# Pricing a power barrier option

I wish to price an option with payoff $$S_T^2{1_{\left\{ {\mathop {\max }\limits_{0 \le t \le T} {S_t} \ge B} \right\}}}$$ in the usual Black Scholes setup with zero interest rate. Now the pricing isn't particularly difficult with some reflection principle results in hand. The part I am stuck at it, is that first we have derived (I didn't get a completely closed form solution; left it at an integral) a function $$h_b^\delta (t) = \Pr \left( {\mathop {\max }\limits_{0 \le s \le t} W_s^\delta \ge b} \right)$$ where $$W_s^\delta = {W_s} - \delta t$$ and $$W_s$$ a SBM under the physical measure. The contention then is that the price at time 0 of the option can be shown to be $$ch_b^\delta \left( T \right)$$ for some choice of the parameters. What I have found is as follows:

$$h_b^\delta (t) = \Pr \left( {\mathop {\max }\limits_{0 \le s \le t} W_s^\delta \ge b} \right) = \int\limits_b^\infty {\int\limits_{ - \infty }^\infty {\exp \left( { - \frac{{{\delta ^2}}}{2}t - \delta w} \right)\frac{{2\left( {2m - w} \right)}}{{t\sqrt {2\pi t} }}{e^{ - \frac{{{{\left( {2m - w} \right)}^2}}}{{2t}}}}dwdm} }$$ $${P_0} = S_0^2\exp \left( { - \sigma T} \right)\int\limits_{{b^'}}^\infty {\int\limits_{ - \infty }^\infty {\exp \left( {2\sigma w} \right)\frac{{2\left( {2m - w} \right)}}{{t\sqrt {2\pi t} }}{e^{ - \frac{{{{\left( {2m - w} \right)}^2}}}{{2t}}}}dwdm} } \\$$ Where do you think I have went wrong?

• Can you edit equation to make it readable? What is the dynamics of $S$? – Gordon May 4 at 15:48