An Hilbert space approach for a class of arbitrage free implied volatilities models

We present an Hilbert space formulation for a set of implied volatility models introduced in \cite{BraceGoldys01} in which the authors studied conditions for a family of European call options, varying the maturing time and the strike price $T$ an $K$, to be arbitrage free. The arbitrage free conditions give a system of stochastic PDEs for the evolution of the implied volatility surface ${\hat\sigma}_t(T,K)$. We will focus on the family obtained fixing a strike $K$ and varying $T$. In order to give conditions to prove an existence-and-uniqueness result for the solution of the system it is here expressed in terms of the square root of the forward implied volatility and rewritten in an Hilbert space setting. The existence and the uniqueness for the (arbitrage free) evolution of the forward implied volatility, and then of the the implied volatility, among a class of models, are proved. Specific examples are also given.Implied volatility; Option pricing; Stochastic SPDE; Hilbert space

Sixty Years of Modern Human Origins in the American Anthropological Association

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/65197/1/aa.2003.105.1.89.pd

Grey wolf genomic history reveals a dual ancestry of dogs

The grey wolf (Canis lupus) was the first species to give rise to a domestic population, and they remained widespread throughout the last Ice Age when many other large mammal species went extinct. Little is known, however, about the history and possible extinction of past wolf populations or when and where the wolf progenitors of the present-day dog lineage (Canisfamiliaris) lived(1-8). Here we analysed 72 ancient wolf genomes spanning the last 100,000 years from Europe, Siberia and North America. We found that wolf populations were highly connected throughout the Late Pleistocene, with levels of differentiation an order of magnitude lower than they are today. This population connectivity allowed us to detect natural selection across the time series, including rapid fixation of mutations in the gene IFT8840,000-30,000 years ago. We show that dogs are overall more closely related to ancient wolves from eastern Eurasia than to those from western Eurasia, suggesting a domestication process in the east. However, we also found that dogs in the Near East and Africa derive up to half of their ancestry from a distinct population related to modern southwest Eurasian wolves, reflecting either an independent domestication process or admixture from local wolves. None of the analysed ancient wolf genomes is a direct match for either of these dog ancestries, meaning that the exact progenitor populations remain to be located.Peer reviewe

Dire wolves were the last of an ancient New World canid lineage

Dire wolves are considered to be one of the most common and widespread large carnivores in Pleistocene America1, yet relatively little is known about their evolution or extinction. Here, to reconstruct the evolutionary history of dire wolves, we sequenced five genomes from sub-fossil remains dating from 13,000 to more than 50,000 years ago. Our results indicate that although they were similar morphologically to the extant grey wolf, dire wolves were a highly divergent lineage that split from living canids around 5.7 million years ago. In contrast to numerous examples of hybridization across Canidae2,3, there is no evidence for gene flow between dire wolves and either North American grey wolves or coyotes. This suggests that dire wolves evolved in isolation from the Pleistocene ancestors of these species. Our results also support an early New World origin of dire wolves, while the ancestors of grey wolves, coyotes and dholes evolved in Eurasia and colonized North America only relatively recently

Author Correction: The Anglo-Saxon migration and the formation of the early English gene pool.

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The Anglo-Saxon migration and the formation of the early English gene pool.

The history of the British Isles and Ireland is characterized by multiple periods of major cultural change, including the influential transformation after the end of Roman rule, which precipitated shifts in language, settlement patterns and material culture1. The extent to which migration from continental Europe mediated these transitions is a matter of long-standing debate2-4. Here we study genome-wide ancient DNA from 460 medieval northwestern Europeans-including 278 individuals from England-alongside archaeological data, to infer contemporary population dynamics. We identify a substantial increase of continental northern European ancestry in early medieval England, which is closely related to the early medieval and present-day inhabitants of Germany and Denmark, implying large-scale substantial migration across the North Sea into Britain during the Early Middle Ages. As a result, the individuals who we analysed from eastern England derived up to 76% of their ancestry from the continental North Sea zone, albeit with substantial regional variation and heterogeneity within sites. We show that women with immigrant ancestry were more often furnished with grave goods than women with local ancestry, whereas men with weapons were as likely not to be of immigrant ancestry. A comparison with present-day Britain indicates that subsequent demographic events reduced the fraction of continental northern European ancestry while introducing further ancestry components into the English gene pool, including substantial southwestern European ancestry most closely related to that seen in Iron Age France5,6

The Anglo-Saxon migration and the formation of the early English gene pool

The history of the British Isles and Ireland is characterized by multiple periods of major cultural change, including the influential transformation after the end of Roman rule, which precipitated shifts in language, settlement patterns and material culture1. The extent to which migration from continental Europe mediated these transitions is a matter of long-standing debate2,3,4. Here we study genome-wide ancient DNA from 460 medieval northwestern Europeans—including 278 individuals from England—alongside archaeological data, to infer contemporary population dynamics. We identify a substantial increase of continental northern European ancestry in early medieval England, which is closely related to the early medieval and present-day inhabitants of Germany and Denmark, implying large-scale substantial migration across the North Sea into Britain during the Early Middle Ages. As a result, the individuals who we analysed from eastern England derived up to 76% of their ancestry from the continental North Sea zone, albeit with substantial regional variation and heterogeneity within sites. We show that women with immigrant ancestry were more often furnished with grave goods than women with local ancestry, whereas men with weapons were as likely not to be of immigrant ancestry. A comparison with present-day Britain indicates that subsequent demographic events reduced the fraction of continental northern European ancestry while introducing further ancestry components into the English gene pool, including substantial southwestern European ancestry most closely related to that seen in Iron Age France5,6

Convalescent plasma in patients admitted to hospital with COVID-19 (RECOVERY): a randomised controlled, open-label, platform trial

SummaryBackground Azithromycin has been proposed as a treatment for COVID-19 on the basis of its immunomodulatoryactions. We aimed to evaluate the safety and efficacy of azithromycin in patients admitted to hospital with COVID-19.Methods In this randomised, controlled, open-label, adaptive platform trial (Randomised Evaluation of COVID-19Therapy [RECOVERY]), several possible treatments were compared with usual care in patients admitted to hospitalwith COVID-19 in the UK. The trial is underway at 176 hospitals in the UK. Eligible and consenting patients wererandomly allocated to either usual standard of care alone or usual standard of care plus azithromycin 500 mg once perday by mouth or intravenously for 10 days or until discharge (or allocation to one of the other RECOVERY treatmentgroups). Patients were assigned via web-based simple (unstratified) randomisation with allocation concealment andwere twice as likely to be randomly assigned to usual care than to any of the active treatment groups. Participants andlocal study staff were not masked to the allocated treatment, but all others involved in the trial were masked to theoutcome data during the trial. The primary outcome was 28-day all-cause mortality, assessed in the intention-to-treatpopulation. The trial is registered with ISRCTN, 50189673, and ClinicalTrials.gov, NCT04381936.Findings Between April 7 and Nov 27, 2020, of 16 442 patients enrolled in the RECOVERY trial, 9433 (57%) wereeligible and 7763 were included in the assessment of azithromycin. The mean age of these study participants was65·3 years (SD 15·7) and approximately a third were women (2944 [38%] of 7763). 2582 patients were randomlyallocated to receive azithromycin and 5181 patients were randomly allocated to usual care alone. Overall,561 (22%) patients allocated to azithromycin and 1162 (22%) patients allocated to usual care died within 28 days(rate ratio 0·97, 95% CI 0·87–1·07; p=0·50). No significant difference was seen in duration of hospital stay (median10 days [IQR 5 to >28] vs 11 days [5 to >28]) or the proportion of patients discharged from hospital alive within 28 days(rate ratio 1·04, 95% CI 0·98–1·10; p=0·19). Among those not on invasive mechanical ventilation at baseline, nosignificant difference was seen in the proportion meeting the composite endpoint of invasive mechanical ventilationor death (risk ratio 0·95, 95% CI 0·87–1·03; p=0·24).Interpretation In patients admitted to hospital with COVID-19, azithromycin did not improve survival or otherprespecified clinical outcomes. Azithromycin use in patients admitted to hospital with COVID-19 should be restrictedto patients in whom there is a clear antimicrobial indication

Dual Swap and Swaption Formulae in the Normal and Lognormal Models

When the Gaussian HJM model is recast in terms of simple forwards, it becomes dual to the LogNormal Forward model in the sense that for almost every formula in one model there is a similar formula in the other which can be obtained by simple transformation or substitution. INTRODUCTION The Gaussian Heath Jarrow Morton (HJM) (see references 2,3 and 4) and LogNormal Forward (see references 1,5 and 6) models are dual in the sense that for almost every formula in one model there is a similar formula in the other which can be obtained by simple transformation or substitution. That is important from both pedagogical and implementation points of view. A common problem in coming to grips with the LogNormal Forward model is understanding the role of the forward measures. But when the widely accepted Gaussian HJM model is recast in terms of simple forwards, it too, just like the lognormal model, is a play on forward measures. So intuition developed on the normal model, can be articulated to th..