Analytical Pricing of Double-Barrier Options on Brownian Motion

Summary

The document formulates a pricing question for a double-barrier knock-out claim on Brownian motion. The underlying follows a driftless diffusion with constant volatility; the payoff depends on the terminal distance from a strike if the path stays within upper and lower barriers, and otherwise pays a rebate. The question asks whether the discounted expected payoff has an analytical expression and suggests a Brownian bridge as a route to explicit integrals.

The answer points to established work on pricing and hedging double-barrier options, particularly in the geometric Brownian motion setting. It also notes that practical extensions involving discrete dividends or local and stochastic volatility may be handled with finite-difference methods or Monte Carlo. The response does not derive a formula for the specific arithmetic Brownian model posed, so the cited reference and numerical methods are guidance rather than a complete solution.

Key ideas

• The payoff depends on whether the underlying path remains between two barriers over the option’s life.
• A Brownian bridge is suggested as a way to derive explicit integral expressions.
• Published analytical treatments are noted for double-barrier options in the geometric Brownian motion setting.
• Finite differences or Monte Carlo may be used for complications such as discrete dividends and local or stochastic volatility.

Tags

# In search of double barrier out option on a BM


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

## Answer by Antoine Conze (score 1, accepted)

https://quant.stackexchange.com/a/46445

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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