三资产乘积的均值与相关

The prices of three assets

The prices $M_{t},N_{t}$ ,and $P_{t}$ of three assets have the following dynamics:

\begin{array}{r l} & {d M_{t} = \mu_{M}M_{t}d t + \sigma_{M}M_{t}d W_{M}(t);}\\ & {d N_{t} = \mu_{N}N_{t}d t + \sigma_{N}N_{t}d W_{N}(t);}\\ & {d P_{t} = \mu_{P}P_{t}d t + \sigma_{P}P_{t}d W_{P}(t).}\\ & {d W_{M}(t)d W_{N}(t) = \rho_{M N}d t;}\\ & {d W_{N}(t)d W_{P}(t) = \rho_{N P}d t}\\ & {d W_{P}(t)d W_{M}(t) = \rho_{M P}d t.} \end{array}

Assume that

\begin{array}{r}dW_{M}(t)dW_{N}(t) = \rho_{MN}dt;\\ dW_{N}(t)dW_{P}(t) = \rho_{NP}dt;\\ dW_{P}(t)dW_{M}(t) = \rho_{MP}dt. \end{array}

What is the mean of $M_{t}N_{t}P_{t}$ ? What is the correlation between $M_{t}$ and $N_{t}P_{t}$ ?

解析

三资产均为相关 GBM：

\frac{dM_t}{M_t}=\mu_Mdt+\sigma_M dW_M,\quad \frac{dN_t}{N_t}=\mu_Ndt+\sigma_N dW_N,\quad \frac{dP_t}{P_t}=\mu_Pdt+\sigma_P dW_P,

且 $dW_M dW_N=\rho_{MN}dt$ 等。

(1) $\mathbb{E}[M_tN_tP_t]$ ：把对数项合并，得到

\boxed{\mathbb{E}[M_tN_tP_t]=M_0N_0P_0\exp\Bigl((\mu_M+\mu_N+\mu_P)t +t(\rho_{MN}\sigma_M\sigma_N+\rho_{NP}\sigma_N\sigma_P+\rho_{MP}\sigma_M\sigma_P)\Bigr)}.

(2) $\operatorname{Corr}(M_t, N_tP_t)$ ：令

v_M=\sigma_M^2 t, \quad v_{NP}=(\sigma_N^2+\sigma_P^2+2\rho_{NP}\sigma_N\sigma_P)t, \quad c=(\rho_{MN}\sigma_M\sigma_N+\rho_{MP}\sigma_M\sigma_P)t.

则对数正态相关系数为

\boxed{\operatorname{Corr}(M_t,N_tP_t)=\frac{e^{c}-1}{\sqrt{(e^{v_M}-1)(e^{v_{NP}}-1)}}}.