Not everything that can be counted...
The following is an excerpt from, Blame it on the copula, published in Analytics Magazine, Spring 2009
As we face the worst financial crisis since the Great Depression, we cannot help but ask what role mathematics played in this. Over the past 20 years the practice of finance has been revolutionized by the widespread adoption of mathematical models for pricing ever more exotic derivative securities. Mortgage-backed securities, which triggered the financial collapse, were priced using the Gaussian copula model. What would Albert Einstein have to say about all this? I suppose Einstein would remind us that to build a mathematical model, it is necessary to make assumptions. At best, these assumptions are idealizations of reality. The art of model building is to choose assumptions so that the model is unencumbered by less relevant considerations in order to illuminate more important phenomena. A good model provides insight and a guide to action. A good user of a model understands its limitations. We have seen all this play out in the financial markets.
Replication.
The granddaddy of financial models is due to Black, Merton and Scholes. This model (catch up on the Black Scholes Merton - Option Pricing in Continuous -Time) begins with the assumption that a stock price evolves continuously in time and has a constant volatility, a parameter characterizing the risk of the stock. A European call on such a stock is the option to buy a share of the stock at a future “expiration” date in exchange for a payment set at the initial time but paid at expiration. Since the owner of the call has a potential gain at expiration, the call has a positive initial value or price Black and Scholes provided a formula for the price of the European call. Merton provided the replication argument we now use to understand it. Consider an investor who invests in the stock and borrows from a money-market account to finance this. It turns out that if the investor can trade continuously, then she can start with a certain initial capital and trade in such a way that the value of her portfolio at expiration agrees with the payoff of the call. The initial capital that permits this must be the initial price of the call, and it is given by the Black-Scholes formula. The Black-Scholes-Merton analysis contains the insight of pricing the call by replication rather than just computing the expected value of its payoff. The flip side of this is hedging. If one owns the call and uses the negative of the replicating strategy, then one has a hedged position. Any loss in the value of the call will be offset by a gain in the value of the portfolio held by this negative replication, and vice versa. An investment bank performing intermediation among parties must hold assets for a time, and during that time hedging protects the bank against loss. The replication argument involves a delicate interplay between the sensitivity of the call price to movements in the stock price and to the ever decreasing time to expiration, and this balancing act is where the stock volatility matters.
Risk-Neutral Pricing.
In the early 1980s, M. Harrison, D. Kreps, and S. Pliska developed a risk-neutral pricing formula that greatly extended the applicability of pricing by replication. They pointed out that once a model is built, one can change the probability measure on the space on which the stock price is defined so that it has mean rate of return equal to the interest rate. This is akin to building a model based on tosses of a fair coin, and then pretending for computational purposes that the coin is biased. Under this socalled risk-neutral measure, both the stock and the money-market account held by the portfolio replicating a call have mean rate of return equal to the interest rate, and so the portfolio itself has this mean rate of return. Therefore, the initial value of the portfolio, which is the Black-Scholes price of the call, can be obtained by discounting the call payoff at the interest rate and taking the expected value under the risk-neutral measure. One can thus build a model with multiple primary assets, change to a risk-neutral measure.
