To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .
To send content items to your Kindle, first ensure firstname.lastname@example.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.
Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Polycystic ovary syndrome (PCOS) is associated with a higher prevalence of sleep disturbances and obesity. Treatment of PCOS includes modifying lifestyle behaviours associated with weight management. However, poor sleep in the non-PCOS population has been associated with poorer lifestyle behaviours.
To investigate whether sleep disturbance confounds or modifies the association between lifestyle factors and PCOS.
Design & Setting:
This study was a cross-sectional analysis from the Australian Longitudinal Study on Women’s Health cohort aged 31-36 years in 2009 were analysed (n=6067, n=464 PCOS, n=5603 non-PCOS).
Main outcome measures:
Self-reported data were collected on PCOS, anthropometry, validated modified version of the Active Australia Physical Activity survey, validated food frequency questionnaire and sleep disturbances through latent class analysis.
Women with PCOS had greater adverse sleep symptoms including severe tiredness (p=0.001), difficulty sleeping (p<0.001) and restless sleep (p<0.001), compared to women without PCOS. Women with PCOS also had higher energy consumption (6911±2453 vs 6654±2215kJ, p=0.017), fibre intake (19.8±7.8 vs 18.9±6.9g, p=0.012) and diet quality (dietary guidelines index (DGI)) (88.1±11.6 vs 86.7±11.1, p=0.008), lower glycaemic index (50.2±4.0 vs 50.7±3.9, p=0.021) and increased sedentary behaviour (6.3±2.8 vs 5.9±2.8 hours, p=0.009). There was a significant interaction between PCOS and sleep disturbances for DGI (p=0.035), therefore only for women who had adequate sleep was PCOS associated with a higher DGI. For women with poorer sleep, there was no association between PCOS and DGI.
The association between PCOS and improved diet quality may only be maintained if women can obtain enough good quality sleep.
To determine clinical characteristics associated with false-negative severe acute respiratory coronavirus virus 2 (SARS-CoV-2) test results to help inform coronavirus disease 2019 (COVID-19) testing practices in the inpatient setting.
A retrospective observational cohort study.
All patients 2 years of age and older tested for SARS-CoV-2 between March 14, 2020, and April 30, 2020, who had at least 2 SARS-CoV-2 reverse-transcriptase polymerase chain reaction tests within 7 days.
The primary outcome measure was a false-negative testing episode, which we defined as an initial negative test followed by a positive test within the subsequent 7 days. Data collected included symptoms, demographics, comorbidities, vital signs, labs, and imaging studies. Logistic regression was used to model associations between clinical variables and false-negative SARS-CoV-2 test results.
Of the 1,009 SARS-CoV-2 test results included in the analysis, 4.0% were false-negative results. In multivariable regression analysis, compared with true-negative test results, false-negative test results were associated with anosmia or ageusia (adjusted odds ratio [aOR], 8.4; 95% confidence interval [CI], 1.4–50.5; P = .02), having had a COVID-19–positive contact (aOR, 10.5; 95% CI, 4.3–25.4; P < .0001), and having an elevated lactate dehydrogenase level (aOR, 3.3; 95% CI, 1.2–9.3; P = .03). Demographics, symptom duration, other laboratory values, and abnormal chest imaging were not significantly associated with false-negative test results in our multivariable analysis.
Clinical features can help predict which patients are more likely to have false-negative SARS-CoV-2 test results.
To examine the rates of antipsychotic prescribing in the Irish paediatric and young adult population enrolled in the Irish General Medical Services Scheme pharmacy claims database from the Health Service Executive Primary Care Reimbursement Services database, with a focus on age and sex differences. To examine concomitant prescribing of certain other related medicines in this population.
Data were obtained from the Irish General Medical Services (GMS) scheme pharmacy claims database from the Health Service Executive (HSE) – Primary Care Reimbursement Services (PCRS). Participants included children aged <16 years and youth aged 16–24 years availing of medicines under the HSE-PCRS GMS scheme between January 2005 and December 2015. Outcome measures included prescribing rates of antipsychotics from 2005 to 2015, differences in prescribing rates between different ages and sexes, and percentage of concomitant prescriptions for antidepressants, psychostimulants, anxiolytics and hypnosedatives.
Overall the trend in prescribing rates of antipsychotic medications was stable at 3.94/1000 in 2005 compared with 3.97/1000 in 2015 for children <16 years, and 48.37/1000 eligible population in 2005 compared to 39.64/1000 in 2015 for those aged 16–24. There was a significant decrease in prescribing rates for males in the 16–24 age group.
While rates of antipsychotic prescribing have decreased or remained stable over the timeframe of the study, we did find a significant proportion of this population were prescribed antipsychotics. This study also shows that co-prescribing of antidepressants increased and highlights the need for guidelines for antipsychotic prescribing in children and youth in terms of clinical indication, monitoring, co-prescribing and treatment duration.
To describe a pilot project infection prevention and control (IPC) assessment conducted in skilled nursing facilities (SNFs) in New York State (NYS) during a pivotal 2-week period when the region became the nation’s epicenter for coronavirus disease 2019 (COVID-19).
A telephone and video assessment of IPC measures in SNFs at high risk or experiencing COVID-19 activity.
SNFs in 14 New York counties, including New York City.
A 3-component remote IPC assessment: (1) screening tool; (2) telephone IPC checklist; and (3) COVID-19 video IPC assessment (ie, “COVIDeo”).
In total, 92 SNFs completed the IPC screening tool and checklist: 52 (57%) were conducted as part COVID-19 investigations, and 40 (43%) were proactive prevention-based assessments. Among the 40 proactive assessments, 14 (35%) identified suspected or confirmed COVID-19 cases. COVIDeo was performed in 26 (28%) of 92 assessments and provided observations that other tools would have missed: personal protective equipment (PPE) that was not easily accessible, redundant, or improperly donned, doffed, or stored and specific challenges implementing IPC in specialty populations. The IPC assessments took ∼1 hour each and reached an estimated 4 times as many SNFs as on-site visits in a similar time frame.
Remote IPC assessments by telephone and video were timely and feasible methods of assessing the extent to which IPC interventions had been implemented in a vulnerable setting and to disseminate real-time recommendations. Remote assessments are now being implemented across New York State and in various healthcare facility types. Similar methods have been adapted nationally by the Centers for Disease Control and Prevention.
This SHEA white paper identifies knowledge gaps and challenges in healthcare epidemiology research related to coronavirus disease 2019 (COVID-19) with a focus on core principles of healthcare epidemiology. These gaps, revealed during the worst phases of the COVID-19 pandemic, are described in 10 sections: epidemiology, outbreak investigation, surveillance, isolation precaution practices, personal protective equipment (PPE), environmental contamination and disinfection, drug and supply shortages, antimicrobial stewardship, healthcare personnel (HCP) occupational safety, and return to work policies. Each section highlights three critical healthcare epidemiology research questions with detailed description provided in supplementary materials. This research agenda calls for translational studies from laboratory-based basic science research to well-designed, large-scale studies and health outcomes research. Research gaps and challenges related to nursing homes and social disparities are included. Collaborations across various disciplines, expertise and across diverse geographic locations will be critical.
For children with normal hearing (NH), early communication skills predict vocabulary, a precursor to grammar. Growth in early communication skills of infants with cochlear implants (CIs) was investigated using the Early Communication Indicator (ECI), a play-based observation measure. Multilevel linear growth modelling on data from six ECI sessions held at three-monthly intervals revealed significant growth overall, with a non-significant slower growth rate than that of children with NH (comparison age centred at 18 months). Analyses of gesture use and of nonword vocalisations revealed the CI group used significantly more of each, with more rapid growth. In contrast, the CI group used significantly fewer single words and multiword utterances, and with slower growth. Maternal education and time to achieve consistent CI use impacted significantly on growth for the CI sample. The results indicate that progression to vocabulary by young CI users can be supported by encouraging their use of prelinguistic communication.
To update current estimates of non–device-associated pneumonia (ND pneumonia) rates and their frequency relative to ventilator associated pneumonia (VAP), and identify risk factors for ND pneumonia.
Academic teaching hospital.
All adult hospitalizations between 2013 and 2017 were included. Pneumonia (device associated and non–device associated) were captured through comprehensive, hospital-wide active surveillance using CDC definitions and methodology.
From 2013 to 2017, there were 163,386 hospitalizations (97,485 unique patients) and 771 pneumonia cases (520 ND pneumonia and 191 VAP). The rate of ND pneumonia remained stable, with 4.15 and 4.54 ND pneumonia cases per 10,000 hospitalization days in 2013 and 2017 respectively (P = .65). In 2017, 74% of pneumonia cases were ND pneumonia. Male sex and increasing age we both associated with increased risk of ND pneumonia. Additionally, patients with chronic bronchitis or emphysema (hazard ratio [HR], 2.07; 95% confidence interval [CI], 1.40–3.06), congestive heart failure (HR, 1.48; 95% CI, 1.07–2.05), or paralysis (HR, 1.72; 95% CI, 1.09–2.73) were also at increased risk, as were those who were immunosuppressed (HR, 1.54; 95% CI, 1.18–2.00) or in the ICU (HR, 1.49; 95% CI, 1.06–2.09). We did not detect a change in ND pneumonia risk with use of chlorhexidine mouthwash, total parenteral nutrition, all medications of interest, and prior ventilation.
The incidence rate of ND pneumonia did not change from 2013 to 2017, and 3 of 4 nosocomial pneumonia cases were non–device associated. Hospital infection prevention programs should consider expanding the scope of surveillance to include non-ventilated patients. Future research should continue to look for modifiable risk factors and should assess potential prevention strategies.
To detect modest associations of dietary intake with disease risk, observational studies need to be large and control for moderate measurement errors. The reproducibility of dietary intakes of macronutrients, food groups and dietary patterns (vegetarian and Mediterranean) was assessed in adults in the UK Biobank study on up to five occasions using a web-based 24-h dietary assessment (n 211 050), and using short FFQ recorded at baseline (n 502 655) and after 4 years (n 20 346). When the means of two 24-h assessments were used, the intra-class correlation coefficients (ICC) for macronutrients varied from 0·63 for alcohol to 0·36 for polyunsaturated fat. The ICC for food groups also varied from 0·68 for fruit to 0·18 for fish. The ICC for the FFQ varied from 0·66 for meat and fruit to 0·48 for bread and cereals. The reproducibility was higher for vegetarian status (κ > 0·80) than for the Mediterranean dietary pattern (ICC = 0·45). Overall, the reproducibility of pairs of 24-h dietary assessments and single FFQ used in the UK Biobank were comparable with results of previous prospective studies using conventional methods. Analyses of diet–disease relationships need to correct for both measurement error and within-person variability in dietary intake in order to reliably assess any such associations with disease in the UK Biobank.
To update current estimates of non–device-associated urinary tract infection (ND-UTI) rates and their frequency relative to catheter-associated UTIs (CA-UTIs) and to identify risk factors for ND-UTIs.
Academic teaching hospital.
All adult hospitalizations between 2013 and 2017 were included. UTIs (device and non-device associated) were captured through comprehensive, hospital-wide active surveillance using Centers for Disease Control and Prevention case definitions and methodology.
From 2013 to 2017 there were 163,386 hospitalizations (97,485 unique patients) and 1,273 UTIs (715 ND-UTIs and 558 CA-UTIs). The rate of ND-UTIs remained stable, decreasing slightly from 6.14 to 5.57 ND-UTIs per 10,000 hospitalization days during the study period (P = .15). However, the proportion of UTIs that were non–device related increased from 52% to 72% (P < .0001). Female sex (hazard ratio [HR], 1.94; 95% confidence interval [CI], 1.50–2.50) and increasing age were associated with increased ND-UTI risk. Additionally, the following conditions were associated with increased risk: peptic ulcer disease (HR, 2.25; 95% CI, 1.04–4.86), immunosuppression (HR, 1.48; 95% CI, 1.15–1.91), trauma admissions (HR, 1.36; 95% CI, 1.02–1.81), total parenteral nutrition (HR, 1.99; 95% CI, 1.35–2.94) and opioid use (HR, 1.62; 95% CI, 1.10–2.32). Urinary retention (HR, 1.41; 95% CI, 0.96–2.07), suprapubic catheterization (HR, 2.28; 95% CI, 0.88–5.91), and nephrostomy tubes (HR, 2.02; 95% CI, 0.83–4.93) may also increase risk, but estimates were imprecise.
Greater than 70% of UTIs are now non–device associated. Current targeted surveillance practices should be reconsidered in light of this changing landscape. We identified several modifiable risk factors for ND-UTIs, and future research should explore the impact of prevention strategies that target these factors.
The aim of this study was to explore the experiences of radiotherapy students on clinical placement, specifically focussing on the provision of well-being support from clinical supervisors.
Materials and methods:
Twenty-five students from the University of the West of England and City University of London completed an online evaluation survey relating to their experiences of placement, involving Likert scales and open-ended questions.
The quantitative results were generally positive; however, the qualitative findings were mixed. Three themes emerged: (1) provision of information and advice; (2) an open, inclusive and supportive working environment; and (3) a lack of communication, understanding, and consistency.
Students’ experiences on placement differed greatly and appeared to relate to their specific interactions with different members of staff. It is suggested that additional training around providing well-being support to students may be of benefit to clinical supervisors.
To determine the burden of skin and soft tissue infections (SSTI), the nature of antimicrobial prescribing and factors contributing to inappropriate prescribing for SSTIs in Australian aged care facilities, SSTI and antimicrobial prescribing data were collected via a standardised national survey. The proportion of residents prescribed ⩾1 antimicrobial for presumed SSTI and the proportion whose infections met McGeer et al. surveillance definitions were determined. Antimicrobial choice was compared to national prescribing guidelines and prescription duration analysed using a negative binomial mixed-effects regression model. Of 12 319 surveyed residents, 452 (3.7%) were prescribed an antimicrobial for a SSTI and 29% of these residents had confirmed infection. Topical clotrimazole was most frequently prescribed, often for unspecified indications. Where an indication was documented, antimicrobial choice was generally aligned with recommendations. Duration of prescribing (in days) was associated with use of an agent for prophylaxis (rate ratio (RR) 1.63, 95% confidence interval (CI) 1.08–2.52), PRN orders (RR 2.10, 95% CI 1.42–3.11) and prescription of a topical agent (RR 1.47, 95% CI 1.08–2.02), while documentation of a review or stop date was associated with reduced duration of prescribing (RR 0.33, 95% CI 0.25–0.43). Antimicrobial prescribing for SSTI is frequent in aged care facilities in Australia. Methods to enhance appropriate prescribing, including clinician documentation, are required.
Despite many interventions aiming to reduce excessive gestational weight gain (GWG), it is currently unclear the impact on infant anthropometric outcomes. The aim of this review was to evaluate offspring anthropometric outcomes in studies designed to reduce GWG. A systematic search of seven international databases, one clinical trial registry and three Chinese databases was conducted without date limits. Studies were categorised by intervention type: diet, physical activity (PA), lifestyle (diet + PA), other, gestational diabetes mellitus (GDM) (diet, PA, lifestyle, metformin and other). Meta-analyses were reported as weighted mean difference (WMD) for birthweight and birth length, and risk ratio (RR) for small for gestational age (SGA), large for gestational age (LGA), macrosomia and low birth weight (LBW). Collectively, interventions reduced birthweight, risk of macrosomia and LGA by 71 g (WMD: −70.67, 95% CI −101.90 to −39.43, P<0.001), 16% (RR: 0.84, 95% CI 0.73–0.98, P=0.026) and 19% (RR: 0.81, 95% CI 0.69–0.96, P=0.015), respectively. Diet interventions decreased birthweight and LGA by 99 g (WMD −98.80, 95% CI −178.85 to −18.76, P=0.016) and 65% (RR: 0.35, 95% CI 0.17–0.72, P=0.004). PA interventions reduced the risk of macrosomia by 51% (RR: 0.49, 95% CI 0.26–0.92, P=0.036). In women with GDM, diet and lifestyle interventions reduced birthweight by 211 and 296 g, respectively (WMD: −210.93, 95% CI −374.77 to −46.71, P=0.012 and WMD:−295.93, 95% CI −501.76 to −90.10, P=0.005, respectively). Interventions designed to reduce excessive GWG lead to a small reduction in infant birthweight and risk of macrosomia and LGA, without influencing the risk of adverse outcomes including LBW and SGA.
Palaeoecology has been prominent in studies of environmental change during the Holocene epoch in Scotland. These studies have been dominated by palynology (pollen, spore and related bio-and litho-stratigraphic analyses) as a key approach to multi- and inter-disciplinary investigations of topics such as vegetation, climate and landscape change. This paper highlights some key dimensions of the pollen- and vegetation-based archive, with a focus upon woodland dynamics, blanket peat, human impacts, biodiversity and conservation. Following a brief discussion of chronological, climatic, faunal and landscape contexts, the migration, survival and nature of the woodland cover through time is assessed, emphasising its time-transgressiveness and altitudinal variation. While agriculture led to the demise of woodland in lowland areas of the south and east, the spread of blanket peat was especially a phenomenon of the north and west, including the Western and Northern Isles. Almost a quarter of Scotland is covered by blanket peat and the cause(s) of its spread continue(s) to evoke recourse to climatic, topographic, pedogenic, hydrological, biotic or anthropogenic influences, while we remain insufficiently knowledgeable about the timing of the formation processes. Humans have been implicated in vegetational change throughout the Holocene, with prehistoric woodland removal, woodland management, agricultural impacts arising from arable and pastoral activities, potential heathland development and afforestation. The viability of many current vegetation communities remains a concern, in that Scottish data show reductions in plant diversity over the last 400 years, which recent conservation efforts have yet to reverse. Palaeoecological evidence can be used to test whether conservation baselines and restoration targets are appropriate to longer-term ecosystem variability and can help identify when modern conditions have no past analogues.
Footprints in Time: The Longitudinal Study of Indigenous Children (LSIC) is a national study of 1759 Australian Aboriginal and Torres Strait Islander children living across urban, regional and remote areas of Australia. The study is in its 11th wave of annual data collection, having collected extensive data on topics including birth and early life influences, parental health and well-being, identity, cultural engagement, language use, housing, racism, school engagement and academic achievement, and social and emotional well-being. The current paper reviews a selection of major findings from Footprints in Time relating to the developmental origins of health and disease for Australian Aboriginal and Torres Strait Islander peoples. Opportunities for new researchers to conduct further research utilizing the LSIC data set are also presented.
Water wave overwash of a step by small steepness, regular incident waves is analysed using a computational fluid dynamics (CFD) model and a mathematical model, in two spatial dimensions. The CFD model is based on the two-phase, incompressible Navier–Stokes equations, and the mathematical model is based on the coupled potential-flow and nonlinear shallow-water theories. The CFD model is shown to predict vortices, breaking and overturning in the region where overwash is generated, and that the overwash develops into fast-travelling bores. The mathematical model is shown to predict bore heights and velocities that agree with the CFD model, despite neglecting the complicated dynamics where the overwash is generated. Evidence is provided to explain the agreement in terms of the underlying agreement of mass and energy fluxes.
A diverse millipede (diplopod) fauna has been recovered from the earliest Carboniferous (Tournaisian) Ballagan Formation of the Scottish Borders, discovered by the late Stan Wood. The material is generally fragmentary; however, six different taxa are present based on seven specimens. Only one displays enough characters for formal description and is named Woodesmus sheari Ross, Edgecombe & Clark gen. & sp. nov. The absence of paranota justifies the erection of Woodesmidae fam. nov. within the Archipolypoda. The diverse fauna supports the theory that an apparent lack of terrestrial animal fossils from ‘Romer's Gap' was due to a lack of collecting and suitable deposits, rather than to low oxygen levels as previously suggested.
Worldwide 350 million people suffer from major depression, with the majority of cases occurring in low- and middle-income countries. We examined the patterns, correlates and care-seeking behaviour of adults suffering from major depressive episode (MDE) in China.
A nationwide study recruited 512 891 adults aged 30–79 years from 10 provinces across China during 2004–2008. The 12-month prevalence of MDE was assessed by the Modified Composite International Diagnostic Interview-short form. Logistic regression yielded adjusted odds ratios (ORs) of MDE associated with socio-economic, lifestyle and health-related factors and major stressful life events.
Overall, 0.7% of participants had MDE and a further 2.4% had major depressive symptoms. Stressful life events were strongly associated with MDE [adjusted OR 14.7, 95% confidence interval (CI) 13.7–15.7], with a dose–response relationship with the number of such events experienced. Family conflict had the highest OR for MDE (18.9, 95% CI 16.8–21.2) among the 10 stressful life events. The risk of MDE was also positively associated with rural residency (OR 1.5, 95% CI 1.4–1.7), low income (OR 2.3, 95% CI 2.1–2.4), living alone (OR 2.6, 95% CI 2.3–3.0), smoking (OR 1.4, 95% CI 1.3–1.6) and certain other mental disorders (e.g. anxiety, phobia). Similar, albeit weaker, associations were observed with depressive symptoms. Among those with MDE, about 15% sought medical help or took psychiatric medication, 15% reported having suicidal ideation and 6% reported attempting suicide.
Among Chinese adults, the patterns and correlates of MDE were generally consistent with those observed in the West. The low rates of seeking professional help and treatment highlight the great gap in mental health services in China.
You work for an airline in the operations department. They ask you to reach into the historical company and industry datasets and figure out the expected loading for every flight in their book for this coming year. This ties directly into projected revenue. In the back of your mind, you know that expenses for the company need to be offset by revenue. It doesn't help to be overly optimistic because the risk of not meeting the projections carries with it the risk of future disappointment.
What should you do? Passenger boardings are cyclical. There are risks of a downturn. But a downturn could impact profitability of the entire company. The price of oil and jet fuel is important. Mergers happen every year which change the competitive landscape and make some routes more efficient.
Time series analysis can certainly help. Just like confidence intervals in statistics (please see also the Appendix), there is a band of uncertainty around any of the projections. The expected airline passenger loadings is a random variable. Passenger boarding is one of several arenas we will explore with the help of tools in R.
Examining Time Series
We begin with a quick survey of the types of times series we will model. We can use the quantmod and PerformanceAnalytics packages, firstly. We will define the vector of symbols to be downloaded. The symbols GSPC, VIX, TNX, refer to the S&P 500 index, the CME volatility index, and the ten-year treasury yield, respectively. Use the getSymbols() function to download the time series for the symbols in sym.vec between the dates of January 3, 2003 and September 10, 2015. If the quantmod and PerformanceAnalytics packages are not yet installed, they can be installed with the commands install.packages(“quantmod”, dependencies=TRUE) and install.packages(“PerformanceAnalytics”, dependencies=TRUE).
> sym.vec <-c(“∧GSPC”,“∧VIX”)
> getSymbols(sym.vec, from = “2005-01-03”, to = “2015-09-16”)
 “GSPC” “VIX”
The first plot is of the S&P 500, shown in Figure 6.1.We see the peak of the market in early 2007 and then the steep decline as the housing crisis hit.We see the market bottom out in mid 2009 and begin a large rally that lasts through mid 2015. We see minor corrections in 2011 and 2012 due to uncertainty surrounding the government debt in the Euro zone, and a significant sell-off in mid 2015 due to uncertainty surrounding China's economy.
When traveling across the agricultural American Midwest, one can hear AM radio stations broadcast the futures prices of corn, soybeans, wheat, and other commodities every weekday at various times of the day. Iowa and Illinois lead the nation in corn production. Kansas leads the nation in wheat production. Listening to these farm reports is entertaining. Included in the broadcasts are very detailed weather reports. Weather is critical to many producers’ livelihoods. After a few weeks of listening to these broadcasts, we learn that nobody truly knows for sure whether the agricultural market prices will go up or down on a trading day. Hedging the production chain price risk, especially for the farmers, who seasonally grow the crops, is achieved with futures and option securities. Producers very often want to lock in a price for delivering corn at the end of the season, or want to be compensated for a significant reduction in the agricultural price in order to guarantee recovering their fixed costs over the upcoming days and months.
From Chapter 14, we have gained more familiarity with the random walk processes assumed by the option models. We continue the option theme from the prior chapter and examine a very popular model for pricing European options. The most famous and widely accepted model of option valuation model is known as the Black–Scholes model of 1973 (Black and Myron, 1973). It revolutionized the pricing and trading of options which, prior to this, were priced in rather arbitrary ways. Black and Scholes relied upon the stochastic calculus. Stochastic calculus was invented by Itô in 1951 to address the need for a calculus for random variables as functions over time, like our stock market prices (Ito, 1951). Together with Merton, Black and Scholes were awarded the 1997 Nobel Prize in Economic Sciences for this invention. We discuss the Black–Scholes model here in order to complete our tour of financial analytics. We will try to make minimal use of stochastic calculus.
We saved this more mathematical material for the end of the book because it involves the more complex type of security: options. As mentioned in Chapter 14, options are derivative securities.