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Saturday, March 1, 2008

The Black-Scholes Formula is Wrong! - Part 6/12 - 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

Recall Axiom #1 of the Black-Scholes Formula:

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

Axiom #1, along with the put/call parity formula, means that the expected return in stock equals the risk-free interest rate. The put/call parity formula guarantees that observed options market prices will follow Axiom #1. There would be an arbitrage opportunity if prices deviated one way or the other. If the expected return priced into the options was higher than the risk-free interest rate, an options trader could borrow money at the Fed Funds Rate (plus a small fee), buy stock, short a call, and buy a put, making a guaranteed profit. If the expected return priced into the options was less than the risk-free interest rate (plus the stock short sale fees), an options trader would short sell stock, deposit that money at the Fed Funds Rate (minus fee), buy a call, and sell a put.

There is another arbitrage argument that justifies Axiom #1. If the expected return of a stock and the risk-free interest rate were different, people would borrow at the risk-free interest rate and buy stocks, or sell their stocks and invest at the risk-free interest rate.

Why does anyone invest in stocks at all then? Why not just lend your money out at the risk-free interest rate and avoid the hassle of dealing with stocks?

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

Justification: In every historic period of reasonable size, this has been true.

Whoops! Axiom #7 contradicts Axiom #1. Does this disturb you? If you do not find this extremely troubling, then I'm not wasting my time writing for you anymore. Go do something where you don't have to think, like watching CNBC.

If you ever wondered why economics is nonsense, now you know the reason. An economist is someone who can believe Axiom #1 and Axiom #7 at the same time, without feeling dirty.

This blows up the whole analysis, doesn't it? It does and does not at the same time. The contradictory Axiom #1 and Axiom #7 should blow up all your confidence in the current economic system. It took me a very long time to figure out the reconciliation of these contradictory axioms.

At this point, all your confidence in the current economic system should have collapsed in contradiction.

2 comments:

sean said...

The error with this thinking is assuming that the risk profiles are identical for the risk-free interest rate and the stock market. Axiom #1 must be assumed, as the author points out, for purposes of arbitrage. Axiom #7 is the correct assumption, empirically speaking over a long enough period of time, but it is not a law. The actual stock market return over any given period of time may be far less than the interest rate, and indeed may be negative. Higher risk accompanies higher rates of return.

From other assumptions I've seen in this series, the Author seems to have a serious blind spot when it comes to risk profile.

Anonymous said...

there is no contradiction. axiom one is not that the expected return equals the risk free rate but that the expected return UNDER RISK NEUTRAL probabilities equals the risk free rate. u should have thought about that for a second before blasting out that nonsense

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