# In search of double barrier out option on a BM

We have a BM $$X_t$$ with $$dX_t=\sigma dB_t$$ ($$X_0$$ not necessarily zero!) under the risk neutral measure $$\Bbb Q$$. Given upper barrier $$U$$, lower barrier $$L$$, "strike" $$K$$ such that $$L, rebate $$b$$, maturity $$T$$, and define $$m:=\min_{0\le t\le T}X_t$$ and $$M:=\max_{0\le t\le T}X_t$$. Suppose the terminal payoff function is $$|X_T - K|I(L\le m \text{ and } M\le U) + bI(\text{otherwise})$$

Suppose in addition a constant discount rate $$r>0$$. Is an analytical formula for this double barrier out option's price, i.e. $$e^{-rT}\Bbb E^{\Bbb Q}\left[|X_T - K|I(L\le m \text{ and } M\le U) + bI(\text{otherwise})\right]$$ possible? Thanks in advance.

EDIT Looks like Brownian Bridge is a good start. At least I can see it lead to explicit integral forms.

This is a well tackled problem in the GBM case. See

Geman/Yor (1996), Pricing and Hedging Double-Barrier Options: A Probabilistic Approach. Mathematical Finance, 6(4), p. 365-378

among other references. Though in practice finite differences or MC would be used to deal with discrete dividends, local and/or stochastic volatility, etc.

• Thanks. However in my case the underlying follows a standard Brownian motion instead of a GBM. So they might be quite different.
– Vim
Jul 4, 2019 at 9:41
• Sorry I didn't see it was for a BM rather than for a GBM. However the barrier in the $\mu$-drifted GBM case translates into a log barrier in the $\mu - \sigma^2/2$ drifted BM case so a similar approach should work, perhaps following that outlined in Pelsser Pricing double barrier options using Laplace transforms core.ac.uk/download/pdf/19187200.pdf. Jul 4, 2019 at 10:14
• Thanks. Yeah the approach provided in your paper also works nicely for Bachelier model and leads to even simpler forms. The results are exponentially fast decaying Fourier series, and I think they should already be good enough for my use.
– Vim
Jul 5, 2019 at 2:21