1. Financial Engineering
Zvi Wiener
FE-W

2. Following Paul Wilmott, Introduces Quantitative Finance Chapter 13
Barrier Options
Following
Paul Wilmott, Introduces Quantitative Finance
Chapter 13
FE-W

3. Types of Barrier Options
Out option pays off if a level is not reached, otherwise it is knocked out.
In option pays as long as a level is reached (knocked in).
Payoff classification is standard: call, put, binary, etc.
Double barrier there are both up and down barriers.
Sometimes a rebate is paid if barrier is reached.
Zvi Wiener
FE-Wilmott-IntroQF Ch13

4. PDE approach to barriers, Out
V=R (rebate)
Out barrier ?
Payoff S R
V=0 T
time
Zvi Wiener
FE-Wilmott-IntroQF Ch13

5. PDE approach to barriers, In
Two stages: first a regular
option above the barrier,
this provides a boundary
condition.
Second the In option is
solved with this boundary
Regular option
In barrier ?
Payoff
V=0 T
time
Zvi Wiener
FE-Wilmott-IntroQF Ch13

6. FE-Wilmott-IntroQF Ch13
Down-and-in call + Down-and-out call = Plain call option
Zvi Wiener
FE-Wilmott-IntroQF Ch13

7. Up-and-out call option
Value of an up-and-out call
barrier S
Zvi Wiener
FE-Wilmott-IntroQF Ch13

8. FE-Wilmott-IntroQF Ch13
Other features
Early exercise, American or Bermudian style
Repeated hitting (both up and down)
Dynamic barrier
Rainbow: payoff determined by asset A, while barrier by asset B.
Soft barriers – (dis)appear gradually
Alon Raviv – default option as a soft barrier.
Zvi Wiener
FE-Wilmott-IntroQF Ch13

9. FE-Wilmott-IntroQF Ch13
Volatility
Most analytical models assume constant volatility.
In practice there is a smile/smirk effect.
It is very important to keep in mind vega as a risk factor!
Bid-ask in terms of vol.
Zvi Wiener
FE-Wilmott-IntroQF Ch13

10. Hedging Barrier Options
Approximate hedge with other options, see 13.7
Delta hedge is problematic at boundaries, since delta is not continuous there (slippage costs).
Solution – smoothing of delta hedging with minimization of tracking error.
Zvi Wiener
FE-Wilmott-IntroQF Ch13

11. Analytic Formulas for Barriers
Up-and-out call/put
Up-and-in call/put
Down-and-out call/put
Down-and-in call/put
See formulas in section 13.8
Zvi Wiener
FE-Wilmott-IntroQF Ch13

12. FE-Wilmott-IntroQF Ch13
Home Assignment
Read chapter 13 in Wilmott.
Classify all structured products that we have discussed as types of barriers (POALIM, LEUMI, etc.)
Plot the appropriate area for PDE and boundary conditions.
Zvi Wiener
FE-Wilmott-IntroQF Ch13