I have a derivative that pays off $S_T^2$ at time $T > 0$ with $S_T$ denoting the price of a non dividend-paying stock at $T$. I came across a question about how one can statically replicate this derivative with vanilla calls and puts.

My guess is that it is impossible to do that on the entire support of $S_T$. Since the square function dominates a linear function eventually and the call option is linear in $S_T$ for $S_T$ large enough, there cannot be a sequence of linear combinations of calls and puts that converges to the payoff of this derivative pointwise. I was also given a hint that I should consider integration. I am aware that $S_T^2$ can be written as $S_T^2 = 2\int_0^{S_T}x\,dx$ but I am not sure if that is what the hint hints at. Any tips/solutions appreciated.

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    $\begingroup$ To add to the excellent answer you've received, note that such a replication is actually possible for any European payout, ie a payout which only depends on the terminal value of the asset, see the Carr Madan 98 paper or this answer: quant.stackexchange.com/questions/27626/carr-madan-formula $\endgroup$ – Quantuple Aug 16 '17 at 7:05

Note that \begin{align*} S_T^2 = 2\int_0^{S_T} k dk. \end{align*} Then \begin{align*} S_T^2 &= 2S_T^2-2\int_0^{S_T} k dk\\ &=2S_T\int_0^{S_T}dk-2\int_0^{S_T} k dk\\ &=2\int_0^{S_T} (S_T-k)dk\\ &=2\int_0^{\infty} (S_T-k)^+dk. \end{align*} For the partition $0=k_0 < k_1 < \cdots < k_n < \infty$, \begin{align*} S_T^2 &=2\int_0^{\infty} (S_T-k)^+dk\\ &\approx 2\sum_{i=1}^n (k_i-k_{i-1})(S_T-k_i)^+. \end{align*} That is, it can be replicated by a portfolio of call options. The replication by put options is similar.

  • $\begingroup$ Thanks. But I am not sure how you make the Riemann sum converge to the improper integral. The "partition" has to grow and get denser simultaneously with increasing $n$. So you need an infinite number of calls. Their values sum up to something finite but still it sounds weird in my head. I will think about this more. $\endgroup$ – Calculon Aug 15 '17 at 20:43
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    $\begingroup$ You can chose a large quantity such as $10E(S_T)$ as the upper limit. For the integral, you can use techniques such as Gaussian quadrature. $\endgroup$ – Gordon Aug 15 '17 at 22:05

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