Abstract  Spatially proximate amino acid positions in a protein tend to co-evolve, so a protein's 3D-structure leaves an echo of correlations in the evolutionary record. Reverse engineering 3D-structures from such correlations is an open problem in structural biology, pursued with increasing vigor as new protein sequences continue to fill the data banks. Within this task lies a statistical stumbling block, rooted in the following: correlation between two amino acid positions can arise from firsthand interaction, but also be network-propagated via intermediate positions; observed correlation is not enough to guarantee proximity. The remedy, and the focus of this thesis, is to mathematically untangle the crisscross of correlations and extract direct interactions, which enables a clean depiction of co-evolution among the positions. Recently, analysts have used maximum-entropy modeling to recast this cause-and-effect puzzle as parameter learning in a Potts model (a kind of Markov random field). Unfortunately, a computationally expensive partition function puts this out of reach of straightforward maximum-likelihood estimation. Mean-field approximations have been used, but an arsenal of other approximate schemes exists. In this work, we re-implement an existing contact-detection procedure and replace its mean-field calculations with pseudo-likelihood maximization. We then feed both routines real protein data and highlight differences between their respective outputs. Our new program seems to offer a systematic boost in detection accuracy.

Abstract Since different patients may have different causes of getting a disease, treating every patient having a certain disease in the same manner is not always be the best way to go. A treatment having effect in one type of patients may not have the same effect in a different type of patients. This makes it possible to partition a patient population into subpopulations in which a drug has distinct expected response. In this thesis the patient population is partitioned into two subpopulations where we have prior knowledge that one of them has a higher expected response to a drug than the other. Based on responses to a drug in Phase II, it has been analyzed in which of the populations Phase III should continue. The results show that the decision is highly dependent on the utility function on which the analysis is based. One interesting case is when the vast majority of the patient population belongs to the subpopulation with the higher expected response and a utility function that takes into account the prevalence of the populations. In that case the simulations show that when the difference in expected response between the subpopulations is large, it is a safer choice in continuing in Phase III in the subpopulation having the higher expected response than in the full population even though the expected utility will be less. This is an expected result which indicates that the approach used to model the situation studied in this report is reasonable

Abstract   An Asian option is a path-depending exotic option, which means that either the settlement price or the strike of the option is formed by some aggregation of underlying asset prices during the option lifetime. This thesis will focus on European style Arithmetic Asian options where the settlement price at maturity is formed by the arithmetic average price of the last seven days of the underlying asset. For this type of option it does not exist any closed form analytical formula for calculating the theoretical option value. There exist closed form approximation formulas for valuing this kind of option. One such, used in this thesis, approximate the value of an Arithmetic Asian option by conditioning the valuation on the geometric mean price. To evaluate the accuracy in this approximation and to see if it is possible to use the well known Black-Scholes formula for valuing Asian options, this thesis examines the bias between Monte-Carlo simulation pricing and these closed form approximate pricings. The bias examination is done for several different volatility schemes. In general the Asian approximation formula works very well for valuing Asian options. For volatility scenarios where there is a drastic volatility shift and the period with higher volatility is before the average period of the option, the Asian approximation formula will underestimate the option value. These underestimates are very significant for OTM options, decreases for ATM options and are small, although significant, for ITM options. The Black-Scholes formula will in general overestimate the Asian option value. This is expected since the Black-Scholes formula applies to standard European options which only, implicitly, considers the underlying asset price at maturity of the option as settlement price. This price is in average higher than the Asian option settlement price when the underlying asset price has a positive drift. However, for some volatility scenarios where there is a drastic volatility shift and the period with higher volatility is before the average period of the option, even the Black-Scholes formula will underestimate the option value. As for the Asian approximation formula, these over-and underestimates are very large for OTM options and decreases for ATM and ITM options. / Sammanfattning En Asiatisk option är en vägberoende exotisk option, vilket betyder att antingen settlement-priset eller strike-priset beräknas utifrån någon form av aggregering av underliggande tillgångens priser under optionens livstid. Denna uppsats fokuserar på Aritmetiska Asiatiska optioner av Europeisk karaktär där settlement-priset vid lösen bestäms av det aritmetiska medelvärdet av underliggande tillgångens priser de sista sju dagarna. För denna typ av option finns det inga slutna analytiska formler för att beräkna optionens teoretiska värde. Det finns dock slutna approximativa formler för värdering av denna typ av optioner. En sådan, som används i denna uppsats, approximerar värdet av en Aritmetisk Asiatisk option genom att betinga värderingen på det geometriska medelpriset. För att utvärdera noggrannheten i denna approximation och för att se om det är möjligt att använda den väl kända Black-Scholes-formeln för att värdera Asiatiska optioner, så analyseras differenserna mellan Monte-Carlo-simulering och dessa slutna formlers värderingar i denna uppsats. Differenserna analyseras utifrån ett flertal olika scenarion för volatiliteten. I allmänhet så fungerar Asiatapproximationsformeln bra för värdering av Asiatiska optioner. För volatilitetsscenarion som innebär en drastisk volatilitetsförändring och där den perioden med högre volatilitet ligger innan optionens medelvärdesperiod, så undervärderar Asiatapproximationen optionens värde. Dessa undervärderingar är mycket påtagliga för OTM-optioner, avtar för ATM-optioner och är små, om än signifikanta, för ITM-optioner. Black-Scholes formel övervärderar i allmänhet Asiatiska optioners värde. Detta är väntat då Black-Scholes formel är ämnad för standard Europeiska optioner, vilka endast beaktar underliggande tillgångens pris vid optionens slutdatum som settlement-pris. Detta pris är i snitt högre än Asiatisk optioners settlement-pris när underliggande tillgångens pris har en positiv drift. Men, för vissa volatilitetsscenarion som innebär en drastisk volatilitetsförändring och där den perioden med högra volatilitet ligger innan optionens medelvärdesperiod, så undervärderar även Black-Scholes formel optionens värde. Som för Asiatapproximationen så är dessa över- och undervärderingar mycket påtagliga för OTM-optioner och avtar för ATM och ITM-optioner.

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