Calibration of stochastic volatility models via second order approximation: the Heston model case

Using a suitable Hull and White type formula we develop a methodology to obtain asecond order approximation to the implied volatility for very short maturities. Using thisapproximation we accurately calibrate the full set of parameters of the Heston model. Oneof the reasons that makes our calibration for short maturities so accurate is that we alsotake into account the term-structure for large maturities. We may say that calibration isnot "memoryless", in the sense that the option's behavior far away from maturity doesinfluence calibration when the option gets close to expiration. Our results provide a wayto perform a quick calibration of a closed-form approximation to vanilla options that canthen be used to price exotic derivatives. The methodology is simple, accurate, fast, andit requires a minimal computational cost.

Option pricing under stochastic volatility and stochastic interest rate in the Spanish case

Among the underlying assumptions of the Black-Scholes option pricingmodel, those of a fixed volatility of the underlying asset and of aconstantshort-term riskless interest rate, cause the largest empirical biases. Onlyrecently has attention been paid to the simultaneous effects of thestochasticnature of both variables on the pricing of options. This paper has tried toestimate the effects of a stochastic volatility and a stochastic interestrate inthe Spanish option market. A discrete approach was used. Symmetricand asymmetricGARCH models were tried. The presence of in-the-mean and seasonalityeffectswas allowed. The stochastic processes of the MIBOR90, a Spanishshort-terminterest rate, from March 19, 1990 to May 31, 1994 and of the volatilityofthe returns of the most important Spanish stock index (IBEX-35) fromOctober1, 1987 to January 20, 1994, were estimated. These estimators wereused onpricing Call options on the stock index, from November 30, 1993 to May30, 1994.Hull-White and Amin-Ng pricing formulas were used. These prices werecomparedwith actual prices and with those derived from the Black-Scholesformula,trying to detect the biases reported previously in the literature. Whereasthe conditional variance of the MIBOR90 interest rate seemed to be freeofARCH effects, an asymmetric GARCH with in-the-mean and seasonalityeffectsand some evidence of persistence in variance (IEGARCH(1,2)-M-S) wasfoundto be the model that best represent the behavior of the stochasticvolatilityof the IBEX-35 stock returns. All the biases reported previously in theliterature were found. All the formulas overpriced the options inNear-the-Moneycase and underpriced the options otherwise. Furthermore, in most optiontrading, Black-Scholes overpriced the options and, because of thetime-to-maturityeffect, implied volatility computed from the Black-Scholes formula,underestimatedthe actual volatility.

Long diagnostic delay in Crohn's disease is associated with complicated disease course and increased operation rate

Background: We have recently shown that the median diagnostic delay to establish Crohn's disease (CD) diagnosis (i.e. the period from first symptom onset to diagnosis) in the Swiss IBD Cohort (SIBDC) was 9 months. Seventy five percent of all CD patients were diagnosed within 24 months. The clinical impact of a long diagnostic delay on the natural history of CD is unknown. Aim: To compare the frequency and type of CD-related complications in the patient groups with long diagnostic delay (>24 months) vs. the ones diagnosed within 24 months. Methods: Retrospective analysis of data from the SIBDCS, comprising a large sample of CD patients followed in hospitals and private practices across Switzerland. The proportions of the following outcomes were compared between groups of patients diagnosed 1, 2-5, 6-10, 11-15, and ≥ 16 years ago and stratified according to the length of diagnostic delay: bowel stenoses, internal fistulas, perianal fistulas, CD-related surgical interventions, and extraintestinal manifestations. Results: Two hundred CD patients (121 female, mean age 44.9 ± 15.0 years, 38% smokers, 71% ever treated with immunomodulators and 35% with anti-TNF) with long diagnostic delay were compared to 697 CD patients (358 female, mean age 39.1 ± 14.9 years, 33% smokers, 74% ever treated with immunomodulators and 33% with anti-TNF) diagnosed within 24 months. No differences in the outcomes were observed between the two patient groups within year one after CD diagnosis. Among those diagnosed 2-5 years ago, CD patients with long diagnostic delay (n = 45) presented more frequently with internal fistulas (11.1% vs. 3.1%, p = 0.03) and bowel stenoses (28.9% vs. 15.7%, p = 0.05), and they more frequently underwent CD-related operations (15.6% vs. 5.0%, p = 0.02) compared to the patients diagnosed within 24 months (n = 159). Among those diagnosed 6-10 years ago, CD patients with long diagnostic delay (n = 48) presented more frequently with extraintestinal manifestations (60.4% vs. 34.6%, p = 0.001) than those diagnosed within 24 months (n = 182). For the patients diagnosed 11-15 years ago, no differences in outcomes were found between the long diagnostic delay group (n = 106) and the one diagnosed within 24 months (n = 32). Among those diagnosed ≥ 16 years ago, the group with long diagnostic delay (n = 71) more frequently underwent CD-related operations (63.4% vs. 46.5%, p = 0.01) compared to the group diagnosed with CD within 24 months (n = 241). Conclusions: A long diagnostic delay in CD patients is associated with a more complicated disease course and higher number of CD-related operations in the years following the diagnosis. Our results indicate that efforts should be undertaken to shorten the diagnostic delay in CD patients in order to reduce the risk for progression towards a complicated disease phenotype.

The equity premium and the risk free rate: A cross country, cross maturity examination

This paper examines the relationship between the equity premium and the risk free rate at three different maturities using post 1973 data fora panel of 7 OECD countries. We show the existence of subsample instabilities,of some cross country differences and of inconsistencies with the expectations theory of the term structure. We perform simulations using a standard consumptionbased CAPM model and demonstrate that the basic features of Mehra and Prescott's(1985) puzzle remain, regardless of the time period, the investment maturity and the country considered. Modifications of the basic setup are also considered.

The optimum quantity of money: Theory and evidence

In this paper we propose a simple and general model for computing the Ramsey optimal inflation tax, which includes several models from the previous literature as special cases. We show that it cannot be claimed that the Friedman rule is always optimal (or always non--optimal) on theoretical grounds. The Friedman rule is optimal or not, depending on conditions related to the shape of various relevant functions. One contribution of this paper is to relate these conditions to {\it measurable} variables such as the interest rate or the consumption elasticity of money demand. We find that it tends to be optimal to tax money when there are economies of scale in the demand for money (the scale elasticity is smaller than one) and/or when money is required for the payment of consumption or wage taxes. We find that it tends to be optimal to tax money more heavily when the interest elasticity of money demand is small. We present empirical evidence on the parameters that determine the optimal inflation tax. Calibrating the model to a variety of empirical studies yields a optimal nominal interest rate of less than 1\%/year, although that finding is sensitive to the calibration.

The impact of interest-rate subsidies on long-term household debt: Evidence from a large program

The responsiveness of long-term household debt to the interest rate is acrucial parameter for assessing the effectiveness of public policies aimedat promoting specific types of saving. This paper estimates the effect ofa reform of Credito Bonificado, a large program in Portugal that subsidizedmortgage interest rates, on long-term household debt. The reform establisheda ceiling in the price of the house that could be financed through theprogram, and provides plausibly exogenous variation in incentives. Usinga unique dataset of matched household survey data and administrative recordsof debt, we document a large decrease in the probability of signing a newloan after the removal of the subsidy.

Rate of convergence of a particle method to the solution of the Mc Kean-Vlasov's equation

This paper studies the rate of convergence of an appropriatediscretization scheme of the solution of the Mc Kean-Vlasovequation introduced by Bossy and Talay. More specifically,we consider approximations of the distribution and of thedensity of the solution of the stochastic differentialequation associated to the Mc Kean - Vlasov equation. Thescheme adopted here is a mixed one: Euler/weakly interactingparticle system. If $n$ is the number of weakly interactingparticles and $h$ is the uniform step in the timediscretization, we prove that the rate of convergence of thedistribution functions of the approximating sequence in the $L^1(\Omega\times \Bbb R)$ norm and in the sup norm is of theorder of $\frac 1{\sqrt n} + h $, while for the densities is ofthe order $ h +\frac 1 {\sqrt {nh}}$. This result is obtainedby carefully employing techniques of Malliavin Calculus.

Business cycles, unemployment insurance and the calibration of matching models

This paper theoretically and empirically documents a puzzle that arises when an RBC economy with a job matching function is used to model unemployment. The standard model can generate sufficiently large cyclical fluctuations in unemployment, or a sufficiently small response of unemployment to labor market policies, but it cannot do both. Variable search and separation, finite UI benefit duration, efficiency wages, and capital all fail to resolve this puzzle. However, either sticky wages or match-specific productivity shocks can improve the model's performance by making the firm's flow of surplus more procyclical, which makes hiring more procyclical too.

Interest rate restrictions in a natural experiment: loan allocation and the change in the usury laws in 1714

This article studies the effects of interest rate restrictions on loan allocation. The British governmenttightened the usury laws in 1714, reducing the maximum permissible interest rate from 6% to5%. A sample of individual loan transactions reveals that average loan size and minimum loan sizeincreased strongly, while access to credit worsened for those with little social capital. Collateralisedcredits, which had accounted for a declining share of total lending, returned to their former role ofprominence. Our results suggest that the usury laws distorted credit markets significantly; we findno evidence that they offered a form of Pareto-improving social insurance.

Resting metabolic rate in six- to ten-year-old obese and nonobese children.

The resting metabolic rate (RMR) and body composition of 130 obese and nonobese prepubertal children, aged 6 to 10 years, were assessed by indirect calorimetry and skin-fold thickness, respectively. The mean (+/- SD) RMR was 4619 +/- 449 kJ.day-1 (164 +/- 31 kJ.kg body weight-1 x day-1) in the 62 boys and 4449 +/- 520 kJ.day-1 (147 +/- 32 kJ.kg body weight-1 x day-1) in the 68 girls. Fat-free mass was the best single predictor of RMR (R2 = 0.64; p < 0.001). Step-down multiple regression analysis, with independent variables such as age, gender, weight, and height, allowed several RMR predictive equations to be developed. An equation for boys is as follows: RMR (kJ.day-1) = 1287 + 28.6 x Weight(kg) + 23.6 x Height(cm) - 69.1 x Age(yr) (R2 = 0.58; p < 0.001). An equation for girls is as follows: RMR (kJ.day-1 = 1552 + 35.8 x Weight (kg) + 15.6 x Height (cm) - 36.3 x Age (yr) (R2 = 0.69; p < 0.001). Comparison between the measured RMR and that predicted by currently used formulas showed that most of these equations tended to overestimate the RMR of both genders, especially in overweight children.

Endogenous labor market participation and the business cycle

Existing models of equilibrium unemployment with endogenous labor market participation are complex, generate procyclical unemployment rates and cannot match unemployment variability relative to GDP. We embed endogenous participation in a simple, tractable job market matching model, show analytically how variations in the participation rate are driven by the cross-sectional density of home productivity near the participation threshold, andhow this density translates into an extensive-margin labor supply elasticity. A calibration of the model to macro data not only matches employment and participation variabilities but also generates strongly countercyclical unemployment rates. With some wage rigidity the model also matches unemployment variations well. Furthermore, the labor supply elasticity implied by our calibration is consistent with microeconometric evidence for the US.

The effect of treatment with calcitonin on vertebral fracture rate in osteoporosis.

The efficacy of treatments for osteoporosis does not become evident when evaluated by fracture incidence (FI). Vertebral FI decreased in all controlled studies on calcitonin, but not significantly. Small sample sizes and short periods of treatment may have masked a possible therapeutic benefit, but longer, controlled studies with sodium fluoride or etidronate in larger groups of patients also failed to show a decrease in FI. The present analysis of nine published, therapeutic studies which indicate the FI per year and the initial prevalence of vertebral fractures, examines the question of whether the initial prevalence of fractures has an effect on the subsequent incidence of new fractures and whether the therapeutic effects have to be evaluated as a function of the initial prevalence of fractures. Bearing in mind the differences in roentgenological evaluation and in the size and quality of the various studies, the analysis revealed (1) that in the control groups there was a higher FI in patients with more than three vertebral fractures at baseline (estimated odds ratio (OR) = 49, p = 0.011); (2) that a similar trend, although not statistically significant, was observed in treated patients; (3) that the groups of control patients treated for more than 1 year showed in general an increase in FI beyond the first year and that the reverse was true in treated patients. In conclusion, failure to allow for the initial prevalence of vertebral fractures at the individual level in therapeutic trials of calcitonin to treat osteoporosis and prevent new fractures might have contributed to the absence of a demonstrable benefit of the treatment in those studies.

Constant interest rate projections without the curse of indeterminacy

Constant interest rate (CIR) projections are often criticized on the grounds that they are inconsistent with the existence of a unique equilibrium in a variety of forward-looking models. This note shows howto construct CIR projections that are not subject to that criticism, using a standard New Keynesian model as a reference framework.

Monetary policy and exchange rate volatility in a small open economy

We lay out a small open economy version of the Calvo sticky price model, and show how the equilibrium dynamics can be reduced to simple representation in domestic inflation and the output gap. We use the resulting framework to analyze the macroeconomic implications of three alternative rule-based policy regimes for the small open economy: domestic inflation and CPI-based Taylor rules, and an exchange rate peg. We show that a key difference amongthese regimes lies in the relative amount of exchange rate volatility that they entail. We also discuss a special case for which domestic inflation targeting constitutes the optimal policy, and where a simple second order approximation to the utility of the representative consumer can be derived and used to evaluate the welfare losses associated with the suboptimal rules.