arXiv subjects
Publications and source records attributed to Frederik S Herzberg.
This short paper proposes a simple general equilibrium approach within a Markov-switching regime to explain how asymmetric information between lenders and speculators may lead to currency crises. The paper concludes by providing necessary as well as sufficient conditions for special cases.
Consider an $L^1$-continuous functional $\ell$ on the vector space of polynomials of Brownian motion at given times, suppose $\ell $ commutes with the quadratic variation in a natural sense, and consider a finite set of polynomials of Brownian motion at rational times, $f_1(\vec b),...,f_m(\vec b)$, mapping the Wiener space to $\mathbb{R}$. In the spirit of Schmüdgen's solution to the finite-dimensional moment problem, we give sufficient conditions under which $\ell$ can be written in the form $\int \cdot dμ$ for some finite measure $μ$ on the Wiener space such that $μ$-almost surely, all the random variables $f_1(\vec b),...,f_m(\vec b)$ are nonnegative.
In a general Feller martingale market with several assets, the existence of optimal exercise regions for multi-dimensional Bermudan options can be established by reference to Neveu's theory of Snell envelopes -- and also, as will be shown, more directly from standard martingale arguments combined with the strong Markov property. Based on this, in the framework of a log-Lévy martingale market, explicit formulae and asymptotic results on the perpetual American-Bermudan (barrier-like) put option price difference (``continuity correction'') near the exercise boundary will be proven, under the -- of course, fairly restrictive -- assumption that the logarithmic optimal exercise region, subject to the barrier, does not depend on the time mesh size and is, up to translation, a half-space. For this sake, Wiener-Hopf type results by Feller will be generalized to higher dimensions. It will be shown that an extrapolation from the exact Bermudan prices to the American price cannot be polynomial in the exercise mesh size in the setting of many common market models, and more specific bounds on the natural scaling exponent of the non-polynomial extrapolation for a number of (both one- and multi-dimensional) market models will be deduced.
Consider a discrete finite-dimensional, Markovian market model. In this setting, discretely sampled American options can be priced using the so-called ``non-recombining'' tree algorithm. By successively increasing the number of exercise times, the American option price itself can be computed; for combinatorial reasons, we shall consider a recursive algorithm that doubles the number of exercise times at each recursion step. First we prove, by elementary arguments, error bounds for the first order differences in this recursive algorithm. From this, bounds on the higher order differences can be obtained using combinatorial arguments that are motivated by the theory of rough paths. We shall obtain an explicit $L^1(C)$ convergence estimate for the recursive algorithm that prices a discretely sampled American $\max$-put option (on a basket of size $d$) at each recursion step, $C$ belonging to a certain class of compact subset of $\RR^d$, in under the assumption of sufficiently small volatilities. In case $d=1$, $L^1(C)$-bounds for an even more natural choice of $C$ will be derived.
A number of Bermudan option pricing methods that are applicable to options on multiple assets are studied in this thesis, one of the dominating questions being the natural scaling needed to extrapolate from Bermudan to American (both approximate and ``exact'') option prices.