Self-Financing Cross-Holdings and Correlated Asset Returns

# 25657


# investor terminal value of portfolio with two risky assets 1) correlated 2)not correlated $\phi_t^1=S^{2}_{t}, \ \phi_t^2=S^{1}_{t}$












I am analyzing a problem where I have two stocks described by the equations $$ \frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t}$$ $$ \frac{dS^{2}_{t}}{S^{2}_{t}}=\mu_{2} dt + \sigma_{2} dW^{2}_{t}$$

where $\rho$ is the correlation between the risky assets.

An investor starts with some capital x and invests in the $\phi_t^1=S^{2}_{t}, \ \phi_t^2=S^{1}_{t}$ strategy. The safe rate is assumed to be zero.

I want to derive the investor wealth $X_t$ in terms of $Y_t$ for two cases

$1) \ \rho=0$ $2) \ \rho \neq 0$

I am getting the following equations for the value of portfolio of the risky assets:

$ Y_{t} = (\phi_t^1 )S^{1}_{t} + (\phi_t^2 )S^{2}_{t} = S^{2}_{t} S^{1}_{t} + S^{1}_{t} S^{2}_{t}$

and

$\frac{dY_t}{Y_t} = \frac{dS^{1}_{t}}{S^{1}_{t}} + \frac{dS^{2}_{t}}{S^{2}_{t}} + \frac{ <S^{1} S^{2}>_t}{S^{1}_{t} S^{1}_{t}}$

My intuition here is that self financing property needs to be applied, so the Xt would be equal to some capital x + end value of risky assets - beg value investment in risky assets and the change in the portfolio would be somehow represent as $dX_t= S_t^1 dS_t^2 + S_t^1 dS_t^2$

I am trying to use the self financing property equation to derive X_t but don't know how to derive the final formulas given in the solutions. I missing some point here, I am stuck. Can anybody explain the how this problem should be analyzed? what should be the starting point and how to proceed further?

the final equations should look like

$1) \rho=0$

$X_t=Y_t - S_0^1 S_0^2 +x $

$2) \ \rho \neq 0$

$dX_t = d(S_t^1 S_t^2) - d\langle S_1, S_2 \rangle_t = \frac{1}{2} dY_t - \frac{1}{2} \rho \sigma_ 1 \sigma_2 Y_tdt$

$X_t=x+ \frac{1}{2} (Y_t -Y_0 - \rho \sigma_ 1 \sigma_2 \int_0^t Y_s ds)$

## Answer by Quantuple (score 1, accepted)

https://quant.stackexchange.com/a/25667

Let $Y_t := 2 S_t^1 S_t^2 $. Applying (multivariate) Itô to the function $f(t,S_t^1,S_t^2)=2 S_t^1 S_t^2$ yields a stochastic differential equation for $Y_t$

$$ \frac{dY_t}{Y_t} = \frac{dS_t^1}{S_t^1} + \frac{dS_t^2}{S_t^2} + \rho \sigma_1 \sigma_2 dt $$

Re-applying Itô's lemma to the function $f(t,Y_t) = \ln(Y_t)$ then yields $$ d\ln Y_t = (\mu_1 + \mu_2 - \frac{\sigma_1^2 + \sigma_2^2}{2}) dt + \sigma_1 dW_t^1 + \sigma_2 dW_t^2 $$

which can be integrated over $[0,T]$ to obtain $$ Y_T = Y_0 e^{(\mu_1+\mu_2-\frac{\sigma_1^2 + \sigma_2^2}{2})T + \sqrt{(\sigma_1^2 + \sigma_2^2 + 2\rho\sigma_1\sigma_2)}\ W_T} $$ where $Y_0 = 2 S_0^1 S_0^2$ and we have replaced $\sigma_1 W_t^1 + \sigma_2 W_t^2$ by $\sqrt{\sigma_1^2 + \sigma_2^2 + \rho \sigma_1 \sigma_2} W_t$ which is a random variable with the exact same distribution (cf. proof here)

Now, assume a self-financing portfolio consisting of holding $S_t^2$ shares of security 1 at time $t$, along with $S_t^1$ shares of security 2: $$X_t := S_t^2 S_t^1 + S_t^1 S_t^2$$ The self-financing conditions gives, over any infinitesimal period of time $$ dX_t = S_t^2 dS_t^1 + S_t^1 dS_t^2$$ which we can rewrite (simple application of multivariate Itô's lemma) $$ dX_t = d(S_t^1 S_t^2) - d\langle S^1 S^2 \rangle_t $$

Now for $\rho=0$ the quadratic variation part is zero, and integrating $$ dX_t = d(S_t^1 S_t^2) $$ over $[0,T]$ yields a final wealth of: \begin{align} X_T &= X_0 + S_T^1 S_T^2 - S_0^1 S_0^2 \\ &= x + \frac{1}{2}(Y_T - Y_0) \end{align}

For $\rho \ne 0$ we write $dX_t = d(S_t^1 S_t^2) - d\langle S^1 S^2 \rangle_t $ as $$ dX_t = \frac{1}{2} dY_t - \frac{1}{2} \rho \sigma_1 \sigma_2 Y_t dt $$ and integrate over $[0,T]$ to obtain $$ X_T = x + \frac{1}{2} (Y_T - Y_0) - \frac{1}{2} \rho \sigma_1 \sigma_2 \int_0^T Y_t dt $$ Note that by setting $\rho = 0$ in the above we fall-back on the previous result.