Finance Without Brownian Motions: An Introduction To Simplified Stochastic Calculus

The paper introduces a simple way of recording and manipulating stochastic processes without explicit reference to a probability measure. In the new calculus, operations traditionally presented in a measure-specific way are instead captured by tracing the behaviour of jumps (also when no jumps are physically present). The new calculus is thus intuitive and compact. The calculus is also fail-safe in that, under minimal assumptions, all formal calculations are guaranteed to yield mathematically well-defined stochastic processes. Several illustrative examples of the new concept are given, among them a novel result on the Margrabe option to exchange one defaultable asset for another

Discrete-Time Quadratic Hedging of Barrier Options in Exponential Lévy Model

We examine optimal quadratic hedging of barrier options in a discretely sampled exponential Lévy model that has been realistically calibrated to reflect the leptokurtic nature of equity returns. Our main finding is that the impact of hedging errors on prices is several times higher than the impact of other pricing biases studied in the literature

Interface Simulation Distances

The classical (boolean) notion of refinement for behavioral interfaces of system components is the alternating refinement preorder. In this paper, we define a distance for interfaces, called interface simulation distance. It makes the alternating refinement preorder quantitative by, intuitively, tolerating errors (while counting them) in the alternating simulation game. We show that the interface simulation distance satisfies the triangle inequality, that the distance between two interfaces does not increase under parallel composition with a third interface, and that the distance between two interfaces can be bounded from above and below by distances between abstractions of the two interfaces. We illustrate the framework, and the properties of the distances under composition of interfaces, with two case studies.Comment: In Proceedings GandALF 2012, arXiv:1210.202

Regression-free Synthesis for Concurrency

While fixing concurrency bugs, program repair algorithms may introduce new concurrency bugs. We present an algorithm that avoids such regressions. The solution space is given by a set of program transformations we consider in for repair process. These include reordering of instructions within a thread and inserting atomic sections. The new algorithm learns a constraint on the space of candidate solutions, from both positive examples (error-free traces) and counterexamples (error traces). From each counterexample, the algorithm learns a constraint necessary to remove the errors. From each positive examples, it learns a constraint that is necessary in order to prevent the repair from turning the trace into an error trace. We implemented the algorithm and evaluated it on simplified Linux device drivers with known bugs.Comment: for source code see https://github.com/thorstent/ConRepai

A counterexample concerning the variance-optimal martingalle measure

The present note addresses an open question concerning a sufficient characterization of the variance-optimal martingale measure

Admissible strategies in semimartingale portfolio selection

The choice of admissible trading strategies in mathematical modelling of financial markets is a delicate issue, going back to Harrison and Kreps [HK79]. In the context of optimal portfolio selection with expected utility preferences this question has been the focus of considerable attention over the last twenty years. We propose a novel notion of admissibility that has many pleasant features { admissibility is characterized purely under the objective measure P; each admissible strategy can be approximated by simple strategies using finite number of trading dates; the wealth of any admissible strategy is a supermartingale under all pricing measures; local boundedness of the price process is not required; neither strict monotonicity, strict concavity nor differentiability of the utility function are necessary; the definition encompasses both the classical mean-variance preferences and the monotone expected utility. For utility functions finite on R, our class represents a minimal set containing simple strategies which also contains the optimizer, under conditions that are milder than the celebrated reasonable asymptotic elasticity condition on the utility function

Pure-jump semimartingales

A taxonomic hierarchy of pure-jump semimartingales is introduced. This hierarchy contains, in particular, the class of sigma-locally finite variation pure-jump processes. The members of this family can be explicitly characterized in terms of the predictable compensators of their jump measures. This family is also closed under stochastic integration and smooth transformations

Simple Explicit Formula for Near-Optimal Stochastic Lifestyling

In lifecycle economics the Samuelson paradigm (Samuelson, 1969) states that optimal investment is in constant proportions out of lifetime wealth (composed of current savings and future income). It is well known that in the presence of credit constraints this paradigm no longer applies. Instead, optimal lifecycle investment gives rise to so-called stochastic lifestyling (Cairns et al., 2006), whereby for low levels of accumulated capital it is optimal to invest fully in stocks and then gradually switch to safer assets as the level of savings increases. In stochastic lifestyling not only does the ratio between risky and safe assets change but also the mix of risky assets varies over time. While the existing literature relies on complex numerical algorithms to quantify optimal lifestyling the present paper provides a simple formula that captures the main essence of the lifestyling effect with remarkable accuracy

A Singular Differential Equation Stemming from an Optimal Control Problem in Financial Economics

We consider the ordinary differential equation x2u′′=axu′+bu−c(u′−1)2,x∈(0,x0), with a∈R,b∈R , c>0 and the singular initial condition u(0)=0, which in financial economics describes optimal disposal of an asset in a market with liquidity effects. It is shown in the paper that if a+b0 then there are infinitely many continuous solutions with indistinguishable asymptotics near 0. Moreover, it is proved that in the latter case there is precisely one solution u corresponding to the choice x 0=∞ which is such that 0≤u(x)≤x for all x>0, and that this solution is strictly increasing and concave

Optimal Trade Execution Under Endogenous Pressure to Liquidate: Theory and Numerical Solutions

We study optimal liquidation of a trading position (so-called block order or meta-order) in a market with a linear temporary price impact (Kyle, 1985). We endogenize the pressure to liquidate by introducing a downward drift in the unaffected asset price while simultaneously ruling out short sales. In this setting the liquidation time horizon becomes a stopping time determined endogenously, as part of the optimal strategy. We find that the optimal liquidation strategy is consistent with the square-root law which states that the average price impact per share is proportional to the square root of the size of the meta-order (Bershova and Rakhlin, 2013; Farmer et al., 2013; Donier et al., 2015; T´oth et al., 2016). Mathematically, the Hamilton-Jacobi-Bellman equation of our optimization leads to a severely singular and numerically unstable ordinary differential equation initial value problem. We provide careful analysis of related singular mixed boundary value problems and devise a numerically stable computation strategy by re-introducing time dimension into an otherwise time-homogeneous task