Small Time Horizons Research Articles

A use of Black-Scholes model in market risk

The aim of this paper is to use the Black-Scholes model for market risk by using the estimated drift and volatility of a stock-return for a relatively small time-horizon, in case where the historical returns are better-fitted to a normal distribution. In this case, we may use the infinitesimal operator of the complete market formulated by the stock and the numeraire by taking the interest rate also constant and equal to the mean reference rate for the turning of theoretical and historical VaR to an economic capital functional.

Comparing Local Risks by Acceptance and Rejection

It is said that risky asset h acceptance dominates risky asset k if any decision maker who rejects the investment in h rejects also the investment in k. As Hart (2011) shows, acceptance dominance is an incomplete order on an ordinary set of gambles. We extend the definition of acceptance dominance order to risky assets whose values follow random processes. We call the risk that arises from investing in such assets, with a short investment time horizon, local risk. We show that for small investment time horizons, the acceptance dominance order is a complete order that can be represented by an index of local risk. Moreover, we show that the measures of riskiness proposed by Aumann & Serrano (2008), Foster & Hart (2009), and Schreiber (2011) all coincide with our index. We use the differential calculus as an analytical tool to present our results.

Simulation method for reliability of TT&C mission with high redundancy and small time horizon

Simulation method for reliability of TT&C mission with high redundancy and small time horizon

On the Theory of Fast Controls under Disturbances

AbstractFast controls are those that act within a very small time horizon. They are treated as bounded approximations of generalized impulse controls (belonging to the class of higher-order distributions). In this paper, we investigate the fast controls in feedback form under unknown-but-bounded disturbances.

Backward stochastic differential equations with time delayed generators—results and counterexamples

We deal with backward stochastic differential equations with time delayed generators. In this new type of equation, a generator at time t can depend on the values of a solution in the past, weighted with a time delay function, for instance, of the moving average type. We prove existence and uniqueness of a solution for a sufficiently small time horizon or for a sufficiently small Lipschitz constant of a generator. We give examples of BSDE with time delayed generators that have multiple solutions or that have no solutions. We show for some special class of generators that existence and uniqueness may still hold for an arbitrary time horizon and for arbitrary Lipschitz constant. This class includes linear time delayed generators which we study in more detail. We are concerned with different properties of a solution of a BSDE with time delayed generator, including the inheritance of boundedness from the terminal condition, the comparison principle, the existence of a measure solution and the BMO martingale property. We give examples in which they may fail.

On Malliavin’s differentiability of BSDEs with time delayed generators driven by Brownian motions and Poisson random measures

We investigate solutions of backward stochastic differential equations (BSDEs) with time delayed generators driven by Brownian motions and Poisson random measures, that constitute the two components of a Lévy process. In these new types of equations, the generator can depend on the past values of a solution, by feeding them back into the dynamics with a time lag. For such time delayed BSDEs, we prove the existence and uniqueness of solutions provided we restrict on a sufficiently small time horizon or the generator possesses a sufficiently small Lipschitz constant. We study differentiability in the variational or Malliavin sense and derive equations that are satisfied by the Malliavin gradient processes. On the chosen stochastic basis this addresses smoothness both with respect to the continuous part of our Lévy process in terms of the classical Malliavin derivative for Hilbert space valued random variables, as well as with respect to the pure jump component for which it takes the form of an increment quotient operator related to the Picard difference operator.

Potential Future Market Risk: Coping with Long Term Model Risk

The recent financial crisis has shown that most market risk models - even if they deliver sufficiently accurate risk figures over short time horizons - are not able to provide reliable forecasts over longer time horizons. This is mainly due to the fact that market risk models are usually tailored for small time horizons and neglect longer horizons like three or twelve months, which are the basis for both limit management and economic capital planning. As a potential remedy we introduce the concept of potential future market risk as a supplement to traditional market risk figures.

Using a radiative-convective model to estimate the temperature potential of greenhouse gases

A comprehensive approach is proposed to the construction of an equilibrium global warming potential with the use of a radiative-convective model of climate and the line-by-line calculations of the characteristics of radiative transfer in the atmosphere on the basis of analysis and comparison of different methods of estimating emission metrics. The studies conducted in the past decade have demonstrated that a widely used method of calculating the global warming potential for methane and other trace greenhouse gases is applicable only to relatively small time horizons (within 100 years). The proposed equilibrium global warming potential makes it possible to consider a set of equilibrium states of the Earth’s climate system under variations in the contents of greenhouse gases in the atmosphere and to estimate the parameters of the system’s response to such variations for arbitrary time intervals. Analysis of a set of different equilibrium states makes it possible to assess a relative contribution of different anthropogenic pollutants to radiation balance and, hence, to a change in the Earth’s climatic regime.

Control of flow using genetic algorithm for a circular cylinder executing rotary oscillation

We propose here a new approach to optimally control incompressible viscous flow past a circular cylinder for drag minimization by rotary oscillation. The flow at Re = 15000 is simulated by solving 2D Navier–Stokes equations in stream function-vorticity formulation. High accuracy compact scheme for space discretization and four stage Runge–Kutta scheme for time integration makes such simulation possible. While numerical solution for this flow field has been reported using a fast viscous-vortex method, to our knowledge, this has not been done at such a high Reynolds number by computing the Navier–Stokes equation before. The importance of scale resolution, aliasing problem and preservation of physical dispersion relation for such vortical flows of the used high accuracy schemes [Sengupta TK. Fundamentals of computational fluid dynamics. Hyderabad, India: University Press; 2004] is highlighted. For the dynamic problem, a novel genetic algorithm (GA) based optimization technique has been adopted, where solutions of Navier–Stokes equations are obtained using small time-horizons at every step of the optimization process, called a GA generation. Then the objective functions is evaluated that is followed by GA determined improvement of the decision variables. This procedure of time advancement can also be adopted to control such flows experimentally, as one obtains time-accurate solution of the Navier–Stokes equation subject to discrete changes of decision variables. The objective function – the time-averaged drag – is optimized using a real-coded genetic algorithm [Deb K. Multi-objective optimization using evolutionary algorithms. Chichester, UK: Wiley; 2001] for the two decision variables, the maximum rotation rate and the forcing frequency of the rotary oscillation. Various approaches to optimal decision variables have been explored for the purpose of drag reduction and the collection of results are self-consistent and furthermore match well with the experimental values reported in [Tokumaru PT, Dimotakis PE. Rotary oscillation control of a cylinder wake. J Fluid Mech 1991;224:77].

Market risk models for intraday data

In this paper, market risk at an intraday time horizon is quantified using normal GARCH, Student GARCH, RiskMetrics and high-frequency duration (log-ACD) models set in the framework of the conditional VaR methodology. Because of the small time horizon of the intraday returns (15 and 30 minute returns in this paper), an evaluation of intraday market risk can be useful to market participants (traders, market makers) involved in frequent trading. As expected, the volatility features an important intraday seasonality, which must be removed prior to using the market risk models. The four models are applied to intraday returns data for three stocks traded on the New York Stock Exchange and it is shown that the Student GARCH model performs best. The use of price durations as a measure of risk on time is commented upon.