# Exotic derivatives - Replication

I would like to replicate the payoff Max(0, Min(S1, K) - S2) with a combination of the following derivatives:

-> option on S1, strike of our choice

-> option on (S1-S2), strike of our choice

-> A residual

S1 and S2 being stocks, and K being fixed.

I'm really stuck, I don't find any solution. Any help?

• Is this a homework? Note that the option payoff is for a rainbow style option, it is difficult to make such decomposition, unless the residual part allows an exotic option payoff itself. – Gordon Jul 27 '18 at 16:28
• Note that \begin{align*} \max(0,\, \min(S_1, \,K)-S_2) &= \max(0, \,\max(S_2-S_1, \,S_2-K))\\ &=\max(S2-S_1,\, S_2-K, \,0)\\ &=S_2 -\min(S_1, \,S_2, \,K). \end{align*} Here $\min(S_1, \,S_2,\, K)$ is the payoff of a rainbow option. – Gordon Jul 27 '18 at 17:00
• As stated the question makes little sense. If you accept a non-zero residual any options on the underlyings would be a solution. So you need to specify criteria on the underlying or the replication. "Option" in your terminology would be any Put or Call but no Barrier or more general function of the underlying, correct? – g g Jul 28 '18 at 8:11
• Furthermore I think that exact replication (i.e. residual is zero function) is not possible. – g g Jul 28 '18 at 8:13