FRM Exam Part I · Exotic Options

Gap, Forward Start and Compound Options Explained

Q: How do you price a gap option in FRM Part I?

Use the Black-Scholes-Merton formula. Put the trigger K2 into d1 and d2. Use the payoff strike K1 in the K e^(−rT) N(d2) term for a call.

Q: What is the difference between a compound option and a chooser option?

A compound option is an option on another option and has two strikes and two expiries. A chooser lets the holder decide at T1 whether to own a call or a put, both with the same strike and expiry.

Q: Why is a forward start option easy to value?

Its strike is set at the money at T1, so its value at T1 is proportional to the stock price then. You price an at-the-money option with life T2 − T1 and scale it.

Q: Can a gap option have a negative payoff?

Yes. If the payoff strike K1 is above the trigger K2, a gap call can pay a negative amount when the stock ends between the two prices.

Updated 11 October 2026 · Fact-checked

A gap option has a trigger price K2 that decides if it pays, and a strike K1 that sets the payoff. A forward start option starts later with a strike set at that date. A compound option is an option on an option. A chooser option lets you pick call or put later.

Understand Gap, Forward Start and Compound Options

These are non-standard options. Each one changes one feature of a plain European option: the payoff rule, the start date, the underlying, or the type.

A gap option has two prices. The trigger price K2 decides whether the option pays. The strike price K1 decides how much it pays. A gap call pays S_T − K1 if S_T > K2, and nothing otherwise. The payoff can be negative when K1 > K2, so the option's price can be negative too. Price it with Black-Scholes-Merton, but use K2 in d1 and d2 and K1 in the strike terms.

A forward start option is paid for today but starts at a future time T1. Its strike is set at T1, usually at the money, so the strike equals S at T1. With no dividends, or a continuous yield, the value at T1 is proportional to S at T1. So today's value is e^(−qT1) times the value of an at-the-money option with life T2 − T1, with S0 scaled to 1 per unit of stock. Volatility is assumed constant.

A compound option is an option on an option. The four types are call on call, call on put, put on call and put on put. It has two strikes and two expiry dates. You exercise the first option by paying the first strike to receive the second option. Pricing needs the bivariate normal distribution, and the critical underlying price at the first expiry is found numerically. Compound options usually have a lower upfront premium than the direct option, but exercising the first option adds the first strike as a further cost. They suit uncertain future hedging needs.

A chooser option (as-you-like-it) lets the holder pick at time T1 whether it becomes a call or a put, both with strike K and expiry T2. Its value at T1 is max(c, p). Put-call parity turns this into a call plus a put with a shorter life, which makes the chooser easy to price.

Key formulas to remember

Gap call price: c = S0 e^(−qT) N(d1) − K1 e^(−rT) N(d2); d1 = [ln(S0/K2) + (r − q + σ²/2)T] ÷ (σ√T); d2 = d1 − σ√T. K2 is the trigger and goes in d1 and d2 only.
Gap put price: p = K1 e^(−rT) N(−d2) − S0 e^(−qT) N(−d1); Same d1 and d2 as the gap call, using K2. Pays K1 − S_T if S_T < K2.
Gap call payoff: S_T − K1 if S_T > K2, otherwise 0; Payoff can be negative if K1 > K2.
Forward start option value: Value today = e^(−qT1) × (value at S0 = 1 of an at-the-money option with life T2 − T1) × S0; Valid for a strike set at the money at T1, constant volatility and a continuous yield q. Value the option with life T2 − T1 using the same r, q and σ. If there are no dividends, q = 0 and the factor e^(−qT1) equals 1.
Chooser option at T1: max(c, p); Both options have strike K and expiry T2.
Chooser decomposition (no income): Chooser = c(K, T2) + p(K e^(−r(T2 − T1)), T1); Call with strike K and expiry T2, plus a put with expiry T1 and strike K e^(−r(T2 − T1)). Uses max(c,p) = c + max(0, p − c) and put-call parity.

How to solve Gap, Forward Start and Compound Options questions

Identify the product from the wording, then map it to a Black-Scholes-Merton variant or a decomposition.

1Name the option: gap, forward start, compound or chooser. Look for two prices, a later start, an option on an option, or a later choice.
2List the inputs: S0, strikes, T, T1, T2, r, q and σ. Check that rates and σ share the same time basis.
3For a gap option, put the trigger K2 in d1 and d2, and the payoff strike K1 in the discount term.
4For a forward start option, price an at-the-money option with life T2 − T1 and scale it by S0 and e^(−qT1).
5For a chooser, split it into a call and a put using put-call parity, or compare max(c, p) at T1.
6For a compound option, find the exercise condition at the first expiry: the second option's value must exceed the first strike.
7Compute N(·) from the tables and combine the terms.
8Sanity check: is the price positive and plausible compared with a plain option?

Quickest way: Shortcut: reuse the Black-Scholes formula

When to use it: Use it when the question gives numbers for a gap or forward start option and asks for a price or a payoff.

For a gap option, compute d1 and d2 using K2 only, then change just the K term to K1.
Gap call price = plain call with strike K2 + (K2 − K1) e^(−rT) N(d2). If K1 < K2, the gap call is worth more than the plain call.
For a forward start, price at-the-money with S = 1 and life T2 − T1, then multiply by S0.
For a chooser, remember it is worth at least as much as either the call or the put alone (strike K, expiry T2), and no more than the sum of the call and the put.
If the gap payoff is only asked at one S_T, just apply the trigger test and the payoff rule.

Common mistakes in Gap, Forward Start and Compound Options

Using K1 in d1 and d2 for a gap option.
Students treat it like a plain option and forget the trigger.
Fix: Trigger K2 drives the probabilities in d1 and d2. Strike K1 appears only in the K e^(−rT) N(d2) term.
Assuming a gap option payoff is never negative.
Plain options have payoffs of at least zero.
Fix: If K1 > K2, the gap call pays S_T − K1 when S_T is above K2, which can be negative.
Treating a forward start option's strike as known today.
The strike looks like any other strike.
Fix: The strike is set at T1. Price using the at-the-money option with life T2 − T1.
Confusing compound and chooser options.
Both involve a decision later.
Fix: A compound option is an option on an option, with two strikes. A chooser picks call or put at T1 with one strike K.
Using the full life T2 for the put in the chooser decomposition.
Students forget that the put expires at T1.
Fix: The put has expiry T1 and strike K e^(−r(T2 − T1)).

Worked examples

Example 1

A gap call has trigger K2 = 50 and strike K1 = 45 on a stock. At expiry, what is the payoff if S_T is 48, 50 or 55?

Show the solution

Rule: it pays S_T − K1 only if S_T > K2 = 50.
If S_T = 48: 48 is not above 50, so the payoff is 0.
If S_T = 50: it is not strictly above 50, so the payoff is 0.
If S_T = 55: 55 > 50, so the payoff is 55 − 45 = 10.

Answer: Payoffs are 0, 0 and 10. Only S_T above the trigger pays, and the amount uses K1.

Example 2

A gap call has trigger K2 = 40 and strike K1 = 50. At expiry, what is the payoff if S_T is 38, 45 or 60?

Show the solution

Rule: it pays S_T − K1 only if S_T > K2 = 40.
If S_T = 38: 38 is not above 40, so the payoff is 0.
If S_T = 45: 45 > 40, so the option pays 45 − 50 = −5.
If S_T = 60: 60 > 40, so the payoff is 60 − 50 = 10.

Answer: Payoffs are 0, −5 and 10. Because K1 > K2, the payoff is negative when K2 < S_T < K1.

Example 3

A chooser option has strike K = 100. At T1 the call is worth 8 and the put is worth 5, both with strike 100 and expiry T2. What is the chooser worth at T1, and which choice is made?

Show the solution

Value at T1 = max(c, p).
c = 8 and p = 5, so max = 8.
The holder picks the call.

Answer: The chooser is worth 8 at T1 and becomes a call.

Exam tips

Check which strike goes where in a gap option: trigger in d1 and d2, payoff strike in the discount term.
Know the chooser decomposition into a call plus a shorter-dated put and be able to state it.
Compound options are usually tested conceptually: four types, two strikes, usually a lower upfront premium than the direct option.
Forward start questions often ask why the value is proportional to S0 and the strike is at the money at T1.
Write d1 and d2 to four decimals before looking up N(·).

Practice questions from Exotic Options

Gap, Forward Start and Compound Options: frequently asked questions

How do you price a gap option in FRM Part I?

Use the Black-Scholes-Merton formula. Put the trigger K2 into d1 and d2. Use the payoff strike K1 in the K e^(−rT) N(d2) term for a call.

What is the difference between a compound option and a chooser option?

A compound option is an option on another option and has two strikes and two expiries. A chooser lets the holder decide at T1 whether to own a call or a put, both with the same strike and expiry.

Why is a forward start option easy to value?

Its strike is set at the money at T1, so its value at T1 is proportional to the stock price then. You price an at-the-money option with life T2 − T1 and scale it.

Can a gap option have a negative payoff?

Yes. If the payoff strike K1 is above the trigger K2, a gap call can pay a negative amount when the stock ends between the two prices.