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How are rebates factored into the Black-Scholes analytical solutions to pricing barrier options?

In Hull's book, he does not have rebates factored into the formulas. Can someone point me to a paper or literature that does this?

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  • $\begingroup$ when is the rebate paid? if at maturity, it's rather easy. $\endgroup$ – Mark Joshi Jun 5 '16 at 23:44
  • $\begingroup$ @MarkJoshi, at expiration but I would also like to know how it is done if it can be paid at any time before as well. $\endgroup$ – user16556 Jun 6 '16 at 8:43
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you can write the pay-off as

$$(S_T-K)_+ I_{\min S_t > L} + RI_{\min S_t < L}$$

for down and out call.

The first term is the standard call. The second is the rebate. Its value is $$ Re^{-rT} P( \min S_t < L). $$
There is a standard formula for this probability. See eg my book Concepts.

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  • $\begingroup$ ,I have a question about the value of the rebate, how can we take it? Is it fixed in the clauses of the contract? I saw that sometimes it is chosen between 0 and 1. In my opinion the choice of rebate depends on the exercise price, right? $\endgroup$ – M. A. Kacef Jan 24 '18 at 20:04
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For payment at the hitting time $\tau$, you basically need to have the density function $\varphi$ of the $\tau$, and then compute the integral $$\int_0^T e^{-rt} \varphi(t) dt.$$ In the case of constant interest rate $r$ and constant volatility $\sigma$, both the density function $\varphi$ and the integral $\int_0^T e^{-rt} \varphi(t) dt$ can be computed analytically. See Paper Closed Form Formulas for Exotic Options and Their Lifetime Distribution by Raphael Douady.

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