Boundary Conditions for Bermudan Up-and-Out Binary Calls

Summary

The document poses a boundary-condition problem for a Bermudan up-and-out binary call. The contract pays a unit amount at maturity when the underlying finishes above the strike; it is knocked out only when the underlying exceeds a barrier on specified observation dates, with the last observation immediately before maturity. The question asks for the value limits as the underlying approaches zero or infinity at times before maturity.

It motivates the problem by comparing the contract with an American-style version, which may exercise as soon as the strike is reached and therefore cannot reach a higher barrier first. That intuition does not directly settle the Bermudan case, because exercise is not continuous and knock-out checks occur only on scheduled dates. The document contains no answer, derivation, pricing model, or numerical evidence, so it identifies a modeling question rather than supplying boundary conditions. Any solution would need to account for the discrete monitoring schedule and terminal payoff.

Key ideas

• The contract pays at maturity only if the underlying finishes above the strike.
• Knock-out occurs only at a finite set of observation dates when the underlying exceeds the barrier.
• The final knock-out observation occurs immediately before maturity.
• Continuous exercise intuition for an American contract does not determine Bermudan boundary values.
• The document asks for limits at zero and infinity but does not provide a solution.

Tags

# What are the boundary conditions for an up-and-out binary call option of Bermudan type?


# What are the boundary conditions for an up-and-out binary call option of Bermudan type?












I know that an up-and-out binary call option of American type will never knock out if barrier is greater than the strike since the option stops immediately if the stock price touches the strike due to unit payoff and thus it will never go up to barrier. But how about finite knock-out observation dates?

I am wondering the boundary conditions for the up-and-out call option of Bermudan type. The structure is as follows. The final payoff is 1 if $S_T > K$ and 0 else. The option will only knock out if $S_t > K_{out} $ observed in finite observation dates which are decided in initiation, say $t_1, t_2, t_3, ..., t_n$, where the last knock out observation date $t_n$ is exactly before the maturity date.

The right boundary condition for all prices at maturity is just its payoff $1_{S_T > K}$. What are the upper and lower condition for all time points when $S_t = 0$ and $S_t = \infty$? Are they all 0?

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