Good day,
A reverse knock-out barrier call option expires worthless if the asset price ever goes above a given barrier level. Calculate the value of this barrier option struck at $K = 3$ with barrier level $B = 9$.
Also, explain why the barrier call option worth less than the vanilla call?
$r=0$
\begin{array}{|c|c|c|c|} \hline & S(t=0,\omega) & S(t=1,\omega)^* & S(t=2,\omega)^* &S(t=3,\omega)^* \\ \hline \omega_1 & 5& 8& 11 &15\\ \hline \omega_2 & 5& 8& 11 &10\\ \hline \omega_3 & 5& 8& 7 &10\\ \hline \omega_4 & 5& 8& 7 &5\\ \hline \omega_5 & 5& 4& 7 &10\\ \hline \omega_6 & 5& 4& 7 &5\\ \hline \omega_7 & 5& 4& 2 &5\\ \hline \omega_8 & 5& 4& 2 &1\\ \hline \end{array}
I found risk neutral probabilities for each path and at each node and I think I am correct (hard to go wrong as $r=0$ and no dividends are paid.I calculated the value of a vanilla call option using dynamic programming but Im not quite sure how to approach the Barrier option valuation. Do I simply put the paths with over $9$ to be equal to $0$ and apply dynamic programming again?
The additional question; its worth less as there is a range of values for which the option is worth anything whereas a vanilla option has only a minimum.