a) Solve the Black-Scholes PDE [ begin{equation*} r g(t, x)=frac{partial g}{partial t}(t, x)+r x frac{partial g}{partial x}(t,
Question:
a) Solve the Black-Scholes PDE
\[
\begin{equation*}
r g(t, x)=\frac{\partial g}{\partial t}(t, x)+r x \frac{\partial g}{\partial x}(t, x)+\frac{\sigma^{2}}{2} x^{2} \frac{\partial^{2} g}{\partial x^{2}}(t, x) \tag{6.38}
\end{equation*}
\]
with terminal condition \(g(T, x)=1, x>0\).
Try a solution of the form \(g(t, x)=f(t)\) and find \(f(t)\).
b) Find the respective quantities \(\xi_{t}\) and \(\eta_{t}\) of the risky asset \(S_{t}\) and riskless asset \(A_{t}=A_{0} \mathrm{e}^{r t}\) in the portfolio with value
\[
V_{t}=g\left(t, S_{t}ight)=\xi_{t} S_{t}+\eta_{t} A_{t}
\]
hedging the contract with claim payoff \(C=\$ 1\) at maturity.
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Related Book For
Introduction To Stochastic Finance With Market Examples
ISBN: 9781032288277
2nd Edition
Authors: Nicolas Privault
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