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.

  • 2
    $\begingroup$ 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
    Commented Aug 16, 2017 at 7:05

1 Answer 1


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
    Commented Aug 15, 2017 at 20:43
  • 1
    $\begingroup$ 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
    Commented Aug 15, 2017 at 22:05
  • $\begingroup$ Hi, thanks so much for your answer. Quick qn on utilizing both calls and puts. As compared to your answer in quant.stackexchange.com/questions/27626/carr-madan-formula/…, how do we relate/equate between using itm calls and using S^2 + otm puts? $\endgroup$
    – chinsoon12
    Commented Feb 14, 2022 at 6:30
  • 1
    $\begingroup$ This can not be shown in a few words. You may ask as another question, and then more people can access and try. $\endgroup$
    – Gordon
    Commented Feb 14, 2022 at 13:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.