Sunday, February 12, 2012

The Black-Scholes pricing formula had nothing to do with the crisis

Ian Stewart published an article today in the Guardian describing how Black-Scholes contributed to the 2008 financial crisis. He concludes that although the option pricing formula is not to blame, its abuses contributed to the crash.
The Guardian: Was an equation to blame for the financial crash, then? Yes and no. Black-Scholes may have contributed to the crash, but only because it was abused.
This is not new. Others have been blaming option valuation techniques for the crisis because it's all about the scary complicated derivatives that were not priced properly due to bad assumptions.

Amazingly this misconception is widespread. There is this image of traders sitting around, making some volatility assumptions that are sometimes incorrect, then plugging these numbers into their formula to come up with the price. Many financial textbooks actually describe it in such a way. And the magic "complexity" of Black-Scholes makes traders go mad in their pursuit of profits. Bad inputs into the formulas make the markets go bad...
The Guardian: The Black-Scholes equation was based on arbitrage pricing theory, in which both drift and volatility are constant. This assumption is common in financial theory, but it is often false for real markets. The equation also assumes that there are no transaction costs, no limits on short-selling and that money can always be lent and borrowed at a known, fixed, risk-free interest rate. Again, reality is often very different.
As an academic, Dr. Stewart probably never traded an option. Dr. Stewar and others may be surprised that option prices are not determined by the Black-Scholes equation. In fact options and other derivatives were traded long before the Black-Scholes formula came about. As is the case with all markets, it is the markets that determine prices, not an equation.

Option pricing formulas are used to compare relative value of options - just as other valuation formulas are used to determine the relative value of other assets. The markets are in fact quite efficient and to the extent there are flaws in theoretical descriptions of price behavior of assets, the market corrects for it. The chart below shows the implied volatility for different strikes of SPY ETF options. By implying a higher volatility away from the current price, particularly to the downside, the market is clearly pricing in "fat tails" of the price movement probability distribution, therefore adjusting for any imperfections in the formula. All option markets tend to exhibit this type of behavior - all adjusting for imperfections in the mathematics describing price behavior.

SPY ETF Option "skew" (Bloomberg)

To demonstrate the madness around Black-Scholes, consider a simple formula used to value a perpetuity.

If one knows the cash flow and the interest rate, one can then obtain the present value of an asset that pays a constant amount in perpetuity. An example of that would be a rental property. Now ask any person who buys and sells rental properties, would they rely on this equation when valuing properties? Unlikely. However they could use it to determine the "implied" discount rate and compare it across properties to get an indicator of relative value of one property vs. another or properties in one geographic region vs. another, etc. As rates decline, one could for example use this equation to argue that rental properties become more attractive - which actually hasn't been the case recently. In fact in recent years the implied r in this equation has been high because of higher implied risk in rental properties. But nobody in their right mind would use this or any other pricing formula by itself to buy and sell properties.

The same applies to the bond yield/spread formulas. The market determines bond prices but the yield and spread formulas allow one to compare bonds with different coupon and other contractual differences. Implied volatility in the Black-Scholes equation is in that sense the same as the yield/spread measure for bonds. It is used to determine relative value, not to come up with a price at which one wants to transact. For convenience many markets quote yield, spread, or implied volatility when trading, simply because the translation from these measures to price has become standardized - i.e. all market participants will obtain the same price using these inputs.

To argue that Black-Scholes contributed to the crisis is like saying the bond/spread formulas or the perpetuity formula above contributed to the mispricing of bonds and properties, thus causing the crisis. Bonds, properties, and options for that matter were priced going into the crisis by the same forces that priced tulip bulbs 1630s in the Dutch Republic. Clearly there wasn't a formula to price tulips back then. Financial bubbles can occur in any market and they will happen again in the future. The Black-Scholes formula or any other mathematical expressions describing price behavior have nothing to do with it.
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