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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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  • $\begingroup$ There's a formula for DOC, and there's one for one-touch coupon/rebate, the sum of these two is the DOC with rebate. A time-dependent rebate is a combination of one-touch rebates with different maturities. For example 1 rebate if hit between 0-1y and 2 rebate if hit between 1-2y is a 2-year one-touch 2 rebate minus a 1-year one-touch 1 rebate. This can be generalised easily. $\endgroup$ – Ivan Mar 11 '18 at 9:22

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