Perpetual American Put Supermartingale 財產
美式看跌期權（永續）的貼現價格過程有 $ dt $ 參與其中，如果當時的價格是負數[Math Processing Error] $ t $ is less than the optimal exercise price. This is the only thing that drags the discounted process down as the time goes on and makes the whole process a supermartingale. So when you don’t exercise the option at it’s stopping time, it has a tendency to go down. However, I do not seem to be understanding the intuition behind here, as the process goes down only when the price is less than the optimal exercise price and shouldn’t having a lower stock price make the option more valuable? So what am I getting wrong?
I would not say there is no link to what you say but here would be my view.
Intuitive explanation
如果您在鍛煉前等待延遲[數學處理錯誤] ，您將失去在[數學處理錯誤]和 $ h $ $ t $ $ t+h $ ，這會導致價值損失。
超鞅屬性證明
（在你的情況下應用它： $ \phi_t=e^{-rt}(L-S_t)^+ $ )
如果我們將[ Math Processing Error ]表示為障礙物，並且 $ \phi $ $ \text{Am}(\phi) $ 假設存在最優策略的美國永續期權收益[數學處理錯誤] $ \phi $ $ \tau^\star(t) $ 行使期權知道您當時購買了期權 $ t $ . 允許的策略是停止時間（意味著您只能根據您當時所知道的情況做出決定）大於或等於 $ t $ .
你得到 ：
[數學處理錯誤]$$ \text{Am}(\phi)t=\mathbb{E}(\phi{\tau^{\star}(t)}|\mathcal{F}t)=\sup{\tau\geq t}\mathbb{E}(\phi_\tau|\mathcal{F}_t) $$ 環境 $ \tau=\tau^\star(t+h) $ 右側將您帶到：
[數學處理錯誤]$$ \text{Am}(\phi)t\geq \mathbb{E}(\phi{\tau^\star(t+h)}|\mathcal{F}_t) $$ 使用條件期望的塔屬性：
[數學處理錯誤]$$ \mathbb{E}(\phi_{\tau^\star(t+h)}|\mathcal{F}t)=\mathbb{E}(\mathbb{E}(\phi{\tau^\star(t+h)}|\mathcal{F}_{t+h})|\mathcal{F}_t) $$ 使用第一個等式 $ t+h $ 而在[數學處理錯誤]中： $ t $
[數學處理錯誤]$$ \mathbb{E}(\phi_{\tau^\star(t+h)}|\mathcal{F}{t+h})=\text{Am}(\phi){t+h} $$ 將此插入先前的不等式會導致您：
[數學處理錯誤]$$ \text{Am}(\phi)t\geq \mathbb{E}(\text{Am}(\phi){t+h}|\mathcal{F}_t) $$