The thesis consists of four chapters. The introductory chapter clarifies different notions of rationality used by economists and gives a summary of the remainder of the thesis. Chapter 2 proposes an explanation for the common empirical observation of the coexistence of infrequently-changing regular price ceilings and promotion-like price patterns. The results derive from enriching an otherwise standard, albeit stylized, general equilibrium model with two elements. First, the consumer-producer interaction is modeled in the spirit of the price dispersion literature, by introducing oligopolistic markets, consumer search costs and heterogeneity. Second, consumers are assumed to be boundedly-rational: In order to incorporate new information about the general price level, they have to incur a small cognitive cost. The decision whether to re-optimize or act according to the obsolete knowledge about prices is itself a result of optimization. It is shown that in this economy, individual retail prices are capped below the monopoly price, but are otherwise flexible. Moreover, they have the following three properties: 1) An individual price has a positive probability of being equal to the ceiling. 2) Prices have a tendency to fall below the ceiling and then be reset back to the cap value. 3) The ceiling remains constant for extended time intervals even when the mean rate of inflation is positive. Properties 1) and 2) can be associated with promotions and properties 1) and 3) imply the emergence of nominal price rigidity. The results do not rely on any type of direct costs of price adjustment. Instead, price stickiness derives from frictions on the consumers’ side of the market, in line with the results of several managerial surveys. It is shown that the developed theory, compared to the classic menu costs-based approach, does better in matching the stylized facts about the reaction of individual prices to inflation. In terms of quantitative assessment, the model, when calibrated to realistic parameter values, produces median price ceiling durations that match values reported in empirical studies.<p><p>The starting point of the essay in Chapter 3 is the observation that the baseline New-Keynesian model, which relies solely on the notion of infrequent price adjustment, cannot account for the observed degree of inflation sluggishness. Therefore, it is a common practice among macro- modelers to introduce an ad hoc additional source of persistence to their models, by assuming that price setters, when adjusting a price of their product, do not set it equal to its unobserved individual optimal level, but instead catch up with the optimal price only gradually. In the paper, a model of incomplete adjustment is built which allows for explicitly testing whether price-setters adjust to the shocks to the unobserved optimal price only gradually and, if so, measure the speed of the catching up process. According to the author, a similar test has not been performed before. It is found that new prices do not generally match their estimated optimal level. However, only in some sectors, e.g. for some industrial goods and services, prices adjust to this level gradually, which should add to the aggregate inflation sluggishness. In other sectors, particularly food, price-setters seem to overreact to shocks, with new prices overshooting the optimal level. These sectors are likely to contribute to decreasing the aggregate inflation sluggishness. Overall, these findings are consistent with the view that price-setters are boundedly-rational. However, they do not provide clear-cut support for the existence of an additional source of inflation persistence due to gradual individual price adjustment. Instead, they suggest that general equilibrium macroeconomic models may need to include at least two types of production sectors, characterized by a contrasting behavior of price-setters. An additional finding stemming from this work is that the idiosyncratic component of the optimal individual price is well approximated by a random walk. This is in line with the assumptions maintained in most of the theoretical literature. <p><p>Chapter 4 of the thesis has been co-authored by Julia Lendvai. In this paper a full-fledged production economy model with Kahneman and Tversky’s Prospect Theory features is constructed. The agents’ objective function is assumed to be a weighted sum of the usual utility over consumption and leisure and the utility over relative changes of agents’ wealth. It is also assumed that agents are loss-averse: They are more sensitive to wealth losses than to gains. Apart from the changes in the utility, the model is set-up in a standard Real Business Cycle framework. The authors study prices of stocks and risk-free bonds in this economy. Their work shows that under plausible parameterizations of the objective function, the model is able to explain a wide set of unconditional asset return moments, including the mean return on risk-free bonds, equity premium and the Sharpe Ratio. When the degree of loss aversion in the model is additionally assumed to be state-dependent, the model also produces countercyclical risk premia. This helps it match an array of conditional moments and in particular the predictability pattern of stock returns. / Doctorat en Sciences économiques et de gestion / info:eu-repo/semantics/nonPublished

In this work we study two problems motivated by Risk Management: the optimal design of financial products from an investor's point of view and the calculation of bounds and approximations for sums involving non-independent random variables. The element that interconnects these two topics is the notion of conditioning, a fundamental concept in probability and statistics which appears to be a useful device in finance. In the first part of the dissertation, we analyse structured products that are now widespread in the banking and insurance industry. These products typically protect the investor against bearish stock markets while offering upside participation when the markets are bullish. Examples of these products include capital guaranteed funds commercialised by banks, and equity linked contracts sold by insurers. The design of these products is complex in general and it is vital to examine to which extent they are actually interesting from the investor's point of view and whether they cannot be dominated by other strategies. In the academic literature on structured products the focus has been almost exclusively on the pricing and hedging of these instruments and less on their performance from an investor's point of view. In this work we analyse the attractiveness of these products. We assess the theoretical cost of inefficiency when buying a structured product and describe the optimal strategy explicitly if possible. Moreover we examine the cost of the inefficiency in practice. We extend the results of Dybvig (1988a, 1988b) and Cox & Leland (1982, 2000) who in the context of a complete, one-dimensional market investigated the inefficiency of path-dependent pay-offs. In the dissertation we consider this problem in one-dimensional Levy and multidimensional Black-Scholes financial markets and we provide evidence that path-dependent pay-offs should not be preferred by decision makers with a fixed investment horizon, and they should buy path-independent structures instead. In these market settings we also demonstrate the optimal contract that provides the given distribution to the consumer, and in the case of risk- averse investors we are able to propose two ways of improving the design of financial products. Finally we illustrate the theory with a few well-known securities and strategies e.g. dollar cost averaging, buy-and-hold investments and widely used portfolio insurance strategies. The second part of the dissertation considers the problem of finding the distribution of a sum of non- independent random variables. Such dependent sums appear quite often in insurance and finance, for instance in case of the aggregate claim distribution or loss distribution of an investment portfolio. An interesting avenue to cope with this problem consists in using so-called convex bounds, studied by Dhaene et al. (2002a, 2002b), who applied these to sums of log-normal random variables. In their papers they have shown how these convex bounds can be used to derive closed-form approximations for several of the risk measures of such a sum. In the dissertation we prove that unlike the log-normal case the construction of a convex lower bound in explicit form appears to be out of reach for general sums of log-elliptical risks and we show how we can construct stop-loss bounds and we use these to construct mean preserving approximations for general sums of log-elliptical distributions in explicit form. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

The goal of this thesis in finance is to combine the use of advanced mathematical methods with a return to foundational economic issues. In that perspective, I study generalized rational expectations and asset pricing in Chapter 2, and a converse comparison principle for backward stochastic differential equations with jumps in Chapter 3. Since the use of stochastic methods in finance is an interesting and complex issue in itself - if only to clarify the difference between the use of mathematical models in finance and in physics or biology - I also present a philosophical reflection on the interpretation of mathematical models in finance (Chapter 4). In Chapter 5, I conclude the thesis with an essay on the history and interpretation of mathematical probability - to be read while keeping in mind the fundamental role of mathematical probability in financial models. / Doctorat en Sciences économiques et de gestion / info:eu-repo/semantics/nonPublished

This thesis presents the main results of my research in the field of computational finance and portfolios optimization. We focus on pricing Asian basket options and portfolio problems in the presence of inflation with stochastic interest rates.<p><p>In Chapter 2, we concentrate upon the derivation of bounds for European-style discrete arithmetic Asian basket options in a Black and Scholes framework.We start from methods used for basket options and Asian options. First, we use the general approach for deriving upper and lower bounds for stop-loss premia of sums of non-independent random variables as in Kaas et al. [Upper and lower bounds for sums of random variables, Insurance Math. Econom. 27 (2000) 151–168] or Dhaene et al. [The concept of comonotonicity in actuarial science and finance: theory, Insurance Math. Econom. 31(1) (2002) 3–33]. We generalize the methods in Deelstra et al. [Pricing of arithmetic basket options by conditioning, Insurance Math. Econom. 34 (2004) 55–57] and Vanmaele et al. [Bounds for the price of discrete sampled arithmetic Asian options, J. Comput. Appl. Math. 185(1) (2006) 51–90]. Afterwards we show how to derive an analytical closed-form expression for a lower bound in the non-comonotonic case. Finally, we derive upper bounds for Asian basket options by applying techniques as in Thompson [Fast narrow bounds on the value of Asian options, Working Paper, University of Cambridge, 1999] and Lord [Partially exact and bounded approximations for arithmetic Asian options, J. Comput. Finance 10 (2) (2006) 1–52]. Numerical results are included and on the basis of our numerical tests, we explain which method we recommend depending on moneyness and time-to-maturity<p><p>In Chapter 3, we propose some moment matching pricing methods for European-style discrete arithmetic Asian basket options in a Black & Scholes framework. We generalize the approach of Curran M. (1994) [Valuing Asian and portfolio by conditioning on the geometric mean price”, Management science, 40, 1705-1711] and of Deelstra G. Liinev J. and Vanmaele M. (2004) [Pricing of arithmetic basket options by conditioning”, Insurance: Mathematics & Economics] in several ways. We create a framework that allows for a whole class of conditioning random variables which are normally distributed. We moment match not only with a lognormal random variable but also with a log-extended-skew-normal random variable. We also improve the bounds of Deelstra G. Diallo I. and Vanmaele M. (2008). [Bounds for Asian basket options”, Journal of Computational and Applied Mathematics, 218, 215-228]. Numerical results are included and on the basis of our numerical tests, we explain which method we recommend depending on moneyness and<p>time-to-maturity.<p><p>In Chapter 4, we use the stochastic dynamic programming approach in order to extend<p>Brennan and Xia’s unconstrained optimal portfolio strategies by investigating the case in which interest rates and inflation rates follow affine dynamics which combine the model of Cox et al. (1985) [A Theory of the Term Structure of Interest Rates, Econometrica, 53(2), 385-408] and the model of Vasicek (1977) [An equilibrium characterization of the term structure, Journal of Financial Economics, 5, 177-188]. We first derive the nominal price of a zero coupon bond by using the evolution PDE which can be solved by reducing the problem to the solution of three ordinary differential equations (ODE). To solve the corresponding control problems we apply a verification theorem without the usual Lipschitz assumption given in Korn R. and Kraft H.(2001)[A Stochastic control approach to portfolio problems with stochastic interest rates, SIAM Journal on Control and Optimization, 40(4), 1250-1269] or Kraft(2004)[Optimal Portfolio with Stochastic Interest Rates and Defaultable Assets, Springer, Berlin].<p><p><p> / Doctorat en Sciences / info:eu-repo/semantics/nonPublished