Dirk Veestraeten

Price dynamics under stochastic process switching: some extensions and an application to EMU1

Abstract In this paper we solve a particular type of stochastic process switching problem where the date of switching is fixed and known but the terminal price may depend on past prices. We derive closed-form solutions for the price dynamics of the asset before the terminal date and deduce the variance and jump components of these dynamics at the announcement. We subsequently extend the model to price dynamics prior to the announcement of the regime switch assuming that markets may have some expectations regarding the occurrence and/or the type of the regime switch. Finally, we apply the general model to discuss the implications of the chosen conversion modalities in the European Monetary Union (EMU) conversion procedure.

Explaining Recent European Exchange-Rate Stability

In this paper we analyse the behaviour of the bilateral exchange rates that were converted into euros on 1 January 1999. Using a model of stochastic regime switching we study the effects of future conversion on current exchange-rate dynamics. We find that exchange rates are to a large extent determined by the discounted (expected) conversion value. The theoretical model is subsequently applied to the currencies that participate in the first wave of the European Monetary Union (EMU). Using a Kalman approach, we find that for most currencies the weight attached to the future conversion value was well over 95%. This pricing characteristic successfully insulated intra-European exchange rates from the turmoil generated by the ongoing crises in Asia, Russia and Latin America. Copyright 1999 by Blackwell Publishers Ltd.

Currency option pricing in a credible exchange rate target zone

This article examines currency option pricing within a credible target zone arrangement where interventions at the boundaries push the exchange rate back into its fluctuation band. Valuation of such options is complicated by the requirement that the reflection mechanism should prevent the arbitrage opportunities that would arise if the exchange rate were to spend finite time on the boundaries. To prevent the latter, we superimpose instantaneously reflecting boundaries upon the familiar geometric Brownian motion (GBM) framework. We derive closed-form expressions for European call and put option prices and show that prices for the GBM model of Garman and Kohlhagen (1983) arise as the limit case for infinitely wide bands. We also illustrate that taking account of boundaries is of considerable economic value as erroneously using the unbounded-domain model of Garman and Kohlhagen (1983) easily overprices options by more than 100%.

On the inverse transform of Laplace transforms that contain (products of) the parabolic cylinder function

The Laplace transforms of the transition probability density and distribution functions for the Ornstein–Uhlenbeck process contain the product of two parabolic cylinder functions, namely and , respectively. The inverse transforms of these products have as yet not been documented. However, the transition density and distribution functions can be obtained by alternatively applying Doobs transform to the Kolmogorov equation and casting the problem in terms of Brownian motion. Linking the resulting transition density and distribution functions to their Laplace transforms then specifies the inverse transforms to the aforementioned products of parabolic cylinder functions. These two results, the recurrence relation of the parabolic cylinder function and the properties of the Laplace transform then enable the calculation of inverse transforms also for countless other combinations in the orders of the parabolic cylinder functions such as , and .

Some integral representations and limits for (products of) the parabolic cylinder function

ABSTRACT Recently, [Veestraeten D. On the inverse transform of Laplace transforms that contain (products of) the parabolic cylinder function. Integr Transf Spec F 2015;26:859–871] derived inverse Laplace transforms for Laplace transforms that contain products of two parabolic cylinder functions by exploiting the link between the parabolic cylinder function and the transition density and distribution functions of the Ornstein–Uhlenbeck process. This paper first uses these results to derive new integral representations for (products of two) parabolic cylinder functions. Second, as the Brownian motion process with drift is a limiting case of the Ornstein–Uhlenbeck process also limits can be calculated for the product of gamma functions and (products of) parabolic cylinder functions. The central results in both cases contain, in stylized form, and such that the recurrence relation of the parabolic cylinder function straightforwardly allows to obtain integral representations and limits also for countless other combinations in the orders such as and .

Measuring Convergence Speed of Asset Prices Toward a Pre-Announced Target

This study examines asset price dynamics (i.e. the convergence speed) in the event of pre-announced conversion values and dates. The theoretical framework for these dynamics has been developed in De Grauwe et al. (1999). Two instances of conversion are examined, notably the 1879-Resumption of Specie Payments in the USA and the conversion of European currencies into the Euro on 1 January, 1999. In the econometric model the underlying fundamentals are treated as unobservable and their evolution is estimated via a Kalman filtering technique. Estimation results reveal values for the rate or speed of convergence that are in line with intuition and amount to levels well below (implicit) estimates listed in the literature.

An integral representation for the product of parabolic cylinder functions

ABSTRACT This paper uses the convolution theorem of the Laplace transform to derive an inverse Laplace transform for the product of two parabolic cylinder functions in which the orders as well as the arguments differ. This result subsequently is used to obtain an integral representation for the product of two parabolic cylinder functions . The integrand in the latter representation contains the Gaussian hypergeometric function or alternatively can be expressed in terms of the associated Legendre function of the first kind.This paper derives new integral representations for products of two parabolic cylinder functions. In particular, expressions are obtained for D_{nu}(x)D_{mu}(y), with x>0 and y>0, that allow for different orders and arguments in the two parabolic cylinder functions. Also, two integral representations are obtained for D_{nu}(-x)D_{mu}(y) by employing the connection between the parabolic cylinder function and the Kummer confluent hypergeometric function. The integral representations are specialized for products of two complementary error functions and of two modified Bessel functions of the second kind of order 1/4, as well as for the product of a parabolic cylinder function and a modified Bessel function of the first kind of order 1/4.

The presence of target zone nonlinearities when narrower bands exist within official zones

The presence of target zone nonlinearities is generally refuted in empirical research. We argue that this may be due to estimation being performed vis-à-vis official limits when monetary authorities are in fact targeting a narrower band. Estimation results for the Belgian and French franc confirm that nonlinearities are present when narrower zones are accounted for.

An alternative integral representation for the product of two parabolic cylinder functions

ABSTRACT Recently, [Veestraeten D. An integral representation for the product of parabolic cylinder functions. Integral Transforms Spec Funct. 2017;28(1):15–21] derived an integral representation for with that was expressed in terms of the Gaussian hypergeometric function. This paper obtains an alternative expression for in which the integrand contains the parabolic cylinder function itself with the condition for convergence being at . The latter property is subsequently used to generate a new integral representation for in which restrictions on the order μ are absent.

On transition and first hitting time densities and moments of the Ornstein-Uhlenbeck process

This paper derives transition and first hitting time densities and moments for the Ornstein–Uhlenbeck Process (OUP) between exponential thresholds. The densities are obtained by simplifying the process via Doob’s representation into Brownian motion between affine thresholds. The densities in this paper also offer easy-to-use and fast small-time approximations for the densities of OUP between constant thresholds given that exponential thresholds are virtually constant for a small time. This is of interest for estimation with high-frequency data given that extant approaches for constant thresholds impose a large demand on computing power. The moments of the transition distribution up to order n are derived within a closed-form recursive formula that offers valuable information for management. Expressions for the moments of the first hitting time distribution are also obtained in closed form by simplifying integrals via series expansions.