Simon Acomb NAG Financial Mathematics Day

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Simon Acomb NAG Financial Mathematics Day"

Transcription

1 1 Why People Who Prce Dervatves Are Interested In Correlaton mon Acomb NAG Fnancal Mathematcs Day

2 Correlaton Rsk What Is Correlaton No lnear relatonshp between ponts Co-movement between the ponts Postve correlaton mon Acomb NAG Fnancal Mathematcs Day

3 Correlaton Rsk tochastcs 3 tandard prcng theory s based on some general stochastc descrpton on the dynamcs of the underlyng asset. Wth two, or more assets dt 1 t d Μ t t d 1 n t n t 1 1 = µ dt + σ dz = µ dt + σ dz n n = µ dt + σ dz dz, dz = ρ dt 1 n Need a correlaton matrx s specfy the dynamcs. mon Acomb NAG Fnancal Mathematcs Day

4 Correlaton Rsk Products Wth Correlaton Exposure 4 1. Quanto Opton N UD JPY 0 JPY 1 + Imagne that the share prce s 100JPY and the delta s 1. In ths case fund of the hedge s done n JPY. If FX rates move then the JPY of the contract moves, but the value of the hedge does not. Weak exposure to the correlaton between FX and Equty. Compostve (Cross) Opton N UD X X JPY UD JPY UD JPY 0 JPY 1 + Lke an opton on a ADR. Every tme we trade delta we fund n UD. Volatlty s the volatlty of measured n UD. trong exposure to FX and Equty correlaton. mon Acomb NAG Fnancal Mathematcs Day

5 mon Acomb NAG Fnancal Mathematcs Day Correlaton Rsk Products Wth Correlaton Exposure 1. pread Opton. Best-of, Worst-Of optons 3. Basket Optons 4. Hmalayan Optons 5. Ranbow Optons 5 + K K x Ma 0 + K Mn 0 + K w 0 + K Max w t 0 remanng + + K Mn w Max w 0 0 1

6 Correlaton Rsk Basket Optons 6 Very commonly traded Can be traded wth the OC market together wth short postons n optons on each of the underlyngs a correlaton product Approxmatons for basket varance σ basket, w w ρ σ σ (Note ths s only an approxmaton as assets are lognormally dstrbuted. Increasng the correlaton ncreases the basket volatlty and makes the optons more expensve Often sold on dverse set of underlyngs to make the opton cheaper More underlyngs n the mx makes the opton cheaper. mon Acomb NAG Fnancal Mathematcs Day

7 Correlaton Rsk Best-Of / Worst-Of 7 General dea s that when the correlaton s low (negatve) there s a more dverse range of outcomes, hence there wll always be one asset whch has outperformed and one that has under performed. (Consder the case of perfect negatve correlaton.) Followng exposure of product to a rse n correlaton Call Put Best-Of - + Worst-Of + - Frequently components of structured products Worst of Call (embedded n a guaranteed product) correlaton used to make the product cheaper hort Worst of Put (embedded n reverse convertble) correlaton used to ncrease the coupon mon Acomb NAG Fnancal Mathematcs Day

8 Correlaton Rsk Hmalayan / Ranbow 8 Hmalayan s lke a combnaton of a basket optons and an asan opton - averagng over both tme and asset. Best assets are fxed early and lose ther tme value. Becomes lke a call on the worst postve exposure to correlaton Ranbow s an nterestng product. Lke a basket ncreasng correlaton, ncreases the value of the product ypcally the product s set up wth the hghest weght appled to the best performng asset, and so has some features of a call on the best ncreasng correlaton, decreases the value of the product Overall product has small correlaton exposure, but t can be ether postve, or negatve. mon Acomb NAG Fnancal Mathematcs Day

9 Correlaton Rsk ypcally Investment Bank Exposure 9 ypcal products that nvestment banks sell. Reverse Convertble on the worst Basket optons Hmalayan / Ranbow Calls on worst All leave the seller short correlaton. Dffcult to manufacture a product whch has correlaton exposure n the other drecton. mon Acomb NAG Fnancal Mathematcs Day

10 Correlaton Rsk Measurng Correlaton 10 Frst attempt would be to use the tme seres of two underlyngs and measure the correlaton of the tme seres. Rollng wndow of 6 months daly data Nkke 5 and &P 500 Note enormous range. For ρ=0. and 6 months daly data statstc confdence nterval s [0.0, 0.36]. Are we pckng up samplng error, or uncertanty n correlaton Correlaton s low due to asynchronous effect. mon Acomb NAG Fnancal Mathematcs Day

11 Correlaton Rsk Measurng Correlaton (Weekly Data) 11 Usng a 1 year rollng wndow Impled correlaton taken from optons markets close to Just as volatlty trades at a premum to hstorc, so does correlaton. Need a mechansm that assess ths premum. mon Acomb NAG Fnancal Mathematcs Day

12 Correlaton Rsk Why Do We Need a Postve em Defnte Matrces 1 Negatve Defnte matrx mples that there are portfolos wth negatve varance. (In portfolo theory can we get away wth ths f we nsst that all assets have postve weght?) Monte Carlo Path Generaton Gven uncorrelated random numbers How do we construct random numbers whch have the correlaton matrx W Z Ρ Fnd pseudo-square root so that and Q R = QQ Algorthms for Q requre R to be postve sem-defnte. Z = k q k W k mon Acomb NAG Fnancal Mathematcs Day

13 Correlaton Rsk Why Do We Get Negatve Defnte Matrces 13 One would magne that f we use tme seres over the same perod and estmated from ths the correlaton matrx would be postve defnte by defnton. Data s not synchronous. Dfferent regons have dfferent holdays. What do you do wth the stock that has only ust been ssued. In realty correlaton matrces suffer from ths problem. For prcng purposes would prefer to use correlatons mpled from opton contracts that I can see. For example baskets and dspersons. Most lkely gong to be mxng hstorc and mpled estmates. Many people want to know what happens when correlaton goes up by 10%. mon Acomb NAG Fnancal Mathematcs Day

14 Correlaton Rsk Correlaton Premum 14 Useful transform ρ ρ + α( 1 ρ) If covarance matrx s postve sem-defnte then ths transform wll mantan ths property Gves calbraton method for correlaton. 1.Estmate correlatons by the best hstorcal method you have avalable.from the small number of observatons that you can observe estmate the correlaton premum 3.Apply ths correlaton premum to all hstorc correlatons that you use. mon Acomb NAG Fnancal Mathematcs Day

15 Correlaton Rsk Correlaton mle 15 Consder an Index such as the Eurotoxx made up of 50 underlyng assets hen to reasonable approxmaton σ I =, w w ρ σ σ I = w If volatlty s a functon of (percentage) strke then ρmust also be a functon of ths strke. I = σ ( K) w w ρ ( K) σ ( K) σ ( K), We know that ndex volatlty smles are steeper than sngle stock volatlty smles so not surprsngly we fnd that correlaton s hgher for low strkes than for hgher strkes mon Acomb NAG Fnancal Mathematcs Day

16 Correlaton Rsk Impled Correlaton 16 mon Acomb NAG Fnancal Mathematcs Day

17 Correlaton Rsk Explanaton of Correlaton mle We know that dstrbutons of assets are not lognormal. o make more sense of the nformaton we should be usng the mpled dstrbutons of each stock.. Gven the dstrbutons of each asset does not determne the ont dstrbuton of two assets. here s a extensve theory of ths called copula. 3. If we used local volatlty as a process we would get very dfferent results. Correlaton smle means many dfferent thngs to dfferent people. If tradng correlaton the most obectve way of dong so wll be wth varance swaps. mon Acomb NAG Fnancal Mathematcs Day

18 Correlaton Rsk Products Exposed to Correlaton mle 18 Dsperson of deep out of the money optons Far out of the money Worst of optons Worst of dgtals Way out of the money altplano products Worst of equty default products mon Acomb NAG Fnancal Mathematcs Day

19 Correlaton Rsk Measurng Explct Correlaton Rsk 19 Common to measure correlaton rsk as ρ ρ +10% What about correlatons at 0.95 What happens f the correlaton matrx s no longer postve defnte Use the α rsk methodology ρ ρ + α( 1 ρ) α = 10% ρ = 90% 91% ρ = 0% 8% maller correlatons move the most (n lne of emprcal observatons) Mantans postve defnteness. Make sure that correlaton scenaro s suffcently large to capture any nonlnearty n correlaton. mon Acomb NAG Fnancal Mathematcs Day

20 Correlaton Rsk Hdden Correlaton Rsk 0 Correlaton exposure wthn dervatves exposure s by no-means unque. Portfolo managers have been dealng wth correlaton exposure for a long tme Common to measure market exposure by movng all assets smultaneously. All the same amount caled by a beta to the market If portfolo only had two assets whch where negatvely correlated then ths rsk measure would be n-approprate Portfolo rsk systems such as Barra break down rsk exposure nto other factors. Named factors related to real varables Prncpal components If you have the break down of delta by asset, worthwhle thnkng about whether portfolo manager technques can help better understand the correlaton of a portfolo whether t has dervatves n t, or not. mon Acomb NAG Fnancal Mathematcs Day

21 Correlaton Rsk Hedgng Correlaton Exposure 1 Most retal structured products leave the seller short correlaton. Reverse Clquet on a Basket + B t Max X 1, 0 t Bt 1 Reverse Convertble on a Basket B Mn B 0,1 mall number of products whch leave the seller long correlaton. mon Acomb NAG Fnancal Mathematcs Day

22 Correlaton Rsk Hedgng Correlaton Rsk - Dsperson Broker market traders dspersons n vanlla optons Max w 1 + w K, 0 w1 Max K,0 wmax, K 1 0 Can go long or short and hence long, or short correlaton. On ntaton product has lttle ndvdual vega, but has exposure to correlaton. When share prces move ths wll no longer be the case. Just as the vega of a vanlla opton dsspates when the opton move away from AM, so the correlaton exposure of a basket opton dsspates as you move away from AM. hese types of trades are popular wth ndces. mon Acomb NAG Fnancal Mathematcs Day

23 Correlaton Rsk Hedgng Correlaton Exposure Varance waps 3 Varance waps provde a smple way of tradng varance wthout an explct strke dependency. Can trade a dsperson of varance swaps Long varance swap on the Eurotoxx 50 hort varance swaps on each of the ndvdual consstuents. Can go long or short. Possble on the ndces wth a small number of assets, but can be approxmated on ndces such as the &P wth trackng portfolos As these are strke-less the correlaton exposure does not dsspate as the underlyng assets move. mon Acomb NAG Fnancal Mathematcs Day

24 Correlaton Rsk Correlaton waps 4 A correlaton swap s an OC product whch has a payoff gven by P = N( N 1) < ρ Gves a drect way of tradng correlaton. Dffcult to hedge product Does t really gve the exposure you requre. If you ms-estmate correlaton between two Geometrc Brownan motons P&L gven by terms ncludng E 0 f 1 1σ 1σ ( ˆ ρ ρ ) dt Covarance swaps wll be easer to prce. mon Acomb NAG Fnancal Mathematcs Day

25 Correlaton Rsk Return to Q-Q Maps 5 Is assumpton correlaton constant correct. hould we be usng a copula based theory. mon Acomb NAG Fnancal Mathematcs Day

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

More information

Portfolio Loss Distribution

Portfolio Loss Distribution Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets hold-to-maturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment

More information

9.1 The Cumulative Sum Control Chart

9.1 The Cumulative Sum Control Chart Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s

More information

Capital asset pricing model, arbitrage pricing theory and portfolio management

Capital asset pricing model, arbitrage pricing theory and portfolio management Captal asset prcng model, arbtrage prcng theory and portfolo management Vnod Kothar The captal asset prcng model (CAPM) s great n terms of ts understandng of rsk decomposton of rsk nto securty-specfc rsk

More information

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ). REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

The Short-term and Long-term Market

The Short-term and Long-term Market A Presentaton on Market Effcences to Northfeld Informaton Servces Annual Conference he Short-term and Long-term Market Effcences en Post Offce Square Boston, MA 0209 www.acadan-asset.com Charles H. Wang,

More information

ErrorPropagation.nb 1. Error Propagation

ErrorPropagation.nb 1. Error Propagation ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then

More information

Describing Communities. Species Diversity Concepts. Species Richness. Species Richness. Species-Area Curve. Species-Area Curve

Describing Communities. Species Diversity Concepts. Species Richness. Species Richness. Species-Area Curve. Species-Area Curve peces versty Concepts peces Rchness peces-area Curves versty Indces - mpson's Index - hannon-wener Index - rlloun Index peces Abundance Models escrbng Communtes There are two mportant descrptors of a communty:

More information

Hedging Interest-Rate Risk with Duration

Hedging Interest-Rate Risk with Duration FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton

More information

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001. Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.

More information

Forecasting the Direction and Strength of Stock Market Movement

Forecasting the Direction and Strength of Stock Market Movement Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems

More information

1. Measuring association using correlation and regression

1. Measuring association using correlation and regression How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a

More information

Pricing Multi-Asset Cross Currency Options

Pricing Multi-Asset Cross Currency Options CIRJE-F-844 Prcng Mult-Asset Cross Currency Optons Kenchro Shraya Graduate School of Economcs, Unversty of Tokyo Akhko Takahash Unversty of Tokyo March 212; Revsed n September, October and November 212

More information

IN THE UNITED STATES THIS REPORT IS AVAILABLE ONLY TO PERSONS WHO HAVE RECEIVED THE PROPER OPTION RISK DISCLOSURE DOCUMENTS.

IN THE UNITED STATES THIS REPORT IS AVAILABLE ONLY TO PERSONS WHO HAVE RECEIVED THE PROPER OPTION RISK DISCLOSURE DOCUMENTS. http://mm.pmorgan.com European Equty Dervatves Strategy 4 May 005 N THE UNTED STATES THS REPORT S AVALABLE ONLY TO PERSONS WHO HAVE RECEVED THE PROPER OPTON RS DSCLOSURE DOCUMENTS. Correlaton Vehcles Technques

More information

Application of Quasi Monte Carlo methods and Global Sensitivity Analysis in finance

Application of Quasi Monte Carlo methods and Global Sensitivity Analysis in finance Applcaton of Quas Monte Carlo methods and Global Senstvty Analyss n fnance Serge Kucherenko, Nlay Shah Imperal College London, UK skucherenko@mperalacuk Daro Czraky Barclays Captal DaroCzraky@barclayscaptalcom

More information

Nasdaq Iceland Bond Indices 01 April 2015

Nasdaq Iceland Bond Indices 01 April 2015 Nasdaq Iceland Bond Indces 01 Aprl 2015 -Fxed duraton Indces Introducton Nasdaq Iceland (the Exchange) began calculatng ts current bond ndces n the begnnng of 2005. They were a response to recent changes

More information

An Alternative Way to Measure Private Equity Performance

An Alternative Way to Measure Private Equity Performance An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

More information

The covariance is the two variable analog to the variance. The formula for the covariance between two variables is

The covariance is the two variable analog to the variance. The formula for the covariance between two variables is Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables.

More information

Pragmatic Insurance Option Pricing

Pragmatic Insurance Option Pricing Paper to be presented at the XXXVth ASTIN Colloquum, Bergen, 6 9th June 004 Pragmatc Insurance Opton Prcng by Jon Holtan If P&C Insurance Company Ltd Oslo, Norway Emal: jon.holtan@f.no Telephone: +47960065

More information

NPAR TESTS. One-Sample Chi-Square Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6

NPAR TESTS. One-Sample Chi-Square Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6 PAR TESTS If a WEIGHT varable s specfed, t s used to replcate a case as many tmes as ndcated by the weght value rounded to the nearest nteger. If the workspace requrements are exceeded and samplng has

More information

Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio

Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio Vascek s Model of Dstrbuton of Losses n a Large, Homogeneous Portfolo Stephen M Schaefer London Busness School Credt Rsk Electve Summer 2012 Vascek s Model Important method for calculatng dstrbuton of

More information

A Model of Private Equity Fund Compensation

A Model of Private Equity Fund Compensation A Model of Prvate Equty Fund Compensaton Wonho Wlson Cho Andrew Metrck Ayako Yasuda KAIST Yale School of Management Unversty of Calforna at Davs June 26, 2011 Abstract: Ths paper analyzes the economcs

More information

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60 BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true

More information

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo

More information

ECONOMICS OF PLANT ENERGY SAVINGS PROJECTS IN A CHANGING MARKET Douglas C White Emerson Process Management

ECONOMICS OF PLANT ENERGY SAVINGS PROJECTS IN A CHANGING MARKET Douglas C White Emerson Process Management ECONOMICS OF PLANT ENERGY SAVINGS PROJECTS IN A CHANGING MARKET Douglas C Whte Emerson Process Management Abstract Energy prces have exhbted sgnfcant volatlty n recent years. For example, natural gas prces

More information

Descriptive Statistics (60 points)

Descriptive Statistics (60 points) Economcs 30330: Statstcs for Economcs Problem Set 2 Unversty of otre Dame Instructor: Julo Garín Sprng 2012 Descrptve Statstcs (60 ponts) 1. Followng a recent government shutdown, Mnnesota Governor Mark

More information

Chapter 4 Financial Markets

Chapter 4 Financial Markets Chapter 4 Fnancal Markets ECON2123 (Sprng 2012) 14 & 15.3.2012 (Tutoral 5) The demand for money Assumptons: There are only two assets n the fnancal market: money and bonds Prce s fxed and s gven, that

More information

Time Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University

Time Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton

More information

The VIX Volatility Index

The VIX Volatility Index U.U.D.M. Project Report :7 he VIX Volatlty Index Mao Xn Examensarbete matematk, 3 hp Handledare och examnator: Macej lmek Maj Department of Mathematcs Uppsala Unversty Abstract. VIX plays a very mportant

More information

The Analysis of Outliers in Statistical Data

The Analysis of Outliers in Statistical Data THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate

More information

Stress test for measuring insurance risks in non-life insurance

Stress test for measuring insurance risks in non-life insurance PROMEMORIA Datum June 01 Fnansnspektonen Författare Bengt von Bahr, Younes Elonq and Erk Elvers Stress test for measurng nsurance rsks n non-lfe nsurance Summary Ths memo descrbes stress testng of nsurance

More information

The impact of hard discount control mechanism on the discount volatility of UK closed-end funds

The impact of hard discount control mechanism on the discount volatility of UK closed-end funds Investment Management and Fnancal Innovatons, Volume 10, Issue 3, 2013 Ahmed F. Salhn (Egypt) The mpact of hard dscount control mechansm on the dscount volatlty of UK closed-end funds Abstract The mpact

More information

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered

More information

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt. Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces

More information

The Analysis of Covariance. ERSH 8310 Keppel and Wickens Chapter 15

The Analysis of Covariance. ERSH 8310 Keppel and Wickens Chapter 15 The Analyss of Covarance ERSH 830 Keppel and Wckens Chapter 5 Today s Class Intal Consderatons Covarance and Lnear Regresson The Lnear Regresson Equaton TheAnalyss of Covarance Assumptons Underlyng the

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

Efficient Project Portfolio as a tool for Enterprise Risk Management

Efficient Project Portfolio as a tool for Enterprise Risk Management Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse

More information

Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc.

Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc. Underwrtng Rsk By Glenn Meyers Insurance Servces Offce, Inc. Abstract In a compettve nsurance market, nsurers have lmted nfluence on the premum charged for an nsurance contract. hey must decde whether

More information

Bond futures. Bond futures contracts are futures contracts that allow investor to buy in the

Bond futures. Bond futures contracts are futures contracts that allow investor to buy in the Bond futures INRODUCION Bond futures contracts are futures contracts that allow nvestor to buy n the future a theoretcal government notonal bond at a gven prce at a specfc date n a gven quantty. Compared

More information

Pricing index options in a multivariate Black & Scholes model

Pricing index options in a multivariate Black & Scholes model Prcng ndex optons n a multvarate Black & Scholes model Danël Lnders AFI_1383 Prcng ndex optons n a multvarate Black & Scholes model Danël Lnders Verson: October 2, 2013 1 Introducton In ths paper, we consder

More information

Multivariate EWMA Control Chart

Multivariate EWMA Control Chart Multvarate EWMA Control Chart Summary The Multvarate EWMA Control Chart procedure creates control charts for two or more numerc varables. Examnng the varables n a multvarate sense s extremely mportant

More information

Course outline. Financial Time Series Analysis. Overview. Data analysis. Predictive signal. Trading strategy

Course outline. Financial Time Series Analysis. Overview. Data analysis. Predictive signal. Trading strategy Fnancal Tme Seres Analyss Patrck McSharry patrck@mcsharry.net www.mcsharry.net Trnty Term 2014 Mathematcal Insttute Unversty of Oxford Course outlne 1. Data analyss, probablty, correlatons, vsualsaton

More information

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression. Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook

More information

Scale Dependence of Overconfidence in Stock Market Volatility Forecasts

Scale Dependence of Overconfidence in Stock Market Volatility Forecasts Scale Dependence of Overconfdence n Stoc Maret Volatlty Forecasts Marus Glaser, Thomas Langer, Jens Reynders, Martn Weber* June 7, 007 Abstract In ths study, we analyze whether volatlty forecasts (judgmental

More information

Portfolio Risk Decomposition (and Risk Budgeting)

Portfolio Risk Decomposition (and Risk Budgeting) ortfolo Rsk Decomposton (and Rsk Budgetng) Jason MacQueen R-Squared Rsk Management Introducton to Rsk Decomposton Actve managers take rsk n the expectaton of achevng outperformance of ther benchmark Mandates

More information

SIMPLE LINEAR CORRELATION

SIMPLE LINEAR CORRELATION SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.

More information

Comparing Class Level Chain Drift for Different Elementary Aggregate Formulae Using Locally Collected CPI Data

Comparing Class Level Chain Drift for Different Elementary Aggregate Formulae Using Locally Collected CPI Data Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng Gareth Clews 1, Anselma Dobson-McKttrck 2 and Joseph Wnton Summary The Consumer Prces Index (CPI) s a measure of consumer

More information

CHAPTER 14 MORE ABOUT REGRESSION

CHAPTER 14 MORE ABOUT REGRESSION CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp

More information

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL

More information

HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION

HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION Abdul Ghapor Hussn Centre for Foundaton Studes n Scence Unversty of Malaya 563 KUALA LUMPUR E-mal: ghapor@umedumy Abstract Ths paper

More information

THE IMPLIED VOLATILITY OF ETF AND INDEX OPTIONS

THE IMPLIED VOLATILITY OF ETF AND INDEX OPTIONS The Internatonal Journal of Busness and Fnance Research Volume 5 Number 4 2011 THE IMPLIED VOLATILITY OF ETF AND INDEX OPTIONS Stoyu I. Ivanov, San Jose State Unversty Jeff Whtworth, Unversty of Houston-Clear

More information

Risk Management and Financial Institutions

Risk Management and Financial Institutions Rsk Management and Fnancal Insttutons By John C. Hull Chapter 3 How Traders manage Ther Exposures... Chapter 4 Interest Rate Rsk...3 Chapter 5 Volatlty...5 Chapter 6 Correlatons and Copulas...7 Chapter

More information

Copulas. Modeling dependencies in Financial Risk Management. BMI Master Thesis

Copulas. Modeling dependencies in Financial Risk Management. BMI Master Thesis Copulas Modelng dependences n Fnancal Rsk Management BMI Master Thess Modelng dependences n fnancal rsk management Modelng dependences n fnancal rsk management 3 Preface Ths paper has been wrtten as part

More information

CS 2750 Machine Learning. Lecture 3. Density estimation. CS 2750 Machine Learning. Announcements

CS 2750 Machine Learning. Lecture 3. Density estimation. CS 2750 Machine Learning. Announcements Lecture 3 Densty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 5329 Sennott Square Next lecture: Matlab tutoral Announcements Rules for attendng the class: Regstered for credt Regstered for audt (only f there

More information

Credit Limit Optimization (CLO) for Credit Cards

Credit Limit Optimization (CLO) for Credit Cards Credt Lmt Optmzaton (CLO) for Credt Cards Vay S. Desa CSCC IX, Ednburgh September 8, 2005 Copyrght 2003, SAS Insttute Inc. All rghts reserved. SAS Propretary Agenda Background Tradtonal approaches to credt

More information

Stock Profit Patterns

Stock Profit Patterns Stock Proft Patterns Suppose a share of Farsta Shppng stock n January 004 s prce n the market to 56. Assume that a September call opton at exercse prce 50 costs 8. A September put opton at exercse prce

More information

Exchange rate volatility and its impact on risk management with internal models in commercial banks

Exchange rate volatility and its impact on risk management with internal models in commercial banks Banks and Bank Systems, Volume, Issue 4, 007 Devjak Sreko (Slovena), Andraž Grum (Slovena) Exchange rate volatlty and ts mpact on rsk management wth nternal models n commercal banks Abstract Fnancal markets

More information

Acta Universitatis Carolinae. Mathematica et Physica

Acta Universitatis Carolinae. Mathematica et Physica Acta Unverstats Carolnae. Mathematca et Physca L. Jarešová: EWMA hstorcal volatlty estmators Acta Unverstats Carolnae. Mathematca et Physca, Vol. 51 (2010), No. 2, 17--28 Persstent URL: http://dml.cz/dmlcz/143654

More information

HÜCKEL MOLECULAR ORBITAL THEORY

HÜCKEL MOLECULAR ORBITAL THEORY 1 HÜCKEL MOLECULAR ORBITAL THEORY In general, the vast maorty polyatomc molecules can be thought of as consstng of a collecton of two electron bonds between pars of atoms. So the qualtatve pcture of σ

More information

An Analysis of Pricing Methods for Baskets Options

An Analysis of Pricing Methods for Baskets Options An Analyss of Prcng Methods for Baskets Optons Martn Krekel, Johan de Kock, Ralf Korn, Tn-Kwa Man Fraunhofer ITWM, Department of Fnancal Mathematcs, 67653 Kaserslautern, Germany, emal: krekel@twm.fhg.de

More information

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

More information

What is Candidate Sampling

What is Candidate Sampling What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble

More information

Financial Mathemetics

Financial Mathemetics Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,

More information

Accurate asset price modeling and related statistical problems under microstructure noise

Accurate asset price modeling and related statistical problems under microstructure noise Accurate asset prce modelng and related statstcal problems under mcrostructure nose José E. Fgueroa-López 1 1 Department of Statstcs Purdue Unversty Explorng Statstcal Scence Research Semnar December 8,

More information

The Application of Fractional Brownian Motion in Option Pricing

The Application of Fractional Brownian Motion in Option Pricing Vol. 0, No. (05), pp. 73-8 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qng-xn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

Using Series to Analyze Financial Situations: Present Value

Using Series to Analyze Financial Situations: Present Value 2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

More information

Basel Committee on Banking Supervision

Basel Committee on Banking Supervision Basel Commttee on Banng Supervson The standardsed approach for measurng counterparty credt rs exposures March 014 (rev. Aprl 014) Ths publcaton s avalable on the BIS webste (www.bs.org). Ban for Internatonal

More information

Introduction: Analysis of Electronic Circuits

Introduction: Analysis of Electronic Circuits /30/008 ntroducton / ntroducton: Analyss of Electronc Crcuts Readng Assgnment: KVL and KCL text from EECS Just lke EECS, the majorty of problems (hw and exam) n EECS 3 wll be crcut analyss problems. Thus,

More information

CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES

CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES In ths chapter, we wll learn how to descrbe the relatonshp between two quanttatve varables. Remember (from Chapter 2) that the terms quanttatve varable

More information

An asymptotic FX option formula in the cross currency Libor market model

An asymptotic FX option formula in the cross currency Libor market model An asymptotc X opton formula n the cross currency Lbor maret model Atsush Kawa Peter Jäcel rst verson: 25th October 26 Ths verson: 3rd ebruary 27 Abstract In ths artcle, we ntroduce analytc approxmaton

More information

Lecture 2: Single Layer Perceptrons Kevin Swingler

Lecture 2: Single Layer Perceptrons Kevin Swingler Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCulloch-Ptts Neuron Ths vastly smplfed model of real neurons s also knon as a Threshold Logc Unt: W 2 A Y 3 n W n. A set of synapses

More information

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy 4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

I. SCOPE, APPLICABILITY AND PARAMETERS Scope

I. SCOPE, APPLICABILITY AND PARAMETERS Scope D Executve Board Annex 9 Page A/R ethodologcal Tool alculaton of the number of sample plots for measurements wthn A/R D project actvtes (Verson 0) I. SOPE, PIABIITY AD PARAETERS Scope. Ths tool s applcable

More information

arxiv:1109.1256v1 [q-fin.pm] 6 Sep 2011

arxiv:1109.1256v1 [q-fin.pm] 6 Sep 2011 WORKING PAPER December 2010 Fnancal Analysts Journal Volume 67, No. 4 July/August 2011, p. 42-49 arxv:1109.1256v1 [q-fn.pm] 6 Sep 2011 Dversfcaton Return, Portfolo Rebalancng, and the Commodty Return Puzzle

More information

Optimal Bidding Strategies for Generation Companies in a Day-Ahead Electricity Market with Risk Management Taken into Account

Optimal Bidding Strategies for Generation Companies in a Day-Ahead Electricity Market with Risk Management Taken into Account Amercan J. of Engneerng and Appled Scences (): 8-6, 009 ISSN 94-700 009 Scence Publcatons Optmal Bddng Strateges for Generaton Companes n a Day-Ahead Electrcty Market wth Rsk Management Taken nto Account

More information

The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets

The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets . The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely

More information

ADVA FINAN QUAN ADVANCED FINANCE AND QUANTITATIVE INTERVIEWS VAULT GUIDE TO. Customized for: Jason (jason.barquero@cgu.edu) 2002 Vault Inc.

ADVA FINAN QUAN ADVANCED FINANCE AND QUANTITATIVE INTERVIEWS VAULT GUIDE TO. Customized for: Jason (jason.barquero@cgu.edu) 2002 Vault Inc. ADVA FINAN QUAN 00 Vault Inc. VAULT GUIDE TO ADVANCED FINANCE AND QUANTITATIVE INTERVIEWS Copyrght 00 by Vault Inc. All rghts reserved. All nformaton n ths book s subject to change wthout notce. Vault

More information

Questions that we may have about the variables

Questions that we may have about the variables Antono Olmos, 01 Multple Regresson Problem: we want to determne the effect of Desre for control, Famly support, Number of frends, and Score on the BDI test on Perceved Support of Latno women. Dependent

More information

Traffic State Estimation in the Traffic Management Center of Berlin

Traffic State Estimation in the Traffic Management Center of Berlin Traffc State Estmaton n the Traffc Management Center of Berln Authors: Peter Vortsch, PTV AG, Stumpfstrasse, D-763 Karlsruhe, Germany phone ++49/72/965/35, emal peter.vortsch@ptv.de Peter Möhl, PTV AG,

More information

Outline. Investment Opportunity Set with Many Assets. Portfolio Selection with Multiple Risky Securities. Professor Lasse H.

Outline. Investment Opportunity Set with Many Assets. Portfolio Selection with Multiple Risky Securities. Professor Lasse H. Portfolo Selecton wth Multple Rsky Securtes. Professor Lasse H. Pedersen Prof. Lasse H. Pedersen Outlne Investment opportunty set wth many rsky assets wth many rsky assets and a rsk-free securty Optmal

More information

Exhaustive Regression. An Exploration of Regression-Based Data Mining Techniques Using Super Computation

Exhaustive Regression. An Exploration of Regression-Based Data Mining Techniques Using Super Computation Exhaustve Regresson An Exploraton of Regresson-Based Data Mnng Technques Usng Super Computaton Antony Daves, Ph.D. Assocate Professor of Economcs Duquesne Unversty Pttsburgh, PA 58 Research Fellow The

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

More information

Lecture 10: Linear Regression Approach, Assumptions and Diagnostics

Lecture 10: Linear Regression Approach, Assumptions and Diagnostics Approach to Modelng I Lecture 1: Lnear Regresson Approach, Assumptons and Dagnostcs Sandy Eckel seckel@jhsph.edu 8 May 8 General approach for most statstcal modelng: Defne the populaton of nterest State

More information

Linear Regression, Regularization Bias-Variance Tradeoff

Linear Regression, Regularization Bias-Variance Tradeoff HTF: Ch3, 7 B: Ch3 Lnear Regresson, Regularzaton Bas-Varance Tradeoff Thanks to C Guestrn, T Detterch, R Parr, N Ray 1 Outlne Lnear Regresson MLE = Least Squares! Bass functons Evaluatng Predctors Tranng

More information

7.5. Present Value of an Annuity. Investigate

7.5. Present Value of an Annuity. Investigate 7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on

More information

Market Risk Evaluation using Monte Carlo Simulation

Market Risk Evaluation using Monte Carlo Simulation 1 Market Rsk Evaluaton usng Monte Carlo mulaton Methodology and Features Dr. Anatoly Antonov 1. Introducton Market Rsk nvolves the uncertanty of future earnngs resultng from changes of varous ndependent

More information

L10: Linear discriminants analysis

L10: Linear discriminants analysis L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss

More information

Chapter 7. Random-Variate Generation 7.1. Prof. Dr. Mesut Güneş Ch. 7 Random-Variate Generation

Chapter 7. Random-Variate Generation 7.1. Prof. Dr. Mesut Güneş Ch. 7 Random-Variate Generation Chapter 7 Random-Varate Generaton 7. Contents Inverse-transform Technque Acceptance-Rejecton Technque Specal Propertes 7. Purpose & Overvew Develop understandng of generatng samples from a specfed dstrbuton

More information

Graph Theory and Cayley s Formula

Graph Theory and Cayley s Formula Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll

More information

Akira Yanagisawa Leader Energy Demand, Supply and Forecast Analysis Group Energy Data and Modelling Center

Akira Yanagisawa Leader Energy Demand, Supply and Forecast Analysis Group Energy Data and Modelling Center Background of Surgng Ol Prces and Market Expectaton Seen n Optons Akra Yanagsawa Leader Energy Demand, Supply and Forecast Analyss Group Energy Data and Modellng Center Summary The crude ol prces (WTI

More information

Rate Monotonic (RM) Disadvantages of cyclic. TDDB47 Real Time Systems. Lecture 2: RM & EDF. Priority-based scheduling. States of a process

Rate Monotonic (RM) Disadvantages of cyclic. TDDB47 Real Time Systems. Lecture 2: RM & EDF. Priority-based scheduling. States of a process Dsadvantages of cyclc TDDB47 Real Tme Systems Manual scheduler constructon Cannot deal wth any runtme changes What happens f we add a task to the set? Real-Tme Systems Laboratory Department of Computer

More information

Analysis of Premium Liabilities for Australian Lines of Business

Analysis of Premium Liabilities for Australian Lines of Business Summary of Analyss of Premum Labltes for Australan Lnes of Busness Emly Tao Honours Research Paper, The Unversty of Melbourne Emly Tao Acknowledgements I am grateful to the Australan Prudental Regulaton

More information

CS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering

CS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering Lecture 7a Clusterng Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Clusterng Groups together smlar nstances n the data sample Basc clusterng problem: dstrbute data nto k dfferent groups such that

More information

MEASURING HISTORICAL VOLATILITY Close-to-Close, Exponentially Weighted, Parkinson, Garman-Klass, Rogers-Satchell and Yang-Zhang Volatility

MEASURING HISTORICAL VOLATILITY Close-to-Close, Exponentially Weighted, Parkinson, Garman-Klass, Rogers-Satchell and Yang-Zhang Volatility Equty Dervatves Europe Madrd, February 3, 01 MEASURIG HISTORICAL VOLATILITY Close-to-Close, Exponentally Weghted, Parknson, Garman-Klass, Rogers-Satchell and Yang-Zhang Volatlty Coln Bennett US nvestors

More information

Thinking about Newton's Laws

Thinking about Newton's Laws Newtonan modellng In ths actvty you wll see how Newton s Laws of Moton are used to connect the moton of an object wth the forces actng on t. You wll practse applyng Newton s three laws n some real contexts.

More information

Risk-based Fatigue Estimate of Deep Water Risers -- Course Project for EM388F: Fracture Mechanics, Spring 2008

Risk-based Fatigue Estimate of Deep Water Risers -- Course Project for EM388F: Fracture Mechanics, Spring 2008 Rsk-based Fatgue Estmate of Deep Water Rsers -- Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn

More information

Linear Regression Analysis for STARDEX

Linear Regression Analysis for STARDEX Lnear Regresson Analss for STARDEX Malcolm Halock, Clmatc Research Unt The followng document s an overvew of lnear regresson methods for reference b members of STARDEX. Whle t ams to cover the most common

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 5.4 Annuities, Present Value, and Amortization Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

More information

An Overview of Financial Mathematics

An Overview of Financial Mathematics An Overvew of Fnancal Mathematcs Wllam Benedct McCartney July 2012 Abstract Ths document s meant to be a quck ntroducton to nterest theory. It s wrtten specfcally for actuaral students preparng to take

More information