2
$\begingroup$

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 $


$2) \ \rho \neq 0$

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

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

$\endgroup$
  • $\begingroup$ Are you sure for the factor $2$ in your $dY_t/Y_t$ equation? Plus, it is not clear at all what you are asking: what is $X_t$ and what is $x$. If you just invest in the portfolio, $Y_t$ isn't your wealth equal to $Y_T - Y_0$ over the period $[0,T]$ ? In that case you just need to find the expression of $Y_T$. If you took this exercise from a book/notes, please provide a reference, because it is very fuzzy as such. $\endgroup$ – Quantuple Apr 27 '16 at 11:13
  • $\begingroup$ @Quantuple An investor starts with some capital x and invests in the $\phi_t^1=S^{2}_{t}, \ \phi_t^2=S^{1}_{t}$ strategy. Safe rate equals zero. $\endgroup$ – Michal Apr 27 '16 at 11:17
  • $\begingroup$ another hint I got is that for $\rho =0 \ \ d(S_t^1 S_t^2)=dY_t$ but I don't get how this applies here $\endgroup$ – Michal Apr 27 '16 at 12:20
  • $\begingroup$ the factor 2 in the $dY_t/Y_t$ was incorrect. it goes away indeed. I corrected that $\endgroup$ – Michal Apr 27 '16 at 12:32
  • $\begingroup$ Since you defined $Y_t := S_t^ 2 S_t^1 + S_t^1 S_t^2 = 2 S_t^1 S_t^2 $, I can see how $dY_t = d(2 S_t^1 S_t^2)$ but not $d(S_t^1 S_t^2) = dY_t$. Where does this question come from? $\endgroup$ – Quantuple Apr 27 '16 at 12:56
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.