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Recognise the characteristics and payoff structures of basket, exchange, Asian, lookback, binary, barrier, chooser, compound, forward start and gap options.

FRM® Part I · Financial Markets and Products · Exotic Options · FRM1.FM.15.e

The idea

Basket: payoff on a portfolio of assets, cheaper than separate options because correlation is below one. Exchange: right to swap one asset for another, with payoff max(V − U, 0). Asian: payoff on the average price, max(average − K, 0) for a call. Lookback: a floating call pays the final price minus the lowest price seen, a fixed call pays max(highest price − K, 0). Binary: pays a fixed cash amount, or the asset itself, if the final price is beyond K. Barrier: comes into existence (knock-in) or ceases to exist (knock-out) if a barrier is touched, and knock-in plus knock-out equals the vanilla option. Chooser: the holder picks call or put at a later date. Compound: an option on an option, with two strikes and two dates. Forward start: starts later, with the strike set at the money on that date. Gap: pays the final price minus K1 only if the final price passes a trigger K2, so the payoff can be negative. Asian, lookback and barrier options are path dependent.

Formulas

In-out parity for barrier options

vanilla option = knock-in option + knock-out option

knock-in option
the barrier option that comes into existence when the barrier is touched
knock-out option
the matching option that ceases to exist when the barrier is touched

Same strike, expiry and barrier. Exactly one of the pair survives on any path, so together they replicate the vanilla option.

Payoff of an exchange option

payoff = max(V − U, 0)

V
the value at expiry of the asset received
U
the value at expiry of the asset given up

The right to swap one asset for another. It depends only on the two terminal values, so it is not path dependent.

Payoff of an average-price (Asian) call

payoff = max(A − K, 0)

A
the average price of the underlying over the averaging period
K
the strike price

Path dependent: every price in the average counts. The average is less volatile than the final price, which makes the option cheaper than a vanilla call.

A practice question

From the question bank, with the reason each wrong option is wrong.

Match each description to an option type: (i) the strike is set equal to the asset price at a future date; (ii) at a future date the holder chooses whether it becomes a call or a put; (iii) an option whose underlying is another option.

  1. (i) forward start, (ii) chooser, (iii) compound - correct
  2. (i) chooser, (ii) forward start, (iii) compoundWhy not: Swaps the first two: the chooser selects call or put, the forward start sets its strike later.
  3. (i) forward start, (ii) compound, (iii) chooserWhy not: Swaps the last two: the compound option is the option on an option; the chooser picks call or put.
  4. (i) gap, (ii) chooser, (iii) compoundWhy not: A gap option has strike and trigger set at inception; the strike set at a future date is a forward start option.

Forward start options fix the strike at the money on a future date, choosers let the holder pick call or put later, and compound options are options on options.

A harder one

An arithmetic average-price call with strike 50 is based on four monthly closing prices: 48, 52, 57 and 55. Its payoff at expiry is:

  1. 5.00Why not: Uses the final price, 55, minus the strike; that is a vanilla call, not an average-price option.
  2. 3.00 - correct
  3. 7.00Why not: Uses the highest price, 57, minus the strike; that is a fixed-strike lookback call.
  4. 2.00Why not: Uses S_T minus the average (55 - 53), the payoff of a floating-strike Asian call.

Average = (48 + 52 + 57 + 55) / 4 = 53. Payoff = max(53 - 50, 0) = 3.00. The path before expiry matters, unlike for a vanilla call.

The hardest tier

A simple chooser option gives the holder, at T1 = 0.5 years, the right to choose a call or a put, both with strike 100 and expiring at T2 = 1.25 years. The asset pays no dividend and the continuously compounded risk-free rate is 5%. It equals a call (strike 100, maturity 1.25) plus a put with:

  1. maturity 0.5 and strike 96.3 - correct
  2. maturity 0.5 and strike 103.8Why not: Compounds the strike forward by e^(r(T2-T1)) instead of discounting it.
  3. maturity 1.25 and strike 93.9Why not: Discounts over the full 1.25 years and uses the wrong maturity.
  4. maturity 0.5 and strike 100.0Why not: Ignores the discounting of the strike over T2 - T1 that put-call parity requires.

At T1 the chooser is worth max(c, p) = c + max(0, p - c). By put-call parity p - c = K e^(-r(T2-T1)) - S, so the extra term is a put maturing at T1 with strike 100 x e^(-0.05 x 0.75) = 96.3.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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