A risky asset St

is assumed to follow the geometric Brownian motion with the following

stochastic differential equation (SDE):

dSt = Stdt + StdWt

In the physical probability measure P, for constant and . The risk-free asset has price Bt satisfying

dBt = rBtdt

A. An option on this asset has payoff P(ST) and value Vt = V(t, St

) at times t before T. i. What does it mean to say that a portfolio consisting of t units of risky assets and t units of the risk-free asset, with a values of t = tBt + tBt

, is a self-financing

strategy? ii. Calculate the hedge ratio t necessary to replicate the option by such a strategy, and deduce that V (St

,t) satisfies the BlackScholes equation (Black-Scholes PDE)