Problems / M05 · Intraday and execution M05 · Problem 10 ★★★ Check
Continuous Almgren–Chriss in closed form Not started Attempted Revisit Solved
For x ( t ) = X sinh ( κ ( T − t ) ) / sinh ( κ T ) x(t) = X\sinh(\kappa(T-t))/\sinh(\kappa T) x ( t ) = X sinh ( κ ( T − t )) / sinh ( κ T )
η ∫ 0 T x ˙ 2 d t = η X 2 κ 2 sinh 2 ( κ T ) ( T 2 + sinh 2 κ T 4 κ ) , σ 2 ∫ 0 T x 2 d t = σ 2 X 2 sinh 2 ( κ T ) ( sinh 2 κ T 4 κ − T 2 ) . \begin{gathered}
\eta\int_0^T\dot x^2dt = \frac{\eta X^2\kappa^2}{\sinh^2(\kappa T)}\Big(\frac T2 + \frac{\sinh 2\kappa T}{4\kappa}\Big),\\
\sigma^2\int_0^Tx^2dt = \frac{\sigma^2X^2}{\sinh^2(\kappa T)}\Big(\frac{\sinh2\kappa T}{4\kappa} - \frac T2\Big).
\end{gathered} η ∫ 0 T x ˙ 2 d t = sinh 2 ( κ T ) η X 2 κ 2 ( 2 T + 4 κ sinh 2 κ T ) , σ 2 ∫ 0 T x 2 d t = sinh 2 ( κ T ) σ 2 X 2 ( 4 κ sinh 2 κ T − 2 T ) . Find the limits as κ → 0 \kappa \to 0 κ → 0 κ T → ∞ \kappa T \to \infty κ T → ∞ 5.7 example with λ = 10 − 6 \lambda = 10^{-6} λ = 1 0 − 6
Solution Try the problem first. Write down where you got stuck before opening it.