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Outline the basic reasoning of static option replication, and how hedges of exotic options can be built with it.

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

The idea

Static options replication hedges an exotic with a fixed portfolio of vanilla options that is not rebalanced. The portfolio is chosen so that its value equals the exotic's value everywhere on a boundary, for instance the barrier. Its payoff must also equal the exotic's at expiry in the region where the exotic survives. If the boundary is hit, the portfolio is sold at the same value as the exotic, so the hedge is unwound at no cost. For example, with the interest rate equal to the dividend yield, put-call symmetry gives a down-and-out call with strike K and barrier H as a long call plus a short position in K ÷ H puts with strike H² ÷ K. Variance swaps are replicated by a strip of out-of-the-money options with weights proportional to 1 ÷ K². A binary option is hedged by a call spread. The benefit is avoiding delta hedging near a barrier, but the approach depends on the model assumptions.

Formulas

Static replication of a down-and-out call

down-and-out call = c(K) − (K ÷ H) × p(H² ÷ K)

c(K)
a vanilla call with strike K
K
the strike of the down-and-out call
H
the barrier, below the current price
p(H² ÷ K)
a vanilla put with strike H² ÷ K and the same expiry

Relies on put-call symmetry, which needs the interest rate to equal the dividend yield. At the barrier the portfolio is worth zero and is sold.

A practice question

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

Which statement best describes static options replication of an exotic option?

  1. Hold a fixed portfolio of vanilla options matching the exotic's value on a boundary and at expiry, unwinding it if the boundary is hit. - correct
  2. Rebalance a position in the underlying and cash continuously to track the option's delta.Why not: Describes dynamic delta hedging, which needs continuous rebalancing.
  3. Hold a fixed position in the underlying sized to the exotic's initial delta.Why not: A static delta hedge in the underlying drifts away as delta changes; it is not a replication with options.
  4. Buy the same exotic from another dealer and hold it to expiry.Why not: Transferring risk through an offsetting trade is not replication using vanilla options.

Static replication uses vanilla options chosen to match the exotic where its value is most awkward, such as on a barrier, and then leaves them alone. If the barrier is hit, the portfolio is liquidated at the exotic's value.

A harder one

A variance swap is replicated statically by a strip of out-of-the-money options in which the number of options at each strike is proportional to 1/K² (for equal strike spacing). The strip holds 100 puts struck at 80. How many calls struck at 100 does it hold?

  1. 80Why not: Uses weights proportional to 1/K, giving 100 x 80/100.
  2. 64 - correct
  3. 156Why not: Inverts the ratio, giving 100 x (100/80)², so higher strikes get more weight.
  4. 100Why not: Treats the weights as equal across strikes.

Weights fall with the square of the strike: 100 x (80 / 100)² = 64. Low strikes carry more options because variance is more sensitive to moves at low prices.

The hardest tier

A down-and-out call has barrier H = 50 and strike K = 55; r = q so put-call symmetry applies. The static replication is a long call plus a short position of K/H = 1.1 puts with strike H²/K = 45.45. When the price reaches 50, the put with strike 45.45 is worth USD 3.10. What is the call with strike 55 worth at that moment?

  1. USD 3.41 - correct
  2. USD 2.82Why not: Multiplies by H/K instead of K/H.
  3. USD 3.25Why not: Multiplies by the square root of K/H instead of by K/H.
  4. USD 3.75Why not: Multiplies by K divided by the put strike (K/K2), omitting the square root in the symmetry formula.

At S = H the replicating portfolio is worth zero: call = (K/H) x put = 1.1 x 3.10 = USD 3.41. That is why the portfolio can be unwound at no cost when the barrier is hit.

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