Exotic Barrier Options

1. Exotic Barrier Options Chu-Lan Kao National Central University Ref. Peter Zhang, “Exotic Option”, 2nd Edition.

2. Introduction • Due to financial needs, different options with different barrier structures arise. • These options, compare to “vanilla” barrier options, is called “exotic” barrier options, and require further pricing computations.

3. Agenda • Forward-Start • Early-Ending • Window • Corridor (Double Barrier)

4. Forward-Start • Definition • The barrier would be effective after a given time τ1 • If S(τ1) is higher then the barrier, it becomes a down-and-in (out) option. • If S(τ1) is lower then the barrier, it becomes a up-and-in (out) option

5. Forward-Start • Pricing – Analytical Form • Simply consider the distribution of S(τ1), and check whether it becomes an up or down option. • For example, for a forward-start-in barrier option:

6. Forward-Start • Pricing – Analytical Form • Substitute the above by where: = • Hence, the price would be given by:

7. Forward-Start • Pricing– Closed Form • But actually, we could plug-in the knock-in (out) options formula, and integrate (11.13) to have a closed-form formula. • The key tool in this integration is the following equality in Appendix 11.1: where N2(μ1,μ2,ρ) is the CDF of bivariate normal.

8. Forward-Start • Pricing– Closed Form • The price for a forward-start-in call would be:

9. Forward-Start • Pricing– Closed Form • The price for a forward-start-in call would be (conti.):

10. Early-Ending • Definition • The barrier would be effective before a given time τe. • Notice that after τe, the option is like a vanilla option.

11. Early-Ending • Pricing • As usual, the price could be achieved once we have the distribution of S(τ). • Our strategy goes like this: • Step 1. We know the distribution of S at τe. • Step 2. We know the distribution of S at τ conditioned on the distribution of S at τe. • Step 3. Combining the previous two, we have the distribution of S at τ.

12. Early-Ending • Pricing • Step 1. Density Function at τe( θ= 1/-1 for down/up ) • Out Options • In Options

13. Early-Ending • Pricing • Step 2. Conditional Density • z part stands for Brownian motion starting from τe.

14. Early-Ending • Pricing • Step 3. Density Function at exercise time θ = 1/-1 for down/up ζ = 1/-1 for out/in • Note that, if θ=1, ζ=-1, then when τe t: • If z>a then: First part  0, second part1 • If z<a then: First part  1, second part0 • Vanilla Down-and-in Option • Similar Argument could see that it converges to vanilla option when τe 0

15. Early-Ending • Pricing • Final Result: ω = 1/-1 for call/put θ = 1/-1 for down/up ζ = 1/-1 for out/in -ζ*

16. Window • Definition • The barrier would be effective within a given period of time • In general, there might be multiple windows.

17. Window • Pricing • The key point is to view the window option as a forward-start “early-ending option”. • For example, for single window: • For N windows, simply replace the early-ending option with the N-1 windows.

18. Corridor Options • Definition • Options with two barriers. • There would be one upper barrier and one lower barrier. • The option is knocked-in / out whether the underlying reaches either one of the barriers. • Payout function: • Out-Corridor • In-Corridor • AKA dual-barrier options.

19. Corridor Options • The Density Function(Cox and Miller, 1965)

20. Corridor Options • Pricing where:

21. Two Curved Boundaries • First studied by Kunitomo & Ikeda (1992) • The case here is two floating barriers: • However, this is not as simple as the vanilla floating barrier options since you couldn’t adjust to both of the barrier at the same time! • Thus, the density function must be re-derived.

22. Two Curved Boundaries • Density Function (Kunitomo & Ikeda,1992) • Warning: This is of the underlying price, not the log-normal price!

23. Two Curved Boundaries • Pricing Formula