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

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,

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

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!

$$\max(B_T,S_T)=\max(0,S_T-B_T)+B_T,$$ so this is just a call option (with strike $$B_T$$) plus $$B_T.$$
• Yes, an integral, but Black and Scholes already solved it in 1973 ;) . Just use the Black Scholes formula for a call with $B_T$ replacing the strike price $X$ and with an additional $B_T$ term added. Dec 11 '19 at 21:11