Put
$$
A_t=\sum_{s=0}^t\frac{C_s^{x,H}}{N_s},
\qquad A_{-1}=0,
$$
and define
$$
K_t=H_t+A_{t-1}\eta_t
\quad(t\geq1).
$$
Because the numéraire strategy is self-financing,
$$
X_t^{x,K}=X_t^{x,H}+A_{t-1}N_t.
$$
Moreover,
$$
\begin{aligned}
C_t^{x,K}
&=X_t^{x,H}+A_{t-1}N_t
-H_{t+1}\cdot P_t-A_t\eta_{t+1}\cdot P_t\\
&=C_t^{x,H}+N_t(A_{t-1}-A_t)=0.
\end{aligned}
$$
This also proves the required wealth formula.


Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-211/1/c/solution