Any mathematical model of reality relies on simplifications and assumptions. 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.In particular, the volatility estimate based on recent trading history will be reliably too low when a market enters a new period of instability and resulting higher volatility. If traders miss this change, then they will see what look like ideal arbitrage opportunities. This is what mostly what sunk LTCM and seems to underlie many more recent failures as well.
Monday, February 13, 2012
Ian Stewart on Black-Scholes
The influential mathematician and writer Ian Stewart has a short article in a recent Guardian that considers the idea that the Black-Scholes model of option pricing contributed to the financial collapse. (Thanks to Ole Hjortland for the link.) As I summarized things back in October 2009, the derivation of the central partial differential equation is quite accessible, and involves the sorts of idealizations that we'd be happy to make in most other areas of science. In my book I discuss the case of Long-Term Capital Management (LTCM) and the ways in which its trading strategies appear to have been misled by the mathematical model itself. One dimension of the problem is noted by Stewart: