Pricing a Power Barrier Option Under Black–Scholes

Summary

The document sets up a barrier contingent claim under the Black–Scholes model with zero interest rates. Its payoff is the squared terminal asset price if the asset has reached an upper barrier at any time before expiry. The author attempts to express the price using the probability that a drifted Brownian motion crosses a level, and writes an integral involving the running maximum and terminal value.

The question is where the derivation may have gone wrong, but no answer or correction is included. Consequently, it provides a useful problem setup and illustrates a possible route using a reflection principle and a change of measure, without validating the displayed integral or completing a closed form price. Care is needed with the model’s drift, volatility scaling, discounting, and the relationship between the barrier event and the squared terminal payoff; those details cannot be resolved from the document alone.

Key ideas

• The claim pays the square of the terminal asset price if the asset has crossed an upper barrier by expiry.
• The setup assumes Black–Scholes dynamics with zero interest rates.
• The author attempts to reduce pricing to a barrier crossing probability for drifted Brownian motion.
• The displayed integral is presented as an unresolved derivation, with no correction or final pricing formula supplied.
• Drift, volatility scaling, and the treatment of the barrier event must be checked when deriving the expectation.

Tags

# Pricing a power barrier option


# 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?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.