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

Question

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.

Answer 1

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 $g(S) = \max(S-K,0)^2$

Answer 2

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 $K$.}