Before I start: Ok, this is something I investigated for a fair amount of time and my question is semi-academic. To simplify, I will introduce the short bit (TLDR) of my question and then lay out specifics.
Short Question: What happens when option pricing model estimate and market price disagree? What is the course of action? What happens to the difference when I take positions on option and delta to create a self replicating portfolio?
It would be great if I get recommended some research (as in research papers, articles etc.) also with the answers.
Important! This is not a speculation or arbitrage question, it is a mere confusion about the fair pricing and use of models in the market.
Now the long story...
- Single asset (say XYZ)
- All options on the are market are European (as if it is an index)
- Assume no dividend paid on XYZ in any form (no need to overcomplicate)
- Both option market and underlying market are liquid enough
- No frictions, no bid-asks, no arbitrage, infinite divisibility, time is continuous, borrow and lend at the same amount from risk free rate $r$ (BS assumptions) and even fix $r$ to zero (practically what it is today for short term rates).
Assume there is a contract (a European Call) A on XYZ priced at the market at \$5 per share.
Let's say my model M1 is a Black-Scholes model using historical volatility (say vol1) and yields fair price as \$4.9. So, there is a clear disagreement with market and model.
I want to use a replicated portfolio. I would take long or short position on the contract and do perfect delta hedging (since my assumptions allow me to do). What happens to the \$0.1 difference? Would it make a difference if I long or short of the option or would delta hedging take care of that difference? (I suspect not)
Here are some of my thoughts on the the problem.
Using Implied Volatilities
One might rightfully say "Use implied volatility, so your model price and market price will be one." OK, I respect that.
I adjusted volatility metric of my model's volatility estimate to the implied volatility (say vol2). Now model M1 prices contract A with vol2 on \$5.
Enter contract B, essentially same as contract A except the strike price (same maturity and EC). Therefore moneyness is different for contract B. Let's say market price of contract B is \$3 and even though it is acceptable with no arbitrage rule it is on a volatility smile.
It means if I use vol1 or vol2, there is a good chance that I will not get the same price estimate as the market's. I can still infer the implied volatility of contract B (say vol3).
But this time I will have two different volatility measures vol2 and vol3 for the same asset and time period (remember maturities are equal for contracts A and B).
It is OK if I'm going to price a synthetic option (an over the counter option, that is not traded on the market). I will find the "interpolated" implied volatility and come up with a fair price estimate where market would probably also agree if the contract were traded on the market. Let's assume I'm not interested in OTC options and I only operate on market options.
So having two different volatility measures for the same period for the same asset is "weird". Sure, technically you can do it. But it says something about your model's functionality and even I know what Fischer Black said about constant volatility assumption.
- We call price estimate of the model (namely Black Scholes) the fair price.
- We also call market price of options as the fair price, since it is determined by the market.
Is the following statement true? "Fair price of an option contract is at the expiration no party would get advantage over the other."
Sure, it can be stated more elegantly. Let's illustrate it with an example. If I put \$5 on an option contract and I get back \$5 at the expiration (remember risk-free rate is assumed zero so no time discount), fair value is \$5.
If it is true it takes the title "fair price" away from both models and market. Fair price would be an idealised state which both market and models try to converge.
Market vs Model
But, we trade in the market. And market is assumed to be the best approximation to fair prices even in the idealised state. I made some remarks about it on this question Is there any other way to measure option pricing model performance than proximity to market prices?
I ask the question in a similar way. Suppose my model price is different from the market price. What should I do?
ps. I can understand using market prices as the best prices in the underlying assets. What I cannot understand is using market option prices as the best prices since it is a derivative product on the future state of the underlying market.