# 2016-Financial and Actuarial Mathematics

Financial and Actuarial Mathematics
Organizer and Chair: Cody Hyndman (Concordia University)
[PDF]

ALEX BADESCU, University of Calgary
Non-Affine GARCH Option Pricing Models, Variance Dependent Kernels, and Diffusion Limits  [PDF]

This paper investigates the pricing and weak convergence of an asymmetric non-affine, non-Gaussian GARCH model when the risk-neutralization is based on a variance dependent exponential linear pricing kernel with stochastic risk aversion parameters. The risk-neutral dynamics are obtained for a general setting and its weak limit is derived. We show how several GARCH diffusions, martingalized via well-known pricing kernels, are obtained as special cases and we derive necessary and sufficient conditions for the presence of financial bubbles. An extensive empirical analysis using both historical returns and options data illustrates the advantage of coupling this pricing kernel with non-Gaussian innovations.

CODY HYNDMAN, Concordia University
Issuing a Convertible Bond with Call-Spread Overlay: Incorporating the Effects of Convertible Arbitrage  [PDF]

Firms may attempt to mitigate some of the negative impacts of issuing convertible bonds, such as the dilution of existing shares, by concurrently entering into call-spread overlays'' or other transactions. Previous empirical studies show the stock prices of convertible bond issuers drops on the issue announcement date due to the activities of convertible bond arbitrageurs. We explore the motivation for using these combined transactions and price the convertible bonds with call-spread subject to default risk. We propose a model to estimate the drop in the stock price due to convertible bond arbitrage activities at the time of planning and designing the security to be offered but before announcement. We examine the features of the model with simulated and real-world data.

ADAM METZLER, Wilfred Laurier University
Importance Sampling for Portfolio Loss Probabilities under Conditional Independence [PDF]

We develop a novel importance sampling algorithm for estimating the probability of large portfolio losses in the conditional independence framework. We apply exponential tilts to (i) the distribution of the natural sufficient statistic of the systematic risk factors and (ii) the conditional distribution of portfolio loss, given the simulated values of the systematic risk factors, and select parameter values by minimizing the Kullback-Leibler divergence of the resulting parametric family from the ideal (zero-variance) importance density. Optimal parameter values are shown to satisfy intuitive moment-matching conditions, and the asymptotic behaviour of large portfolios is used to approximate the requisite moments. In a sense we extend Glasserman and Li (2005) to allow for heavy-tailed risk factors and/or PD-LGD correlation.