Replicating the Integrated Proportional Price Change with Stock

# Strategy of replicating a portfolio with payoff $\int_0^T \frac{dS_t}{S_t}$


# Strategy of replicating a portfolio with payoff $\int_0^T \frac{dS_t}{S_t}$












Given the asset price $S_t$ which is defined as follows $$\frac{dS_t}{S_t}= r_tdt+\sigma_tdW_t$$ where $r_t$ is not necessarily deterministic.

What is the strategy of replication of the portfolio with the payoff $\int_0^T \frac{dS_t}{S_t}$ ?

My attempt:

In fact, I can solve this problem only for the special case where $r_t$ is deterministic. For simplicity's sake, I provide the solution for an easier case where $r_t =r$ constant.

Let's $V_t$ the replicating porfolio of $\int_0^T \frac{dS_t}{S_t}$, we have \begin{align} V_t &=e^{-r(T-t)}E^{\Bbb Q}[\int_0^T \frac{dS_u}{S_u}|\mathcal{F}_t] \\ &=e^{-r(T-t)}\int_0^t \frac{dS_u}{S_u}+e^{-r(T-t)}E^{\Bbb Q}[\int_t^T (rdu+\sigma_udW_u)|\mathcal{F}_t] \tag{1}\\ &=e^{-r(T-t)}(\int_0^t \frac{dS_u}{S_u}+r(T-t)) \tag{2}\\ \end{align}

From (2), by applying the Ito's lemma, we obtain easily that \begin{align} dV_t &= re^{-r(T-t)}(\int_0^t \frac{dS_u}{S_u}+r(T-t))dt +e^{-r(T-t)}(\frac{dS_t}{S_t}-rdt) \\ &= r(V_t-e^{-r(T-t)})dt+e^{-r(T-t)} \frac{dS_t}{S_t} \tag{3}\\ \end{align}

From (3), we obseve that we can replicate $V_t$ (which is equal to $\frac{e^{-r(T-t)}}{S_t}S_t+ \frac{V_t-e^{-r(T-t)}}{B_t}B_t$) by investing $e^{-r(T-t)}$ in the asset $S_t$ at time $t$ and the rest of the portfolio $(V_t-e^{-r(T-t)})$ in cash.

Problem:

For the general case where $r_t$ is stochastic, I don't know how to deduce (2) from (1), or (3) from (2).

I guess the strategy in the general case must be investing $P(t,T)$ in the asset $S_t$ ($P(t,T)$ is the zero-coupon bond price between $t$ and $T$) and the rest of the portfolio in cash. But I don't know how to prove that.

The zero-coupon bond $P(t,T)$ is specified by

$$\frac{dP(t,T)}{P(t,T)} = r_tdt + \gamma_t dB_t$$

For simplicity's sake, let's suppose the correlation between $B_t$ and $W_t$ be zero ($\left<dB_t,dW_t\right> = 0$)

## Answer by siou0107 (score 1)

https://quant.stackexchange.com/a/61558

It is actually fairly simple: just hold $\frac{1}{S_t}$ units of the stock at all time! Then, no matter if rates or volatilities are stochastic, the change in value of your portfolio is $\frac{dS_t}{S_t}$ at all time and the terminal value of your portfolio is therefore $$ \int_0^T{\frac{\mathrm{d}S_t}{S_t}} $$

## Answer by user34971 (score 1)

https://quant.stackexchange.com/a/61580

So it is sufficient to assume the existence of zero coupon bonds. We do not have to specify anything else.

Define first

$$ X_t = \int_0^t \frac{dS_u}{S_u} $$

Then $$ dX_t = \frac{dS_t}{S_t} $$

Note that $$ X_T = \frac{X_T}{P_T} = \int_0^T d \left( \frac{X_t}{P_t} \right) $$ since $P_T = 1$ and $X_0 = 0$.

Under the $T$-forward measure $X_t/P_t$ is a martingale, which again shows the current price of the claim is $0$.

We are mainly interested in the expression in the integrand: $$ d \left( \frac{X_t}{P_t} \right) = \frac{1}{P_t} dX_t - \frac{X_t}{P_t^2} dP_t + O(dt) $$ We are not interested in the Ito terms as they add up to zero (pricing PDE).

So the replication should be $$ \frac{1}{P_t S_t} dS_t - \frac{1}{P_t^2} \left( \int_0^t \frac{dS_u}{S_u} \right) dP_t $$

I think this is the way to do this.