Barrier Options

Barrier options are path-dependent options that are either activated or cancelled if the underlying price hits a specified barrier level.

Types of Barrier Options

Type	Activated/cancelled when...
Down-and-out	Price falls below barrier → option cancelled (knocked out)
Down-and-in	Price falls below barrier → option activated (knocked in)
Up-and-out	Price rises above barrier → option cancelled
Up-and-in	Price rises above barrier → option activated

Down-and-Out Call

The most common barrier option. It behaves like a vanilla call, but is cancelled if the stock price ever touches or drops below the barrier B (where B < S).

$\text{Payoff} = \begin{cases} \max(S_T - K, 0) & \text{if } S_t > B \text{ for all } t \\ 0 & \text{if } S_t \leq B \text{ for any } t \end{cases}$

Monte Carlo Algorithm

For each path:

Simulate step by step
At each step, check if the price has hit or crossed the barrier
If knocked out, the payoff is 0
Otherwise, payoff = max(S_T - K, 0)

Effect of the Barrier

Barrier far below current price: rarely triggered → price ≈ vanilla call
Barrier close to current price: frequently triggered → price < vanilla call
As B → 0: down-and-out call → vanilla call
As B → S: down-and-out call → 0 (immediately knocked out)

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