Normalize the self-financing strategy $\eta$ by defining
$$
L_t=\frac{\eta_t}{\pi_{t-1}^\eta}.
$$
Part e makes this well-defined, and
$$
\pi_t^L=\frac{\eta_{t+1}\cdot P_t}{\pi_t^\eta}=1.
$$
Moreover,
$$
\xi_t^L
=\frac{\eta_t\cdot(P_t+\delta_t)}{\pi_{t-1}^\eta}-1
=\frac{\pi_t^\eta}{\pi_{t-1}^\eta}-1.
$$
Both $K$ and $L$ have constant unit price and predictable dividends. Their difference has zero price and predictable dividend $\xi^K-\xi^L$. If that dividend were nonzero with positive probability, taking its known sign would produce an arbitrage. Therefore $\xi^K=\xi^L$, proving
$$
\xi_t^K=\frac{\pi_t^\eta}{\pi_{t-1}^\eta}-1.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 211 / 1 / f / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-211/1/f/solution