Multivariate financial models and partial integro-differential equations for the valuation of financial options

Due to their versatility, multi-name or rainbow options are popular financial products, which are traded increasingly since the opening of the Chicago Board Option Exchange (CBOE) in 1973. An option is a so-called financial derivative, a contract whose value depends on some underlying asset(s). Determining a fair value for these types of financial contracts is a central question in financial mathematics. Black and Scholes were the first to derive, under some ideal assumptions on the underlying asset price, a time-dependent partial differential equation (PDE) for the fair value of a European option, and a semiclosed-form solution to it. This famous Black-Scholes formula was published only one month after the opening of the CBOE and caused a boom in the number of traded options. However, the assumptions under the Black-Scholes model are not in line with market observations, and over the years many extensions and alternatives to the Black-Scholes model, both single-name and multi-name, have been developed. Such a multi-name financial model should be flexible enough to explain the stylized facts of asset log-returns and at the same time remain computationally tractable, such that the model can be calibrated on market ​data. ​Once such a model is established, the fair option value can be obtained by computing an expected value under some equivalent martingale measure. However, due to the increasing complexity of the financial models for the underlying asset prices, the option value can often no longer be computed exactly and numerical methods are required to approximate the fair option value. The development of fast, accurate and stable methods for obtaining the fair value of all sorts of options forms a central topic in the field of computational finance. A prominent technique is to numerically solve the time-dependent PDE that holds for the option value using finite difference methods, among which the Method-of-Lines (MOL) approach is popular. This approach consists of two consecutive steps, where first the PDE is discretized in space and subsequently in time. When the process driving the underlying assets exhibit jumps, the option value is described by a time-dependent partial integro-differential equation (PIDE), which includes a nonlocal integral term. This integral term poses an extra challenge for numerical methods. ​This thesis focuses on the two central questions within financial mathematics described above, namely the development and calibration of tractable and flexible multivariate financial models and the valuation of rainbow options under multi-name financial models exhibiting jumps.​

Publication year:2019

Keywords:Doctoral thesis

Accessibility:Open