FRM Exam Part I · Exotic Options

Binary and Lookback Options for FRM Part I

Q: What is the difference between cash-or-nothing and asset-or-nothing options?

A cash-or-nothing option pays a fixed amount if it finishes in the money. An asset-or-nothing option pays the value of the asset instead. Their BSM prices use N(d2) and N(d1) respectively.

Q: How are floating strike and fixed strike lookbacks different?

A floating strike lookback sets the strike at the lowest price (call) or highest price (put) during the life. A fixed strike lookback has a set strike, and the payoff uses the maximum price (call) or minimum price (put).

Q: Why are binary options hard to hedge?

The payoff jumps at the strike, so delta becomes very large near the strike close to expiry. Small price moves force large hedge trades. Sellers often hedge with a tight call spread that overstates the payoff slightly.

Q: How do lookback options differ from barrier options?

A barrier option is activated or cancelled if the price touches a barrier level. A lookback option has no barrier. It uses the maximum or minimum price over the period directly in its payoff.

Updated 11 October 2026 · Fact-checked

Binary options pay a fixed cash amount (cash-or-nothing) or the asset itself (asset-or-nothing) if the underlying finishes beyond the strike. Lookback options pay based on the maximum or minimum price during the life. To solve questions, write the payoff, then match it to the pricing or hedging rule.

Understand Binary and Lookback Options

A binary (digital) option has a discontinuous payoff. A cash-or-nothing call pays a fixed amount Q if the asset price at expiry is above the strike K, and zero otherwise. An asset-or-nothing call pays the asset price S_T if S_T > K, and zero otherwise. The puts pay in the same way when S_T < K.

Under Black-Scholes-Merton, the price is the discounted risk-neutral expected payoff. For a cash-or-nothing call, N(d2) is the risk-neutral probability of finishing above K, so the price is Q × e^(-rT) × N(d2). For an asset-or-nothing call, the price is S0 × e^(-qT) × N(d1). A vanilla call is a long asset-or-nothing call and a short position in K cash-or-nothing calls paying 1. That is why d1 and d2 both appear in the vanilla formula.

Binary options are hard to hedge. Near expiry and near the strike, delta becomes very large, and the payoff jumps from 0 to Q over a tiny price move. A common approach is to replicate the binary with a tight call spread, which gives a payoff that is close to the binary but not identical. To be safe, a seller buys a spread that pays at least as much as the binary, accepting a higher cost.

A lookback option gives the holder the benefit of hindsight. A floating strike lookback call pays S_T − S_min, so you effectively buy at the lowest price. A floating strike lookback put pays S_max − S_T, so you sell at the highest price. Both payoffs are never negative. A fixed strike lookback call pays max(S_max − K, 0), and a fixed strike lookback put pays max(K − S_min, 0).

Do not confuse lookbacks with barrier options. A barrier option switches on or off if the price touches a level. A lookback uses the extreme price itself in the payoff. Lookbacks cost more than comparable vanilla options because they remove the regret of poor timing. Monitoring the extreme only at discrete dates makes them cheaper than continuous monitoring.

Key formulas to remember

Cash-or-nothing call payoff: Payoff = Q if S_T > K; 0 otherwise; The put pays Q if S_T < K. The payoff is fixed and does not depend on how far the option is in the money.
Asset-or-nothing call payoff: Payoff = S_T if S_T > K; 0 otherwise; The put pays S_T if S_T < K.
Cash-or-nothing call price (BSM): c = Q × e^(-rT) × N(d2); The put is Q × e^(-rT) × N(-d2). Here d2 = d1 − σ√T.
Asset-or-nothing call price (BSM): c = S0 × e^(-qT) × N(d1); The put is S0 × e^(-qT) × N(-d1). Use q = 0 for a non-dividend stock.
Vanilla decomposition: Vanilla call = Asset-or-nothing call − K × Cash-or-nothing call (Q = 1); For the put: vanilla put = K × cash-or-nothing put (Q = 1) − asset-or-nothing put.
Floating strike lookback payoffs: Call = S_T − S_min; Put = S_max − S_T; Both are always ≥ 0, so they are always exercised at expiry.
Fixed strike lookback payoffs: Call = max(S_max − K, 0); Put = max(K − S_min, 0); S_max and S_min are the highest and lowest prices over the monitoring period.

How to solve Binary and Lookback Options questions

Use this method for any binary or lookback question. It starts with the payoff, because most errors come from misreading it.

1Identify the type: cash-or-nothing, asset-or-nothing, floating strike lookback or fixed strike lookback, and whether it is a call or a put.
2Write the payoff in plain symbols, with its condition (S_T > K for calls, S_T < K for puts).
3For a path question, pick out S_T, S_max and S_min from the given prices and check which of them the payoff uses.
4For a pricing question, choose the right BSM term: N(d2) with Q × e^(-rT) for cash, N(d1) with S0 × e^(-qT) for asset.
5Use N(-x) = 1 − N(x) for puts. Check that your put price makes sense against the call.
6For a decomposition question, build the vanilla option from the two binaries and subtract or add as needed.
7For a hedging question, remember delta spikes near the strike close to expiry, and a call spread can replicate the binary approximately.
8Sanity check: a binary cash call price must be between 0 and Q × e^(-rT), and a floating lookback must cost at least as much as a vanilla option.

Quickest way: Payoff-first shortcut

When to use it: Use this when time is short and the question is a payoff calculation or a conceptual comparison.

Scan the answer options for the required number and sign. A floating lookback payoff is never negative, so reject any negative option.
For path-based payoffs, take S_T, S_max and S_min straight from the data, then plug in. Floating call: last minus lowest. Floating put: highest minus last.
For cash binaries, multiply Q by e^(-rT) by N(d2). Use a calculator with e^x and compute in one pass.
For vanilla decomposition, remember: call = asset binary − K × cash binary.

Common mistakes in Binary and Lookback Options

Using N(d1) for a cash-or-nothing option.
Both d1 and d2 appear in the vanilla formula, and students mix up which goes where.
Fix: Cash-or-nothing uses N(d2) as the probability of finishing in the money. Asset-or-nothing uses N(d1).
Forgetting to discount the cash payoff.
Students treat N(d2) as the final price.
Fix: Multiply by Q × e^(-rT). N(d2) alone is a probability, not a price.
Using S0 instead of S_T in the lookback payoff, or swapping S_max and S_min between the call and the put.
The names are similar and the formulas are close.
Fix: Floating call: S_T − S_min. Floating put: S_max − S_T. The call buys at the low and the put sells at the high.
Treating a lookback as a barrier option.
Both depend on the path.
Fix: A barrier option switches on or off when a level is hit. A lookback uses the maximum or minimum itself in the payoff.
Saying a binary option is easy to delta hedge.
The payoff looks simple.
Fix: Delta and gamma become very large near the strike close to expiry. Hedging is unstable, and a call spread replication is used instead.
Assuming a floating lookback can expire worthless.
Students apply the logic of vanilla options.
Fix: S_T is always ≥ S_min and ≤ S_max, so floating lookback payoffs are ≥ 0. They can be zero, but they are never negative.

Worked examples

Example 1

A cash-or-nothing call on a non-dividend stock pays $100 if the stock is above the strike in one year. The risk-free rate is 5% (continuously compounded) and N(d2) = 0.5398. What is its price?

Show the solution

Use c = Q × e^(-rT) × N(d2).
Q = 100, r = 0.05, T = 1, so e^(-0.05) = 0.951229.
Discounted payoff = 100 × 0.951229 = 95.1229.
Multiply by N(d2): 95.1229 × 0.5398 = 51.35.

Answer: The option is worth about $51.35.

Example 2

Over the life of an option, an asset starts at 100 and its lowest price is 92, highest is 118, and final price is 110. Find the payoffs of a floating strike lookback call, a floating strike lookback put, a fixed strike lookback call with K = 104 and a fixed strike lookback put with K = 104.

Show the solution

Floating call = S_T − S_min = 110 − 92 = 18.
Floating put = S_max − S_T = 118 − 110 = 8.
Fixed call = max(S_max − K, 0) = max(118 − 104, 0) = 14.
Fixed put = max(K − S_min, 0) = max(104 − 92, 0) = 12.

Answer: Floating call 18, floating put 8, fixed call 14, fixed put 12.

Exam tips

Write the payoff before you touch any formula. Most questions test whether you know which price (S_T, S_max, S_min) enters the payoff.
Remember the d1 and d2 link: asset-or-nothing is the N(d1) piece, cash-or-nothing is the N(d2) piece.
Conceptual questions often ask why binaries are hard to hedge or why lookbacks are more expensive than vanillas. Answer with delta spikes and hindsight benefit.
Expect to compare lookbacks with barriers. Barrier means a trigger level is hit, lookback means the extreme price sets the payoff.
On a phone or a calculator, compute e^(-rT) first and keep at least four decimals until the final step.

Practice questions from Exotic Options

Binary and Lookback Options: frequently asked questions

What is the difference between cash-or-nothing and asset-or-nothing options?

A cash-or-nothing option pays a fixed amount if it finishes in the money. An asset-or-nothing option pays the value of the asset instead. Their BSM prices use N(d2) and N(d1) respectively.

How are floating strike and fixed strike lookbacks different?

A floating strike lookback sets the strike at the lowest price (call) or highest price (put) during the life. A fixed strike lookback has a set strike, and the payoff uses the maximum price (call) or minimum price (put).

Why are binary options hard to hedge?

The payoff jumps at the strike, so delta becomes very large near the strike close to expiry. Small price moves force large hedge trades. Sellers often hedge with a tight call spread that overstates the payoff slightly.

How do lookback options differ from barrier options?

A barrier option is activated or cancelled if the price touches a barrier level. A lookback option has no barrier. It uses the maximum or minimum price over the period directly in its payoff.