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 underlying Mathematics is wrong. The axioms don't model reality. Options are priced as if the expected return in the underlying asset equals the risk-free interest rate. The expected return of stocks exceeds the risk-free interest rate. The expected return of bonds and commodities also exceeds the risk-free interest rate.

The entire options market feeds off negative real interest rates, sponsored by the Federal Reserve. In an options trade, the two parties to the transaction experience different interest rates. Banks can borrow cheaper than retail customers. Banks can lend at a better rate than retail customers. Banks get much more favorable terms for short sales and investing the short sale proceeds.

Under a gold standard, you can't have a large derivatives market. Negative real interest rates are needed to funnel money to banks and hedge funds. Complicated derivative transactions allow this interest rate subsidy to be packaged and sold.

Options traders do not notice that every listed option is mispriced because they adopt a hedged position and the errors in the pricing model work in favor of professional traders. If you are willing to adopt an unhedged position, you can profit from this structural defect in the economic system.

As a retail customer, you shouldn't make a short stock bet unless you have specific information that a stock is going to tank. Similarly, you should avoid writing calls or buying puts. Statistically, the odds for those trades are very poor because you are betting AGAINST inflation. A retail customer shouldn't short naked puts, because of the margin rules. Long-term deep out-of-the-money calls are extremely attractively priced for a retail customer.

A retail customer can't normally borrow at the Fed Funds Rate to buy stock. When you buy a call option, the interest rate priced into the call option is the Fed Funds Rate, making the purchase very lucrative. This trade is useless to banks and hedge funds, who have other more reliable ways to make money.

Buying out-of-the-money call options risks a 100% loss certain times. A banker or hedge fund manager won't risk losing his job by adopting a strategy that might lose money. Their managers and shareholders want a trading system that *ALWAYS* makes money. The Kelly Criterion tells individual investors approximately how much of their savings they can risk buying options.

When you buy an unhedged call option, it's like you're making a leveraged bet that there WILL be inflation. According to my calculations, some options are mispriced by as much as 200%. The edge from the trade is a lot greater than the transaction costs and capital gains taxes.

Commodity futures are mispriced, because the interest rate priced into the future is the Fed Funds Rate and not the true inflation rate. Bond options and commodity options are also similarly mispriced, although I haven't analyzed the details fully. I couldn't find a reliable online resource for prices and quotes. There are many online brokers that offer stock option trades. I couldn't find a decent online broker for commodity option trades. I've been considering buying some gold, silver, or oil commodity options, but haven't found a good broker or information source.

Analyzing the Black-Scholes formula and its contradictory axioms led me to discover the fundamental structural flaw in the US monetary system. This is a good information source, because all the "official" axioms for the Black-Scholes formula can be found in many economics books. Mainstream economists have a strange ability to simultaneously believe contradictory axioms. I'm lucky that I didn't study economics much in school. I avoided economics because I sensed that there was some funny business going on. I learned to think before I started studying economics.

## Saturday, March 22, 2008

### The Black-Scholes Formula is Wrong! - Part 12/12 - Summary

Posted by FSK at 12:00 PM

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## 4 comments:

"Options are priced as if the expected return in the underlying asset equals the risk-free interest rate. The expected return of stocks exceeds the risk-free interest rate. The expected return of bonds and commodities also exceeds the risk-free interest rate."

This is complete rubbish. The B-S model makes no assumptions whatsoever about the expected return on the underlying asset. It only makes the assumption that the return distribution is normal. The risk-free rate factor is there for an entirely different reason, unrelated to the underlying asset return.

The Black Scholes formula is for idiots who want to lose their money. I have a roulette betting system that "guarantees" you'll win! Using the risk free rate as the expected return makes as much sense as using a "risk free volatility" as the stock's volatility. "But you can't use the historical return because that leads to arbitrage!" Yes, but in a free market, that just forces the interest rate to converge to the historical return.

And there is no such thing as a "risk free rate" in a free market. Well there is, but it's called 0%. In a free market the interest rate naturally coincides with the return on capital, and that cannot be risk free. So what does a nonzero "risk free rate" mean? It means that if you lose your money, the State will use its monopoly of force to steal it from someone else. A nonzero "risk free rate" is an artifact of a centrally planned economy.

Remember that they got the 1997 Swedish Central Bankster Prize (aka "Nobel Prize in Economics) for their formula.

"The B-S model makes no assumptions whatsoever about the expected return on the underlying asset."

And that's exactly why it's BS. Let's do a little thought experiment.

Asset A is a printing press where you can print as much money as you want. Asset A has a historical return javascript:void(0)(expected return) of 999999999999% (more if you're in Weimar Germany or Zimbabwe).

Asset B is cash and has a historical return (expected return) of 0%.

Asset C is my fireplace that I can put my cash in (I always have a chance of gluing ashes back together). Asset C usually loses almost all of its value annually and has a historical return of -99.999999999999% (but somehow never completely goes bust. I always have some chance of gluing ashes back together).

All 3 assets go for $1 today, because $1 will buy you $1 in cash for any of those purposes. The BS formula (Bull-Shit formula) will give the same price for a $1 call a year from now for all 3 assets, and that is obviously BS.

Ok, it's possible that "cash" doesn't have nearly the same volatility as the other two. So replace "cash" with "continuously and randomly switching from printing money to burning it up". Basically, you're continuously investing in Asset A (for a random amount of time), liquidating, investing in Asset B, liquidating...

And the printing press prints enough money on average to cancel out everything that's burned in the fireplace. One process is the TIME REVERSAL of the other (so they have the SAME VOLATILITY!) If two options have the same time to maturity, same strike price, same spot price, and same volatility, the the options are the same price. But how can an option for a soon to be (but not yet) worthless underlying be the same price as one where you literally print as much profit as you want? You want "arbitrage", then sell the $1 strike call for Asset C and use the proceeds to buy a $1 strike for Asset A. Cost of transaction: free, because they have the same price. At expiry, Asset C is $0.000000000001, Asset A is $999999999999999. So you make $999999999999999.000000000001. That's arbitrage. Now just find a company where you might as well put money in the fireplace and one that's as good as owning a printing a press.

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