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Suppose we sell a put option with maturity $T$, strike $K$ and fee $P_t=v(t, S_t, T, K, ...)$. The replicating portfolio consists of holding $\alpha_t = \frac{\partial{P}}{\partial{S}}=:\Delta_t$ units of stock, $S_t$, and $\beta_t = (P_t-\frac{\partial{P}}{\partial{S}} S_t)/M_t$ units of the money-market account or risk-free bond $M_t$, where $dM_t=rM_t dt$. For a put option, the delta is negative so going long the replicating portfolio means shorting the stock and holding cash.

Suppose now that we cannot directly short-sell any stock but we can instead buy a highly inversely correlated stock. Is it possible to find a replication strategy still? Informally, I think one could take the total dollar value of the short-position divided by the inversely correlated stock price: $\alpha_2 =-\Delta \cdot S/S_2$ to obtain the number of units to buy of the second inversely correlated stock, assuming the correlation is $\rho=-1$ and otherwise we would need to factor in the correlation somehow... I suppose rigorously one would try to solve for $(\alpha, \beta)$ in $$\alpha_t \, dS_2(t)+ \beta_t \, dM(t)=dP(t),$$ assuming $\pi_t:= \alpha_t S_2(t)+\beta_t M(t)$ is a self-financing portfolio, where $$dS_2(t)=\mu_2 S_2(t) dt +\sigma_2S_2(t)\left(\rho dB_1(t)+\sqrt{1-\rho^2}dZ(t)\right),$$ and $B=(B_t)$ and $Z=(Z_t)$ are independent Brownian motions with the former driving $S_1(t)$, i.e. $dS_1(t)=\mu_1 S_1(t)dt+\sigma_1 S_1(t) dB_1(t)$. Now when we can actually hedge with the underlying, $S_1$, without restrictions, we simply use the pricing PDE to write $dP=(P-S_1\Delta)rdt+\Delta dS_1$ and can identify $(\alpha, \beta)$ as given above. As far as I can see, this does not work in the new scenario and here is where I am stuck.

In short, my question is how, if even possible, can we replicate a put option when short-selling the underlying is not allowed but we may trade inversely correlated stocks? Please comment if I've made any mistakes in my setup or assumptions, thanks.

Update 3/17/2020

Numerical experiments in R under Black-Scholes dynamics suggest that $\alpha_t = \gamma\frac{\sigma_1 S_1(t) \Delta_t}{\sigma_2 S_2(t) \rho}$ for some proportion $\gamma$ is an ad-hoc approach that works, with $\beta=\gamma(P_t-\Delta_t S_1(t))/M_t$. But I have not verified this is self-financing, etc nor have formal proof it replicates $P_t$, or any objective criteria of selecting $\gamma$ yet. So my doubts/question still remains.

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    $\begingroup$ Is it possible to trade other options on the underlying? If it is, if there are other options on the asset that are traded already then I can give you an answer that does not depend on other correlated assets. $\endgroup$ – ilovevolatility Mar 29 at 14:29
  • $\begingroup$ @ilovevolatility Yes, calls and puts. I would be interested in the answer and will edit the question to include such case. Thank you for commenting. $\endgroup$ – Nap D. Lover Mar 29 at 17:58
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how, if even possible, can we replicate a put option when short-selling the underlying is not allowed but we may trade inversely correlated stocks?

Ok first thing first. Your self-financing portfolio fails because you have two different risk factors (brownian motions), so you will never be able to replicate the put option without involving the underlying. So the portfolio will not be riskless no matter how much you try. While you can not be riskless, you can still reduce risk by using a negatively correlated stock, but you will still bleed PnL by not being able to hedge perfectly (far from it). This is because with $\rho \neq -1$, you will only partially be able to substitute the dynamics of $S_1$ with $S_2$.

You know that you can perfectly hedge the Option with $n_1=\Delta_t$ units of it. How may units of $S_2$ do we need to hedge/substitute $S_1$?

$$ dV = n_1 \cdot dS_1 - n_2 \cdot dS_2 = \{\text{insert equatons for dS_1 and dS_2}\} $$ Take variance of both sides, differentiate wrt $n_2$ and find minimal variance. You get that $$ n_2^*(t) = \frac{n_1(t)\sigma_1S_1(t)}{\sigma_2S_2(t)}\rho = \frac{\sigma_1S_1(t)\Delta_t}{\sigma_2S_2(t)}\rho $$

Now insert this solution in your differential equation for your portfolio consisting of the Option, money account and the stock $S_2$, and you will get the dynamics for this portfolio (which will not be riskless).

You will notice that, you need a rather large negative correlation in order to hedge well, which will be difficult to find.

An alternative approach, which is also mostly used in practise t hedge market risk, is to use either single stock futures or index futures to hedge. They are both cheaper (commission wise) (esp. index) and do not have any short-selling constraint, and provide leverage since you mainly only fork up cash for the margin ~20%.

As has been mentioned in the comments, you can also use call options on the same underlying in order to constrain your risk on the put. For instance, you can buy a call with the same expiry and strike, using put-call parity you recover the stock, and you hedge with an equity futures, which means you no longer need to dynamically hedge as frequently due to removing most convexity.

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