This Blog Has Moved!

My blog has moved. Check out my new blog at

Your Ad Here

Saturday, March 8, 2008

The Black-Scholes Formula is Wrong! - Part 8/12 - The Kelly Criterion

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

Large banks know that the Federal Reserve will bail them out with an interest rate cut during an economic bust. Insiders may receive advance warning of changes to the Fed Funds Rate. Individual investors have no such assurances.

One rule that individuals can use to manage their investments is the Kelly Criterion.

The Kelly Criterion comes from gambling.

Suppose you can pick NFL games well enough to make a profit. When you wager on an NFL game, you are usually wagering $110 to win $100, because you pay a fee to the bookmaker. The line in an NFL game is set so that half of the people bet on either side, and the bookmaker will make a profit of $5 per $100 bet. If you can pick correctly more than 55% of the time, you can profit wagering on NFL games.

If you can profitably wager on NFL games, how much should you risk on each game? Suppose you are correct 60% of the time. If you wager 100% of your bankroll on each game, you will go bankrupt eventually. If you wager only 1% of your bankroll on each game, you aren't optimizing your profit. You won't go bankrupt, but you aren't optimizing the growth rate of your bankroll.

The Kelly Criterion tells you how much of your bankroll you can profitably wager on each game.

When investing, you don't have a binary outcome. You have a range of possible outcomes: lose 100%, lose 25%, break even, gain 25%, etc. However, you can still use the Kelly Criterion to tell you what percentage of your savings you should invest in any given trading system.

Suppose your trading system has a risk of 100% loss, but has a high expected return. This is possible. Suppose your trading system gains 300% half the time and loses 100% half the time. Your expected gain is 100%, but if you risk all your savings you will soon go bankrupt. The Kelly Criterion says that you should risk some percentage of your savings on this system.

The options market misprices EVERY SINGLE option. The put/call parity formula says that options are priced as if the expected gain in a stock equals the risk-free interest rate. Historically, stocks have outperformed the risk-free interest rate by 2%-5%/year. The Federal Reserve still provides government subsidized negative real interest rates, so this spread should continue to hold in the future.

If you are willing to adopt an unhedged option position, you can profit from the fact that EVERY LISTED EQUITY OPTION IS MISPRICED. From the point of view of a bank or hedge fund, options are not mispriced, because banks can already borrow at the risk-free interest rate. From the point of view of an individual, who cannot borrow at the risk-free interest rate, these mispriced options are a REALLY GOOD DEAL.

Options are mispriced, but if you risk ALL your savings buying unhedged long calls, you will go bankrupt during the next bust phase of the economic cycle. The Kelly Criterion says that you should risk some % of your savings on unhedged long call options.

Based on a heuristic estimate, I decided that I'm willing to risk 10% of my savings each year on my options trading system.

The Kelly Criterion can help individuals with portfolio allocation to risky trading strategies. However, you don't have the full benefit of the massive government subsidy that large banks and hedge funds receive. Still, by investing some of your savings in unhedged long call options, this will help you outperform inflation in the long-run.

As an individual investor, your goal should be to slightly outperform inflation. With an inflation rate of 15% or more, this is a hard task indeed!

No comments:

This Blog Has Moved!

My blog has moved. Check out my new blog at