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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? $\endgroup$ – MAKacef Jan 24 '18 at 20:04
  • $\begingroup$ I though KO options could pay rebate as soon as the barrier is crossed, hence not only at maturity, would the crossing have taken place. In which case the payoff is slightly different, as the payment time of the rebate could be random would the option pay it at crossing, for instance. $\endgroup$ – Olorin Nov 10 '20 at 13:00
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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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