I was working through the pricing of some binary option and after changing measure and doing some clean up, I have arrived at the following quantity:
$$\mathbb{E}_\mathbb{P}\left[ \exp(cW_T) \cdot 1\{\sup_{t \leq T} W_t \geq a\}\right]$$
Where above $W_t$ is $\mathbb{P}$-brownian motion, and $a, c$ and $T$ are fixed constants.
I was attempting it as follows: Let $\tau_a$ denote the first time $W_t$ reaches $a$. Then, we know the density for $f_{\tau_a}(t)$:
$$f_{\tau_a}(t) = \frac{a \exp\left( -\frac{a^2}{2t}\right)}{\sqrt{2\pi t^3}}$$
Then:
$$\mathbb{E}_\mathbb{P}\left[ \exp(cW_T) \cdot 1\{\sup_{t \leq T} W_t \geq a\}\right] = \int_0^T \mathbb{E}_\mathbb{P}\left[\exp(cW_T)|W_t = a\right] f_{\tau_a}(t) dt $$
we also have:
$$\mathbb{E}_\mathbb{P}\left[\exp(cW_T)|W_t = a\right] = \exp(ca + c^2(T-t) / 2)$$
to get the above we can use the strong markov property and the reflection property.
Putting everything together:
$$I:=\int_0^T \frac{\exp\left(ca + \frac{c^2 (T-t)}{2} - \frac{a^2}{2t} \right)}{\sqrt{2\pi t^3}}dt $$
At this point I started reading some material and I saw that my desired quantity was equal to
\begin{align} J:=e^{ca} \int_0^\infty \left( e^{cx} + e^{-cx}\right) e^{-(x+a)^2 /(2T)} \cdot \frac{1}{\sqrt{2\pi T}}dx \end{align}
This integral is much easier to do by hand and we get:
$$ J = e^{ca} \left( \exp\left(\frac{c^2 T - 2ac}{2}\right) \Phi\left(\frac{cT - a}{\sqrt{T}}\right) + \exp\left(\frac{c^2 T + 2ac}{2}\right) \Phi\left(\frac{-cT - a}{\sqrt{T}}\right) \right)$$
where $\Phi(\cdot)$ is the CDF for the standard normal distribution.
I have checked (doing numeric integration with a few different set of values) that the integrals in $J$ and $I$ do indeed match and that my closed formula for $J$ is correct. My question is:
- How can one go from the first equation to the integral set up in $J$? The notes I found say Use the reflection principle and the Strong Markov property to justify the identity
- Is there a way to solve the integral $I$ directly? If so, how?
References:
First hitting time density can be viewed here: Density of first hitting time of Brownian motion with drift
Expected value of an exponential brownian motion here: Mean of exponential Brownian motion
Notes that I was reading, bottom of page 8(Careful -- lots of typos!): https://galton.uchicago.edu/~lalley/Courses/390/Lecture8.pdf Note that their $\theta$ is equal to $-c$ in my notation.
Numerical integration: numerical integrations
