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Saturday, February 16, 2008

The Black-Scholes Formula is Wrong! - Part 2/12 - Axioms

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

The Mathematics itself of the Black-Scholes formula is 100% correct. The problem is that its axioms do not model the actual market. I review the axioms of the Black-Scholes formula.

What happens when the entire market is following a pricing algorithm that does not match reality, yet they make a ton of money at the same time? The answer is that you have an economic system with a fundamental fatal flaw. These profits aren't free. They're paid by everyone else as inflation. The CPI understates the true inflation rate, so the average person doesn't realize how badly he is being screwed over by inflation.

The Black-Scholes pricing formula for equity options make a certain number of assumptions about how the stock market behaves.

Axiom #1: The expected return of a stock equals the risk-free interest rate. (Currently, the overnight risk-free interest rate is 3.0%. The overnight risk-free interest rate is directly set by the Federal Reserve. The long-term risk-free interest rate is indirectly set by the Federal Reserve, by predicting its future policy moves. The long-term risk-free interest rate can be determined by the expected average future Fed Funds Rate. For example, the 2 year risk free interest rate equals the expected average Fed Funds Rate over the next 2 years. If they were substantially different, traders at large banks would have an arbitrage opportunity.)

Justification: If the expected return of the stock were greater than the risk-free interest rate, then people would borrow at the risk-free interest rate and buy stocks. If the risk-free rate is higher, then people will sell their stocks and lend at the risk-free interest rate.

The "risk free interest rate" and the "Fed Funds Rate" are synonymous. Large banks can lend or borrow as much as they want at the Fed Funds Rate. Customers who want to borrow pay more than the Fed Funds Rate, depending on their size and credit rating. Customers who want to lend collect less than the Fed Funds Rate.

SPOILER ALERT! Think about this axiom carefully! If you smell a rat already, then you understand the key defect in the current economic system. If you can give the correct explanation, which I give in a later post, then you get a PhD in Understanding How Badly The Current Economic System Is Screwed Up from FSK University.

Axiom #2: It is possible to short sell the underlying asset.

Pretty much any financial instrument can be short sold nowadays.

Axiom #3: The distribution of prices over time follows a log-normal distribution. This distribution is determined by two components. First, there is the expected gain, which is determined by Axiom #1. Second, there is the volatility, which determines the width of the distribution.

The reason it is log-normal rather than normal is that, when buying a stock, you only care about the % gain or loss, and not the absolute gain or loss. In other words, the log of the price follows a normal distribution.

Axiom #4: The stock price is continuous over time.

This is obviously false. Everyone knows that stocks open up or down 10% or more in one day. However, this is not my fundamental objection. Some people claim that Axiom #4's defect is corrected by the "volatility smile". The "volatility smile" is basically one huge fudge factor that accounts for all differences between prices in the Black-Scholes model and prices observed in the market.

Axiom #5: Transaction costs and taxes are zero. No fees are paid when short selling.

Again, this is false. However, this is not my fundamental objection.

Axiom #6: We can trade in arbitrarily small increments, such as a millionth of a share.

This is only needed for the mathematical derivation. In practice, a serious options trader will only adjust his hedge when his current position deviates by more than a certain amount from his target hedge. An options trader who tries to continuously trade single shares to perfect his hedge will find himself with a very annoyed broker and paying a bundle in transaction costs.

In my next post, I explain how the Black-Scholes formula is derived.

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