Ross Model Research Articles

Solutions of two-factor models with variable interest rates

The focus of this work is on numerical solutions to two-factor option pricing partial differential equations with variable interest rates. Two interest rate models, the Vasicek model and the Cox–Ingersoll–Ross model (CIR), are considered. Emphasis is placed on the definition and implementation of boundary conditions for different portfolio models, and on appropriate truncation of the computational domain. An exact solution to the Vasicek model and an exact solution for the price of bonds convertible to stock at expiration under a stochastic interest rate are derived. The exact solutions are used to evaluate the accuracy of the numerical simulation schemes. For the numerical simulations the pricing solution is analyzed as the market completeness decreases from the ideal complete level to one with higher volatility of the interest rate and a slower mean-reverting environment. Simulations indicate that the CIR model yields more reasonable results than the Vasicek model in a less complete market.

A method for computing the transition probability density associated with a multifactor Cox–Ingersoll–Ross model of the term structure of interest rates with no drift term

We consider an n -dimensional square root process and we obtain a formula involving series expansions for the associated transition probability density. The process mentioned previously can be used to model forward rates, future prices, forward prices and, as a consequence, can be used to price derivatives on these underlyings. The formula that we propose for the transition probability density has been obtained using appropriately a perturbative expansion in the correlation coefficients of the square root process, the Fourier transform and the method of characteristics to solve first-order hyperbolic partial differential equations. The computational effort needed to evaluate this formula is polynomial with respect to the dimension n of the space spanned by the square root process when the order where all the series involved in the transition probability density formula are truncated is fixed. This strategy gives an accuracy that some numerical tests show approximately constant for a wide range of values of n . Some examples of prices of financial derivatives whose evaluation involves integrals in two, twenty and one hundred dimensions (i.e. n = 2 , 20 , 100 ), that is derivatives on two, twenty and one hundred assets, where accurate results can be obtained are shown. An experiment shows that the formula derived here for the transition probability density is well suited for parallel computing. This feature makes the formula computationally very attractive to price derivatives of the LIBOR market such as caplets or swaptions since the use of parallel computing and the formula makes it possible to evaluate derivatives on several tens of underlyings in negligible times. The website http://www.econ.univpm.it/recchioni/finance/w1 contains an interactive tool that helps with the understanding of this paper and a portable software library that makes it possible to the user to exploit the formula derived in this paper to evaluate the transition probability densities of its own models and the prices of the associated financial derivatives.

The consumption-based determinants of the term structure of discount rates

The rate of return of a zero-coupon bond with maturity T is determined by our expectations about the mean (+), variance (-) and skewness (+) of the growth of aggregate consumption between 0 and T. The shape of the yield curve is thus determined by how these moments vary with T. We first examine growth processes in which a higher past economic growth yields a first-degree dominant shift in the distribution of the future economic growth, as assumed for example by Vasicek (J. Financ. Econ. 5, 177–188, 1977). We show that when the growth process exhibits such a positive serial dependence, then the yield curve is decreasing if the representative agent is prudent (\(u^{\prime \prime \prime } > 0\)), because of the increased risk that it yields for the distant future. A similar definition is proposed for the concept of second-degree stochastic dependence, as observed for example in the Cox–Ingersoll–Ross model, with the opposite comparative static property holding under temperance (\(u^{\prime \prime \prime \prime } < 0\)), because the change in downside risk (or skweness) that it generates. Finally, using these theoretical results, we propose two arguments in favor of using a smaller rate to discount cash-flows with very large maturities, as those associated to global warming or nuclear waste management.

Implications of Two Well-Known Models For Instructional Designers In Distance Education:Dick-Carey Versus Morrison-Ross-Kemp

This paper first summarizes, and then compares and contrasts two well-known instructional design models: Dick and Carey Model (DC) and Morrison, Ross and Kemp model (MRK). The target audiences of both models are basically instructional designers. Both models have applications for different instructional design settings. They both see the instructional design as a means to problem-solving. However, there are also differences between the two models. Applications of each model for instructional design and technology are discussed, and a reference to instructional designers in distance education was made.

Evaluation of Two Computational Models Based on Different Effective Core Potentials for Use in Organocesium Chemistry.

The performance of two computational models was evaluated in describing some aggregated structures, the bond lengths and dimerization energies of cesium halides, aquation energies of the cesium cation, and protonation energies of a range of organocesium compounds. One model used the Hay-Wadt (HW) effective core potential (ECP) and a double-ζ valence basis set on Cs; the other used the Ross ECP with two polarization functions on Cs. In both models, the standard 6-31+G** basis was used for the other atoms. At the Hartree-Fock (HF) level, the Ross ECP was found to give geometries and energies in good agreement with experimental results. Second-order Møller-Plesset calculations with this model gave only modestly improved results compared to HF; the B3LYP level gave variable results with unsatisfactory energies. Although the HW model is generally less satisfactory, it often shows comparable trends to those of the Ross model.

Arbitrage with Fixed Costs and Interest Rate Models

AbstractWe study securities market models with fixed costs. We first characterize the absence of arbitrage opportunities and provide fair pricing rules. We then apply these results to extend some popular interest rate and option pricing models that present arbitrage opportunities in the absence of fixed costs. In particular, we prove that the quite striking result obtained by Dybvig, Ingersoll, and Ross (1996), which asserts that under the assumption of absence of arbitrage long zero-coupon rates can never fall, is no longer true in models with fixed costs, even arbitrarily small fixed costs. For instance, models in which the long-term rate follows a diffusion process are arbitrage-free in the presence of fixed costs (including arbitrarily small fixed costs). We also rationalize models with partially absorbing or reflecting barriers on the price processes. We propose a version of the Cox, Ingersoll, and Ross (1985) model which, consistent with Longstaff (1992), produces yield curves with realistic humps, but does not assume an absorbing barrier for the short-term rate. This is made possible by the presence of (even arbitrarily small) fixed costs.

Empirical evaluation of the market price of risk using the CIR model

We describe a simple but effective method for the estimation of the market price of risk. The basic idea is to compare the results obtained by following two different approaches in the application of the Cox–Ingersoll–Ross (CIR) model. In the first case, we apply the non-linear least squares method to cross sectional data (i.e., all rates of a single day). In the second case, we consider the short rate obtained by means of the first procedure as a proxy of the real market short rate. Starting from this new proxy, we evaluate the parameters of the CIR model by means of martingale estimation techniques. The estimate of the market price of risk is provided by comparing results obtained with these two techniques, since this approach makes possible to isolate the market price of risk and evaluate, under the Local Expectations Hypothesis, the risk premium given by the market for different maturities. As a test case, we apply the method to data of the European Fixed Income Market.

Valuation and hedging of life insurance liabilities with systematic mortality risk

This paper considers the problem of valuating and hedging life insurance contracts that are subject to systematic mortality risk in the sense that the mortality intensity of all policy-holders is affected by some underlying stochastic processes. In particular, this implies that the insurance risk cannot be eliminated by increasing the size of the portfolio and appealing to the law of large numbers. We propose to apply techniques from incomplete markets in order to hedge and valuate these contracts. We consider a special case of the affine mortality structures considered by Dahl [Dahl, M., 2004. Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts. Insurance Math. Econom. 35, 113–136], where the underlying mortality process is driven by a time-inhomogeneous Cox–Ingersoll–Ross (CIR) model. Within this model, we study a general set of equivalent martingale measures, and determine market reserves by applying these measures. In addition, we derive risk-minimizing strategies and mean-variance indifference prices and hedging strategies for the life insurance liabilities considered. Numerical examples are included, and the use of the stochastic mortality model is compared with deterministic models.

Empirically Confronting Stochastic Singularity: An Application to the Cox, Ingersoll, and Ross Model

We just-identify a no-arbitrage term structure model in estimation and then test it using both a classical orthogonality restriction test and a test of conditional predictive ability. We treat the error structure as unmodeled heterogeneity so that the model is estimated without errors, and the statistical question is whether using the model to characterize the dynamics and patterns in historical data is either useful or optimal as a forecasting tool. The data we use are from the transparent Greenspan regime at the Fed (1989-2005), and we also use a rolling estimation format so that regime shifts are not a likely cause of the model's performance. Substantively we find that the model is not a good forecasting device for short rates which in this period are strongly affected by changes in the target Fed Funds rate. For longer term rates, especially at longer forecast horizons where Fed policy has no effect, the model is more informative.

The entomological inoculation rate and Plasmodium falciparum infection in African children

Malaria is an important cause of global morbidity and mortality. The fact that some people are bitten more often than others has a large effect on the relationship between risk factors and prevalence of vector-borne diseases. Here we develop a mathematical framework that allows us to estimate the heterogeneity of infection rates from the relationship between rates of infectious bites and community prevalence. We apply this framework to a large, published data set that combines malaria measurements from more than 90 communities. We find strong evidence that heterogeneous biting or heterogeneous susceptibility to infection are important and pervasive factors determining the prevalence of infection: 20% of people receive 80% of all infections. We also find that individual infections last about six months on average, per infectious bite, and children who clear infections are not immune to new infections. The results have important implications for public health interventions: the success of malaria control will depend heavily on whether efforts are targeted at those who are most at risk of infection.

A Fuzzy Set Approach for Generalized CRR Model: An Empirical Analysis of S&amp;P 500 Index Options

This paper applies fuzzy set theory to the Cox, Ross and Rubinstein (CRR) model to set up the fuzzy binomial option pricing model (OPM). The model can provide reasonable ranges of option prices, which many investors can use it for arbitrage or hedge. Because of the CRR model can provide only theoretical reference values for a generalized CRR model in this article we use fuzzy volatility and fuzzy riskless interest rate to replace the corresponding crisp values. In the fuzzy binomial OPM, investors can correct their portfolio strategy according to the right and left value of triangular fuzzy number and they can interpret the optimal difference, according to their individual risk preferences. Finally, in this study an empirical analysis of S&P 500 index options is used to find that the fuzzy binomial OPM is much closer to the reality than the generalized CRR model.

Multivariate Jacobi process with application to smooth transitions

Abstract We introduce the multivariate Jacobi process as a representation for the dynamics of a stochastic discrete probability distribution. Its domain of application is dynamic analysis of switching regimes in asset return volatility, business cycle and corporate credit ratings. The paper shows how the multivariate Jacobi process is derived from the multivariate Cox–Ingersoll–Ross (CIR) model by time deformation and presents the main distributional properties. For illustration, selected continuous time models of prices and returns on financial assets are extended to smooth transitions processes featuring regimes of different volatilities and persistence. In this framework the effects of transitions between the regimes on derivative prices and long memory are examined.

Wiener chaos and the Cox–Ingersoll–Ross model

In this paper we recast the Cox–Ingersoll–Ross (CIR) model of interest rates into the chaotic representation recently introduced by Hughston and Rafailidis. Beginning with the ‘squared Gaussian representation’ of the CIR model, we find a simple expression for the fundamental random variable X ∞ . By use of techniques from the theory of infinite–dimensional Gaussian integration, we derive an explicit formula for the n th term of the Wiener chaos expansion of the CIR model, for n = 0,1,2,…. We then derive a new expression for the price of a zero coupon bond which reveals a connection between Gaussian measures and Ricatti differential equations.

Jackknifing Bond Option Prices

In continuous time specifications, the prices of interest rate derivative securities depend crucially on the mean reversion parameter of the associated interest rate diffusion equation. This parameter is well known to be subject to estimation bias when standard methods like maximum likelihood (ML) are used. The estimation bias can be substantial even in very large samples and it translates into a bias in pricing bond options and other derivative securities that is important in practical work. The present paper proposes a very general method of bias reduction for pricing bond options that is based on Quenouille's (1956) jackknife. We show how the method can be applied directly to the options price itself as well as the coefficients in continuous time models. The method is implemented and evaluated here in the Cox, Ingersoll and Ross (1985) model, although it has much wider applicability. A Monte Carlo study shows that the proposed procedure achieves substantial bias reductions in pricing bond options with only mild increases in variance that do not compromise the overall gains in mean squared error. Our findings indicate that bias correction in estimation of the drift can be more important in pricing bond options than correct specification of the diffusion. Thus, even if ML or approximate ML can be used to estimate more complicated models, it still appears to be of equal or greater importance to correct for the effects on pricing bond options of bias in the estimation of the drift. An empirical application to U.S. interest rates highlights the differences between bond and option prices implied by the jackknife procedure and those implied by the standard approach. These differences are large and suggest that bias reduction in pricing options is important in practical applications.

Statistical Surveillance of the Parameters of a One-Factor Cox–Ingersoll–Ross Model

In the paper, the tools of statistical process control, and in particular control charts, are applied to sequentially detect a change in the parameters of the Cox–Ingersoll–Ross term-structure model. Different types of multivariate EWMA control charts are introduced and constructed, taking into account the peculiarities of the Cox– Ingersoll–Ross model. The effectiveness of the control charts is analysed on the basis of extensive Monte Carlo simulations for a parameter constellation obtained from a data set on the term-structure in USA. In addition, the paper includes a practical example of a detection of a change in an initially estimated Cox–Ingersoll–Ross model.

PRICING S&amp;P 500 INDEX OPTIONS UNDER STOCHASTIC VOLATILITY WITH THE INDIRECT INFERENCE METHOD

This paper studies the price of SP in the second step, the risk premium, λ, the spot variance, vt, and the correlation coefficient between the asset return and its volatility, ρ, are estimated by a nonlinear least-squares method that minimizes the sum of the squares of the error between the cross-sectional option price and the corresponding model price. The model performance is assessed by directly comparing the computed option model price with the market price. We find that both the Black–Scholes model and the Heston model overprice the out-of-the-money options and underprice the in-the-money options, but the degree of the bias is different. The Heston model significantly outperforms the Black–Scholes model in almost all moneyness-maturity groups. On average, the Heston model can reduce pricing errors by about 25%. However, pricing bias still exists in the Heston model. In particular, the Heston model always overprices short-term options, indicating that some other factors, such as the random jump, may also be needed to explain the option price.

Inference for Observations of Integrated Diffusion Processes

Abstract. Estimation of parameters in diffusion models is investigated when the observations are integrals over intervals of the process with respect to some weight function. This type of observations can, for example, be obtained when the process is observed after passage through an electronic filter. Another example is provided by the ice‐core data on oxygen isotopes used to investigate paleo‐temperatures. Finally, such data play a role in connection with the stochastic volatility models of finance. The integrated process is not a Markov process. Therefore, prediction‐based estimating functions are applied to estimate parameters in the underlying diffusion model. The estimators are shown to be consistent and asymptotically normal. The theory developed in the paper also applies to integrals of processes other than diffusions. The method is applied to inference based on integrated data from Ornstein–Uhlenbeck processes and from the Cox–Ingersoll–Ross model, for both of which an explicit optimal estimating function is found.

The Term Structure of Interest Rates: A Test of the Cox, Ingersoll and Ross Model on Italian Treasury Bonds

This paper tests the Cox, Ingersoll and Ross model using the prices of Italian Treasury bonds in the secondary market. The model is estimated daily for the period 30 December 1983 to 13 March 1989. The resulting term structures of interest rates are compared with those obtained using interpolation techniques (the cubic splines method). The daily estimation of the yield curves also makes it possible to analyze the changes in Treasury bond prices, determine the turning points and obtain useful indications regarding the efficiency of the secondary market and the consistency between the primary and the secondary markets.

The Valuation of Puttable Bonds: An Application of the Cox, Ingersoll and Ross Model to Italian Treasury Option Certificates

The Italian Treasury's puttable bonds (Certificati del Tesoro con opzione di rimborso anticipato - CTOs) are the first example in Italy of retractable/extendible bonds, which have been used on the Canadian market for some time and recently been adopted on the Euromarket. In this paper the single-factor version of the Cox, Ingersoll and Ross model is used to determine the equilibrium value of CTOs at issue. The simulation of the effects of changes in their features provides useful information on the optimal design of CTOs.

A field and numerical study of the evolution of sea-ice thickness in the Ross Sea, Antarctica, 1998-99

AbstractDuring two cruises in 1998 and 1999, we examined drift and ridging characteristics of sea ice in the Ross Sea, Antarctica. Mean ice thickness in the western Ross Sea in autumn was 0.5 m, while higher level-ice thicknesses, greater areal coverages of ridges and higher sails were found in the central and eastern Ross Sea in summer. Near the continent, ice drifted westward near the coast and turned eastward further north. We use a regional sea-ice−mixed-layer−pycnocline model to initiate backward trajectories at the time and location of field observations and examine the dynamic and thermodynamic processes that determine ice thickness along these trajectories. Model results agree with previously published field data to indicate that thermodynamic and dynamic thickening and snow-ice formation each contribute significantly to the ice mass of the summer ice field in the central and eastern Ross Sea. For first-year ice in the western Ross Sea, model results and field data both indicate that thermodynamic thickening is the dominant process that determines ice thickness, with dynamic thickening also contributing 20% to the net ice-thickening rate. However, model results fail to reproduce the prevalence of snow- ice formation that was seen in field data.