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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 $L<X_0<U, L<K < U$, 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.

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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.

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  • $\begingroup$ Thanks. However in my case the underlying follows a standard Brownian motion instead of a GBM. So they might be quite different. $\endgroup$ – Vim Jul 4 at 9:41
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    $\begingroup$ 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. $\endgroup$ – Antoine Conze Jul 4 at 10:14
  • $\begingroup$ 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. $\endgroup$ – Vim Jul 5 at 2:21

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