Exotic options

Three families with closed-form prices under the same assumptions as Black-Scholes. Each is shown next to the vanilla option with the same terms, and the parity relations that tie them together are checked live.

Contract

Spot price (S)

Strike (K)

Volatility (σ)

Annualised, 0.20 = 20%

Time to maturity (T)

Years

Risk-free rate (r)

Annualised, 0.05 = 5%

Dividend yield (q)

Annualised, continuous

Call

Put

Try this

0 of 3 done

Knock-in plus knock-out

Price a down-and-out call and note it. Price a down-and-in call with the same barrier.

What to look for: The two prices add up to the vanilla call. One of them pays if the barrier is hit and the other if it is not, so together they always pay.

Move the barrier away

Take the down-and-out call and lower the barrier far below the spot.

What to look for: The knock-out price rises towards the vanilla. A barrier that is almost never touched barely matters.

A digital is N(d2)

Price a cash-or-nothing call with a payout of 1, then open the Black-Scholes page with the same contract and read N(d2).

What to look for: The digital price is e^(-rT) times N(d2): the discounted risk-neutral probability of finishing in the money. That is the same N(d2) that appears in the vanilla formula.

Exotic

Type

Knocked out (dies) or knocked in (comes alive) if the spot touches the barrier before expiry. Continuous monitoring, no rebate.

Barrier type

Barrier level (H)

Barrier call

$8.6655

Vanilla call

$10.4506

Ratio to vanilla

82.9%

Parity checks

Down-and-in	1.785112
Down-and-out	8.665472
In + out (should equal vanilla)	10.450584
Vanilla	10.450584

Value against spot

Knock-out values fall to zero at the barrier, so their Delta changes sign near it.

Value against barrier level

Knock-in and knock-out always sum to the vanilla, whatever the barrier.