arXiv subjects

Annalena Mickel

Publications and source records attributed to Annalena Mickel.

5 recordsLinked to original sources

arXiv2024

Worst-Case Optimal Investment in Incomplete Markets

We study and solve the worst-case optimal portfolio problem as pioneered by Korn and Wilmott (2002) of an investor with logarithmic preferences facing the possibility of a market crash with stochastic market coefficients by enhancing the martingale approach developed by Seifried in 2010. With the help of backward stochastic differential equations (BSDEs), we are able to characterize the resulting indifference optimal strategies in a fairly general setting. We also deal with the question of existence of those indifference strategies for market models with an unbounded market price of risk. We therefore solve the corresponding BSDEs via solving their associated PDEs using a utility crash-exposure transformation. Our approach is subsequently demonstrated for Heston's stochastic volatility model, Bates' stochastic volatility model including jumps, and Kim-Omberg's model for a stochastic excess return.

q-fin.MF↗

arXiv2024

On the convergence order of the Euler scheme for scalar SDEs with Hölder-type diffusion coefficients

We study the Euler scheme for scalar non-autonomous stochastic differential equations, whose diffusion coefficient is not globally Lipschitz but a fractional power of a globally Lipschitz function. We analyse the strong error and establish a criterion, which relates the convergence order of the Euler scheme to an inverse moment condition for the diffusion coefficient. Our result in particular applies to Cox-Ingersoll-Ross-, Chan-Karolyi-Longstaff-Sanders- or Wright-Fisher-type stochastic differential equations and thus provides a unifying framework.

math.NA↗

arXiv2023

Sharp $L^1$-Approximation of the log-Heston SDE by Euler-type methods

We study the $L^1$-approximation of the log-Heston SDE at equidistant time points by Euler-type methods. We establish the convergence order $ 1/2-ε$ for $ε>0$ arbitrarily small, if the Feller index $ν$ of the underlying CIR process satisfies $ν> 1$. Thus, we recover the standard convergence order of the Euler scheme for SDEs with globally Lipschitz coefficients. Moreover, we discuss the case $ν\leq 1$ and illustrate our findings by several numerical examples.

math.NA↗

arXiv2023

The order barrier for the $L^1$-approximation of the log-Heston SDE at a single point

We study the $L^1$-approximation of the log-Heston SDE at the terminal time point by arbitrary methods that use an equidistant discretization of the driving Brownian motion. We show that such methods can achieve at most order $ \min \{ ν, \tfrac{1}{2} \}$, where $ν$ is the Feller index of the underlying CIR process. As a consequence Euler-type schemes are optimal for $ν\geq 1$, since they have convergence order $\tfrac{1}{2}-ε$ for $ε>0$ arbitrarily small in this regime.

math.NA↗

arXiv2022

The weak convergence order of two Euler-type discretization schemes for the log-Heston model

We study the weak convergence order of two Euler-type discretizations of the log-Heston Model where we use symmetrization and absorption, respectively, to prevent the discretization of the underlying CIR process from becoming negative. If the Feller index $ν$ of the CIR process satisfies $ν>1$, we establish weak convergence order one, while for $ν\leq 1$, we obtain weak convergence order $ν-ε$ for $ε>0$ arbitrarily small. We illustrate our theoretical findings by several numerical examples.

math.NA↗