B-spline techniques for volatility modeling

This paper is devoted to the application of B-splines to volatility modeling, specifically the calibration of the leverage function in stochastic local volatility models and the parameterization of an arbitrage-free implied volatility surface calibrated to sparse option data. We use an extension of classical B-splines obtained by including basis functions with infinite support. We first come back to the application of shape-constrained B-splines to the estimation of conditional expectations, not merely from a scatter plot but also from the given marginal distributions. An application is the Monte Carlo calibration of stochastic local volatility models by Markov projection. Then we present a new technique for the calibration of an implied volatility surface to sparse option data. We use a B-spline parameterization of the Radon-Nikodym derivative of the underlying's risk-neutral probability density with respect to a roughly calibrated base model. We show that this method provides smooth arbitrage-free implied volatility surfaces. Finally, we sketch a Galerkin method with B-spline finite elements to the solution of the partial differential equation satisfied by the Radon-Nikodym derivative.Comment: 25 page

Nonparametric Estimation of Risk-Neutral Densities

This chapter deals with nonparametric estimation of the risk neutral density. We present three different approaches which do not require parametric functional assumptions on the underlying asset price dynamics nor on the distributional form of the risk neutral density. The first estimator is a kernel smoother of the second derivative of call prices, while the second procedure applies kernel type smoothing in the implied volatility domain. In the conceptually different third approach we assume the existence of a stochastic discount factor (pricing kernel) which establishes the risk neutral density conditional on the physical measure of the underlying asset. Via direct series type estimation of the pricing kernel we can derive an estimate of the risk neutral density by solving a constrained optimization problem. The methods are compared using European call option prices. The focus of the presentation is on practical aspects such as appropriate choice of smoothing parameters in order to facilitate the application of the techniques.Risk neutral density, Pricing kernel, Kernel smoothing, Local polynomials, Series methods

Nonparametric Option Pricing under Shape Restrictions

Frequently, economic theory places shape restrictions on functional relationships between economic variables. This paper develops a method to constrain the values of the first and second derivatives of nonparametric locally polynomial estimators. We apply this technique to estimate the state price density (SPD), or risk-neutral density, implicit in the market prices of options. The option pricing function must be monotonic and convex. Simulations demonstrate that nonparametric estimates can be quite feasible in the small samples relevant for day-to-day option pricing, once appropriate theory-motivated shape restrictions are imposed. Using S&P500 option prices, we show that unconstrained nonparametric estimators violate the constraints during more than half the trading days in 1999, unlike the constrained estimator we propose.

Arbitrage-Free Smoothing of the Implied Volatility Surface

The pricing accuracy and pricing performance of local volatility models crucially depends on absence of arbitrage in the implied volatility surface: an input implied volatility surface that is not arbitrage-free invariably results in negative transition probabilities and/ or negative local volatilities, and ultimately, into mispricings. The common smoothing algorithms of the implied volatility surface cannot guarantee the absence arbitrage. Here, we propose an approach for smoothing the implied volatility smile in an arbitrage-free way. Our methodology is simple to implement, computationally cheap and builds on the well-founded theory of natural smoothing splines under suitable shape constraints. Unlike other methods, our approach also works when input data are scarce and not arbitrage-free. Thus, it can easily be integrated into standard local volatility pricers.Arbitrage-Free Smoothing, Volatility, Implied Volatility Surface

Bayesian Analysis of Hazard Regression Models under Order Restrictions on Covariate Effects and Ageing

We propose Bayesian inference in hazard regression models where the baseline hazard is unknown, covariate effects are possibly age-varying (non-proportional), and there is multiplicative frailty with arbitrary distribution. Our framework incorporates a wide variety of order restrictions on covariate dependence and duration dependence (ageing). We propose estimation and evaluation of age-varying covariate effects when covariate dependence is monotone rather than proportional. In particular, we consider situations where the lifetime conditional on a higher value of the covariate ages faster or slower than that conditional on a lower value; this kind of situation is common in applications. In addition, there may be restrictions on the nature of ageing. For example, relevant theory may suggest that the baseline hazard function decreases with age. The proposed framework enables evaluation of order restrictions in the nature of both covariate and duration dependence as well as estimation of hazard regression models under such restrictions. The usefulness of the proposed Bayesian model and inference methods are illustrated with an application to corporate bankruptcies in the UK

Bayesian Analysis of Hazard Regression Models under Order Restrictions on Covariate Effects and Ageing

We propose Bayesian inference in hazard regression models where the baseline hazard is unknown, covariate effects are possibly age-varying (non-proportional), and there is multiplicative frailty with arbitrary distribution. Our framework incorporates a wide variety of order restrictions on covariate dependence and duration dependence (ageing). We propose estimation and evaluation of age-varying covariate effects when covariate dependence is monotone rather than proportional. In particular, we consider situations where the lifetime conditional on a higher value of the covariate ages faster or slower than that conditional on a lower value; this kind of situation is common in applications. In addition, there may be restrictions on the nature of ageing. For example, relevant theory may suggest that the baseline hazard function decreases with age. The proposed framework enables evaluation of order restrictions in the nature of both covariate and duration dependence as well as estimation of hazard regression models under such restrictions. The usefulness of the proposed Bayesian model and inference methods are illustrated with an application to corporate bankruptcies in the UK.Bayesian nonparametrics; Nonproportional hazards; Frailty; Age-varying covariate e¤ects; Ageing

Dynamics of state price densities

State price densities (SPDs) are an important element in applied quantitative finance. In a Black-Scholes world they are lognormal distributions but in practice volatility changes and the distribution deviates from log-normality. In order to study the degree of this deviation, we estimate SPDs using EUREX option data on the DAX index via a nonparametric estimator of the second derivative of the (European) call pricing function. The estimator is constrained so as to satisfy no-arbitrage constraints and corrects for the intraday covariance structure in option prices. In contrast to existing methods, we do not use any parametric or smoothness assumptions

Rating Crop Insurance Policies with Efficient Nonparametric Estimators that Admit Mixed Data Types

The identification of improved methods for characterizing crop yield densities has experienced a recent surge in activity due in part to the central role played by crop insurance in the Agricultural Risk Protection Act of 2000 (estimates of yield densities are required for the determination of insurance premium rates). Nonparametric kernel methods have been successfully used to model yield densities; however, traditional kernel methods do not handle the presence of categorical data in a satisfactory manner and have therefore tended to be applied on a county-by-county basis. By utilizing recently developed kernel methods that admit mixed data types, we are able to model the yield density jointly across counties, leading to substantial finite sample efficiency gains. Findings show that when we allow insurance companies to strategically reinsure with the government based on this novel approach they accrue significant rents.discrete data, insurance rating, kernel estimation, yield distributions, Risk and Uncertainty,