dstefan
07-08-2008, 02:17 PM
Chapter 5, Problem 8
Consider an ATM put option with strike 40 on a non-dividend paying asset with volatility 30%, and assume zero interest rates. Compute the relative approximation error of the approximation P \approx \sigma S \sqrt{\frac{T}{2 \pi}}
if the put option expires in 1, 3, 5, 10, and 20 years.
Solution:
We expect the precision of the approximation formula for ATM options to decrease as the maturity of the option increases. This is, indeed, the case:
\begin{tabular}{|c|c|c|c|}\hline T & P_{approx} & P_{BS} & Approximation Error \\ \hline 1 & 4.787307 & 4.769417 & 0.38% \\ \hline 3 & 8.291860 & 8.199509 & 1.13% \\ \hline 5 & 10.704745 & 10.507368 & 1.88% \\ \hline 10 & 15.138795 & 14.589748 & 3.76% \\ \hline 20 & 21.409489 & 19.906608 & 7.55% \\ \hline \end{tabular}
Here, the Approximation Error is the relative approximation error defined as
\frac{|P_{BS}-P_{approx}|}{P_{BS,r=0,q=0}}
Consider an ATM put option with strike 40 on a non-dividend paying asset with volatility 30%, and assume zero interest rates. Compute the relative approximation error of the approximation P \approx \sigma S \sqrt{\frac{T}{2 \pi}}
if the put option expires in 1, 3, 5, 10, and 20 years.
Solution:
We expect the precision of the approximation formula for ATM options to decrease as the maturity of the option increases. This is, indeed, the case:
\begin{tabular}{|c|c|c|c|}\hline T & P_{approx} & P_{BS} & Approximation Error \\ \hline 1 & 4.787307 & 4.769417 & 0.38% \\ \hline 3 & 8.291860 & 8.199509 & 1.13% \\ \hline 5 & 10.704745 & 10.507368 & 1.88% \\ \hline 10 & 15.138795 & 14.589748 & 3.76% \\ \hline 20 & 21.409489 & 19.906608 & 7.55% \\ \hline \end{tabular}
Here, the Approximation Error is the relative approximation error defined as
\frac{|P_{BS}-P_{approx}|}{P_{BS,r=0,q=0}}