Barrier Option Pricing within the Black-Scholes Model
Barrier options are a particular type of exotic option in which a certain "barrier" level is specified, and then the option either "knocks in" (becomes exercisable) or "knocks out" (becomes worthless) if the underlying asset price crosses this level from above (for "down" types) or below (for "up" types). There are four types of barriers, varying according to how the barrier affects the price: "up and in", "up and out", "down and in", and "down and out". Additionally, as with regular vanilla options, barrier options come in "call" and "put" form. This Demonstration illustrates the pricing formulas for these options within the Black–Scholes framework.
The closed-form pricing formulas for barrier options used in this Demonstration can be found in, e.g., P. Wilmott, Paul Wilmott on Quantitative Finance, New York: Wiley, 2006. In the Demonstration we assume zero dividend yield, an interest rate of 5%, and a strike price of 100.
Snapshot 1: ticking the "show vanilla" box causes the price of the corresponding vanilla option to be shown in the plot; for cases like the "up and in" call this option makes it easier to see how the barrier option differs from the vanilla one
Snapshot 2: when the time to expiry is zero, the price curves reduce to the final payoff, which is a piecewise function
Snapshot 3: for certain combinations of strike price and barrier level, the option can be worthless; in this case, an "up and out" call with strike > barrier could never be exercised because it would "knock out" if the spot price ever went above the barrier (which would be necessary since it would only make sense to exercise if spot > strike)