Pricing squared derivative: Equating S^2 + a strip of otm calls + a strip of otm puts = only calls

Question

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.

Answer 1

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*}