# Replicating the square of an option $C^2 (S,K,t,T)$

Given a vanilla options market, i.e. $$C(S,K,t, T)$$ for all strikes $$K$$, is it possible to replicate $$C^2 (S,K,t,T)$$? So I am looking for a self-financing portfolio which has a price equal to $$C^2(S,K,t,T)$$ for a fixed strike $$K$$ and for all $$t$$.

Thanks.

There is no terminal $$\mathcal{F}_T$$ mesurable payoff $$g$$ such that $$e^{-r(T-t)} E_t[g] = C(S_t, t, T, K)^2$$, simply because $$E_t[g]$$ must be a martingale and $$e^{r(T-t)} C(S_t, t, T, K)^2$$ is not.

So any deal that has npv $$C(S_t, t, T, K)^2$$ must involve a stream of intermediary payoffs $$h(S_t,t) dt$$, which you can solve for by plugging $$V(S,t) = C(S, t, T, K)^2$$ in the BS PDE $$\frac{\partial V}{\partial t} + r S \frac{\partial V}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} -rV + h(S,t) = 0$$ to obtain $$h(S,t) = -\left(\frac{\partial V}{\partial t} + r S \frac{\partial V}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - rV\right)$$ along with the terminal payoff $$g(S) = \max(S-K,0)^2$$

• Ok, I was wondering if there wasn't a $-rV$ term missing. But it's the discounted payoff $e^{-r(T-t)}E_t[g]$ that must be a payoff, why do you write $\color{blue}{e^{r(T-t)}}C(S_t,t,T,K)^2$ must be a martingale, and not $C(S_t,t,T,K)^2$? – Daneel Olivaw Mar 19 at 13:39
• Yes I fixed some typos, sorry about that. $e^{-rt} C(S_t, t, T, K)^2$, or equivalently $e^{r(T-t)} C(S_t, t, T, K)^2$, must be a martingale if $C(S_t, t, T, K)^2$ is to represent the npv of a single terminal payoff. – Antoine Conze Mar 19 at 13:44
• Ok. Just to add to the excellent answer, note that the expression for $h(S,t)$ can thus be decomposed in 1) the (Black-Scholes) hedging P&L for a vanilla option, 2) a funding cost $rV$, and 3) a third term $(\sigma S\Delta_t^{BS})^2$. Not sure how to interpret/hedge this last term... – Daneel Olivaw Mar 19 at 13:59

I assume your trade $$V(S,K,t,T)$$ is European. Its payoff is: \begin{align} V(S,K,T,T)&=C^2(S,K,T,T) \\[3pt] &=\max(S_T-K,0)^2 \\[3pt] &=\boldsymbol{1}_{\{S_T\geq K\}}(S_T-K)^2 \\[3pt] &=\boldsymbol{1}_{\{S_T\geq K\}}f(S_T) \end{align} where $$f(x)=(x-K)^2$$. By Carr-Madan's static replication formula (see this question or this paper), we have(1): \begin{align} f(S_T)&=f(K)+f'(K)(S_T-K)+\int_0^{K}f''(k)(k-S_T)^+\text{d}k+\int_{K}^{\infty}f''(k)(S_T-k)^+\text{d}k \\[3pt] &=2\int_0^{K}(k-S_T)^+\text{d}k+2\int_{K}^{\infty}(S_T-k)^+\text{d}k \end{align} where $$(x)^+=\max(x,0)$$. Multiplying by $$\boldsymbol{1}_{\{S_T\geq K\}}$$: $$\boldsymbol{1}_{\{S_T\geq K\}}f(S_T)=2\int_{K}^{\infty}(S_T-k)^+\text{d}k$$ Multiplying by the discount factor $$D(t,T)$$ and taking the conditional expectation under the risk-neutral measure $$Q$$, we get the following theoretical replicating strategy: $$V(S,K,t,T)=2\int_{K}^{\infty}C(S,k,t,T)\text{d}k$$ Given in practice there is no availability of a continuum of call options, the following approximation is made: $$V(S,K,t,T)\approx2\sum_{i=0}^nC(S,k_i,t,T)\delta_i$$ where $$\{k_i:i=0,\dots,n\}$$ are the quoted strikes with $$k_0=K$$ and $$\delta_i=k_{i+1}-k_i$$.

(1) We have chosen as threshold value the strike $$K$$.

• Thanks, but I probably did not state my question as clearly as I should have. What I am looking for is the replicating portfolio for $\left( E_t \left[ (S_T-K)_+ \right] \right)^2$. What you gave is an upper bound, namely $E_t \left[ (S_T - K)^2_+ \right]$. At maturity they will of course be the same. I may actually have found the answer myself, but still checking a few things before I dare post it. In the meanwhile of course if you or anyone else has the solution would be great to know. – ilovevolatility Mar 16 at 18:27
• @ilovevolatility What is the payoff of your trade? From your description it seems the only cash flow is $\max(S_T-K,0)^2$ at maturity, as you state in your comment, with no intermediate payment thus by no arbitrage the price of your “squared option” must be the same as the price of the claim I describe and hence the hedging strategy. Can you be more specific regarding the structure of your deal? – Daneel Olivaw Mar 16 at 21:07
• It's true that both have the same value at maturity but they are still different. Take for example the special case $K=0$, then the difference $E_t[ S_T^2] - (E_t[S_T])^2$ is related to the volatility of $S$. – ilovevolatility Mar 17 at 7:18
• So you are asking: is there a payoff whose price is the square of the regular option price. – dm63 Mar 17 at 9:15
• @dm63 Exactly, so the question is what is $g(S_{T'})$ such that for all $t < T' < T$ we have $E_t [g(S_{T'})] = \left( C(S_t,t,K,T) \right)^2$. I suspect I'd need options on options but would like to avoid that if there is any other way e.g. by dynamic trading in vanilla options. Since $dC^2 = 2CdC + (dC)^2$ I need to hold $2C$ options at any time, plus something that has a p/l of $(dC)^2$ over a time interval $dt$. It's the last part that I am struggling with, maybe I am missing something very simple though. – ilovevolatility Mar 17 at 10:44