# Essay about Exam

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

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9. Now focus only on the American option and estimate by how much the BBS method and the BBSR method can reduce the error relative to the simple tree. As fair value, you can use the CV-BBS price with n=100. Compute the absolute dollar difference to fair value (the error) for the simple tree price, the BBS price and the BBSR price. Now take the ratio of the absolute BBS error over the absolute simple tree error (as percentage), repeat for BBSR over simple tree error. This should give you an idea of the general reduction in error for both methods. Submit the table and describe your results.

Barrier Options - Analytical Solutions vs. MC Simulation

1. You will notice that depending on the order of S, H and X there can be lots of rows where all options pay off zero. Why can we not use the same extension of importance sampling that your lecturer used in the example spreadsheet that simulated European option prices alone?

2. Now use a concrete case: S= $100, X= $105, H= $95,