Stochastic Volatility Modeling

Introduction 21

price. The price we charge must thus be equal to the Black-Scholes price of the barrier

option augmented by the dierence between market and Black-Scholes prices of the

double European digital: this extra charge covers the cost of actually purchasing the

European hedge.

14

If we reach maturity without hitting the barrier

L = 120

, the payos of the

barrier option and the static European hedge exactly match: the hedge is perfect.

What if instead

S

hits the barrier? When

S

hits

L

at time

τ

, the barrier option ex-

pires worthlessly and we need to unwind our static European hedge. By construction,

in the Black-Scholes model, its value for S = L approximately ...