The Variance Gamma Self-Decomposable Process in Actuarial Modelling

A scaled self-decomposable stochastic process put forward by Carr, Geman, Madan
and Yor (2007) is used to model long term equity returns and options prices. This
parsimonious model is compared to a number of other one-dimensional continuous time
stochastic processes (models) that are commonly used in finance and the actuarial
sciences. The comparisons are conducted along three dimensions: the models ability to
fit monthly time series data on a number of different equity indices; the models ability to
fit the tails of the times series and the models ability to calibrate to index option prices
across strike price and maturities. The last criteria is becoming increasingly important
given the popularity of capital gauranteed products that contain long term imbedded
options that can be (at least partially) hedged by purchasing short term index options
and rolling them over or purchasing longer term index options. Thus we test if the
models can reproduce a typical implied volatility surface seen in the market.