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

Show how a zero-cost version of any derivative can be created.

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

The idea

Any derivative can be made zero cost at inception by changing how or whether the premium is paid. The simplest route is deferring payment, so the holder pays nothing today and pays at expiry whether or not the option is exercised. The deferred payment is the premium c compounded at the risk-free rate r over the life T. Deferred payment = c × e^(rT). A variant, the contingent-premium option, charges the premium at expiry only if the option ends in the money, so it must be larger. Contingent payment = c × e^(rT) ÷ Q, where Q is the risk-neutral probability of finishing in the money. A second route is to combine options so the premiums offset. A zero-cost collar, or range forward, buys a put at a lower strike and sells a call at a higher strike, with the strikes chosen so the two premiums are equal. The usual mistake is forgetting to compound the deferred premium. Zero cost is not zero risk.

Formulas

Deferred premium of a pay-later option

deferred premium = c × e^(r × T)

c
the premium of the option paid today
r
the continuously compounded risk-free rate
T
the time to expiry, in years

Paid at expiry whether or not the option is exercised. Forgetting to compound the premium is the usual mistake.

Premium of a contingent-premium option

contingent premium = c × e^(r × T) ÷ Q

c
the premium of the option paid today
r
the continuously compounded risk-free rate
T
the time to expiry, in years
Q
the risk-neutral probability that the option ends in the money

Paid only if the option ends in the money, so it is always larger than the pay-later premium.

A practice question

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

Which technique can turn any option into a zero-cost structure at inception?

  1. Choosing a strike equal to the forward price, which is said to make the premium zero.Why not: An at-the-forward option still has a positive premium; only the intrinsic value is zero.
  2. Delta hedging the option continuously from inception.Why not: Hedging manages risk but does not remove the premium.
  3. Deferring payment of the premium to expiry, owing the premium compounded at the risk-free rate. - correct
  4. Buying the option for twice the notional so the time value is covered.Why not: Scaling the notional scales the premium; it does not remove the premium.

Deferring the premium to expiry costs nothing today, and the holder owes c x e^(rT) at expiry. Equally, combining a bought and sold option with equal premiums yields a zero-cost package.

A harder one

A 2-year at-the-money call costs USD 6.50 today. A dealer offers a pay-later version: nothing upfront and a fixed payment at expiry, whether or not the option is exercised. The continuously compounded risk-free rate is 4%. The fair payment is closest to:

  1. USD 6.77Why not: Compounds for one year only (6.50 x e^0.04) instead of two years.
  2. USD 6.00Why not: Discounts the premium (6.50 x e^-0.08) instead of compounding it forward.
  3. USD 6.58Why not: Adds the interest rate as an amount, 6.50 + 2 x 0.04, instead of compounding the premium.
  4. USD 7.04 - correct

Deferred premium = c x e^(rT) = 6.50 x e^(0.04 x 2) = 6.50 x 1.0833 = USD 7.04, the future value of the premium at the risk-free rate.

The hardest tier

A 6-month call costs USD 5.00. A contingent-premium version requires no upfront payment; at expiry the holder pays a fixed amount X only if the option finishes in the money. The risk-neutral probability of finishing in the money is 55%, and the continuously compounded risk-free rate is 4%. The fair X is closest to:

  1. USD 5.10Why not: Compounds the premium but ignores that payment is made only with probability 55%.
  2. USD 9.09Why not: Divides by the probability but forgets to compound the premium to expiry.
  3. USD 9.27 - correct
  4. USD 8.91Why not: Divides by the probability but discounts the premium (e^-0.02) instead of compounding it.

The expected present value of the payment must equal the premium: 0.55 x X x e^(-0.04 x 0.5) = 5.00, so X = 5.00 x e^0.02 / 0.55 = USD 9.27. It exceeds the pay-later premium of 5.10.

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