# Lower bound for Bermudan Option Price

i have the following question. The price of an Bermudan option is given by \begin{align*} V_{0} = \sup_{\tau \in \mathcal{T}(0,\dots, T)} \mathbb{E}[f_{\tau}(X_{\tau})]. \end{align*}

It is possible to approximate this price using Monte-Carlo and the continuation values defined as \begin{align*} q_{t}(x) = \sup_{\tau \in \mathcal{T}(t+1, \dots, T)}\mathbb{E}[f_{\tau}(X_{\tau})\mid X_{t} = x]. \end{align*}

My question is now, why do I get a lower bound for the Bermudan option price when calculating the continuation values recursively via \begin{align*} q_{t}(x) = \mathbb{E}[\max\{f_{t+1}(X_{t+1}), q_{t+1}(X_{t+1})\} \mid X_{t} = x]? \end{align*}

Is it because the $$supremum$$ of the continuation values is always smaller than the $$supremum$$ of the actual stopping problem, because the range of stopping times is a subset of the other?

Best regards,

Peter

• I guess you can prove by induction starting from the terminal nodes rolling back.
– Vim
Feb 5, 2020 at 17:10