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  • (2023) Boglioni Beaulieu, Guillaume
    Accurately capturing the dependence between risks, if it exists, is an increasingly relevant topic of actuarial research. In recent years, several authors have started to relax the traditional 'independence assumption', in a variety of actuarial settings. While it is known that 'mutual independence' between random variables is not equivalent to their 'pairwise independence', this thesis aims to provide a better understanding of the materiality of this difference. The distinction between mutual and pairwise independence matters because, in practice, dependence is often assessed via pairs only, e.g., through correlation matrices, rank-based measures of association, scatterplot matrices, heat-maps, etc. Using such pairwise methods, it is possible to miss some forms of dependence. In this thesis, we explore how material the difference between pairwise and mutual independence is, and from several angles. We provide relevant background and motivation for this thesis in Chapter 1, then conduct a literature review in Chapter 2. In Chapter 3, we focus on visualising the difference between pairwise and mutual independence. To do so, we propose a series of theoretical examples (some of them new) where random variables are pairwise independent but (mutually) dependent, in short, PIBD. We then develop new visualisation tools and use them to illustrate what PIBD variables can look like. We showcase that the dependence involved is possibly very strong. We also use our visualisation tools to identify subtle forms of dependence, which would otherwise be hard to detect. In Chapter 4, we review common dependence models (such has elliptical distributions and Archimedean copulas) used in actuarial science and show that they do not allow for the possibility of PIBD data. We also investigate concrete consequences of the 'nonequivalence' between pairwise and mutual independence. We establish that many results which hold for mutually independent variables do not hold under sole pairwise independent. Those include results about finite sums of random variables, extreme value theory and bootstrap methods. This part thus illustrates what can potentially 'go wrong' if one assumes mutual independence where only pairwise independence holds. Lastly, in Chapters 5 and 6, we investigate the question of what happens for PIBD variables 'in the limit', i.e., when the sample size goes to infi nity. We want to see if the 'problems' caused by dependence vanish for sufficiently large samples. This is a broad question, and we concentrate on the important classical Central Limit Theorem (CLT), for which we fi nd that the answer is largely negative. In particular, we construct new sequences of PIBD variables (with arbitrary margins) for which a CLT does not hold. We derive explicitly the asymptotic distribution of the standardised mean of our sequences, which allows us to illustrate the extent of the 'failure' of a CLT for PIBD variables. We also propose a general methodology to construct dependent K-tuplewise independent (K an arbitrary integer) sequences of random variables with arbitrary margins. In the case K = 3, we use this methodology to derive explicit examples of triplewise independent sequences for which no CLT hold. Those results illustrate that mutual independence is a crucial assumption within CLTs, and that having larger samples is not always a viable solution to the problem of non-independent data.

  • (2023) Zarban, Ashwaq Ali Y
    This thesis aims to price the exchange option in two different scenarios theoretically. First, we consider a continuous-time, finite-state regime-switching framework to price such an option under double regime-switching jump-diffusion models. We derive the characteristic function as a matrix exponential, which, using matrix notation, allows us to derive the compensators for the synchronous jumps and find the expected values for the price. Using a unit vector representation of the regime-switching process, we can simplify the calculation from a nonlinear function of the value of Markov chain $X_{t}$ into a linear function. Second, as this type of option is being traded in an over-the-counter market, which encounters default risk, we price them considering this risk; these are called defaultable claims, and these options are known as vulnerable options. In the literature, structural and reduced-form models are two models to price such an option. However, we choose the structural model as the reduced form model will never encounter the problem we are addressing in this thesis. The new idea we incorporate in the structural model is the incomplete information setup. We assume that the information about the firm value is only observable at specific times. This approach can be used to price risky debt, deal with the problem of credit spreads approaching zero as the risky bond approaches its maturity, and can be extended nicely to price any derivative; we also study the pricing of these under the regime-switching framework which surprisingly under this incomplete information idea looks like a problem that requires the synchronised jump methodology in deriving the characteristic function.