Symmetric Normal Mixture GARCH

Normal mixture (NM) GARCH models are better able to account for leptokurtosis in financial data and offer a more intuitive and tractable framework for risk analysis and option pricing than student’s t-GARCH models. We present a general, symmetric parameterisation for NM-GARCH(1,1) models, derive the analytic derivatives for the maximum likelihood estimation of the model parameters and their standard errors and compute the moments of the error term. Also, we formulate specific conditions on the model parameters to ensure positive, finite conditional and unconditional second and fourth moments. Simulations quantify the potential bias and inefficiency of parameter estimates as a function of the mixing law. We show that there is a serious bias on parameter estimates for volatility components having very low weight in the mixing law. An empirical application uses moment specification tests and information criteria to determine the optimal number of normal densities in the mixture. For daily returns on three US Dollar foreign exchange rates (British pound, euro and Japanese yen) we find that, whilst normal GARCH(1,1) models fail the moment tests, a simple mixture of two normal densities is sufficient to capture the conditional excess kurtosis in the data. According to our chosen criteria, and given our simulation results, we conclude that a two regime symmetric NM-GARCH model, which quantifies volatility corresponding to ‘normal’ and ‘exceptional’ market circumstances, is optimal for these exchange rate data.Volatility regimes, conditional excess kurtosis, normal mixture, heavy trails, exchange rates, conditional heteroscedasticity, GARCH models.

Mortality risk and the valuation of annuities with guaranteed minimum death benefit options: application to the Italian population

In this note, we describe the payoff of Guaranteed Minimum Death Benefit options (GMDB) embedded in annuity contracts and discuss their valuation using data for the Italian male population as a case study. These put options have stochastic maturity dates due to the involuntary exercise at the moment of death. We value the GMDB as a weighted average price of a set of deterministic put options with different maturity dates, where the weights are the probability of death at every date. We take into account the mortality risk and investigate the sensitivity of the price of the option to changes in mortality probability using both deterministic and stochastic approaches

New copula models in quantitative finance

Imperial Users onl

The flexible coefficient multinomial logit (FC-MNL) model of demand for differentiated products

We show FC-MNL is flexible in the sense of Diewert (1974), thus its parameters can be chosen to match a well-defined class of possible own- and cross-price elasticities of demand. In contrast to models such as Probit and Random Coefficient-MNL models, FC-MNL does not require estimation via simulation; it is fully analytic. Under well-defined and testable parameter restrictions, FC-MNL is shown to be an unexplored member of McFadden’s class of Multivariate Extreme Value discrete-choice models. Therefore, FC-MNL is fully consistent with an underlying structural model of heterogeneous, utility-maximizing consumers. We provide a Monte-Carlo study to establish its properties and we illustrate the use by estimating the demand for new automobiles in Italy

Implied Calibration of Stochastic Volatility Jump Diffusion Models

In the context of arbitrage-free modelling of financial derivatives, we introduce a novel calibration technique for models in the affine- quadratic class for the purpose of contingent claims pricing and risk- management. In particular, we aim at calibrating a stochastic volatility jump diffusion model to the whole market volatility surface at any given time. We numerically implement the algorithm and show that the proposed approach is both stable and accurate.Affine-quadratic models, Option pricing, Model Calibration

Proxy simulation schemes using likelihood ratio weighted Monte Carlo for generic robust Monte-Carlo sensitivities and high accuracy drift approximation (with applications to the LIBOR Market Model)

We consider a generic framework for generating likelihood ratio weighted Monte Carlo simulation paths, where we use one simulation scheme K° (proxy scheme) to generate realizations and then reinterpret them as realizations of another scheme K* (target scheme) by adjusting measure (via likelihood ratio) to match the distribution of K° such that E( f(K*) | F_t ) = E( f(K°) w | F_t ). This is done numerically in every time step, on every path. This makes the approach independent of the product (the function f) and even of the model, it only depends on the numerical scheme. The approach is essentially a numerical version of the likelihood ratio method [Broadie & Glasserman, 1996] and Malliavin's Calculus [Fournie et al., 1999; Malliavin, 1997] reconsidered on the level of the discrete numerical simulation scheme. Since the numerical scheme represents a time discrete stochastic process sampled on a discrete probability space the essence of the method may be motivated without a deeper mathematical understanding of the time continuous theory (e.g. Malliavin's Calculus). The framework is completely generic and may be used for high accuracy drift approximations and the robust calculation of partial derivatives of expectations w.r.t. model parameters (i.e. sensitivities, aka. Greeks) by applying finite differences by reevaluating the expectation with a model with shifted parameters. We present numerical results using a Monte-Carlo simulation of the LIBOR Market Model for benchmarking.Monte-Carlo, Likelihood Ratio, Malliavin Calculus, Sensitivities, Greeks