Barrier Option In/Out Parity

Worked Solution

How to Think About It: The key insight is purely about path partitioning. Every possible path of the underlying from time $0$ to $T$ either hits the barrier $H$ at some point, or it does not. These two events are mutually exclusive and exhaustive. A knock-in option activates when the barrier is hit; a knock-out option dies when the barrier is hit. So for any given path, exactly one of the two barrier options is "alive" at expiry and delivers the vanilla payoff. Adding them together reconstructs the vanilla call payoff on every path.

Quick Sanity Checks: If the barrier is set far away (say $H \to \infty$ for a down-and-in call with $H < S_0$, or $H \to 0$), the knock-out call should approach the vanilla call and the knock-in should approach zero -- and the parity $0 + C = C$ holds trivially. Conversely, if the barrier is set right at the current spot, the knock-in immediately activates and equals the vanilla call while the knock-out is immediately dead.

Derivation:

Part 1 -- Base parity (no rebates, continuous monitoring):

Define $\tau_H$ as the first hitting time of barrier $H$. Partition the set of all paths $\Omega$ into: - $A = \{\omega : \tau_H(\omega) \leq T\}$ (barrier is hit before or at expiry) - $A^c = \{\omega : \tau_H(\omega) > T\}$ (barrier is never hit)

These sets are disjoint and $A \cup A^c = \Omega$.

The knock-in call pays $(S_T - K)^+$ on paths in $A$ and $0$ on $A^c$: $C_{\text{in}} = e^{-rT}\, E^Q\!\left[(S_T - K)^+ \mathbf{1}_A\right]$

The knock-out call pays $(S_T - K)^+$ on paths in $A^c$ and $0$ on $A$: $C_{\text{out}} = e^{-rT}\, E^Q\!\left[(S_T - K)^+ \mathbf{1}_{A^c}\right]$

Adding: $C_{\text{in}} + C_{\text{out}} = e^{-rT}\, E^Q\!\left[(S_T - K)^+ (\mathbf{1}_A + \mathbf{1}_{A^c})\right] = e^{-rT}\, E^Q\!\left[(S_T - K)^+\right] = C$

This is the vanilla call price. $\square$

Part 2 -- Effect of rebates:

Suppose the knock-out option pays rebate $R$ at the moment the barrier is hit (time $\tau_H$), and the knock-in option pays rebate $R$ at expiry $T$ if the barrier is never hit. Define the present values of these rebate payments:

• Knock-out rebate value: $R_{\text{out}} = E^Q\!\left[e^{-r\tau_H} R \cdot \mathbf{1}_A\right]$
• Knock-in rebate value: $R_{\text{in}} = E^Q\!\left[e^{-rT} R \cdot \mathbf{1}_{A^c}\right]$

The parity with rebates becomes: $C_{\text{in}} + C_{\text{out}} = C + R_{\text{out}} + R_{\text{in}}$

The rebate terms do not cancel because they are paid at different times: $R_{\text{out}}$ is paid at $\tau_H \leq T$ (discounted at the random stopping time) while $R_{\text{in}}$ is paid at $T$ (discounted at the fixed maturity). Only if $R = 0$ does the clean parity hold.

If both rebates are paid at the same fixed time $T$, then $R_{\text{out}} + R_{\text{in}} = e^{-rT}R$ (a constant), and the parity simplifies to $C_{\text{in}} + C_{\text{out}} = C + e^{-rT}R$.

Part 3 -- Discrete monitoring:

With discrete monitoring (e.g., daily closes $t_1, t_2, \ldots, t_m$), the barrier is only checked at those times. The path partition argument still holds -- every path either triggers the barrier at some monitoring date or it does not -- so the structural parity $C_{\text{in}}^{\text{disc}} + C_{\text{out}}^{\text{disc}} = C$ still holds exactly, assuming no rebates.

However, the individual prices $C_{\text{in}}^{\text{disc}}$ and $C_{\text{out}}^{\text{disc}}$ differ from their continuously monitored counterparts. Discrete monitoring is "less likely" to detect a barrier crossing (the price can breach and recover between observation dates), so: - A discretely monitored knock-out is worth more than its continuous counterpart (it survives more often) - A discretely monitored knock-in is worth less (it activates less often)

Broadbent and Glasserman showed that the leading-order correction is to shift the barrier by $H \to H e^{\pm \beta \sigma \sqrt{\Delta t}}$, where $\beta \approx 0.5826$ (related to the Riemann zeta function $\zeta(1/2)/\sqrt{2\pi}$), $\sigma$ is the volatility, and $\Delta t$ is the monitoring interval.

Practical Interpretation: In practice, barrier options are monitored discretely (daily closes, or even less frequently). Traders know that a discretely monitored knock-out is cheaper to hedge and more valuable to own than the textbook continuous version. The in/out parity is the fundamental no-arbitrage relationship that lets you price a knock-in from a knock-out (or vice versa) plus the vanilla -- which is useful because knock-out options are more liquid. If you can price the vanilla and the knock-out, you get the knock-in for free.

Answer: The parity $C_{\text{in}} + C_{\text{out}} = C$ follows from the fact that barrier hit/no-hit partitions all paths into two mutually exclusive, exhaustive sets, and the two barrier options reconstruct the vanilla payoff on every path. Rebates break the clean parity by adding path-dependent cash flows paid at different times. Discrete monitoring preserves the structural parity but shifts the individual option values because the barrier is less likely to be detected.