Sinem Kozpınar, Ph.D.
Department of Financial Mathematics
September 2018, 82 pages

Supervisor: Ömür Uğur (Institute of Applied Mathematics, Middle East Technical University, Ankara)
Co-supervisor: Zehra Eksi-Altay (Institute for Statistics and Mathematics, WU Vienna University of Economics and Business)

### Abstract

This thesis aims to study the evaluation of spread and basket options under the classical Markov-modulated framework, for which a transition in the Markov process leads to a switch in the parameters. In this regard, we provide approximations to the exact option prices based on ideas from the literature without regime switching.

We start with pricing spread options when risky assets follow Markov-modulated geometric Brownian motions (MMGBMs). In this context, we focus on the regimes witching version of Kirk’s formula. For that reason, a change of numeraire technique is introduced which allows to associate the spread option price with the value of a European call option. Since the underlying asset of this European call follows a MMGBM for relatively small strikes, we evaluate the spread option by using Markov modulated Black-Scholes formula. Then, we discuss the valuation of spread options when the underlying asset prices are driven by Markov-modulated Lévy processes (MMLPs). Under this modeling set-up, we approximate the spread option price by means of a lower bound, which is obtained via a univariate Fourier inversion. For this method, we only require the joint characteristic function; and therefore, our approximation becomes valid for many regime-switching models.

Afterwards, we concentrate on the valuation of basket options for which we provide lower and upper bounds considering the MMLP framework. We first obtain an accurate lower bound by using a univariate Fourier inversion combined with an optimization procedure. However, this optimization procedure increases the computational cost. Therefore, we then derive faster analogous bounds by using the arithmetic-geometric mean inequality and univariate Fourier inversion without an optimization. As in the case of spread options, the approaches we followed for basket options are applicable to several MMLPs under which the joint characteristic functions of the underlying assets are known analytically.

Furthermore in this thesis we aim to price spread and basket options under a more generalized framework, in which a transition in the Markov process may induce a switch in the parameters as well as synchronous jumps in the asset prices. For this purpose, we extend the results obtained under the classical MMLP framework, which does not take the synchronous jumps into account, to this generalized framework.

Finally, in order to verify the accuracy of proposed approximations presented in this thesis, we include several numerical experiments.

Keywords: Regime-switching, Spread options, Kirk’s formula, Basket options, Fourier inversion, Synchronous jumps