 4, respectively. Multiplying by P(X) to get expected value, we have,
Instead of computing E(f(X)), suppose that we compute f(E(X)).
Notice that in this example, f(E(X)) does NOT equal E(f(X)). That is because of convexity.
Draw a graph of the function y = x2. Take two points on the graph, say (1,1) and (3,9). If you draw a straight line between them, does the midpoint of the line lie above or below the graph of the f(X) where X = 2?
The midpoint of the straight line is an approximation of E(f(X)) between 1 and 3. The point on the graph of the function represents f(E(X)) between 1 and 3.
Here are some remarks on convexity.
In terms of calculus, f"(X) is positive. When f(X) is convex, E(f(X)) is greater than f(E(X)).
In terms of calculus, f"(X) = 0. When f(X) is linear, E(f(X)) is equal to f(E(X)).
In terms of calculus, f"(X) is negative. When f(X) is negatively convex, E(f(X)) is less than f(E(X)).
Suppose that I own a savings bond that will be worth $1000 in one year. How much is that bond worth today? To make this calculation, we use a discount rate, or interest rate. If the discount rate is 8 percent (.08), then the value of the bond is $1000/(1 + .08), or $925.93
Next, suppose that the discount rate is a random variable. Depending on what Alan Greenspan does at the Federal Reserve this afternoon, the discount rate will turn out to be either 6 percent or 10 percent. He will flip a coin to decide which way to move interest rates.
The expected value of the random variable is 8 percent (.08). So you might think that the value of the bond is $925.93, as before. But that would be incorrect.
The present value of the bond is a nonlinear function of the discount rate. If the discount rate is 6 percent, then the value is $1000/(1 + .06), which is $943.40; If the discount rate is 10 percent, the value of the bond is $909.09; The expected present value of the bond is the average of $943.40 and $909.09, which makes it $926.24. Before Greenspan flips his coin, the fair price of the bond is $926.24, not $925.93
In this instance, the expected present value function has positive convexity. The value of the bond is higher if the discount rate is random.
For many securities, valuation functions exhibit strong positive or negative convexity. For example, if you own a call option on a stock, this option has strong positive convexity with respect to the random variable of the stock price. On the other hand, if you are short such an option, then your position has strong negative convexity.
The phenomenon of convexity means that the value of securities often depends on the variance of the random variable. Financial engineers refer to variance as volatility. They will say that you are "long volatility" when you hold securities with positive convexity. That is because other things equal, greater volatility increases the value of securities with positive convexity.
Conversely, you are "short volatility" if you hold securities with negative convexity. Other things equal, greater volatility reduces the expected present value of securities with negative convexity.
The main point that you should take away from this lecture is this:
When X is a random variable and f(X) is a nonlinear function, E(f(X)) does not equal f(E(X)).
This is a mathematical fact about expected value. In pricing financial securities, where it is particularly important, this phenomenon is referred to as convexity.