具有兩種風險資產的投資組合的投資者終值 1) 相關 2) 不相關φ1噸=小號2噸,φ2噸=小號1噸ϕt1=St2,ϕt2=St1phi_t^1=S^{2}{t}, phi_t^2=S^{1}{t}
我正在分析一個問題，其中我有兩隻股票由方程式描述
$$ \frac{dS^{1}{t}}{S^{1}{t}}=\mu_{1} dt + \sigma_{1} dW^{1}{t} $$ $$ \frac{dS^{2}{t}}{S^{2}{t}}=\mu{2} dt + \sigma_{2} dW^{2}_{t} $$ 在哪裡 $ \rho $ 是風險資產之間的相關性。
投資者從一些資本 x 開始投資於 $ \phi_t^1=S^{2}{t}, \ \phi_t^2=S^{1}{t} $ 戰略。假定安全率為零。
我想獲得投資者的財富 $ X_t $ 按照 $ Y_t $ 兩種情況
$ 1) \ \rho=0 $
$ 2) \ \rho \neq 0 $
我得到以下風險資產投資組合價值的方程式：
$ 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} $
和
$ \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}} $
我的直覺是需要應用自籌資金財產，因此 Xt 將等於一些資本 x + 風險資產的最終價值 - 乞求風險資產的價值投資，投資組合的變化將以某種方式表示為 $ dX_t= S_t^1 dS_t^2 + S_t^1 dS_t^2 $
我正在嘗試使用自籌資金財產方程來推導 X_t 但不知道如何推導解決方案中給出的最終公式。我在這裡錯過了一些觀點，我被卡住了。誰能解釋這個問題應該如何分析？應該從什麼出發點以及如何進一步進行？
最終方程應該看起來像
$ 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) $
讓 $ Y_t := 2 S_t^1 S_t^2 $ . 將（多變數）Itô應用於函式 $ f(t,S_t^1,S_t^2)=2 S_t^1 S_t^2 $ 產生一個隨機微分方程 $ Y_t $
$$ \frac{dY_t}{Y_t} = \frac{dS_t^1}{S_t^1} + \frac{dS_t^2}{S_t^2} + \rho \sigma_1 \sigma_2 dt $$ 將伊藤引理重新應用於函式 $ f(t,Y_t) = \ln(Y_t) $ 然後產生
$$ d\ln Y_t = (\mu_1 + \mu_2 - \frac{\sigma_1^2 + \sigma_2^2}{2}) dt + \sigma_1 dW_t^1 + \sigma_2 dW_t^2 $$ 可以集成在 $ [0,T] $ 獲得
$$ Y_T = Y_0 e^{(\mu_1+\mu_2-\frac{\sigma_1^2 + \sigma_2^2}{2})T + \sqrt{(\sigma_1^2 + \sigma_2^2 + 2\rho\sigma_1\sigma_2)}\ W_T} $$ 在哪裡 $ 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.