How to replicate a Call EUR /PUT USD performance pay off CCY 1 = Nominal CCY * Max( Spot final - K) / K) > vanilla pay-off

[DavidFX]

Hi everybody,

I try to replicate a performance Call EUR / Put USD K 1.10 maturity 1 year.
Pay-off : CCY 1 = Nominal CCY * Max( Spot final - K) / K) > vanilla pay-off.
Step 1 :
You need the valuation of the vanilla Call Strike 1.10 mat 1 year.

Step 2 :
You need to convert the pay-off the call at strike rate. The amount is the EUR notional * (K - Spot Final).
Problem : we don't know the final pay-off of the vanilla call.
But we can evaluate with a risk neutral probability at maturity.
N * (K - Spot Final) * 1(ST>K)
N * K1(ST>K) - S* 1(ST>K)

Expected Cash flow
N * K * P(ST>K) = N * K * N(d2) - E(ST / ST>K) * P(ST>K)
With discounting
N * K e-it * N(d2) - N * Se-rt * N(d1)

The amount of the call to buy is given by this equation.
It 's not a perfect hedge but It allows to convert the pay off of the call at strike price.

PV performance Call = PV Vanilla call + PV vanilla call 1.10 amount (N * K e-it * N(d2) - N * Se-rt * N(d1))

Do you agree with that ?

Many thanks for your help
David