Pricing Down-and-Out Calls with Time-Dependent Rebates

Summary

The document asks whether a closed-form valuation exists for an American down-and-out call whose principal amortizes linearly with elapsed time if the option is knocked out. It frames the payoff as a possible time-dependent rebate and compares the problem with published work on BOOST options and double-barrier step options.

The discussion highlights a key modeling distinction: the cited examples concern double barriers or rebates with different payoff rules, while the question concerns a single barrier and a one-touch knock-out. No pricing formula, derivation, or empirical evidence is provided, so the document does not establish that a closed form exists. It is useful as a statement of a specialized derivatives-pricing problem and the limitations of two nearby references, rather than as a worked valuation method.

Key ideas

• The pricing problem concerns an American down-and-out call with linearly amortizing principal upon knock-out.
• The proposed interpretation is a rebate that varies with the time elapsed before barrier crossing.
• The cited BOOST and step-option work addresses related but different double-barrier structures.
• The document poses the question but gives no solution or evidence that a closed form exists.

Tags

# Barrier Option with Time-Dependent Rebate


# Barrier Option with Time-Dependent Rebate












Is there a closed form solution for American Single-Barrier Options (specifically Down-and-Out Calls) which undergo linear principal amortization based on the amount of time passed before being KO'ed?

The Paper Closed Form Formulas for Exotic Options and Their Lifetime Distribution by Raphael Douady discusses BOOST options which are somewhat similar. I suppose the principal amortization could be thought of as a linear time-dependent rebate / coupon, but the BOOST options mentioned are Double Barrier Options with both an upper and lower barrier and the rebate formulae aren't linear.

Also, the Paper Structuring, Pricing and Hedging Double-Barrier Step Options by Dmitry Davydov and Vadim Linetsky discusses Simple (Arithmetic) Double-Barrier Step Options, but from what I understand, this only comes into effect when the Stock Price is at or below the barrier and I am trying to value one-touch options which will cease to exist once the threshold is crossed.

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