Derivative pricing, and in particular the pricing of options, is an important area of current research in financial mathematics. Experts debate on the best method of pricing and the most appropriate model of a price process to use. In this thesis, a ``Switching Black-Scholes'' model of a price process is proposed. This model is based on the standard geometric Brownian motion (or Black-Scholes) model of a price process. However, the drift and volatility parameters are permitted to vary between a finite number of possible values at known times, according to the state of a hidden Markov chain. This type of model has been found to replicate the Black-Scholes implied volatility smiles observed in the market, and produce option prices which are closer to market values than those obtained from the traditional Black-Scholes formula. As the Markov chain incorporates a second source of uncertainty into the Black-Scholes model, the Switching Black-Scholes market is incomplete, and no unique option pricing methodology exists. In this thesis, we apply the methods of mean-variance hedging, Esscher transforms and minimum entropy in order to price options on assets which evolve according to the Switching Black-Scholes model. C programs to compute these prices are given, and some particular numerical examples are examined. Finally, filtering techniques and reference probability methods are applied to find estimates of the model parameters and state of the hidden Markov chain. / Thesis (Ph.D.)--Applied Mathematics, 2003.

Cette thèse consiste en une série d'applications du calcul stochastique aux mathématiques financières. Elle est composée de quatre chapitres. Dans le premier on étudie le rapport entre la complétude du marché et l'extrémalité des mesures martingales equivalentes dans le cas d'une infinité d'actifs. Dans le deuxième on trouve des conditions équivalentes à l'existence et unicité d'une mesure martingale equivalente sous la quelle le processus des prix suit des lois n-dimensionnelles données à n fixe. Dans le troisième on étend à un marché admettant une infinité dénombrable d'actifs une charactérisation de la stratégie de couverture optimale (pour le critère moyenne-variance) basé sur une technique de changement de numéraire et extension artificielle. Enfin, dans le quatrième on s'occupe du problème de couverture d'un actif contingent dans un marché avec information asymetrique.

<p>Hedge funds are a debated subject in today’s financial industry. During 2008, despite hedge funds absolute return target, the global hedge fund industry showed a negative performance whilst the Swedish hedge fund market performed relatively well in comparison. Many studies have been made investigating the effect on incorporating hedge funds in a traditional portfolio though none focused separately on the Swedish market. In a global perspective it is quite easy to invest in hedge fund portfolios due to the existence of investable indices. To invest on the Swedish market is a more complex matter. SIX Harcourt HFXS Index is a Swedish hedge fund index representing the Swedish hedge fund market though it is not investable. Hence it would be interesting to see if it is possible to create an investable version of SIX Harcourt HFXS. When creating an investable index, several administrative costs will arise and in order to cover these costs it would be interesting to see whether or not it possible to optimize SIX Harcourt HFXS Index in purpose of achieving a outperformance which could cover any administrative costs for setting up the investable version. Also, since the optimized version must replicate the standard SIX Harcourt HFXS Index it must maintain a certain level of correlation.</p><p>This thesis, which is based on a positivistic epistemology, is built upon a quantitative case study where SIX Harcourt HFXS Index is optimized in purpose of achieving an outperformance in terms of the risk-adjusted return. The optimization uses an adjusted mean-variance methodology and is limited to a maintained correlation above 0,9 towards the standard SIX Harcourt HFXS Index. The optimization is created through the use of an Excel application created by Harcourt Investment Consulting.</p><p>Also, based on the outperformance by Swedish hedge funds compared to global hedge funds, this study aims to show the effect of incorporating Swedish hedge funds in a traditional portfolio consisting of equities and bonds. This effect is analyzed by the use of several performance-and risk measures.</p><p>The study shows that it is possible to optimize SIX Harcourt HFXS Index and produce an outperformance of approximately 1,5% per annum with a maintained correlation above 0,9. It also shows that the effect of incorporating Swedish hedge funds to a traditional portfolio is positive in regards to both risk and return.</p>

<p><strong>Syfte: </strong>En komparativ studie och beskriva möjligheterna för att skapa en effektiv portfölj av olika aktieportföljer, där det undersöks om faktorerna som företagens omsättningsstorlek, branschen de är aktiva i och valet att ha utländska aktier, har betydelse eller inte för att skapa en effektiv portfölj med hänsyn till dess korrelation, avkastning och risk. </p><p><strong>Metod: </strong>Uppsatsen utgår från en kvantitativ studie, som sträcker sig från april 2008 till april 2010, då historiska aktiekurspriser används för olika sorters uträkningar, som baseras på uppsatsens huvudteorier; CAPM, Sharpekvot och portföljteori. Utgångspunkten är deduktiv, där slutsatserna har dragits från teorierna.       </p><p><strong>Slutsats: </strong>Utifrån uträkningarna, kunde det inte dras generella slutsatser där de undersökta faktorerna inte utmärkte sig i något mönster. Däremot visade sig att det blev högre avkastning till liknande risk eller lägre risk till liknade avkastning när man väljer aktier som är olika varandra, då korrelationen inte samvarierar. Det bästa resultatet är när utländska aktier blandas in i portföljen.    </p>

Hedge funds are a debated subject in today’s financial industry. During 2008, despite hedge funds absolute return target, the global hedge fund industry showed a negative performance whilst the Swedish hedge fund market performed relatively well in comparison. Many studies have been made investigating the effect on incorporating hedge funds in a traditional portfolio though none focused separately on the Swedish market. In a global perspective it is quite easy to invest in hedge fund portfolios due to the existence of investable indices. To invest on the Swedish market is a more complex matter. SIX Harcourt HFXS Index is a Swedish hedge fund index representing the Swedish hedge fund market though it is not investable. Hence it would be interesting to see if it is possible to create an investable version of SIX Harcourt HFXS. When creating an investable index, several administrative costs will arise and in order to cover these costs it would be interesting to see whether or not it possible to optimize SIX Harcourt HFXS Index in purpose of achieving a outperformance which could cover any administrative costs for setting up the investable version. Also, since the optimized version must replicate the standard SIX Harcourt HFXS Index it must maintain a certain level of correlation. This thesis, which is based on a positivistic epistemology, is built upon a quantitative case study where SIX Harcourt HFXS Index is optimized in purpose of achieving an outperformance in terms of the risk-adjusted return. The optimization uses an adjusted mean-variance methodology and is limited to a maintained correlation above 0,9 towards the standard SIX Harcourt HFXS Index. The optimization is created through the use of an Excel application created by Harcourt Investment Consulting. Also, based on the outperformance by Swedish hedge funds compared to global hedge funds, this study aims to show the effect of incorporating Swedish hedge funds in a traditional portfolio consisting of equities and bonds. This effect is analyzed by the use of several performance-and risk measures. The study shows that it is possible to optimize SIX Harcourt HFXS Index and produce an outperformance of approximately 1,5% per annum with a maintained correlation above 0,9. It also shows that the effect of incorporating Swedish hedge funds to a traditional portfolio is positive in regards to both risk and return.

Many optimal stochastic control problems in finance can be formulated in the form of Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs). In this thesis, a general framework for solutions of HJB PDEs in finance is developed, with application to asset allocation. The numerical scheme has the following properties: it is unconditionally stable; convergence to the viscosity solution is guaranteed; there are no restrictions on the underlying stochastic process; it can be easily extended to include features as needed such as uncertain volatility and transaction costs; and central differencing is used as much as possible so that use of a locally second order method is maximized. In this thesis, continuous time mean variance type strategies for dynamic asset allocation problems are studied. Three mean variance type strategies: pre-commitment mean variance, time-consistent mean variance, and mean quadratic variation, are investigated. The numerical method can handle various constraints on the control policy. The following cases are studied: allowing bankruptcy (unconstrained case), no bankruptcy, and bounded control. In some special cases where analytic solutions are available, the numerical results agree with the analytic solutions. These three mean variance type strategies are compared. For the allowing bankruptcy case, analytic solutions exist for all strategies. However, when additional constraints are applied to the control policy, analytic solutions do not exist for all strategies. After realistic constraints are applied, the efficient frontiers for all three strategies are very similar. However, the investment policies are quite different. These results show that, in deciding which objective function is appropriate for a given economic problem, it is not sufficient to simply examine the efficient frontiers. Instead, the actual investment policies need to be studied in order to determine if a particular strategy is applicable to specific investment problem.

In view of the current situation of fluctuating high crude oil prices, it is now more important than ever for petroleum refineries to operate at an optimal level in the present dynamic global economy. Acknowledging the shortcomings of deterministic models, this work proposes a hybrid of stochastic programming formulations for an optimal midterm refinery planning that addresses three factors of uncertainties, namely price of crude oil and saleable products, product demand, and production yields. An explicit stochastic programming technique is utilized by employing compensating slack variables to account for violations of constraints in order to increase model tractability. Four approaches are considered to ensure both solution and model robustness: (1) the Markowitz’s mean–variance (MV) model to handle randomness in the objective coefficients of prices by minimizing variance of the expected value of the random coefficients; (2) the two-stage stochastic programming with fixed recourse approach via scenario analysis to model randomness in the right-hand side and left-hand side coefficients by minimizing the expected recourse penalty costs due to constraints’ violations; (3) incorporation of the MV model within the framework developed in Approach 2 to minimize both the expectation and variance of the recourse costs; and (4) reformulation of the model in Approach 3 by adopting mean-absolute deviation (MAD) as the risk metric imposed by the recourse costs for a novel application to the petroleum refining industry. A representative numerical example is illustrated with the resulting outcome of higher net profits and increased robustness in solutions proposed by the stochastic models.

Many optimal stochastic control problems in finance can be formulated in the form of Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs). In this thesis, a general framework for solutions of HJB PDEs in finance is developed, with application to asset allocation. The numerical scheme has the following properties: it is unconditionally stable; convergence to the viscosity solution is guaranteed; there are no restrictions on the underlying stochastic process; it can be easily extended to include features as needed such as uncertain volatility and transaction costs; and central differencing is used as much as possible so that use of a locally second order method is maximized. In this thesis, continuous time mean variance type strategies for dynamic asset allocation problems are studied. Three mean variance type strategies: pre-commitment mean variance, time-consistent mean variance, and mean quadratic variation, are investigated. The numerical method can handle various constraints on the control policy. The following cases are studied: allowing bankruptcy (unconstrained case), no bankruptcy, and bounded control. In some special cases where analytic solutions are available, the numerical results agree with the analytic solutions. These three mean variance type strategies are compared. For the allowing bankruptcy case, analytic solutions exist for all strategies. However, when additional constraints are applied to the control policy, analytic solutions do not exist for all strategies. After realistic constraints are applied, the efficient frontiers for all three strategies are very similar. However, the investment policies are quite different. These results show that, in deciding which objective function is appropriate for a given economic problem, it is not sufficient to simply examine the efficient frontiers. Instead, the actual investment policies need to be studied in order to determine if a particular strategy is applicable to specific investment problem.

Syfte: En komparativ studie och beskriva möjligheterna för att skapa en effektiv portfölj av olika aktieportföljer, där det undersöks om faktorerna som företagens omsättningsstorlek, branschen de är aktiva i och valet att ha utländska aktier, har betydelse eller inte för att skapa en effektiv portfölj med hänsyn till dess korrelation, avkastning och risk.  Metod: Uppsatsen utgår från en kvantitativ studie, som sträcker sig från april 2008 till april 2010, då historiska aktiekurspriser används för olika sorters uträkningar, som baseras på uppsatsens huvudteorier; CAPM, Sharpekvot och portföljteori. Utgångspunkten är deduktiv, där slutsatserna har dragits från teorierna.        Slutsats: Utifrån uträkningarna, kunde det inte dras generella slutsatser där de undersökta faktorerna inte utmärkte sig i något mönster. Däremot visade sig att det blev högre avkastning till liknande risk eller lägre risk till liknade avkastning när man väljer aktier som är olika varandra, då korrelationen inte samvarierar. Det bästa resultatet är när utländska aktier blandas in i portföljen.

The combination of two or more portfolio rules is theoretically convex in return-risk space, which provides for a new class of portfolio rules that gives purpose to the Mean-Variance framework out-of-sample. The author investigates the performance loss from estimation risk between the unconstrained Mean-Variance portfolio and the out-of-sample Global Minimum Variance portfolio. A new two-fund rule is developed in a specific class of combined rules, between the equally weighted portfolio and a mean-variance portfolio with the covariance matrix being estimated by linear shrinkage. The study shows that this rule performs well out-of-sample when covariance estimation error and bias are balanced. The rule is performing at least as good as its peer group in this class of combined rules.