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Tuesday, February 26, 2008

The Black-Scholes Formula is Wrong! - Part 5/12 - The Volatility Smile

Table of Contents

Part 1 - Overview and Background
Part 2 - Axioms
Part 3 - Formula Derivation
Part 4 - The Put/Call Parity Formula
Part 5 - The Volatility Smile
Part 6 - The Contradiction
Part 7 - Resolving the Contradiction
Part 8 - The Kelly Criterion
Part 9 - How FSK Trades Options
Part 10 - Only Fools and Hedge Funds Write Covered Calls
Part 11 - Other Options
Part 12 - Summary

According to the theory of the Black-Scholes formula, the volatility component of an option price should be the same for all strikes, the same for all times to expiration, and the same for calls and puts. For a long time, option prices did have the same volatility for all strikes, until October 1987.

At this point, anyone who owned deep out-of-the money put options made a fortune. Anyone who owned in-the-money calls lost less than someone who had made an equivalent long stock position.

The October 1987 crash caused options traders to reevaluate how they priced options. Now, different strikes are priced with different volatilities. This phenomenon is usually referred to as the "volatility smile". It is smile-shaped, because at-the-money strikes have the lowest volatility, and strikes further from the current underlying price have higher volatility.

There are people who will come up with all sorts of reasons to explain the "volatility smile". Basically, the volatility smile is a fudge factor that accounts for all discrepancies between the options theory and prices actually observed in the market.

This is also explained as the "fat tails problem". Stock prices do *NOT* follow a log-normal distribution. Extreme events such as the October 1987 crash occur a LOT more often than they would if stock prices followed a log-normal distribution, as predicted by the theory. The problem is that most stock price volatility is actually money supply volatility. Due to the Compound Interest Paradox, there are times when the money supply abruptly expands or contracts. This is the true explanation of the fat tails problem.

If the money supply abruptly contracts, the Federal Reserve will lower interest rates, causing the money supply to expand. If the money supply abruptly expands, the Federal Reserve will raise interest rates, causing the money supply to contract. However, there still is the occasional extreme expansion or contraction in the money supply.

The "volatility smile" is a fudge-factor workaround. It's easier for economists to justify the "volatility smile" as "compensation for small errors in the model" instead of "the US economy is not really a free market".

In my next post, I give a detailed explanation of the contradiction.

1 comment:

phubaba said...

please continue this 12 part article, I'm finding it very interesting. (along with the rest of your articles)

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