Tuesday, March 29, 2011

Why doesn't Black-Scholes model work in long period option pricing?

Buffett doesn't like derivatives and calls it as the financial weapon of mass destruction. Yet he wrote Billions of European Put Options of various stock indexes. The term is very long (10-20 years) and no collateral needs to be posted.

Apparently he thinks this is a good deal and he mentioned in his letter to share holders this is a way to show his belief that the Black-Scholes model doesn't work in long term option pricing. In this case, the price given by the formula is just too high.

There are a list of limitations and assumptions for deriving BSM. The number one is that the stock price follows Wiener process and it is log-normal distribution. In order to arrive at this conclusion, a big assumption here is that the price has same likelihood of going down and going up, no matter what the current price is. In another word, there is no such thing as "over-bought" or "over-sold". A large decline of the stock or index doesn't mean it will be more likely to rise later.

The primary source behind this assumption is again "efficient market assumption". If market is efficient, the current price is "always" justified, and of course, there is no such thing as "cheap" or "expensive" stock.

This may be true for many short term stock movement. But when we get to the stock index in 20 years term, it becomes more obvious that this assumption is not aligned with common sense. In history, we have observed that the price of stock or index eventually revert to their economic earning power, despite the possibility of long depressing period and bubble period (sometimes that period could last 5-10 years, but usually it doesn't last more than 2-3 years).

In a long time scale, it becomes obvious that the market is far from efficient, as we can find so many unbelievable and ridiculous bubbles in the history.

Put it in another way, we can invent a process called "economic arbitrage" (the term is to differentiate it from "trading arbitrage"). If a company's stock is very cheap comparing to its earning power, business people will find it compelling to buy the company's stock instead of investing on starting a new company in that industry. The other argument could be true, if the company's stock is too expensive, people could borrow money to sell short the stock or dump existing portfolio and use the cash proceed to fund a new company on their own.

Now the question is: if we have economic arbitrage, why could the stock price still go out of the reasonable range sometimes? The answer is simple, the market force that are willing to trade and focus on the short term is much bigger than the pure business force (after all, the ordinary Joes can't buy a company or start a new company). But eventually, business force (the weighing machine) will win, when the market force returns to their senses, or when the power of their mania is gradually worn off by the business force.

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