Option pricing models relation between theoretical and actual price

Question

I have trying to figure out the relationship between theoretical option price and actual market price spotted from market which is determined by supply and demand.

I yet cannot understand how to interpret that "theoretical price" which is given by for example binomial option pricing model. In what kind of world that is the the "true option price"? Does this mean, that if all assumptions HOLDS in real world, the true price should be the theoretical option price given by model and reflect the price spotted on market?

And if there is mispricing, ie. the market price deviates from price given by model, there is chance for arbitrage because if assumptions hold, the price will drive up/down to the theoretical price and one could make arbitrage profit in theory.

As illustrate my confusion, let take the example, where binomial option pricing model is used to price option with these details:

Let S=$100, K=$95,r=8%,T =0.5,and δ=0.Let u=1.3, d=0.8, and n=1. The price given by model for European call is then $16.196.

Now, the b) ask you tell, what kind of arbitrage possibility there is, if spotted price on market is 17 dollars, ie. the option is overpriced.

Okay, this is clear as you can sell the option and create synthetic call worth 16,196 dollars. We buy 0,7 share and borrow 53,804 dollars. We make then risk free profit by 0,804 dollars.

And this is the thing I don't understand. In what kind of world is this possible? I mean, if I price the call with model, where do I know if the option price is the "true" value that I can make risk free arbitrage? What is this worth, because if there is not certainty that call "true" price is 16,196 dollars how can I suppose to go in arbitrage?

So am I right if i say that if market would be really EFFICIENT and all assumptions regarding the model holds, this kind of arbitrage is possible in THEORY. But because this is just mathematical formula we can never be sure if it is the theoretical value and so we cannot gain risk free arbitrage by these transactions?

So the last point is, do I understand this right, or how should I interpret theoretical option value and martket value? What is the benefit after all to use these kind of pricing models?

I am bit confused with the thing that books are all the time telling about risk free arbitrage and theoretical values, but in what kind of world it is possible to make this risk free profit as we cannot know the true value for sure and the fact that if the price will be the theoretical value in future.

Thank you for answering, I appreciate it.

Answer 1

I think you have the correct understanding. The arbitrage is only possible if the risk-neutral probability distribution of the stock is perfectly known, as it is in your simple binomial model. In the real world you can never know the precise distribution, so you cannot create a true arbitrage between an option and its underlying stock in this manner.

That is not to say arbitrages are impossible in the real world. For example, if market prices are such that put/call parity is violated, there is an arbitrage that can be exploited regardless of the assumed probability distribution.