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Tuesday, February 12, 2008

The Black-Scholes Formula is Wrong! - Part 1/12 - Overview and Background

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 problem with the Black-Scholes formula is not that the Mathematics is wrong. The problem is that the axioms and assumptions don't match the actual behavior of the market. Stock options are priced as if the expected return in stock equals the risk-free interest rate, but everyone knows that the expected return in stock exceeds the risk-free interest rate. Once I fully understood the Black-Scholes formula, my reaction was "I call shenanigans!"

In 1973, Black and Scholes published a paper describing how to price options using a hedging strategy. In 1997, they received a Nobel prize for their work. Only a handful of people read an obscure blog like this. It is interesting to observe that the Black-Scholes formula is simultaneously completely wrong and completely accurate. Of course, I don't expect any type of award for a post in an obscure blog, even if my post contains information that Black and Scholes completely ignored.

The publication date of 1973 is no coincidence. The gold standard was abandoned in 1971. Under a gold standard, you can't have derivative markets. It doesn't make sense to trade options under an international gold standard. It wasn't until after the abandonment of the gold standard that derivatives markets took off. Central bank subsidized negative real interest rates feed the derivatives markets. With fiat debt-based money, central banks can subsidize low interest rates and show a bookkeeping profit at the same time, via their "monetizing the debt" trick.

Black and Scholes are frequently credited with being responsible for the explosion in derivative markets. That is complete and utter nonsense. It was the abandonment of the gold standard, coupled with central bank subsidized negative real interest rates. The money made by options traders is paid by everyone else as inflation.

All Black and Scholes did was provide a mechanism whereby banks and professional traders could reliably take in the firehose of money provided by the Federal Reserve. Hedged options transactions allow them to package up some of this firehose money and sell it to others. In an options market, the professional trader can borrow or lend at the Fed Funds Rate. The counter-party to the trade is a retail customer who has to borrow at a much higher rate or can only lend at a much lower rate. Most serious options trades are made due to different interest rates experienced by each party. The options transaction packages up the favorable interest rate received by the large bank.

Similarly, leveraged buyouts and hedge funds didn't become popular until after the abandonment of the gold standard. Leveraged buyout firms and hedge funds take advantage of the same massive Federal Reserve subsidy that banks and options traders receive.

I was confused by the contradictions of the Black-Scholes formula. This led to my discovery of the fundamental flaw in the current economic system and the injustice of the Federal Reserve. That's also the reason no mainstream economics journal will touch this topic. An economics professor at a major university is never going to say "The current economic system is fatally flawed", because he's dependent on government grants to keep his job.

I realized that there's no risk of some large bank or hedge fund stealing my options trading system. They already receive a massive subsidy from the Federal Reserve in the form of negative real interest rates. From their point of view, buying options is equivalent to directly borrowing from the Federal Reserve at the risk-free rate and buying stock. I figured out a way for small investors to get a slice of the action.

I doubt that someone would read this and buy enough call options to substantially move market prices. In my options trading system, I'm not exploiting the professional options traders. They still make their expected profit from their trade with me. In my options trading system, I'm arbitraging the Federal Reserve and its monetary policy of negative real interest rates. Such a policy cannot be abandoned without reforming the financial system, which is my ultimate goal anyway!

You might as well take advantage of the massive subsidy from the Federal Reserve to the financial industry. If the Federal Reserve is providing this geyser of money to the financial industry, there's no reason for the average person to not claim their share. I consider this money spent by the Federal Reserve to be unclaimed property. If other people interested in counter-economics and agorism can claim their fair share, then good for them. Further, only someone who really understands the defect in the current economic system will be able to trade options the way I do.

Here, I am primarily discussing equity options. Other types of options exist as well, most notably bond options (technically, bond futures options), but equity options illustrate all the points. For a bond option, you aren't buying an option to buy a bond; you're buying an option to buy a bond futures contract. This maximizes the amount of leverage; in other words, this maximizes the size of the subsidy big bond traders (banks and hedge funds) receive from the Federal Reserve.

A call option on a stock is the right to buy 100 shares of stock on or before a certain date at a specific price. A put option is the right to sell 100 shares of stock on or before a certain date at a specific price. I'm referring to American-style options, where you can exercise before the expiration date. There also are European-style options, which can only be exercised on the expiration date. The analysis in this series of posts applies to both types of options, but I'm sticking with American-style options, which covers almost all single stock options in the USA. Based on my analysis, index options and ETF options aren't as lucrative as single-stock options.

If a professional options trader owns a call option, he will hedge with a short stock position. Suppose the options is deeply in-the-money and the correct hedge is 100 shares short. Suppose that a stock is priced at $10/share and the options trader short sells 100 shares. He now has $1000 cash. However, he can't just spend that money, because it is collateral for the short position. That money is put in a special bank account where it earns interest at a slight discount to the Fed Funds Rate. If the stock goes down, then that money is free to be spent on other things. The options trader will have lost money on his call, but made money on his short trade. If the stock goes up, the options trader will have made money on his call, but will lose money on his short trade. Additional cash collateral must be provided for the short position, but the options trader can borrow against his long call position.

The professional options trader is mostly indifferent to a rise or decline in the stock price, unless the stock price tanks and the call option is out-of-the money and there is a huge windfall on the short position. The options trader paid money when he bought the call, but he will collect interest on his short stock position. If the options trader priced the call properly, he will almost definitely show a small profit.

Notice that the retail investor cannot short sell on favorable terms like the options trader. First, he may have already owned the stock and was writing a covered call (fool!). Second, he cannot borrow as favorably as the professional options trader if he really did desire to sell short.

The Black-Scholes formula is dependent on a bunch of axioms/assumptions, which I enumerate in the next article.


JEK said...

An economist was giving a series of lectures for small investors in some community center. Now, there weren't many people attending this lectures, so the man was very surprised to find overcrowded room , waiting for lecture on "Problems with Black-Scholes".
The whole crowd stared at him throgh the lecture, and then left without asking a single question. Only latter did he learn that the publisher of the info leteer for the community center misspeled his lecture as "Problems of Black Schools".

for some reason (well, i can think of number of them) , people find cheap socialism far more intresting than concrete economics.

Joshua said...


Your article is interesting, as you have observed something most people miss:

Provided the stock market outperforms the risk free rate (which tends to be the case), the expected return of calls is significantly above the risk free rate and the expected return of puts is significantly below the risk free rate. This is particularly true of further out of the money puts/calls.

This does _NOT_, however, mean that equity options are mispriced. Black-Scholes is based on an arbitrage argument, not directly on the expected return. The relationship you observe:

price = exp(-rT) * (Risk-neutral expectation)

is a theorem regarding risk-neutral valuation, but not the primary motivation for the Black-Scholes equation.

Indeed, if options were priced so that their expected return is zero (I suspect you think this is the "fair" price), then there would in fact be an arbitrage opportunity whereby one could earn more than the risk free rate by buying/selling options and delta hedging them.

Also, note that calls and puts are far more leveraged investment vehicles than the underlying instrument itself. If one were to invest by buying calls (or shorting puts), you would indeed outperform the risk free rate provided the underlying did also, but you would do this at the expense of assuming more risk (variance in your returns). Thus you are merely collecting a risk premium. You aren't really getting anything for free. You can verify this yourself by looking at the Sharpe ratio for these investment techniques.


Anonymous said...

here is an online BS calculator

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