Determining the No Arbitrage price of max[B(T), S(T)]

Question

Following is given,

$dB(t)=rB(t)dt$

$dS(t)= (r-\delta)S(t)dt+\sigma S(t)dW(t)$

where, $r$ is the risk-free interest rate, $\delta$ the continous dividend yield $\sigma$ is the stock asset volatility and $W$ brownian motion.

By solving the SDE that is the money account B and applying Itô's lemma on the stock dynamic S I get,

$B(T)=e^{r(T-t)}$ and

$S(T)=S(t)e^{(r-\delta-\sigma^2/2)+\sigma(W(T)-W(t))}$

I know further that the No Arbitrage price is defined as,

$\Pi(t;X)=1/B(T)*E^Q_t[max[B(T),S(T)]]$

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Question

Can I somehow cancel out B(T) since it is deterministic? How do I calculate the expectation of a maximum with a brownian motion W?

I am new to probability theory, financial mathematics and stochastic calculus. Would appreciate step-wise guidance. Thank you!

Answer 1

$\max(B_T,S_T)=\max(0,S_T-B_T)+B_T,$ so this is just a call option (with strike $B_T$) plus $B_T.$