# 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.

• Could you give an example with specific numbers? If the current price is 100 and you've somehow reverse computed a current price of 80, I'd like to see how. Also, explain a little bit about how you would hedge? Remember, r is set by the market.
– user59
May 5, 2016 at 14:48

Well, hopefully your calculations are right. There are a few things to remember:

1. 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.

2. 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.

3. Transaction fees, slippage, etc.

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.