\documentclass[11pt]{article}

\author{Jeremy Morris}
\title{Homework 5}

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\begin{enumerate}
\item \textit{Building an $N$-period binomial model}

The condition needed to impose on $\Delta t$ to make it a probability measure would be that $0 \leq \tilde{p} \leq 1$.  We solve for $\Delta t$ in order to determine the condition.
\begin{align*}
0 \leq \frac{1}{2} \left( 1 + \left(\frac{r}{\sigma} - \frac{\sigma}{2}\right)\sqrt{\Delta t} \right) \leq 1 \\
0 \leq 1 + \left(\frac{r}{\sigma} - \frac{\sigma}{2}\right)\sqrt{\Delta t} \leq 2 \\
-1 \leq \left(\frac{2r - \sigma^{2}}{2\sigma}\right)\sqrt{\Delta t} \leq 1 \\
-\frac{2\sigma}{2r - \sigma^{2}} \leq \sqrt{\Delta t} \leq \frac{2\sigma}{2r - \sigma^{2}}
\end{align*}
Of course, this does not work when $2r = \sigma^{2}$.

The code for the program is attached.

\item \textit{Black-Scholes and simulations}

The simulations mentioned have been coded and attached here.  The results are below.  (The simulations were coded to provide output in \LaTeX.)

\begin{longtable}{|l|l|l|l|l|l|} \hline
$K$ & $\sigma$ & $N$ & $V_{N}$ & Black-Scholes & Error \\ \hline
20  &  0.1  &  10  &  3.2485  &  3.2488  &  -0.00032185\\
20  &  0.1  &  100  &  3.2487  &  3.2488  &  -4.4583e-05\\
20  &  0.1  &  500  &  3.2488  &  3.2488  &  -6.4861e-06\\\hline
20  &  0.15  &  10  &  3.2579  &  3.2610  &  -0.0030249\\
20  &  0.15  &  100  &  3.2609  &  3.2610  &  -8.1121e-05\\
20  &  0.15  &  500  &  3.2609  &  3.2610  &  -3.6200e-05\\\hline
20  &  0.2  &  10  &  3.3044  &  3.3077  &  -0.0032947\\
20  &  0.2  &  100  &  3.3067  &  3.3077  &  -0.00097663\\
20  &  0.2  &  500  &  3.3077  &  3.3077  &  4.3604e-05\\\hline
21  &  0.1  &  10  &  2.2675  &  2.2685  &  -0.0010533\\
21  &  0.1  &  100  &  2.2684  &  2.2685  &  -0.00017330\\
21  &  0.1  &  500  &  2.2685  &  2.2685  &  -5.4881e-06\\\hline
21  &  0.15  &  10  &  2.3195  &  2.3236  &  -0.0041325\\
21  &  0.15  &  100  &  2.3230  &  2.3236  &  -0.00060395\\
21  &  0.15  &  500  &  2.3237  &  2.3236  &  7.2629e-05\\\hline
21  &  0.2  &  10  &  2.44  &  2.431  &  0.0090116\\
21  &  0.2  &  100  &  2.4320  &  2.431  &  0.00098223\\
21  &  0.2  &  500  &  2.4309  &  2.431  &  -2.8946e-05\\\hline
22  &  0.1  &  10  &  1.3480  &  1.3442  &  0.0037936\\
22  &  0.1  &  100  &  1.3447  &  1.3442  &  0.00044204\\
22  &  0.1  &  500  &  1.3441  &  1.3442  &  -8.5592e-05\\\hline
22  &  0.15  &  10  &  1.4827  &  1.4896  &  -0.006851\\
22  &  0.15  &  100  &  1.4886  &  1.4896  &  -0.00099292\\
22  &  0.15  &  500  &  1.4898  &  1.4896  &  0.00019904\\\hline
22  &  0.2  &  10  &  1.6825  &  1.6691  &  0.013388\\
22  &  0.2  &  100  &  1.6697  &  1.6691  &  0.00057887\\
22  &  0.2  &  500  &  1.6688  &  1.6691  &  -0.00030761\\\hline
23  &  0.1  &  10  &  0.60124  &  0.61291  &  -0.011670\\
23  &  0.1  &  100  &  0.61173  &  0.61291  &  -0.0011819\\
23  &  0.1  &  500  &  0.61267  &  0.61291  &  -0.00023662\\\hline
23  &  0.15  &  10  &  0.8189  &  0.83607  &  -0.017167\\
23  &  0.15  &  100  &  0.83433  &  0.83607  &  -0.0017387\\
23  &  0.15  &  500  &  0.83572  &  0.83607  &  -0.0003481\\\hline
23  &  0.2  &  10  &  1.0387  &  1.0614  &  -0.022713\\
23  &  0.2  &  100  &  1.0591  &  1.0614  &  -0.0023006\\
23  &  0.2  &  500  &  1.0610  &  1.0614  &  -0.00046058\\\hline
24  &  0.1  &  10  &  0.19949  &  0.19650  &  0.0029896\\
24  &  0.1  &  100  &  0.19654  &  0.19650  &  4.2369e-05\\
24  &  0.1  &  500  &  0.19660  &  0.19650  &  0.00010660\\\hline
24  &  0.15  &  10  &  0.39609  &  0.40291  &  -0.0068213\\
24  &  0.15  &  100  &  0.40276  &  0.40291  &  -0.00014946\\
24  &  0.15  &  500  &  0.4031  &  0.40291  &  0.00019262\\\hline
24  &  0.2  &  10  &  0.63732  &  0.62196  &  0.015359\\
24  &  0.2  &  100  &  0.62144  &  0.62196  &  -0.0005228\\
24  &  0.2  &  500  &  0.62212  &  0.62196  &  0.00015936\\\hline
25  &  0.1  &  10  &  0.040776  &  0.042011  &  -0.0012350\\
25  &  0.1  &  100  &  0.041777  &  0.042011  &  -0.00023406\\
25  &  0.1  &  500  &  0.041975  &  0.042011  &  -3.6215e-05\\\hline
25  &  0.15  &  10  &  0.16567  &  0.16507  &  0.00060862\\
25  &  0.15  &  100  &  0.16555  &  0.16507  &  0.0004814\\
25  &  0.15  &  500  &  0.16515  &  0.16507  &  8.8202e-05\\\hline
25  &  0.2  &  10  &  0.34145  &  0.33519  &  0.0062645\\
25  &  0.2  &  100  &  0.33476  &  0.33519  &  -0.00042605\\
25  &  0.2  &  500  &  0.33534  &  0.33519  &  0.00015331\\\hline
\end{longtable}

\end{enumerate}

\end{document}
