AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 14.
More info at http://summerschool.ssa.org.ua
The Delta Of An Arithmetic Asian Option Via The Pathwise MethodAnna Borisova
In the slides present a structured description of the methods that can be used to calculate the delta for an asian option. Only European options are considered. The reference list has added as the last slide. Enjoy the presentation!
60 Years Birthday, 30 Years of Ground Breaking Innovation: A Tribute to Bruno...Antoine Savine
The RiO 2018 conference in mathematical finance was held in Buzios, Rio de Janeiro, Brazil, 24-28 November 2018, to celebrate the 60th birthday of Bruno Dupire, one of the most influential figures in the history of financial derivatives.
This presentation, given by Antoine Savine, one of Bruno Dupire's original alumni and a lecturer in Volatility at Copenhagen University, celebrates Dupire's most influential contributions to mathematical finance and puts in perspective the history and main results of volatility modeling.
The Delta Of An Arithmetic Asian Option Via The Pathwise MethodAnna Borisova
In the slides present a structured description of the methods that can be used to calculate the delta for an asian option. Only European options are considered. The reference list has added as the last slide. Enjoy the presentation!
60 Years Birthday, 30 Years of Ground Breaking Innovation: A Tribute to Bruno...Antoine Savine
The RiO 2018 conference in mathematical finance was held in Buzios, Rio de Janeiro, Brazil, 24-28 November 2018, to celebrate the 60th birthday of Bruno Dupire, one of the most influential figures in the history of financial derivatives.
This presentation, given by Antoine Savine, one of Bruno Dupire's original alumni and a lecturer in Volatility at Copenhagen University, celebrates Dupire's most influential contributions to mathematical finance and puts in perspective the history and main results of volatility modeling.
A Probability Distribution is a way to shape the sample data to make predictions and draw conclusions about an entire population. It refers to the frequency at which some events or experiments occur. It helps finding all the possible values a random variable can take between the minimum and maximum statistically possible values.
A Probability Distribution is a way to shape the sample data to make predictions and draw conclusions about an entire population. It refers to the frequency at which some events or experiments occur. It helps finding all the possible values a random variable can take between the minimum and maximum statistically possible values.
Advanced macroeconomics, 4th edition. Romer.
Chapter12.
12.1. The stability of fiscal policy. (Blinder and Solow, 1973.) By definition, the budget deficit equals the rate of change of the amount of debt outstanding: δ(t) ≡ D ̇(t). Define d(t) to be the ratio of debt to output: d(t) = D(t)/Y(t). Assume that Y(t) grows at a constant rate g > 0.
(a) Suppose that the deficit-to-output ratio is constant: δ(t)/Y(t) = a, where a > 0.
̇
(i) Find an expression for d(t) in terms of a, g, and d(t). ̇
(ii) Sketch d(t) as a function of d(t). Is this system stable?
(b) Suppose that the ratio of the primary deficit to output is constant and equal to a > 0. Thus the total deficit at t, δ(t), is given by δ(t) = aY(t) + r(t)D(t), where r(t) is the interest rate at t. Assume that r is an increasing function of the debt-to-output ratio: r(t) = r(d(t)), where r′(•) > 0, r′′(•) > 0, limd→−∞ r(d) < g, limd→∞ r(d) > g.
̇
(i) Find an expression for d(t) in terms of a, g, and d(t). ̇
(ii) Sketch d(t) as a function of d(t). In the case where a is sufficiently small that d ̇ is negative for some values of d, what are the stability properties of the system? What about the case where a is sufficiently large that d ̇ is positive for all values of d ?
12.2. Precautionary saving, non-lump-sum taxation, and Ricardian equivalence.
(Leland, 1968, and Barsky, Mankiw, and Zeldes, 1986.) Consider an individual who lives for two periods. The individual has no initial wealth and earns labor incomes of amounts Y1 and Y2 in the two periods. Y1 is known, but Y2 is random; assume for simplicity that E[Y2] = Y1. The government taxes income at rate τ1 in period 1 and τ2 in period 2. The individual can borrow and lend at a fixed interest rate, which for simplicity is assumed to be zero. Thus second-period consumption is C2 = (1 − τ1)Y1 − C1 + (1 − τ2)Y2. The individualchoosesC1 tomaximizeexpectedlifetimeutility,U(C1)+E[U(C2)].
(a) Find the first-order condition for C1.
(b) Show that E[C2] = C1 if Y2 is not random or if utility is quadratic.
(c) Show that if U ′′′(•) > 0 and Y2 is random, E[C2] > C1.
(d) Suppose that the government marginally lowers τ1 and raises τ2 by the same amount, so that its expected total revenue, τ1Y1 + τ2E[Y2], is un- changed. Implicitly differentiate the first-order condition in part (a) to find an expression for how C1 responds to this change.
(e) Show that C1 is unaffected by this change if Y2 is not random or if utility is quadratic.
(f) Show that C1 increases in response to this change if U ′′′(•) > 0 and Y2 is random.
12.3
Consider the Barro tax-smoothing model. Suppose that output, Y, and the real interest rate, r, are constant, and that the level of government debt out- standing at time 0 is zero. Suppose that there will be a temporary war from time 0 to time τ. Thus G(t) equals GH for 0 ≤ t ≤ τ, and equals GL there- after,whereGH >GL.Whatarethepathsoftaxes,T(t),andgovernmentdebt outstanding, D(t)?
12.4
Consider the Barro tax-smoothing model. Supp.
Scalable inference for a full multivariate stochastic volatilitySYRTO Project
Scalable inference for a full multivariate stochastic volatility
P. Dellaportas, A. Plataniotis and M. Titsias UCL(London), AUEB(Athens), AUEB(Athens)
Final SYRTO Conference - Université Paris1 Panthéon-Sorbonne
February 19, 2016
Slides FIS5.pdfOutline1 Fixed Income DerivativesThe .docxbudabrooks46239
Slides FIS5.pdf
Outline
1 Fixed Income Derivatives
The Forward-Risk Adjusted Measure
2 Example
Dr Lara Cathcart () 2015 2 / 28
The problem
Consider a fixed-income derivative with a single payo↵ at time T which depends
on the term-structure. In particular, we will look at options on zero-coupon
bonds. For a call option on a zero-coupon bond maturing at time T
1
, the time T
payo↵ and hence the value of the derivative is given by
V
T
= max(P(T, T
1
) � K, 0) (1)
Dr Lara Cathcart () 2015 3 / 28
The problem
By the no-arbitrage theorem, the price today (t=0)is
V
0
= EQ
0
[e�
R
T
0
rsds
V
T
] (2)
where the expectation is taken under the risk-neutral distribution (also called the
Q measure). Thus the price depends on the stochastic process for the short rate
and the contractual specification of the security (i.e how the payo↵ is linked to the
term structure).
Dr Lara Cathcart () 2015 4 / 28
The problem
The price V
0
in equation (2) is given by the expectation of the product of two
dependent random variables, and calculating this expectation is often quite
di�cult. The purpose of this note is presenting a change-of measure technique
which considerably simplifies the evaluation of V
0
.
Dr Lara Cathcart () 2015 5 / 28
The problem
Specifically we are going to calculate V
0
as
V
0
= P(0, T)EQ
T
0
(V
T
) (3)
where QT is a new probability measure (distribution), the so-called forward-risk
adjusted measure. This technique was introduced in the fixed-income literature by
Jamishidian (1991).
Dr Lara Cathcart () 2015 6 / 28
Model setup and notation
Our term-structure is a general one-factor HJM model see Heath, Jarrow and
Morton (1992). Under the Q-measure, forwards rates are governed by
df (t, T) = ��(t, T)�
P
(t, T)dt + �(t, T)dW Q
t
(4)
where
�
P
(t, T) = �
Z
T
t
�(t, u)du (5)
Dr Lara Cathcart () 2015 7 / 28
The problem
Bond prices evolve according to the SDE
dP(t, T) = r
t
P(t, T)dt + �
P
(t, T)P(t, T)dW Q
t
(6)
so �
P
(t, T) is the time t volatility of the zero maturing at time T.
Dr Lara Cathcart () 2015 8 / 28
The Forward-Risk Adjusted Measure
The price of derivative security follows the SDE
dV
t
= r
t
V
t
dt + �
V
(t)V
t
dW
Q
t
(7)
This means that, under the risk-neutral distribution, the expected rate of return
equals the short rate (just like any other security), and the return volatility is
�
V
(t). So far neither V
t
nor �
V
(t) are known, but this is not essential for the
following arguments. In fact, the only thing that matters is that the process has
the form (7) since this facilitates pricing by the forward-risk adjusted measure.
Dr Lara Cathcart () 2015 9 / 28
The Forward-Risk Adjusted Measure
We begin by defining the deflated price process
F
t
⌘ V
t
/P(t, T) (8)
for t 2 [0, T]. We can interpret F
t
as the price of V
t
in units of the T-maturity
bond price (i.e., as a relative price).
Dr Lara Cathcart () 2015 10 / 28
The Forward-Ri.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!