Why Option Mispricing Does Not Guarantee a Risk-Free Return
Summary
The document examines a proposed options trade: if a trader believes volatility is known and constant, a Black–Scholes price below the market price might appear to imply an unusually high return from delta hedging. It asks why traders use implied volatility rather than treating the difference as a dependable hedging premium. The replies emphasize that the conclusion depends on the model and on the ability to hedge as assumed; practical carry, slippage, and trading costs also affect outcomes.
A second reply points to nonconstant volatility and return distributions with fat tails as limits of the model. Implied volatility is the value that makes the model match the observed option price, while a volatility forecast is an input whose accuracy is uncertain. If the forecast and assumptions truly held, a mispricing could invite trading and competition that erodes it. The short exchange offers conceptual cautions, not a derivation or empirical evidence that implied volatility always exceeds realized volatility or that arbitrage opportunities vanish immediately.
Key ideas
- Solving for a model input that matches an option price produces an implied parameter, not a guaranteed return.
- Delta hedging outcomes depend on volatility forecasts and the assumptions behind the pricing model.
- Changing volatility and fat-tailed returns can make constant-volatility model results unreliable.
- Carry, slippage, and other trading frictions reduce the payoff from hedging strategies.
- Persistent apparent mispricing may attract traders, but the discussion does not prove that arbitrage is risk-free or immediate.
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Full text
# Solving for r in the Black Scholes equation # Solving for r in the Black Scholes equation Could you please correct which parts of my reasoning are wrong? Let's suppose that I know for sure that my estimate for a stock volatility is right (I have a crystal ball) and that it will be for sure constant until the maturity of the option. I plugged my volatility, the risk free interest rate and the other variables into the Black Scholes equation and got a price of 80. The market price is 100. The same way people derive an implicit volatility by solving for sigma to match the BS with the market price, I solve for r, the risk free interest rate, and I get a value of .6. My conclusion is that I can take advantage of this mispricing by properly hedging in almost continuous time and having an annualized return on these transactions of .6, ignoring fees and liquidity issues. Is this correct? If so, why people give so much value to implied volatility instead of trusting in their volatility estimates and realizing that mispricing should in fact exist in the market as a sort of premium to reward the work of hedging continuously, the fees and liquidity issues and the risk of not seeing the model assumptions working in practice? Ignore issues like transaction problems and fees (I'm a robot, I don't make mistakes and the broker is my friend, there are no fees). The real point is the last paragraph. Why option pricing seems to be so much focused in assuming that arbitrage is absolutely impossible, even considering fees and assumption problems, always trying to find a pricing method that will make the market 'right'. It seems so clear to me that high risk free returns should exist to compensate the million problems involving continuous hedging. ## Answer by JoshK (score 1) https://quant.stackexchange.com/a/25852 Well, hopefully your calculations are right. There are a few things to remember: - The carry can be higher than what you are thinking. Very often you will get charged if you are long or short. That can cost a lot depending on the name. - Implied is theoretically always higher than realized. You are selling insurance. You should collect a premium more often than not, but when you are losing, you loose a little bigger. Big gaps will be painful. - Transaction fees, slippage, etc. ## Answer by e.mal (score 1) https://quant.stackexchange.com/a/25860 To my point of view, the answer is hidden in your question. You correctly stated some of the BS assumptions and empirically it is proven that they are not true (volatility is not constant and the assumption regarding the distribution of returns is unrealistic due to fat tails). The model is as good as its assumptions are. Given that volatility is the unobserved factor of the model, we can solve for it and find its value so that the model holds. In case your volatility estimate and model assumptions are correct, you must be able to exploit the arbitrage opportunity leading to the correction of the mispricing. What every equilibrium model says is that arbitrage opportunities do not persist.
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