What is the nature of a model price? November 7, 2007 at 7:25 am

In an otherwise excellent post on mathematical finance and its fallacies, Epicurean Dealmaker says something that might be confusing:

Black-Scholes works not because it describes some external ontological fact about how pricing relationships between securities and their derivatives have to work; it works because everyone agrees, more or less, that that’s how prices should work. It is a convention, not a physical or financial law.

Black-Scholes is a convention for quoting prices. In particular when we say ’6000 strike 5 year FTSE puts are trading at 21 vol’ what we mean is ‘if we put 21% into Black-Scholes along with all the other parameters, we get the right price for this option’, that is our expected cost of hedging it. What we do not mean is that the dynamics of the FTSE follows a log normal diffusion as Black-Scholes assume.

Things get dangerous when we go from interpolation to extrapolation. Using Black Scholes to deduce the price of the 5950 strike if we know market price of the the 5900 and the 6000 strike options is fairly safe. Using it to deduce the price of a ten year option when we only know the prices of five year instruments is more dangerous, especially in the presence of persistent fat-tails and autocorrelation. Using it (or indeed anything else) to bet large numbers of dollars on the cost of delta hedging a path dependent exotic option is alarming (unless you are leaving before the real cost of the hedge strategy becomes clear).

Now, about those $100 strike oil digitals…

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