PDE models for pricing fixed rate morgages and their insurance and coinsurance
Carmen Calvo-Garrido, Carlos Vázquez*
In the pricing of fixed rate mortgages with prepayment and default options, we introduce jump-diffusion models for the house price evolution. These models take into account sudden changes in the price (jumps) during bubbles and crisis situations in real estate markets. After posing the models based on partial-integro differential equations (PIDE) problems for the contract, insurance and the fraction of the total loss not covered by the insurance (coinsurance), we propose appropriate numerical methods to solve them. Among this methods are semilagrangian schemes for time discretization combined with finite elements, ALAS algorithm for inequality constraints and quadrature formulas for nonlocal terms.
Randomised Heston models
Inspired by recent works on the behaviour of the forward implied volatility smile, we introduce a new class of stochastic volatility models. The dynamics are the same as the classical Heston model, but the random starting point of the variance process is randomly distributed. We show how to choose the initial distribution (i) to fit the short end of the smile—traditionally mis-calibrated in classical stochastic volatility models, and (ii) to estimate past realisation of the volatility time series. This is a joint work with Fangwei Shi (Imperial College London).
Pricing Bermudan options under local Levy models with default
Anastasia Borovykh, Andrea Pascucci* and Cornelis W. Oosterlee
We consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential Levy-type martingale. This class of models allows for local volatility, local default intensity and a locally dependent Levy measure. We present a pricing method for Bermudan options based on an analytical approximation of the characteristic function combined with the COS method. We derive the adjoint expansion of the characteristic function using a Taylor expansion of the coecients. Due to a special form of the obtained characteristic function the price can be computed using a Fast Fourier Transform- based algorithm resulting in a fast and accurate calculation.
Backtesting Lambda Value at Risk
Jacopo Corbetta*, Ecole des Ponts – ParisTech; Ilaria Peri, University of Greenwich
A new risk measure, the lambda value at risk ($Lambda VaR$), has been recently proposed from a theoretical point of view as an immediate generalization of the value at risk ($VaR$). The $Lambda VaR$ appears to be attractive for its potential ability to solve several problems of the $VaR$.In this paper we propose three nonparametric backtesting methodologies for the $Lambda VaR$ which exploit different features. Two of these tests directly assess the correctness of the level of coverage predicted by the model. One of these tests is bilateral and provides an asymptotic result. A third test assess the accuracy of the $Lambda VaR$ that depends on the choice of the P$&$L distribution. However, this test requires the storage of more information.Finally, we perform a backtesting exercise and we compare our results with the ones from Hitaj and Peri (2015).