The Black-Scholes Formula is Wrong! - Part 4/12 - The Put/Call Parity Formula

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

When I first saw the put/call parity formula, my immediate reaction was "I call shenanigans!"

There's a formula frequently cited by options traders as justification for using "expected gain in stock equals the risk-free interest rate". This is called the put/call parity formula. I ignore the value of early exercising an option, which is typically worth only a few cents, about 1% of the option price.

Suppose I own a 1 year call option with a strike of $50 and I am short a 1 year put option with a strike of $50. What happens at expiration? If the stock is above $50, I will exercise my call option and my short put will expire worthless. If the stock is below $50, I will be assigned on my put and let my call expire worthless. I ignore the singularity when the stock is exactly at the strike at expiration. In that case, I will let my call expire worthless, and I don't know what the person who owns the put will do. If I get assigned on my put, I will immediately sell the stock for at most a small loss. In practice, this is a risky situation because I will be assigned on Friday evening and won't be able to sell until Monday morning. By then, all sorts of factors could have caused the stock to move.

In other words, if I own a 1 year call option and am short a 1 year put option, with the same expiration date and same strike of $50, I have a 1 year futures contract to buy the stock at $50. If I buy a call, short a put, and short 100 shares of stock, I have a riskless position. What is my cashflow? I will pay $50/share at expiration when I exercise my long call or am assigned on my short put. From now until expiration, I will collect the stock price times the risk-free interest rate in interest on my short stock position (minus any short sale fees). I am going to collect the current stock price when I short sell the stock right now. I am going to spend money buying a call and collect money when I short sell a put. (If the stock paid a dividend, I would also owe dividends on the short stock position, but I ignore dividends here.)

Summarizing,

C - P = S - X + basis

where
C is the call price
P is the put price
S is the current stock price
X is the strike price
and basis is the expected interest I will collect on the short stock position, plus the interest I will collect by delaying payment of the strike until expiration

In order to understand the put/call parity formula, just track the cashflow of the above riskless trade. Make sure you include all interest payments.

The above formula is called the put/call parity formula. There is only one degree of freedom in call and put prices, the volatility. If you know the call price, you can figure out the put price from the formula. If the observed options market price deviates too much from this formula, some market maker will perform arbitrage and make a guaranteed profit.

In the actual market, there are short transaction fees. This allows some wiggle room in real world options prices. If it's easy to borrow shares for short selling, the put/call parity formula holds pretty closely.

These riskless positions are given a special name "reversal" and "conversion". A conversion is when you are short a call, long a put, and own the stock. A reversal is when you are long a call, short a put, and short sell the stock. (I may have reversed the names, calling a "conversion" a "reversal" and vice versa.)

The put/call parity formula explains why professional options traders price options under the assumption that the expected gain in the stock equals the risk-free interest rate. If they priced options differently, other professional traders would conduct arbitrage against them.

When I first saw the put/call parity formula, my immediate reaction was "I call shenanigans!" Anyone who understands the put/call parity formula and does *NOT* call shenanigans is clueless. Many mainstream economists understand the put/call formula. It is standard for an introductory course on options theory.

Next time, I will write about the "volatility smile".