Barrier Options

A barrier option is a standard call or put with one extra condition: it comes into existence, or ceases to exist, if the underlying price touches a set level, the barrier, during the option's life. Knock out options die if the barrier is hit; knock in options only come alive if it is hit. Because the extra condition removes some scenarios, barrier options are cheaper than equivalent vanilla options. They are the most widely traded exotic options, especially in foreign exchange.

The four basic types#

Each can be a call or a put, giving eight combinations. Some knock out options pay a rebate, a fixed amount, if they are knocked out.

In out parity#

A knock in and a knock out with the same strike, barrier and expiry add up to a vanilla option:

knock in + knock out = vanilla option

If the barrier is hit, the knock out dies and the knock in comes alive; if not, the reverse. Exactly one of them is alive at expiry. This makes it easy to price one from the other.

Worked example#

Why use barrier options#

• Cheaper hedges: a company hedging currency risk may buy a knock out option if it believes the rate is unlikely to move past the barrier.
• Targeted views: "I think the euro will rise but not above 1.20" fits an up and out call.
• Structured products: many reverse convertibles and autocallables contain down and in puts sold by the investor, which become active only if the underlying falls below a level, often 60% to 70% of the starting price. See Exotic Options Explained.

Monitoring#

How the barrier is checked matters:

• Continuous monitoring: any touch at any moment counts.
• Discrete monitoring: only closing prices on set dates count. Discrete barriers are less likely to be hit and are worth more for knock outs.

Contract terms also specify which price source is used, an important detail in fast markets.

Pricing#

Under Black Scholes assumptions, continuously monitored barrier options have closed form formulas (first published by Merton in 1973 and extended by Reiner and Rubinstein in 1991). In practice, banks use local volatility, stochastic volatility or local stochastic volatility models, because barrier prices are very sensitive to the volatility smile and its dynamics. See Local Volatility and Stochastic Volatility and the Heston Model. Monte Carlo and finite differences handle discrete monitoring and complex features. See Monte Carlo Option Pricing.

Hedging problems near the barrier#

Close to the barrier, especially with a large payoff at stake, Greeks can swing wildly:

• A knock out call deep in the money just below the barrier has large negative gamma: a small rise could wipe out a large value.
• Delta can flip sign near the barrier.
• Dealers hedging large barrier positions may trade heavily around barrier levels, which is sometimes said to cause price behaviour near popular barriers in FX markets.

See Gamma and Charm, Vanna and Volga.

Common mistakes#

• Buying knock outs too close to the barrier, where small moves end the option.
• Underestimating the chance of touching the barrier before expiry, which is much higher than the chance of ending beyond it.
• Selling down and in puts in structured products without understanding the crash risk.

Frequently asked questions#

What is a barrier option?#

An option that comes into existence or ceases to exist if the underlying touches a set price level during its life.

What is the difference between knock in and knock out options?#

A knock out option dies if the barrier is touched; a knock in option only becomes active if the barrier is touched.

Why are barrier options cheaper than vanilla options?#

Because the barrier condition removes some outcomes in which a vanilla option would pay, reducing the option's expected payoff.

Next, learn fixed payout options in Binary Options.