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Currently I'm thinking about how to calculate the expectation of the product of two euro option, which is

$E[(S_T-K_1)^+(S_T-K_2)^+]$

I can fit some parametric vol model from the market listed option price. The confused part is when the volatility is a function of $ln(F/K)$ (for example, SVI), how to calculate this expectation.

Many Thanks

OneDayMemo
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    For more complicated European payoffs, when the payoff does not depend on terminal (realized) volatility but only terminal stock price, you can always use the Carr Madan formula. It is model-free, meaning basically you can always use it (vol smile or no vol smile). Take a look here for the formule: https://quant.stackexchange.com/questions/27626/carr-madan-formula –  Apr 01 '22 at 08:57
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    If both calls written on same asset and same expiry, then you only get a positive payoff if ST>max(K1,K2). Assume K1<K2, i think this looks like a strip of calls, starting at K2 with first option notional K2-K1 and then higher strikes have more positive notional. – James Spencer-Lavan Apr 01 '22 at 11:40
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    Let $K=\max(K_1, K_2)$. Then \begin{align} (S_T-K_1)^+(S_T-K_1)^+ &= (S_T-K)(S_T-K)^+\ &= S_T(S_T-K)^+ - K(S_T-K)^+. \end{align} The calculation should now be straightforward assuming a constant volatility approximated by the volatility corresponding to $K$. – Gordon Apr 01 '22 at 19:05
  • Thank you all! Very helpful – OneDayMemo Apr 08 '22 at 06:25

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