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In Peter Carr, Dilip Madan, Towards a Theory of Volatility Trading, 1998, (also derived here by Gordon), both calls and puts are used to replicate any twice differentiable payoff. I suppose one would choose the atm fwd to be the kappa and then use otm options to price the derivative.

In the answer to Replicating a square derivative with calls and puts, the payoff of the squared derivative is replicated with only call options.

How do we show that the original formula in Carr and Madan paper and the replication using only call options are equal?

This feels like its related to put-call parity and integrating the spot to get the first term on the original formula. But I am lost with how the strike term in put-call parity are handled.

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For a sufficiently smooth function $f$ and positive constant $a$, \begin{align*} f(x) &= f(a) + f'(a) (x-a) + \int_a^{\infty}(x-u)^+f''(u)du + \int_{0}^a(u - x)^+f''(u)du. \end{align*} Then \begin{align*} S_T^2 &= a^2 + 2a(S_T-a) + 2\int_a^{\infty}(S_T-u)^+du + 2\int_{0}^a(u - S_T)^+du\\ &=a^2 + 2a(S_T-a) + 2\int_0^{\infty}(S_T-u)^+du + 2\int_{0}^a\Big[(u - S_T)^+-(S_T-u)^+\Big]du\\ &=a^2 + 2a(S_T-a) + 2\int_0^{\infty}(S_T-u)^+du +2\int_{0}^a (u - S_T)du\\ &=a^2 + 2a(S_T-a) + 2\int_0^{\infty}(S_T-u)^+du + a^2-2aS_T\\ &=2\int_0^{\infty}(S_T-u)^+du. \end{align*}

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  • $\begingroup$ In practice though, this should not be used for a squared payoff because of this quant.stackexchange.com/questions/65437/… $\endgroup$
    – Peter A
    Commented Feb 16, 2022 at 12:47
  • $\begingroup$ This is of course not a usual option payoff. However, it may be used for some other purpose, for example, to relate a single volatility to a volatility surface, by valuing using both a single vol and a surface. $\endgroup$
    – Gordon
    Commented Feb 16, 2022 at 13:22

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