Question $1$: A market has to assets A and $B.$ A portfolio is made of these two assets with $V_n=$

${}_n{A}_n+{}_n{B}_n.$ Write down an equation which enforces the self$-$financing condition on this portfolio. Use

this equation to see that the appropriate continuous$-$time analogue is

$d{V}_t={}_{t}d{A}_t+{}_{t}d{B}_t$

Now let $A=S$ be a risky stock with dynamics $d{S}_t=S_{t}dt+S_{t}d{W}_t$ and let $B=M$ be a risk$-$free money

market account with $d{M}_t=r{M}_{t}dt.$ Derive the Black$-$Scholes equation for a path$-$independent European

option that pays $(S_T)$ at time $T.$