Charta is on Google Play: practise on the go. Get the app
Charta

Show how a nonstandard American option can be engineered from a standard one.

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

The idea

A standard American option can be exercised at any time up to expiry at one fixed strike. Over-the-counter and employee-warrant versions often change those rules. A Bermudan option can be exercised only on specified dates. A lock-out period forbids exercise for an initial part of the life. The strike can also step up or down over time. These nonstandard American options are engineered from the standard one by restricting where early exercise is allowed or by changing the strike schedule. In a binomial tree this is easy. The test of continuation value against intrinsic value is applied only at nodes where exercise is permitted, using the strike in force at that date, and the terminal payoff is always included. The value always lies between the European value, which allows no early exercise, and the unrestricted American value. Applying the exercise test at every node is the classic error.

A practice question

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

An employee option cannot be exercised during the first year but can be exercised at any time in the following two years. Which feature of a nonstandard American option does this describe?

  1. A barrier, because exercise is knocked out in year one.Why not: No price level is involved; exercise is simply forbidden for a period of time.
  2. A chooser feature, because the holder decides later whether to exercise.Why not: A chooser decides call or put; here the question is when exercise is permitted.
  3. A gap, because the payoff differs from the strike in year one.Why not: A gap option has a trigger separate from its strike; here there is no payoff difference, only restricted timing.
  4. A lock-out period at the start of its life. - correct

Early exercise restricted during an initial part of the life is a lock-out period. It is a standard way to engineer a nonstandard American option from a standard one.

A harder one

A binomial tree values a standard American put. To value a put that can be exercised only on three specified dates before expiry, which modification is correct?

  1. Apply the test at every node, then scale the result by the share of dates on which exercise is permitted.Why not: Scaling is ad hoc; exercise is either allowed or forbidden at a given node.
  2. Drop the terminal payoff and value only the permitted exercise dates.Why not: The terminal payoff at expiry is always needed to start the backward induction.
  3. Apply the exercise test only at nodes on the permitted dates, using the continuation value elsewhere. - correct
  4. Take the European value and add the intrinsic value at the first permitted date.Why not: Adding intrinsic value ignores the optimal exercise decision and double counts.

Backward induction is unchanged except that max(continuation, intrinsic) is applied only at permitted dates. At other nodes the option cannot be exercised, so its value is the discounted expected continuation value.

The hardest tier

A 3-year put has S = 50 and K = 52. Each year the price moves up 20% or down 20%; the continuously compounded risk-free rate is 5%, so the risk-neutral up probability is 0.628. Early exercise is permitted only at the end of year 2 (and at expiry in year 3). The value is closest to:

  1. 5.26Why not: Applies early exercise at every node (the unrestricted American value) instead of only at year 2.
  2. 4.54 - correct
  3. 4.22Why not: Ignores the year-2 exercise right and gives the European value.
  4. 4.74Why not: Takes the midpoint of the European and American values instead of building the restriction into the tree.

Work back from the year-3 payoffs, discounting with p = 0.628. Test max(continuation, intrinsic) only at the year-2 nodes and nowhere else. The result is about 4.54, between the European 4.22 and the American 5.26.

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

Start free →

More in Exotic Options