E[Φ(Wt)]\mathbb{E}[\Phi(W_t)]E[Φ(Wt)] Cumulative Brownian 专题 Finance / 金融 难度 L4 来源 QuantQuestion 题目详情 Calculate E(Φ(Wt))\mathbb{E}\left(\Phi \left(W_t\right)\right)E(Φ(Wt)) where WtW_tWt a brownian motion and Φ\PhiΦ the standard normal cumulative distribution. 解析 由对称性 Wt=d−WtW_t\overset{d}{=}-W_tWt=d−Wt： E[Φ(Wt)]=E[Φ(−Wt)]=E[1−Φ(Wt)]=1−E[Φ(Wt)],\mathbb{E}[\Phi(W_t)]=\mathbb{E}[\Phi(-W_t)]=\mathbb{E}[1-\Phi(W_t)]=1-\mathbb{E}[\Phi(W_t)],E[Φ(Wt)]=E[Φ(−Wt)]=E[1−Φ(Wt)]=1−E[Φ(Wt)], 因此 E[Φ(Wt)]=12.\boxed{\mathbb{E}[\Phi(W_t)]=\frac12}.E[Φ(Wt)]=21.