Valuing Rebates in Barrier Options

Summary

The document explains two ways to value rebates in barrier options under Black–Scholes assumptions. For a down-and-out call with a rebate paid at maturity, the payoff can be separated into the ordinary surviving call payoff and a rebate paid if the barrier is breached. The rebate value is its discounted amount multiplied by the probability that the underlying reaches the barrier during the option’s life.

If the rebate is paid at the barrier-hitting time instead, the relevant quantity is the hitting-time density. Discount each possible payment time and integrate that discounted density over the option term. The response says that with constant interest rates and volatility, both the density and its discounted integral admit analytical calculation, and points to a reference on closed-form exotic-option formulas and lifetime distributions. The discussion is concise and assumes the stated constant-parameter setting; it does not provide the formulas themselves or address more general models.

Key ideas

• A maturity-paid rebate can be valued separately from the surviving option payoff.
• The maturity rebate value depends on the discounted probability of breaching the barrier.
• A rebate paid at the hitting time requires discounting across the distribution of hitting times.
• With constant rates and volatility, the response states that the hitting-time calculation has an analytical solution.

Tags

# Pricing Barrier Options with Rebates


# 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?

## Answer by Mark Joshi (score 2)

https://quant.stackexchange.com/a/27461

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.

## Answer by Gordon (score 0)

https://quant.stackexchange.com/a/29509

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