International Journal of Theoretical and Applied Finance

This paper provides a new approach for modeling and calculating premiums for unemployment insurance products. The innovative modeling concept consists of combining the benchmark approach with its real-world pricing formula and Markov chain techniques in a doubly stochastic setting. We describe individual insurance claims based on a special type of unemployment insurance contracts, which are offered on the private insurance market. The pricing formulas are first given in a general setting and then specified under the assumption that the individual employment-unemployment process of an employee follows a time-homogeneous doubly stochastic Markov chain. In this framework, formulas for the premiums are provided depending on the ℙ-numéraire portfolio of the benchmark approach. Under a simple assumption on the ℙ-numéraire portfolio, the model is tested on its sensitivities to several parameters. With the same specification the model's employment and unemployment intensities are estimated on public data of the Federal Employment Office in Germany.

We study the term structure of the implied volatility in the presence of a symmetric smile. Exploiting the result by Tehranchi (2009) that a symmetric smile generated by a continuous martingale necessarily comes from a mixture of normal distributions, we derive representation formulae for the at-the-money (ATM) implied volatility level and curvature in a general symmetric model. As a result, the ATM curve is directly related to the Laplace transform of the realized variance. The representation formulae for the implied volatility and its curvature take semi-closed form as soon as this Laplace transform is known explicitly. To deal with the rest of the volatility surface, we build a time dependent SVI-type (Gatheral, 2004) model which matches the ATM and extreme moneyness structure. As an instance of a symmetric model, we consider uncorrelated Heston: in this framework, the SVI approximation displays considerable performances in a wide range of maturities and strikes. All these results can be applied to skewed smiles by considering a displaced model. Finally, a noteworthy fact is that all along the paper we avoid dealing with any complex-valued function.

We propose an approach for pricing and hedging weather derivatives based on including forward looking information about the temperature available to the market. This is achieved by modeling temperature forecasts by a finite dimensional factor model. Temperature dynamics are then inferred in the short end. In analogy to interest rate theory, we establish conditions which guarantee consistency of a factor model with the martingale dynamics of temperature forecasts. Finally, we consider a specific two-factor model and examine in more detail pricing and hedging of weather derivatives in this context.

In this work we propose a new approach to build multivariate pure jump processes. We introduce linear and nonlinear dependence, without restrictions on marginal properties, by imposing a multi-factorial structure separately on both positive and negative jumps. Such a new approach provides higher flexibility in calibrating nonlinear dependence than in other comparable Lévy models in the literature. Using the notion of multivariate subordinator, this modeling approach can be applied to the class of univariate Lévy processes which can be written as the difference of two subordinators. A common example in the financial literature is the variance gamma process, which we extend to the multivariate (multi-factorial) case. The model is tractable and a straightforward multivariate simulation procedure is available. An empirical analysis documents an accurate multivariate fit of stock index returns in terms of both linear and nonlinear dependence. An example of multi-asset option pricing emphasizes the importance of the proposed multivariate approach.

For purposes of Value-at-Risk estimation, we consider several multivariate families of heavy-tailed distributions, which can be seen as multidimensional versions of Paretian stable and Student's t distributions allowing different marginals to have different indices of tail thickness. After a discussion of relevant estimation and simulation issues, we conduct a backtesting study on a set of portfolios containing derivative instruments, using historical US stock price data.

The present study addresses the problem of designing a catastrophe derivative that insurers can use to hedge catastrophe-related losses in an incomplete market. The losses are modeled as a doubly stochastic compound Poisson process with shot-noise intensity. The hedging capability of a derivative is measured by the reduction of the mean squared hedging error resulting from optimal trading in the derivative. A general form of this measure is obtained in terms of the coefficients in the martingale dynamics of the loss process and the price process of the derivative. Six specific derivatives, with pay-offs depending in different ways on available catastrophe indices and portfolio data, are compared by the proposed criterion.

We present a new numerical method to price vanilla options quickly in time-changed Brownian motion models. The method is based on rational function approximations of the Black-Scholes formula. Detailed numerical results are given for a number of widely used models. In particular, we use the variance-gamma model, the CGMY model and the Heston model without correlation to illustrate our results. Comparison to the standard fast Fourier transform method with respect to accuracy and speed appears to favour the newly developed method in the cases considered. We present error estimates for the option prices. Additionally, we use this method to derive a procedure to compute, for a given set of arbitrage-free European call option prices, the corresponding Black-Scholes implied volatility surface. To achieve this, rational function approximations of the inverse of the Black-Scholes formula are used. We are thus able to work out implied volatilities more efficiently than one can by the use of other common methods. Error estimates are presented for a wide range of parameters.