This document discusses challenges in teaching quantitative finance concepts and introduces threshold concepts as a framework. It provides examples of difficult concepts in quantitative finance like Ito's lemma. Graphs show distributions of student learning styles, from surface to deep. Definitions of threshold concepts are given. Examples illustrate mapping concepts during expert practitioner sessions, which helped students integrate knowledge in a transformative way. Assessment strategies aim to discourage surface learning and encourage reasoning and interpretation.
The complexity of the epistemological genesis of mathematical proof (V.2 comp...Nicolas Balacheff
Early learning of mathematics is first rooted in pragmatic evidences or learners’ confidence in the facts and procedures taught. Nonetheless, learners develop a true knowledge which works as a tool in significant problem situations, and which is accessible to falsification and argumentation. As teachers know, they could validate what they claim to be true, but based on means in general not conforming to mathematical standards. Teaching these standards requires an evolution of their understanding of what can count as a proof in the mathematical classroom, as well as an evolution of their mathematical knowing. This claim is discussed from the perspective of modelling the learners ways of knowing (the model cK¢), within the framework of the theory of didactical situations, bridging the semiotic system they use, the type of actions they perform and the controls they implement either to construct or to validate the solutions they propose to a problem.
The complexity of the epistemological and didactical genesis of mathematical ...Nicolas Balacheff
Students’ mathematical knowledge is first rooted in pragmatic evidences and in the effort to make sense of the content and procedures taught. They develop a true knowledge which works as a tool in problem situations, and is accessible to falsification and argumentation. They can validate what they claim to be true, but based on means which may not conform to current mathematical standards. The theory of didactical situations (TSD) is based on the recognition of the existence of this true knowledge and the analysis of the specific complexity of the teaching situations from an epistemological perspective. It is in this framework that I propose to address the problems raised by the teaching and learning of mathematical proof. The main issue which I will discuss is that the evolution of the students understanding of what count as proof in mathematics implies – and is constitutive of – an evolution of their knowing of mathematical concepts. This discussion will support the claim that the “situation of validation” conceptualized by the TSD must be the starting point of any didactical engineering.
The complexity of the epistemological genesis of mathematical proof (V.2 comp...Nicolas Balacheff
Early learning of mathematics is first rooted in pragmatic evidences or learners’ confidence in the facts and procedures taught. Nonetheless, learners develop a true knowledge which works as a tool in significant problem situations, and which is accessible to falsification and argumentation. As teachers know, they could validate what they claim to be true, but based on means in general not conforming to mathematical standards. Teaching these standards requires an evolution of their understanding of what can count as a proof in the mathematical classroom, as well as an evolution of their mathematical knowing. This claim is discussed from the perspective of modelling the learners ways of knowing (the model cK¢), within the framework of the theory of didactical situations, bridging the semiotic system they use, the type of actions they perform and the controls they implement either to construct or to validate the solutions they propose to a problem.
The complexity of the epistemological and didactical genesis of mathematical ...Nicolas Balacheff
Students’ mathematical knowledge is first rooted in pragmatic evidences and in the effort to make sense of the content and procedures taught. They develop a true knowledge which works as a tool in problem situations, and is accessible to falsification and argumentation. They can validate what they claim to be true, but based on means which may not conform to current mathematical standards. The theory of didactical situations (TSD) is based on the recognition of the existence of this true knowledge and the analysis of the specific complexity of the teaching situations from an epistemological perspective. It is in this framework that I propose to address the problems raised by the teaching and learning of mathematical proof. The main issue which I will discuss is that the evolution of the students understanding of what count as proof in mathematics implies – and is constitutive of – an evolution of their knowing of mathematical concepts. This discussion will support the claim that the “situation of validation” conceptualized by the TSD must be the starting point of any didactical engineering.
Slides in support of a talk at the conference "Explanation and Proof in Mathematics: Philosophical and Educational Perspective" held in Essen in November 2006.
Abstract:
The learning of mathematics starts early but remains far from any theoretical considerations: pupils' mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which they can apply in significant problem situations, and which is amenable to falsification and argumentation. They can validate what they claim to be true but using means generally not conforming to mathematical standards. Here, I analyze how this situation underlies the epistemological and didactical complexities of teaching mathematical proof. I show that the evolution of the learners' understanding of what counts as proof in mathematics implies an evolution of their knowing of mathematical concepts. The key didactical point is not to persuade learners to accept a new formalism but to have them understand how mathematical proof and statements are tightly related within a common framework; that is, a mathematical theory. I address this aim by modeling the learners' way of knowing in terms of a dynamic, homeostatic system. I discuss the roles of different semiotic systems, of the types of actions the learners perform and of the controls they implement in constructing or validating knowledge. Particularly with modern technological aids, this model provides a basis designing didactical situations to help learners bridge the gap between pragmatics and theory.
Pricing vulnerable European options when the option’s payoff can increase the risk of financial distressPeter Klein, Michael InglisJournal of Banking & Finance
Mathematical argumentation as a precursor of mathematical proofNicolas Balacheff
Along history or across educational traditions, the space given to mathematical proof in compulsory school curricula varies from a quasi-absence to a formal obligation which for some has turned into an obstacle to mathematics learning. The contemporary evolution is to give to proof the space it deserves in the learning of mathematics. This is for example witnessed in different ways by The national curriculum in England (2014), the Common Core State Standards for Mathematics (2010) in the US or the recent Report on the teaching of mathematics (1918) commissioned by the French government; the latter asserts: The notion of proof is at the heart of mathematical activity, whatever the level (this assertion is valid from kindergarten to university). And, beyond mathematical theory, understanding what is a reasoned justification approach based on logic is an important aspect of citizen training. The seeds of this fundamentally mathematical approach are sown in the early grades. These are a few examples of the current worldwide consensus on the centrality proof should have in the compulsory school curricula. However, the institutional statements share difficulty to express this objective. The vocabulary includes words such as argument, justification and proof without clear reasons for such diversity: are these words mere synonymous or are there differences that we should pay attention to? What are the characteristics of the discourse these words may refer to in the mathematics classroom? Eventually, how can be addressed the problem of assessing the truth value of a mathematical statement at the different grades all along compulsory school? I shall explore these questions, starting from questioning the meaning of these words and its consequences. Then, I shall shape the relations between argumentation and proof from an epistemological and didactical perspective. In the end, the participants will be invited to a discussion on the benefit and relevance of shaping the notion of mathematical argumentation as a precursor of mathematical proof.
International journal of engineering and mathematical modelling vol2 no2_2015_1IJEMM
Management of the portfolios containing low liquidity assets is a tedious problem. The buyer proposes the price that can differ greatly from the paper value estimated by the seller, so the seller can not liquidate his portfolio instantly and waits for a more favorable offer. To minimize losses and move the theory towards practical needs one
can take into account the time lag of the liquidation of an illiquid asset. Working in the Merton’s optimal consumption framework with continuous time we consider an optimization problem for a portfolio with an illiquid, a risky and a risk-free asset. While a standard Black-Scholes market describes the liquid part of the investment the illiquid asset is sold at an exogenous random moment with prescribed liquidation time distribution. The investor has the logarithmic utility function as a limit case of a HARA-type utility. Different distributions of the liquidation time of the illiquid asset are under consideration - a classical exponential distribution andWeibull distribution that is more practically relevant. Under certain conditions we show the existence
of the viscosity solution in both cases. Applying numerical methods we compare classical Merton’s strategies and the optimal consumption-allocation strategies for portfolios with different liquidation time distributions of an illiquid asset.
Strategic capability - strategic human resource managementmanumelwin
Strategic capability is a concept that refers to the ability of an organization to develop and implement strategies that will achieve sustained competitive advantage. It is therefore about the capacity to select the most appropriate vision, to define realistic intentions, to match resources to opportunities and to prepare and implement strategic plans.
Core competency is a concept in management theory introduced by, C. K. PRAHALAD and GARY HAMEL.
It can be defined as "a harmonized combination of multiple resources and skills that distinguish a firm in the marketplace“
Core competency are the skills, characteristics, and assets that set your company apart from competitors.
They are the fuel for innovation and the roots of competitive advantage.
The engine for new business development, underlying component of a company’s competitive advantage created from the coordination, integration and harmonization of diverse skills and multiple streams of technologies.
Slides in support of a talk at the conference "Explanation and Proof in Mathematics: Philosophical and Educational Perspective" held in Essen in November 2006.
Abstract:
The learning of mathematics starts early but remains far from any theoretical considerations: pupils' mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which they can apply in significant problem situations, and which is amenable to falsification and argumentation. They can validate what they claim to be true but using means generally not conforming to mathematical standards. Here, I analyze how this situation underlies the epistemological and didactical complexities of teaching mathematical proof. I show that the evolution of the learners' understanding of what counts as proof in mathematics implies an evolution of their knowing of mathematical concepts. The key didactical point is not to persuade learners to accept a new formalism but to have them understand how mathematical proof and statements are tightly related within a common framework; that is, a mathematical theory. I address this aim by modeling the learners' way of knowing in terms of a dynamic, homeostatic system. I discuss the roles of different semiotic systems, of the types of actions the learners perform and of the controls they implement in constructing or validating knowledge. Particularly with modern technological aids, this model provides a basis designing didactical situations to help learners bridge the gap between pragmatics and theory.
Pricing vulnerable European options when the option’s payoff can increase the risk of financial distressPeter Klein, Michael InglisJournal of Banking & Finance
Mathematical argumentation as a precursor of mathematical proofNicolas Balacheff
Along history or across educational traditions, the space given to mathematical proof in compulsory school curricula varies from a quasi-absence to a formal obligation which for some has turned into an obstacle to mathematics learning. The contemporary evolution is to give to proof the space it deserves in the learning of mathematics. This is for example witnessed in different ways by The national curriculum in England (2014), the Common Core State Standards for Mathematics (2010) in the US or the recent Report on the teaching of mathematics (1918) commissioned by the French government; the latter asserts: The notion of proof is at the heart of mathematical activity, whatever the level (this assertion is valid from kindergarten to university). And, beyond mathematical theory, understanding what is a reasoned justification approach based on logic is an important aspect of citizen training. The seeds of this fundamentally mathematical approach are sown in the early grades. These are a few examples of the current worldwide consensus on the centrality proof should have in the compulsory school curricula. However, the institutional statements share difficulty to express this objective. The vocabulary includes words such as argument, justification and proof without clear reasons for such diversity: are these words mere synonymous or are there differences that we should pay attention to? What are the characteristics of the discourse these words may refer to in the mathematics classroom? Eventually, how can be addressed the problem of assessing the truth value of a mathematical statement at the different grades all along compulsory school? I shall explore these questions, starting from questioning the meaning of these words and its consequences. Then, I shall shape the relations between argumentation and proof from an epistemological and didactical perspective. In the end, the participants will be invited to a discussion on the benefit and relevance of shaping the notion of mathematical argumentation as a precursor of mathematical proof.
International journal of engineering and mathematical modelling vol2 no2_2015_1IJEMM
Management of the portfolios containing low liquidity assets is a tedious problem. The buyer proposes the price that can differ greatly from the paper value estimated by the seller, so the seller can not liquidate his portfolio instantly and waits for a more favorable offer. To minimize losses and move the theory towards practical needs one
can take into account the time lag of the liquidation of an illiquid asset. Working in the Merton’s optimal consumption framework with continuous time we consider an optimization problem for a portfolio with an illiquid, a risky and a risk-free asset. While a standard Black-Scholes market describes the liquid part of the investment the illiquid asset is sold at an exogenous random moment with prescribed liquidation time distribution. The investor has the logarithmic utility function as a limit case of a HARA-type utility. Different distributions of the liquidation time of the illiquid asset are under consideration - a classical exponential distribution andWeibull distribution that is more practically relevant. Under certain conditions we show the existence
of the viscosity solution in both cases. Applying numerical methods we compare classical Merton’s strategies and the optimal consumption-allocation strategies for portfolios with different liquidation time distributions of an illiquid asset.
Strategic capability - strategic human resource managementmanumelwin
Strategic capability is a concept that refers to the ability of an organization to develop and implement strategies that will achieve sustained competitive advantage. It is therefore about the capacity to select the most appropriate vision, to define realistic intentions, to match resources to opportunities and to prepare and implement strategic plans.
Core competency is a concept in management theory introduced by, C. K. PRAHALAD and GARY HAMEL.
It can be defined as "a harmonized combination of multiple resources and skills that distinguish a firm in the marketplace“
Core competency are the skills, characteristics, and assets that set your company apart from competitors.
They are the fuel for innovation and the roots of competitive advantage.
The engine for new business development, underlying component of a company’s competitive advantage created from the coordination, integration and harmonization of diverse skills and multiple streams of technologies.
Presentation for researchED maths and science on June 11th 2016. References at the end (might be some extra references from slides that were removed later on, this interesting :-)
Interested in discussing, contact me at C.Bokhove@soton.ac.uk or on Twitter @cbokhove
I of course tried to reference all I could. If you have objections to the inclusion of materials, please let me know.
Design & Technology and Computer Science in the CAMAU Project: The Genesis of...David Morrison-Love
Wales in currently undergoing significant and ambitious educational reform on a national scale. This presentation outlines some of the work undertaken by the CAMAU Project which seeks to place learning progression at the heart of the new curriculum for Wales. Here, the focus is on the work done in phase 1 of the project in the curricular areas of Design & Technology and Computing Science.
The CAMAU Project is large-scale, 3-year, collaborative R&D project (£500,000) commission by the Welsh Government and funded by the Welsh Government and University of Wales Trinity Saint David. This work was presented as part of the PATT36 Conference in Malta (June, 2019).
Dr David Morrison-Love, July 2019.
Suggested Enrichment Program Using Cinderella (DGS) in Developing Geometric C...Mohamed El-Demerdash
Suggested Enrichment Program Using Cinderella (DGS) in Developing Geometric Creativity
Presentation at the University of Education - Schwaebisch Gmuend
Date: Nov. 27th 2008
A Suggested Enrichment Program Using Cinderella (DGS) in Developing Geometric Creativity. Mathematics Seminar of the University of Education Schwaebisch Gmuend, Germany, November 27, 2008
Suggested Enrichment Program Using Cinderella (DGS) in Developing Geometric C...Mohamed El-Demerdash
Herbsttagung des Arbeitskreis Mathematikunterricht und Informatik (AKMUI), from 26th till 28th September 2008. I contribute with a talk entitled: "Suggested Enrichment Program Using Cinderella (DGS) in Developing Geometric Creativity"
Lecture slide titled Fraud Risk Mitigation, Webinar Lecture Delivered at the Society for West African Internal Audit Practitioners (SWAIAP) on Wednesday, November 8, 2023.
What price will pi network be listed on exchangesDOT TECH
The rate at which pi will be listed is practically unknown. But due to speculations surrounding it the predicted rate is tends to be from 30$ — 50$.
So if you are interested in selling your pi network coins at a high rate tho. Or you can't wait till the mainnet launch in 2026. You can easily trade your pi coins with a merchant.
A merchant is someone who buys pi coins from miners and resell them to Investors looking forward to hold massive quantities till mainnet launch.
I will leave the telegram contact of my personal pi vendor to trade with.
@Pi_vendor_247
1. Elemental Economics - Introduction to mining.pdfNeal Brewster
After this first you should: Understand the nature of mining; have an awareness of the industry’s boundaries, corporate structure and size; appreciation the complex motivations and objectives of the industries’ various participants; know how mineral reserves are defined and estimated, and how they evolve over time.
The European Unemployment Puzzle: implications from population agingGRAPE
We study the link between the evolving age structure of the working population and unemployment. We build a large new Keynesian OLG model with a realistic age structure, labor market frictions, sticky prices, and aggregate shocks. Once calibrated to the European economy, we quantify the extent to which demographic changes over the last three decades have contributed to the decline of the unemployment rate. Our findings yield important implications for the future evolution of unemployment given the anticipated further aging of the working population in Europe. We also quantify the implications for optimal monetary policy: lowering inflation volatility becomes less costly in terms of GDP and unemployment volatility, which hints that optimal monetary policy may be more hawkish in an aging society. Finally, our results also propose a partial reversal of the European-US unemployment puzzle due to the fact that the share of young workers is expected to remain robust in the US.
How Does CRISIL Evaluate Lenders in India for Credit RatingsShaheen Kumar
CRISIL evaluates lenders in India by analyzing financial performance, loan portfolio quality, risk management practices, capital adequacy, market position, and adherence to regulatory requirements. This comprehensive assessment ensures a thorough evaluation of creditworthiness and financial strength. Each criterion is meticulously examined to provide credible and reliable ratings.
BYD SWOT Analysis and In-Depth Insights 2024.pptxmikemetalprod
Indepth analysis of the BYD 2024
BYD (Build Your Dreams) is a Chinese automaker and battery manufacturer that has snowballed over the past two decades to become a significant player in electric vehicles and global clean energy technology.
This SWOT analysis examines BYD's strengths, weaknesses, opportunities, and threats as it competes in the fast-changing automotive and energy storage industries.
Founded in 1995 and headquartered in Shenzhen, BYD started as a battery company before expanding into automobiles in the early 2000s.
Initially manufacturing gasoline-powered vehicles, BYD focused on plug-in hybrid and fully electric vehicles, leveraging its expertise in battery technology.
Today, BYD is the world’s largest electric vehicle manufacturer, delivering over 1.2 million electric cars globally. The company also produces electric buses, trucks, forklifts, and rail transit.
On the energy side, BYD is a major supplier of rechargeable batteries for cell phones, laptops, electric vehicles, and energy storage systems.
Turin Startup Ecosystem 2024 - Ricerca sulle Startup e il Sistema dell'Innov...Quotidiano Piemontese
Turin Startup Ecosystem 2024
Una ricerca de il Club degli Investitori, in collaborazione con ToTeM Torino Tech Map e con il supporto della ESCP Business School e di Growth Capital
The secret way to sell pi coins effortlessly.DOT TECH
Well as we all know pi isn't launched yet. But you can still sell your pi coins effortlessly because some whales in China are interested in holding massive pi coins. And they are willing to pay good money for it. If you are interested in selling I will leave a contact for you. Just telegram this number below. I sold about 3000 pi coins to him and he paid me immediately.
Telegram: @Pi_vendor_247
where can I find a legit pi merchant onlineDOT TECH
Yes. This is very easy what you need is a recommendation from someone who has successfully traded pi coins before with a merchant.
Who is a pi merchant?
A pi merchant is someone who buys pi network coins and resell them to Investors looking forward to hold thousands of pi coins before the open mainnet.
I will leave the telegram contact of my personal pi merchant to trade with
@Pi_vendor_247
US Economic Outlook - Being Decided - M Capital Group August 2021.pdfpchutichetpong
The U.S. economy is continuing its impressive recovery from the COVID-19 pandemic and not slowing down despite re-occurring bumps. The U.S. savings rate reached its highest ever recorded level at 34% in April 2020 and Americans seem ready to spend. The sectors that had been hurt the most by the pandemic specifically reduced consumer spending, like retail, leisure, hospitality, and travel, are now experiencing massive growth in revenue and job openings.
Could this growth lead to a “Roaring Twenties”? As quickly as the U.S. economy contracted, experiencing a 9.1% drop in economic output relative to the business cycle in Q2 2020, the largest in recorded history, it has rebounded beyond expectations. This surprising growth seems to be fueled by the U.S. government’s aggressive fiscal and monetary policies, and an increase in consumer spending as mobility restrictions are lifted. Unemployment rates between June 2020 and June 2021 decreased by 5.2%, while the demand for labor is increasing, coupled with increasing wages to incentivize Americans to rejoin the labor force. Schools and businesses are expected to fully reopen soon. In parallel, vaccination rates across the country and the world continue to rise, with full vaccination rates of 50% and 14.8% respectively.
However, it is not completely smooth sailing from here. According to M Capital Group, the main risks that threaten the continued growth of the U.S. economy are inflation, unsettled trade relations, and another wave of Covid-19 mutations that could shut down the world again. Have we learned from the past year of COVID-19 and adapted our economy accordingly?
“In order for the U.S. economy to continue growing, whether there is another wave or not, the U.S. needs to focus on diversifying supply chains, supporting business investment, and maintaining consumer spending,” says Grace Feeley, a research analyst at M Capital Group.
While the economic indicators are positive, the risks are coming closer to manifesting and threatening such growth. The new variants spreading throughout the world, Delta, Lambda, and Gamma, are vaccine-resistant and muddy the predictions made about the economy and health of the country. These variants bring back the feeling of uncertainty that has wreaked havoc not only on the stock market but the mindset of people around the world. MCG provides unique insight on how to mitigate these risks to possibly ensure a bright economic future.
USDA Loans in California: A Comprehensive Overview.pptxmarketing367770
USDA Loans in California: A Comprehensive Overview
If you're dreaming of owning a home in California's rural or suburban areas, a USDA loan might be the perfect solution. The U.S. Department of Agriculture (USDA) offers these loans to help low-to-moderate-income individuals and families achieve homeownership.
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The Role of Non-Banking Financial Companies (NBFCs)
Threshold Concepts in Quantitative Finance - DEE 2011 Presentation
1. !
Threshold Concepts
in Quantitative Finance
by Richard Diamond and Holly Smith
for Developments in Economics Education Conference
September 2011
2. In This Presentation...
• Challenges of quantitative instruction. Evidence for
surface-to-deep learning dynamics
• Introduction to the threshold concept as a
paradigm
• Quantitative Finance: the volume of procedural
knowledge and ‘a maths sweet spot’
• Illustrations and ideas for knowledge integration
(special learning situations) in quantitative finance
3. Quantitative Instruction:
The Problem(s)
• Students acquire formal knowledge but seem
unable to use it when making sense of experience
• Students struggle with underpinning theory and
resort to verbatim learning of isolated aspects of a
subject, being unable to use them in conjunction
• Students are unable to transfer their skills outside
specific, structured problems
If struggling with gaps in what students actually understood
please consult A Handbook on Threshold Concepts in Economics:
Implications for teaching, learning and assessment
5. ‘Surface ‘Deep
Learners’ Learners’
Multi-Modal Distribution
Reflects a distribution of learning styles chosen by
students. Only several start with an in-depth style
6. Definitions
“The threshold concept
“A threshold concept is akin to a approach is concerned with
portal, opening up a new and how students can be helped
previously inaccessible way of to acquire integrating
thinking about something. It ideas.” (Davies & Mangan 2007b)
represents a transformed way of
understanding, or interpreting, or
viewing something without which “The threshold concept
the learner cannot approach helps to anticipate
progress.” (Meyer & Land 2005) challenges of quantitative
teaching.” (Diamond & Smith 2011)
A well-maintained collection of resources can be found on
The Threshold Concept Portal
at www.ee.ucl.ac.uk/~mflanaga/thresholds.html
7. ‘One Line’ Examples
• Maths: complex number, a limit, the Fourier transform
• Economics: opportunity cost, price elasticity
• Quantitative Finance: Ito's lemma, change of
measure, risk neutrality, incomplete markets
A couple of questions:
• Which concepts did your students experience the most
difficulty with?
• Were you satisfied with progress of your students in
understanding of those concepts and disciplinary modelling
(e.g., econometrics)?
8. A Big Box of Tools for
Quantitative Finance
• The discipline utilises techniques of pure and applied
mathematics (analysis & measure theory; stochastic
calculus & PDE solution methods) and statistics
• There are discipline-specific modelling techniques:
replication of instrument with a portfolio and pricing
via expectations (FATF & Feynman-Kac)
Option pricing formulae were known explcitly from 1960 but
real significance was in how they were derived by B-S. Delta-
hedging and independence of expected rate of return were the
great discoveries awarded the Nobel Prize (Haug 2007: 39-40).
Finance maths is simple: PDEs are parabolic and
numerical methods are well-specified
9. Example: Black’76 Formula for
Fixed Income Derivatives
The proof of Black’s (1976) formula involved the following
mathematics: differential equations, Brownian Motion,
stochastic calculus and Ito’s integral (Ito’s lemma), double
change of measure (Girsanov theorem) to obtain a futures
martingale measure, Feynman-Kac theorem, and assumptions
allowing to use the Normal Distribution.
The following techniques specific to quantitative finance were
also utilised: forward price (given discounting in continuos
time), self-financing trading strategy, no arbitrage valuation,
FAPF and Black-Scholes solutions formulae.
That was an extremely brief overview of a dense
8-page proof
10. A Maths Sweet Spot
• The idea is advocated by Paul Wilmott (2009)
“The models should not be too elementary so as to make
it impossible to invent new structured products, but nor
should they be so abstract as to be easily misunderstood
by all except their inventor (and sometimes even by him),
with the obvious and financially dangerous consequences.”
• Example of CDO and CDO^2 products that show ‘bizarre’
correlation behaviours (to default) that are different for
each tranche, leading to ‘off the cliff’ scenario for value
• Tendency to make presentation of finance research overly
complex (unlike experts, students cannot discern a paper)
Outside of the sweet spot, model risk is increased
11. ‘Live’ Integration
• The following Illustrations show how expert
practitioners, who were faculty and tutors on CQF
programme, mapped concepts during their sessions
• Programme participants reported ‘a transformative
impact’ on their knowledge, generated by experiences
of dynamic mapping of disciplinary concepts
Understanding of the Illustrations requires familiarity
with the type of modelling done in quantitative finance
12. Illustration 1 compares Black-Scholes pricing equation
that applies locally (pricing over the next small period) to
FAPF that applies globally (over the life of a product)
13. Illustration 2 aims to show a critical difference between
risk-free and forward rates: forward rate curve is not an
expected path of a mean-reverting risk-free rate
14. Popular one-factor spot-rate models
The real spot rate r satisfies the stochastic differential equation
dr = u(r, t)dt + w(r, t)dX. (4)
Model u(r, t) − λ(r, t)w(r, t) w(r, t)
Vasicek a − br c
CIR a − br cr1/2
Ho & Lee a(t) c
Hull & White I a(t) − b(t)r c(t)
Hull & White II a(t) − b(t)r c(t)r1/2
General affine a(t) − b(t)r (c(t)r − d(t))1/2
Here λ(r, t) denotes the market price of risk. The function u − λw
is the risk-adjusted drift.
For all of these models the zero-coupon bond value is of the
form Z(r, t; T ) = eA(t,T )−rB(t,T ).
Certificate in Quantitative Finance
36
Illustration 3 relates the parameters (e.g, drift and
process volatility) of several named stochastic interest rate
models to the same parameters in one generalised model
15. Illustration 4 is a flow chart that utilises ‘black boxes’ to
present sophisticated statistical procedures for testing and
modelling of long-term relationships in a simple form
16. Learning Situations Design
• Go through mathematical proofs ‘by hand’ and make
learners comfortable with volumes of it
• Teach initial modelling skills without the formal
disciplinary apparatus
• Structure situations in which insight comes from the
re-working of prior knowledge and replacing of
simple ways of understanding
For an example, see DEE 2007 Keynote on Teaching
Undergraduate Econometrics by Prof. David Hendry
http://www.economicsnetwork.ac.uk/dee2007/
17. Assessment in
Quantitative Subjects
• Provide scaffolding for knowledge integration early in
the module: hand out past exam papers at the first
lecture and suggest portable textbooks
• Design assessment so as to remove incentives for
surface learning: include questions on reasoning,
model derivation and output interpretation
• Be realistic about what is possible. In cases such as a
six-week Masters module, we can only bring specific
mathematical and modelling skills up to a standard
• If things are tight, choose a project over exam
18. This was the first glance at pedagogic
value of the threshold concept approach.
With feedback and suggestions for
further enquiry, please get in touch via
richard.diamond@ucl.ac.uk