Gerhard Larcher
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View article: Valuation of American Lookback-Options and the Distribution of Kronecker Sequences
Valuation of American Lookback-Options and the Distribution of Kronecker Sequences Open
In this paper, we determine the complexity of calculating the fair value of an American lookback-option in a binomial n–step model via backwardation. It will turn out, that the complexity also depends on diophantine properties of a certain…
View article: On the quasi-uniformity properties of quasi-Monte Carlo lattice point sets and sequences
On the quasi-uniformity properties of quasi-Monte Carlo lattice point sets and sequences Open
The discrepancy of a point set quantifies how well the points are distributed, with low-discrepancy point sets demonstrating exceptional uniform distribution properties. Such sets are integral to quasi-Monte Carlo methods, which approximat…
View article: Supervised Machine Learning Classification for Short Straddles on the S&P500
Supervised Machine Learning Classification for Short Straddles on the S&P500 Open
In this paper, we apply machine learning models to execute certain short-option strategies on the S&P500. In particular, we formulate and focus on a supervised classification task which decides if a plain short straddle on the S&P500 shoul…
View article: Discrepancy bounds for normal numbers generated by necklaces in arbitrary base
Discrepancy bounds for normal numbers generated by necklaces in arbitrary base Open
Mordechay B. Levin has constructed a number $λ$ which is normal in base 2, and such that the sequence $(\left\{2^n λ\right\})_{n=0,1,2,\ldots}$ has very small discrepancy $D_N$. Indeed we have $N\cdot D_N = \mathcal{O} \left(\left(\log N\r…
View article: The exact order of discrepancy for Levin's normal number in base 2
The exact order of discrepancy for Levin's normal number in base 2 Open
Mordechay Levin has constructed a number $α$ which is normal in base 2, and such that the sequence $\left\{2^n α\right\}_{n=0,1,2,\ldots}$ has very small discrepancy $D_N$. Indeed we have $N\cdot D_N = \mathcal{O} \left(\left(\log N\right)…
View article: Supervised machine learning classification for short straddles on the S&P500
Supervised machine learning classification for short straddles on the S&P500 Open
In this working paper we present our current progress in the training of machine learning models to execute short option strategies on the S&P500. As a first step, this paper is breaking this problem down to a supervised classification tas…
View article: Analysis of Option Trading Strategies Based on the Relation of Implied and Realized S&P500 Volatilities
Analysis of Option Trading Strategies Based on the Relation of Implied and Realized S&P500 Volatilities Open
In this paper, we examine the performance of certain short option trading strategies on the S&P500 with backtesting based on historical option price data. Some of these strategies show significant outperformance in relation to the S&P500 i…
View article: 7. On pair correlation of sequences
7. On pair correlation of sequences Open
We give a survey on the concept of Poissonian pair correlation (PPC) of sequences in the unit interval, on existing and recent results and we state a list of open problems. Moreover, we present and discuss a quite recent multi-dimensional …
View article: Two concrete FinTech applications of QMC
Two concrete FinTech applications of QMC Open
I present the basics and numerical result of two (or three) concrete applications of quasi-Monte-Carlo methods in financial engineering. The applications are in: derivative pricing, in portfolio selection, and in credit risk management.
View article: Sets of Bounded Remainder for The Billiard on A Square
Sets of Bounded Remainder for The Billiard on A Square Open
We study sets of bounded remainder for the billiard on the unit square. In particular, we note that every convex set S whose boundary is twice continuously differentiable with positive curvature at every point, is a bounded remainder set f…
View article: On Bounded Remainder Sets and Strongly Non-Bounded Remainder Sets for Sequences $(\{a_nα\})_{n\geq 1}$
On Bounded Remainder Sets and Strongly Non-Bounded Remainder Sets for Sequences $(\{a_nα\})_{n\geq 1}$ Open
We give some results on the existence of bounded remainder sets (BRS) for sequences of the form $(\{a_nα\})_{n\geq 1}$, where $(a_n)_{n\geq 1}$ - in most cases - is a given sequence of distinct integers. Further we introduce the concept of…
View article: On Bounded Remainder Sets and Strongly Non-Bounded Remainder Sets for Sequences $(\{a_n\alpha\})_{n\geq 1}$
On Bounded Remainder Sets and Strongly Non-Bounded Remainder Sets for Sequences $(\{a_n\alpha\})_{n\geq 1}$ Open
We give some results on the existence of bounded remainder sets (BRS) for sequences of the form $(\{a_n\alpha\})_{n\geq 1}$, where $(a_n)_{n\geq 1}$ - in most cases - is a given sequence of distinct integers. Further we introduce the conce…
View article: Pair correlation of sequences $(\lbrace a_n \alpha \rbrace)_{n \in \mathbb{N}}$ with maximal order of additive energy
Pair correlation of sequences $(\lbrace a_n \alpha \rbrace)_{n \in \mathbb{N}}$ with maximal order of additive energy Open
We show for sequences $\left(a_{n}\right)_{n \in \mathbb{N}}$ of distinct positive integers with maximal order of additive energy, that the sequence $\left(\left\{a_{n} \alpha\right\}\right)_{n \in \mathbb{N}}$ does not have Poissonian pai…
View article: Pair correlation of sequences $(\lbrace a_n α\rbrace)_{n \in \mathbb{N}}$ with maximal order of additive energy
Pair correlation of sequences $(\lbrace a_n α\rbrace)_{n \in \mathbb{N}}$ with maximal order of additive energy Open
We show for sequences $\left(a_{n}\right)_{n \in \mathbb{N}}$ of distinct positive integers with maximal order of additive energy, that the sequence $\left(\left\{a_{n} α\right\}\right)_{n \in \mathbb{N}}$ does not have Poissonian pair cor…
View article: Remark on a result of Bourgain on poissonian pair correlation
Remark on a result of Bourgain on poissonian pair correlation Open
We show for a class of sequences $(a_n)_{n\geq 1}$ of distinct positive integers, that for no $α$ the sequence $(\left\{a_n α\right\})_{n \geq 1}$ does have Poissonian pair correlation. This class contains for example all strictly increasi…
View article: On Quasi-Energy-Spectra, Pair Correlations of Sequences and Additive Combinatorics
On Quasi-Energy-Spectra, Pair Correlations of Sequences and Additive Combinatorics Open
The investigation of the pair correlation statistics of sequences was initially motivated by questions concerning quasi-energy-spectra of quantum systems. However, the subject has been developed far beyond its roots in mathematical physics…
View article: Additive Energy and Irregularities of Distribution
Additive Energy and Irregularities of Distribution Open
We consider strictly increasing sequences (a n ) n≥1 of integers and sequences of fractional parts ({a n α}) n≥1 where α ∈ R. We show that a small additive energy of (a n ) n≥1 implies that for almost all α the sequence ({a n α}) n≥1 has l…
View article: On pair correlation and discrepancy
On pair correlation and discrepancy Open
We say that a sequence $$\left( x_n\right) _{n \ge 1}$$ in [0, 1) has Poissonian pair correlations if $$\begin{aligned} \lim _{N \rightarrow \infty } \frac{1}{N} \# \left\{ 1 \le l \ne m \le N{:}\,\left||x_l-x_m\right|| < \frac{s}{N} \righ…
View article: On evil Kronecker sequences and lacunary trigonometric products
On evil Kronecker sequences and lacunary trigonometric products Open
An important result of Weyl states that for every sequence (n k ) k≥1 of distinct positive integers the sequence of fractional parts of (n k α) k≥1 is u.d. mod 1 for almost all α. However, in this general case it is usually extremely diffi…
View article: On pair correlation and discrepancy
On pair correlation and discrepancy Open
We say that a sequence $\{x_n\}_{n \geq 1}$ in $[0,1)$ has Poissonian pair correlations if \begin{equation*} \lim_{N \rightarrow \infty} \frac{1}{N} \# \left\{ 1 \leq l \neq m \leq N \, : \, \left\lVert x_l-x_m \right\rVert < \frac{s}{N} \…
View article: Additive Energy and the Hausdorff dimension of the exceptional set in\n metric pair correlation problems
Additive Energy and the Hausdorff dimension of the exceptional set in\n metric pair correlation problems Open
For a sequence of integers $\\{a(x)\\}_{x \\geq 1}$ we show that the\ndistribution of the pair correlations of the fractional parts of $\\{ \\langle\n\\alpha a(x) \\rangle \\}_{x \\geq 1}$ is asymptotically Poissonian for almost all\n$\\al…
View article: An Improved Bound for the Star Discrepancy of Sequences in the Unit Interval
An Improved Bound for the Star Discrepancy of Sequences in the Unit Interval Open
It is known that there is a constant c > 0 such that for every sequence x 1 , x 2 , . . . in [0, 1) we have for the star discrepancy D N * $D_N^* $ of the first N elements of the sequence that ND N * ≥ c ⋅ log N $ND_N^* \ge c \cd…
View article: Sets of bounded remainder for the continuous irrational rotation on\n $[0,1)^2$
Sets of bounded remainder for the continuous irrational rotation on\n $[0,1)^2$ Open
We study sets of bounded remainder for the two-dimensional continuous\nirrational rotation $(\\{x_1+t\\}, \\{x_2+t\\alpha \\})_{t \\geq 0}$ in the unit\nsquare. In particular, we show that for almost all $\\alpha$ and every starting\npoint…
View article: Sets of bounded remainder for a continuous irrational rotation on $[0,1]^2$
Sets of bounded remainder for a continuous irrational rotation on $[0,1]^2$ Open
We study sets of bounded remainder for the two-dimensional continuous irrational rotation $(\{x_1+t\}, \{x_2+tα\})_{t \geq 0}$ in the unit square. In particular, we show that for almost all $α$ and every starting point $(x_1, x_2)$, every …