This Blog Has Moved!

My blog has moved. Check out my new blog at

Your Ad Here

Tuesday, March 4, 2008

The Black-Scholes Formula is Wrong! - Part 7/12 - Resolving the Contradiction

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

We have arrived at a contraction.

Axiom #1: The expected return of a stock equals the risk-free interest rate.

Axiom #7: The expected return of a stock is greater than the risk-free interest rate.

If this isn't obvious enough for you:

Axiom #1: A=B
Axiom #7: A>B

Axiom #7 is the one that models the actual stock market.

If you observe the options market, prices do indeed follow Axiom #1. If Axiom #1 is violated, a professional options trader will perform the arbitrage indicated by the put/call parity formula.

In a truly free market, when someone borrows at the risk-free interest rate, the risk-free interest rate rises. A bank can borrow as much as they want at the risk-free interest rate, without changing the risk-free interest rate. This money isn't free; it's paid by everyone else as inflation. When a bank borrows from the Federal Reserve at the Fed Funds Rate (the risk-free rate), the money supply increases. Those economic models assume that the money borrowed came from someone else, not that it was new money borrowed into existence.

Wall Street has a bunch of quants and traders chasing profits in the options market. They aren't chasing money directly lost by other people. They're chasing this massive Federal Reserve subsidy of the financial industry! The quality of financial software is lousy overall. The value of this massive Federal Reserve subsidy is far greater than the advantage anyone could get by writing good software! Who needs to write good financial software when the government is, literally, giving money away!

Axiom #1 assumes that if the risk-free rate and the expected gain in stocks are different, then people will borrow at the risk-free rate to buy stocks. Hedge funds do this, but the effect is that the money supply increases. Stocks are inflation-hedged, so when hedge funds increase the money supply by borrowing, they are literally causing inflation. Hedge funds cause inflation, which makes their borrowing even more lucrative! You can never perform enough arbitrage to make Axiom #1 hold. The act of borrowing to buy stock increases the money supply, causing more inflation.

Occasionally, you hear Ben Bernanke say "We are concerned that hedge funds or banks will try to take advantage of our interest rate policy." That is pointless, because the US monetary system was *DESIGNED* to be exploited by banks and hedge funds!

A large bank doesn't want to borrow to buy risky assets such as stocks. If a bank maximized its leverage borrowing to buy stocks, it would be bankrupted during the next bust cycle. A bank prefers to borrow to buy illiquid "Level 3 Assets", so it doesn't have to mark-down to market during the bust phase.

In other words, people who are actually able to borrow at the risk-free interest rate have FAR MORE attractive investment options than stocks. Banks prefer to issue bonds or loans, where they have collateral and can use higher leverage ratios. Banks prefer to buy assets that they don't have to "mark to market", so they can hold onto them during an economic bust and sell them during the next economic boom.

If you price options using Axiom #7 instead of Axiom #1, you conclude that calls are underpriced and puts are overpriced. Everyone who writes a call is giving away money. Everyone who buys a put is giving away money.

Retail customers typically buy stock, write a covered call, and use the proceeds to buy a put. This leaves them with stock ownership, but a low-risk position. Most people won't mind having a 15% gain when they could have had a 30% gain. That is faulty reasoning, because the tail is the most valuable part of the distribution! Most of the benefit from owning stock comes from the possibility of getting a 50% gain in two years. With inflation over 15%, realizing a 50% gain isn't that hard! People would rather have the insurance of not losing, even though they're giving away a ton of their expected value by writing a call and buying a put.

Professional options traders have the opposite position of retail customers. They are long calls and short puts. This is a great deal for them, because the calls are underpriced and the puts are overpriced. However, after buying a call, which is a great deal, the professional option trader short sells stock, which is a lousy deal. Buying a call, you are effectively borrowing at the Fed Funds Rate and buying stock, which is a bargain. When an options trader short sells stock, he is lending the proceeds at the Fed Funds Rate minus the short sale fee. The options trader has an expected gain of 5%-10% on his call position, but he then has an equal expected loss of 5%-10% on his short stock position.

The gain from buying a call is approximately equal to the loss from hedging with short stock. Put options are analogous. The extent to which the market misprices calls is approximately equal to the losses traders realize on their short stock position.

Professional options traders are making two huge mistakes in their calculations! Their errors approximately offset, so their actual profits match what their model predicts. By using Axiom #1 instead of Axiom #7, options traders are miscalculating their hedge. They are hedging to delta-neutral on a mark-to-market basis, but not in accordance with the true risk of their options. All the errors they're making wind up in their favor, and options traders wind up with actual profits in excess of what their model predicts.

Summarizing, the fallacies of axiom #1 are:

- In a truly free market, when someone borrows at the risk-free interest rate, the risk-free interest rate should rise. This does not happen, because the Federal Reserve has an unlimited budget. The Federal Reserve can fix interest rates at an artificially low level without cost.

- When a bank or hedge fund borrows at the risk-free interest rate, it isn't taking money from some other source. The bank is literally borrowing new money into existence. That prevents the arbitrage argument that justifies Axiom #1 from occurring.

- Professional options traders don't notice this huge mistake in their model because they're hedged. The errors in their model are actually errors in their favor, because professional options traders tend to be long calls, which are underpriced, and short puts, which are overpriced. Everyone is following the same pricing model, so on a daily mark-to-market basis, everyone experiences approximately the returns they expect. Over a period of months, the cumulative error means that options traders realize profits in excess of what their model predicts.

- If you're willing to take an unhedged position, buying call options is a great deal. Large banks and hedge funds don't bother to do this, because they have other investment options that are more attractive. Therefore, I can afford to explain my options trading system without being concerned that someone would steal it. Even if you followed my options trading system perfectly, you would not adversely impact my ability to execute it myself.

If the expected gain priced into the option deviated too much from the risk-free interest rate, some professional trader would perform the indicated arbitrage and make a guaranteed profit. If the volatility priced into the option deviated too much from the expected future volatility, some professional trader would make an option trade and delta-hedge it, making a practically guaranteed profit. It is *IMPOSSIBLE*, even with a budget of billions of dollars, to move options prices substantially away from this equilibrium. (If you bought a lot of call options for a stock, that would temporarily move up the price. Eventually, other options traders would gladly write options and delta-hedge them.)

- From the point of view of a large bank, Axiom #1 is good enough. If they wanted to accept risk of an asset price changing, they have more attractive investments. Level 3 assets are particularly lucrative for banks, because they don't have to mark them down during the bust phase of the business cycle.

- The price of an option is different, depending on whether you want to adopt a hedged position or an unhedged position.

- The fundamental unit of value, the dollar, has an intrinsic value of zero. There's a hidden division by zero error whenever an economist does a calculation.

From my point of view, EVERY listed equity option is mispriced, because they use Axiom #1 instead of Axiom #7. I don't have the special perk of borrowing at the Fed Funds Rate. From the point of view of a large bank or hedge fund, this information is irrelevant. Large banks or hedge funds have other, more lucrative, investment options. Large banks only want to trade options when they are making a practically guaranteed profit, so they always hedge their positions.

However, the average person is not able to borrow at the Fed Funds rate. When the average person buys a call option, they are able to indirectly borrow at the Fed Funds rate. In other words, buying call options is a good deal for the average person. It's a way to bet that there will be inflation!

No comments:

This Blog Has Moved!

My blog has moved. Check out my new blog at