Testing the forward rate unbiasedness hypothesis during the 1920s

This paper reexamines the forward rate unbiasedness hypothesis (FRUH) during the 1920s and it contributes to the literature as follows: first, it utilizes a database that includes currencies not studied before, as well as the 3 month forward rates; second, it applies three different approaches to test for cointegration and it shows that the choice of the technique is not of crucial importance; third, it tests for the temporal stability of the cointegration results; finally, it tests for the existence of the FRUH in the short run, by means of error correction models, whereas previous studies focused on cointegrated vectors only. Our analysis shows that for countries that did not undergo major financial turmoil during that period, there exists more favorable evidence for the FRUH.     

On the inversion of moving averages

Abstract

1. Let an infinite sequence of real numbers be given by (1) {η_t } =[…, η _t-2, η _t-1, η _t , η _t+1, η _t+2, . . . ] and let (b) = (b ₀, b ₁,.., b_h ) represent real constants. The sequence(2) {ζ _t } = […, ζ _t-2, ζ _t-1, ζ_t , ζ _t+1, ζ _t+2,…] defined for every t by the relation(3) ζ _t = b ₀ · η _t + b ₁ · η _t-1 — . . . + b _h-1 · η _t-h+1 + b _h · η _t-h is said to be a moving average of {η _t } with weights (b _i ) The variable t, which is restricted to integral values 0, ± 1, ± 2 etc., will in the sequel be spoken of as representing time.     

A Fractional Cointegration Approach to Testing Mean Reversion Between Spot and Forward Exchange Rates: A Case of High Frequency Data with Low Frequency Dynamics

This paper applies a relatively new but generalised concept of fractional cointegration to shed some light on the validity of a long-run relationship between high frequency daily spot and the lagged forward Australian-US dollar exchange rate. An investigation of the stochastic properties of these rates reveals that, while the relationship is not cointegrated in their logs, they appear to be fractionally cointegrated if we allow for mean reverting processes that are CI (1, d ) with 0< d <1. The paper demonstrates that relaxing the condition that the residual from the cointegration equation must be a I (0) process, captures a much wider class of mean-reversion behaviour. This result is interpreted in the context of the speculative EMH between the spot and forward exchanges rates, as having some empirical support. Furthermore, an analysis of the short-run dynamics propelling the long-run relationship tends to imply that in both the short- and long-term, the forward rate is led by the spot rate. In the longer term, the spot rate is found to be the initial receptor of any exogenous shock to the equilibrium and it is the forward exchange rate that bears the brunt of short-run adjustment to re-establish the long-run equilibrium relationship. The approach illustrated in this paper is shown to hold enormous potential for tests of mean reversion involving hypotheses popular to financial econometrics in general, where the dynamics of high frequency data are under scrutiny.     

The effect of interventions on realignment probabilities

The expected future change of the exchange rate within its currency band and the expected realignment rate are estimated using the Regime–Switching Model. There exists an unobserved variable s_t, which characterises the equilibrium state of the expected future change of the exchange rate within its currency band at any time t with certain probabilities. Different values of s_t correspond to states with high and low risk of realignment, respectively. The probabilities of switching between one regime and another depend on central bank intervention in the foreign exchange market. Daily data on intervention by Norges Bank are used. The data contain relatively few actual realignments, and the sample distribution of realignments may not be representative enough to capture the discrete changes in the exchange rate caused by a non-zero subjective probability of realignment (even when no realignment has in fact taken place). This causes the very well known peso problem in the estimation.     

Le XIImecongrès international d'actuaires

Abstract

We study the following inverse thinning problem for renewal processes: for which completely monotone functions f is f/(p+qf), 0?p?1, q=1-p, completely monotone? A characterisation of such f's is given. We also study the case when f comes from a gamma distribution, and present some ideas for more general results.

The intention of this note is to add some information to a paper by Yannaros (1985), in which thinned renewal processes are considered. Let X_n , n?1, be i.i.d. non-negative random variables, distributed according to a probability measure µ, and let S_n = X ₁+...+X_n (with S ₀=0) be the corresponding renewal process. Replacing µ by the probability measure ν=∑_n?1 pq^n-1 µ^n* (µ^n* =µ* ... µ*, n times) we get a new renewal process, obtained from the original one by independently at each stage preserving the process with probability p. Here and below q= 1-p, and to avoid trivialities we assume that 0 Let µ^(s) = ∫[0,∞) exp (-sx)µ(dx) , s?0, denote the Laplace transform of µ. Then ν^=pµ/(1-µ^). We will study the inverse problem: given a completely monotone function ψ, when does ψ(p+qψ) define a completely monotone function. A complete characterisation, and some of its consequences, is given in §§ 1–3 below. In §§ 4–5 we study the gamma distribution. It is proved that the inverse problem has a negative solution when the parameter a > 1, i.e. 1/(p + q(1 + s) ^a ) is not completely monotone then. In Yannaros (1985) this was proved for a=2, 3, ... with entirely different methods. (That 1/(p+q(1+s)^a is completely monotone for 0?a?1 is easily seen; cf. Yannaros (1985). Finally, in § 6 we give some suggestions to more general results related to thinning. Perhaps the most interesting problem is to find sufficiently general conditions for an absolutely monotone function to have a Bernstein function as its inverse.     

On a class of law invariant convex risk measures

We consider the class of law invariant convex risk measures with robust representation r_h,p(X)=sup_fò₀¹ [AV@R_s(X)f(s)-f^p(s)h(s)] ds\rho_{h,p}(X)=\sup_{f}\int_{0}^{1} [AV@R_{s}(X)f(s)-f^{p}(s)h(s)]\,ds, where 1≤p<∞ and h is a positive and strictly decreasing function. The supremum is taken over the set of all Radon–Nikodym derivatives corresponding to the set of all probability measures on (0,1] which are absolutely continuous with respect to Lebesgue measure. We provide necessary and sufficient conditions for the position X such that ρ _h,p (X) is real-valued and the supremum is attained. Using variational methods, an explicit formula for the maximizer is given. We exhibit two examples of such risk measures and compare them to the average value at risk.     

Does the stock market predict real activity? Time series evidence from the G-7 countries

This paper extends one aspect of the US stock market study of Fama (1990) and Schwert (1990). We examine the relationship between industrial production (IP) growth rates and lagged real stock returns for the G-7 countries using both in-sample cointegration and error-correction models and the out-of-sample forecast-evaluation procedure of Ashley et al. (1980). The cointegration tests show a long-run equilibrium relationship between the log levels of IP and real stock prices, while the error-correction models indicate a correlation between IP growth and lagged real stock returns for all countries except Italy. The out-of-sample tests show that in several sub-periods the US, UK, Japanese, and Canadian stock markets enhance predictions of future IP.     

A characterization of the exponential distribution and of the geometric distribution

Abstract

Cook (1978) has proved that n positive random variables X ₁ ..., X _n are independent and follow the same exponential distribution iff the random vectors (X ₁ ..., X _s ) and (X _s+1, ..., X _n ) are independent for some s ∈ {1, ..., n-l} and E(Π} _j=1 ⁿ max {X _j -a _j , 0}) is a function of Σ _j=1 ⁿ a _j for a ₁, ..., a _n ∈ dR ⁺. In this paper a generalization of this characterization of the exponential distribution and an analogous characterization of the geometric distribution are given.     

Semi-analytical MBS Pricing

This paper presents a multi-factor valuation model for fixed-rate callable mortgage backed securities (MBS). The model yields semi-analytic solutions for the value of MBS in the sense that the MBS value is found by solving a system of ordinary differential equations. Instead of modelling the conditional prepayment rate (CPR), as is customary, the pool size is the primary modelling object. It is shown that the value of a single MBS payment due at time t _n can be found by computing two expectations of the pool size at time t _n–1 and t _n respectively. This is a general result independent of any interest rate model. However, if the pool size is specified in a way that makes the expectations solvable using transform methods, semi-analytic pricing formulas are achieved. The affine and quadratic pricing frameworks are combined to get flexible and sophisticated prepayment functions. We show that the model has no problem of generating negative convexity as the spot rate falls, and still be close to a similar non-callable bond when the spot rate rises.     

Eksamen i Forsikringsvidenskab og Statistik ved Københavns Universitet

Abstract

Let X ₁, X ₂,... be a sequence of independent, identically distributed random variables with P(X?0)=0, and such that pκ = ?₀ ^∞ x ^κ dP(x)<∞, k= 1, 2, 3, 4. Assume that {N(t), t?0} is a Poission stochastic process, independent of the X ₁ with E(N(t))=t. For λ ? 0, let Z _T= max {Σ _t?1 ^N(t) X _t ?t(p ₁+λ)}. Expressions 0 ?t?T for E(Z _T ), E(Z _T ²), and P(Z _T =0) are derived. These results are used to construct an approximation for the finite-time ruin function Ψ(u, T) = P(Z _T >u) for u?0. An alternate method of approximating Ψ(u, T) was presented in [10] by Olof Thorin and exemplified in [11] by Nils Wikstad. One of the purposes of this paper is to compare the two methods for two distributions of claims where the number of claims is a Poisson variate. The paper will also discuss the advantages and disadvantages of the two methods. We will also present a comparison of our approximate figures with the exact figures for the claim distribution      

Mean square error for the Leland–Lott hedging strategy: convex pay-offs

Leland’s approach to the hedging of derivatives under proportional transaction costs is based on an approximate replication of the European-type contingent claim V _T using the classical Black–Scholes formula with a suitably enlarged volatility. The formal mathematical framework is a scheme of series, i.e., a sequence of models with transaction cost coefficients k _n =k ₀ n ^−α , where α∈[0,1/2] and n is the number of portfolio revision dates. The enlarged volatility [^(s)]_n\widehat{\sigma}_{n} in general depends on n except for the case which was investigated in detail by Lott, to whom belongs the first rigorous result on convergence of the approximating portfolio value Vⁿ_TV^{n}_{T} to the pay-off V _T . In this paper, we consider only the Lott case α=1/2. We prove first, for an arbitrary pay-off V _T =G(S _T ) where G is a convex piecewise smooth function, that the mean square approximation error converges to zero with rate n ^−1/2 in L ² and find the first order term of the asymptotics. We are working in a setting with non-uniform revision intervals and establish the asymptotic expansion when the revision dates are t_iⁿ=g(i/n)t_{i}^{n}=g(i/n), where the strictly increasing scale function g:[0,1]→[0,1] and its inverse f are continuous with their first and second derivatives on the whole interval, or g(t)=1−(1−t) ^β , β≥1. We show that the sequence n^1/2(V_Tⁿ-V_T)n^{1/2}(V_{T}^{n}-V_{T}) converges in law to a random variable which is the terminal value of a component of a two-dimensional Markov diffusion process and calculate the limit. Our central result is a functional limit theorem for the discrepancy process.     

Nonparametric estimation for a stochastic volatility model

Consider discrete-time observations (X _{? δ} )_1≤?≤n+1 of the process X satisfying $dX_{t}=\sqrt{V_{t}}dB_{t}Consider discrete-time observations (X _{ℓ δ} )_{1≤ℓ≤n+1} of the process X satisfying dX_t=?{V_t}dB_tdX_{t}=\sqrt{V_{t}}dB_{t} , with V a one-dimensional positive diffusion process independent of the Brownian motion B. For both the drift and the diffusion coefficient of the unobserved diffusion V, we propose nonparametric least square estimators, and provide bounds for their risk. Estimators are chosen among a collection of functions belonging to a finite-dimensional space whose dimension is selected by a data driven procedure. Implementation on simulated data illustrates how the method works.     

Local time and the pricing of time-dependent barrier options

A time-dependent double-barrier option is a derivative security that delivers the terminal value φ(S _T ) at expiry T if neither of the continuous time-dependent barriers b _±:[0,T]→ℝ₊ have been hit during the time interval [0,T]. Using a probabilistic approach, we obtain a decomposition of the barrier option price into the corresponding European option price minus the barrier premium for a wide class of payoff functions φ, barrier functions b _± and linear diffusions (S _t ) _t∈[0,T]. We show that the barrier premium can be expressed as a sum of integrals along the barriers b _± of the option’s deltas Δ _±:[0,T]→ℝ at the barriers and that the pair of functions (Δ ₊,Δ ₋) solves a system of Volterra integral equations of the first kind. We find a semi-analytic solution for this system in the case of constant double barriers and briefly discus a numerical algorithm for the time-dependent case.     

Relationship between Treasury bills and Eurodollars: Theoretical and Empirical Analyses

In this paper, we derive an equilibrium relationship between the yields on Eurodollar and Treasury bills based on equivalent martingale results derived by Harrison and Kreps (1979) and Harrison and Pliska (1981, 1983) as well as the corporate debt pricing model developed by Merton (1974). The derived equilibrium relationship incorporates the models used by Booth and Tse (1995) and Shrestha and Welch (2001) as special cases. The equilibrium relationship indicates that the conditional volatility of the yield on Eurodollars explains the variation in the TED spread. We empirically test the equilibrium relationship using a GARCH-M model and the concept of fractional cointegration. We use both the ex ante data implied by the respective futures contracts as well as the ex post spot data with daily, weekly and monthly frequencies. We find empirical support for the Equilibrium relationship.

Numerical evaluation of the compound Poisson distribution: Recursion or Fast Fourier Transform?

1. The problem

The finite vector p=(p ₁,p ₂, ...,p_s ) defines a probability distribution on the integers 1,2, ...,s.     

On the informational content of spot and forward exchange rates

In foreign exchange markets, efficiency tests have typically been applied to the forward rate on the argument that the forward rate should be a good proxy for the unobservable market expectations of future spot rates. The present study offers innovations in two directions. First we utilize a data set which consists of daily observations on spot and forward exchange rates. This allows us to match the forward contract with the exact settlement date and to create a large number of non-overlapping data sets. Second, and more importantly, we show that in general the current spot rate is a ‘better’ predictor of the future spot rate than is the current forward rate of appropriate maturity.     

Value-at-Risk analysis for long-term interest rate futures: Fat-tail and long memory in return innovations

This article uses the FIGARCH(1,d,1) models to calculate daily Value-at-Risk (VaR) for T-bond interest rate futures returns of long and short trading positions based on the normal, Student-t, and skewed Student-t innovations distributions. The empirical results show that based on Kupiec LR failure rate tests, in-sample and out-of-sample VaR values calculated using FIGARCH(1,d,1) model with skewed Student-t innovations are more accurate than those generated using traditional GARCH(1,1) models. Moreover, we find that the in-sample values of VaR are subject to a significant positive bias, as pointed out by Inui et al. [Inui, K., Kijima, M., Kitano, A., 2003. VaR is subject to a significant positive bias, working paper].     

Adaptive Learning in an Expectational Difference Equation with Several Lags: Selecting among Learnable REE

It is demonstrated that adaptive learning in the least squares sense may be incapable of satisfactorily reducing the number of attainable equilibria in a rational expectations model when focusing on the forward‐solutions to the model. The model examined, as an illustration, is a basic asset pricing model for exchange rate determination that is augmented with technical trading in the currency market in the form of moving averages since it is the most commonly used technique according to questionnaire surveys. The forward‐solutions to such a model are preferable to the backward‐solutions that are normally utilized since announcement effects is an important feature in currency trade. Because of technical trading in foreign exchange, the current exchange rate depends on j _max lags of the exchange rate, meaning that the model has j _max+1 rational expectations equilibria, where several of them are adaptively learnable in the least squares sense. However, since past exchange rates should not affect the current exchange rate when technical trading is absent, it is possible to single out a unique equilibrium among the adaptively learnable equilibria that is economically meaningful. It is worth noting that the model examined can also be viewed as a model for stock price determination in which the forward‐solutions to the model are preferable to the backward‐solutions since the importance of announcement effects is a common characteristic for currency and stock markets.     

Solutions and Simulations of Some One-Dimensional Stochastic Differential Equations

We consider a one dimensional SDE dX _t  = μ(X _t )dt + σ(X _t )dB _t . We give a new general formula for solutions that involves solving an associated ordinary differential equation. Explicit solutions are obtained in cases where the ODE has such. This recovers linear case but also some non-linear cases. In any case our approach leads to a new simulation scheme that returns positive values for processes on I R⁺{{\hskip 0.02in \hbox{\rm I}\hskip -.02in \hbox{\rm R}{^+}}}, which is advantageous when modelling prices or rates.