Implied Volatilities & Volatility Smiles

1. Why does the target cell in the Solver minimization reference the control variate estimate of the American Put option instead of the value as implied by the tree?

It is because that control variate estimate is more accurate than the implied value by the tree. The error of the binominal tree can be reduced by using it only to calculate the difference between the price of the American and the equivalent European options with the same strike and the same time to maturity.

2. Use Solver to find the implied volatilities for all put options with strike prices between $70 and $100 that are divisible by 5 and that are available for a given option chain. Save your implied volatility results in a separate worksheet along with maturity and strike price or add them to the data file. Once you have done this for all 3 option chains, you will need to create a graph in Excel that depicts the three volatility smiles as a function of strike price.

The shortest maturity option shows the steepest smile

The shortest maturity option shows the steepest smile

3. Describe what you see in that graph. What would you expect to see in a Black-Scholes world? Which option chain exhibits the steepest volatility smile? Why is this the case?

We see a kinked smile like a smirk.

This type of smile are observed in real financial markets.

On the other hand, a normal horizontal straight line is can be seen under the assumption of Black-Sholes model.

The option with the shortest maturity exhibits the steepest smile. The reason is that in the short run, the volatility of options with short maturity tend to response to short term shocks more drastically compared to the ones with longer maturity. On a short term scale, the variance of volatility of options overwhelms the mean reversion property of volatility.

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4. Fix the