# Pricing Barrier Options with Rebates

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?

• when is the rebate paid? if at maturity, it's rather easy. – Mark Joshi Jun 5 '16 at 23:44
• @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. – user16556 Jun 6 '16 at 8:43

$$(S_T-K)_+ I_{\min S_t > L} + RI_{\min S_t < L}$$
The first term is the standard call. The second is the rebate. Its value is $$Re^{-rT} P( \min S_t < L).$$
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.