Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il 02-588-3049 FE-W http://pluto.mscc.huji.ac.il/~mswiener/zvi.html

Following Paul Wilmott, Introduces Quantitative Finance Chapter 13 Barrier Options Following Paul Wilmott, Introduces Quantitative Finance Chapter 13 FE-W http://pluto.mscc.huji.ac.il/~mswiener/zvi.html

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

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

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

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

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

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

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

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

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

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