Bachelier (1900) model. Consider a market made of a riskless asset valued \(A_{t}=A_{0}\) with zero interest rate, \(t \geqslant 0\), and a risky asset whose price \(S_{t}\) is modeled by a standard Brownian motion as \(S_{t}=B_{t}, t \geqslant 0\).

a) Show that the price \(g\left(t, B_{t}ight)\) of the option with claim payoff \(C=\left(B_{T}ight)^{2}\) satisfies the heat equation

\[ -\frac{\partial g}{\partial t}(t, y)=\frac{1}{2} \frac{\partial^{2} g}{\partial y^{2}}(t, y) \]

with terminal condition \(g(T, x)=x^{2}\).

b) Find the function \(g(t, x)\) by solving the PDE of Question (a).

Try a solution of the form \(g(t, x)=x^{2}+f(t)\).

c) Find the risky asset allocation \(\xi_{t}\) hedging the claim payoff \(C=\left(B_{T}ight)^{2}\), and the amount \(\eta_{t} A_{t}=\eta_{t} A_{0}\) invested in the riskless asset.