Option Model Prices, Implied Volatility, and Delta-Hedged Trading

# Model Price vs Market Price in terms of Fair Price (Options)


# Model Price vs Market Price in terms of Fair Price (Options)












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...

## Problem Specifications

- 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.

### Fair Price

- 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.

## Answer by berkorbay (score 2, accepted)

https://quant.stackexchange.com/a/22646

I think after spending some time on the topic with research papers, I can come up with a satisfying answer. I will list them item by item so (I hope) it would be more clear. Starting from the most obvious.

- Theoretically, adjusting your model to the market price has no benefit except to calibrate your greeks (prominently Delta and Gamma). It literally says "I believe the market has the fairest price of all." Otherwise any divergence from the market would not be labeled as "error".

- If you use implied volatility per contract, you literally say "I believe market movement is best represented by Geometric Brownian Motion and Black Scholes model". Because what you do is basically inverting BS formula (unless you are using another model to calculate volatilities, then you are bound by that model)

- If you are somehow logically using the volatility smile in order not to fall for different volatility estimates for the same maturity but different strike values; congratulations, now you have your own opinion about the contract price. No matter how small, your model disagrees with the market (at least part of it). That's OK, since volatility smile itself is an argument against the BS.

- Merton (1973) claims if you can continuously delta hedge your position, you will be protected from small changes in the underlying. So if your model is true and the market is wrong. You will be scalping the price difference (as @mbison said) and be protected from all other influences.

- Efficient Markets Hypothesis dictates that you cannot make risk adjusted excess profits (or in other terms, beat the market). Samuelson (1965) and Fama (1970) started this discussion and it became a hot topic. (side note: Grossman and Stiglitz (1980) roughy asked if 'we cannot make an extra buck, why bother trading?'.)

- EMH has three different types based on the information level (past price changes, public info such as earnings/splits/etc and private information such as insider trading/access to limit order books) and the definitions changed later.

- Of course it didn't stop there. Many researchers either tested or attacked EMH. There was much bread to be made from academic articles on this discussion. Many tests and surveys were written (some of us had to read most of that stuff, it becomes unfortunate sometimes). The general attack method was: "I found a model/trading strategy, made lots of paper profits with it. Your hypothesis is wrong!". Many also said: "I tried models, tested and saw that even if there are a few bucks to be made with the models, the gains are all wiped out by transaction costs. So don't bother, market is efficient." See Jensen (1978) Fama (1991), Fama (1998), Lo (2007) and Jarrow and Larsson (2012) for detailed surveys and overviews. There are also some points about rationality and behavioral finance if you like.

- I managed to find some studies theoretically extending EMH to options markets, albeit few. Jarrow (2012) and Jarrow (2013) combine option theory with market efficiency. Apart from loads of mathematical symbols, what I understand is market efficiency is strongly tied to no arbitrage condition (unsurprisingly). Also, there might be times of "asset bubbles" temporarily disrupting the efficiency. Surely there were efficiency tests on options markets. See Black and Scholes (1972) and Galai (1978) for instance.

- Long story short, at some point Joint Hypothesis Problem term is coined. The situation is also called Bad Model Problem. It says that "In order to disprove EMH, you need to bring an equilibrium model which allegedly covers market risks. But there is no guarantee that your model covers all the risks in the market. Any excess profits you made during your trades might be attributed to the uncovered risks." In other words, it says "You might not be hedging properly. Reality will be different from your model."



## Answer by mbison (score 3)

https://quant.stackexchange.com/a/22616

If the market prices the option at USD5 and your model says the price is USD4.9. Assuming all parameters between model and market, then the difference comes from the vol you have and the implied.

If you sell the option at USD5 and you delta hedge "continuously", then you will scalp the difference between realized and implied vol. You will realize a pnl that depends on the difference between daily realized and the IV and the gamma of your option.

There are many threads on this forum that discuss gamma scalping.

## Answer by Amiro (score 0)

https://quant.stackexchange.com/a/21488

I guess your question is more about model risk and model valuation. well you already mention all the ways that you can adjust your price. from my humble opinion, you should use implied volatility to get the correct value or do some calibrations on your parameters to get exact same price as market.

Think of swaption for example. you can use either use black 76 model (log normal vol) or bachelier model (normal vol) and both works fine so It does not matter which model you use, you should always get the same exact price. in both models everything is identical such as K, S_0, T, r except volatilities. then you can say you have one to one relationship between price of option and volatility. right?

what I mean and I hope that I did not confuse more is that, adjust your model (volatility) to market price.

## Answer by user12348 (score 0)

https://quant.stackexchange.com/a/21624

you are confusing too much with the future state of the market. It is easy to confuse. NO one can price totally uncertain future. Future or forward prices are arbitrage free projection of the current prices. No magic. It means you can buy a forward product and hedge it using products that defined the forward curve.

Fair price is model price, meaning it is not based on bid and ask or market price. The fair is not fair as in fair dealing.

## Answer by Misha Fomytskyi (score 0)

https://quant.stackexchange.com/a/85801

There are two conceptually different uses of models here.

- You are a dealer (a market maker or a bank). You use a model to estimate the cost of hedging the risks with listed products. You calibrate a vol surface to listed options, then price any option off this vol surface. For exotic options, you calibrate a model, such as a local volatility model, to the implied vol surface, then price with that model. The price in this case is an estimate of the cost of hedging the risks using listed products (replication cost). You sell above this price, buy below it, and hedge the residual risks.

- You are a risk taker (e.g., a hedge fund). You calibrate a model to the listed market to determine the market-implied value of a particular risk (e.g., variance, correlation, dividends, etc.). That's your Q-measure estimate. Then you estimate/forecast the real-world, P-measure value of the same risk (e.g., future realized volatility). Finally, you determine the optimal portfolio to structure your bet and estimate whether the expected P&L justifies the risk.

For more explanation of implied volatility surfaces, see this video: Implied volatility surface video.

For more information on the trading workflow, see this video: Trading workflows video.