Genetics and/of basket options
|
|
- Juniper Parker
- 6 years ago
- Views:
Transcription
1 Genetics and/of basket options Wolfgang Karl Härdle Elena Silyakova Ladislaus von Bortkiewicz Chair of Statistics Humboldt-Universität zu Berlin
2 Motivation 1-1 Basket derivatives Let us consider a basket of N assets with value at time t defined by B(t) = N i=1 a is i (t). Then payoffs of some basket options: Basket call: {B(T ) K B } [ + ] + Rainbow (best-of-n): max {S i(t )} K B 1 i N Atlas (Mountain range): + N N 1 2 S j (T ) N (N 1 + N 2 ) S j (0) K B j=1+n 1 where S i (t) - price of the i-th basket constituent at time t, a i - quantity of the i-th asset, K B - exercise price (strike) of a basket option, T - time of the option s expiry, N 1,N 2 - number of best and worst performing stocks.
3 Motivation 1-2 Research questions 1. Which pricing model is suitable for multiasset options? 2. How to estimate dependence (correlation) between assets in the basket? 3. How to estimate correlations in large dimensional baskets?
4 Outline 1. Motivation 2. Basket dynamics in the Black-Scholes framework 3. Estimating correlation matrix Historical (time series) correlation Implied correlation 4. From equicorrelation to block correlation 5. Conclusion
5 Basket dynamics in the Black-Scholes framework 2-1 Price dynamics of basket constituents The price dynamic of the i-th stock in a basket is given by: ds i (t) S i (t) = (r q i)dt + σ i dw i (t) (1) ρ ij dt = dw i (t)dw j (t) (2) where r - interest rate, q i - dividend yield of a stock i, σ i - constant volatility of the i-th stock, ρ ij - constant correlation between the i-th and the j-th stock, W - Brownian motion.
6 Basket dynamics in the Black-Scholes framework 2-2 Dynamics of the basket s value The dynamics of the basket s value is then given by: db(t) N B(t) = (r q i=1 B)dt + w is i (t)σ i dw i (t) N i=1 w = (3) is i (t) = (r q B )dt + dz(t) where q B is the dividend yield of the basket and the relative weight w i of the i-th constituent varies over time and is given by: w i = a i S i (t) N l=1 a ls l (t) (4)
7 Basket dynamics in the Black-Scholes framework 2-3 Dynamics of correlated basket constituents Let ρ 11 ρ 1N Σ =..... ρ N1 ρ NN the correlation matrix of a basket. By Cholesky decomposition Σ = MM we obtain M = (m i,j ) 1 i N,1 j N, a lower triangular matrix, a square root of Σ. The process for every individual asset S i is then defined by: ds i (t) S i (t) = (r q)dt + σ i N m i,l dw l (t) (5) l=1
8 Basket dynamics in the Black-Scholes framework 2-4 Finally applying Itô s lemma we obtain the closed-form expression for simulation of the i-th stock process on a time interval t = [t 1, t 2 ]: S i (t 2 ) = S i (t 1 ) exp { (r d σ2 i 2 ) t + σ i } N m i,l tgl l=1 (6) where g l N(0, 1), i.i.d.
9 Estimating correlation matrix: historical (time series) correlation 3-1 Historical correlation X i (t) = log S i (t) log S i (t 1), log returns: T k=0 ρ ij = λk {X i (t k) X i (t)}{x j (t k) X j (t)} T k=0 λk {X i (t k) X i (t)} 2 T k=0 λk {X j (t k) X j (t)} 2 to obtain the historical correlation matrix 1 ρ 12 ρ 1N ρ 12 1 ρ 2N ρ 1N ρ N2 1 Here X i (t) the arithmetic mean of the i-th log return calculated at time t, λ - decay parameter (RiskMetrics: λ = 0.94).
10 Estimating correlation matrix: historical (time series) correlation 3-2 Equicorrelation matrix Basket variance σ 2 Basket = N i=1 w 2 i σ 2 i + 2 N N i=1 j=i+1 w i w j σ i σ j ρ ij (7) 1 ρ 12 ρ 1N 1 ρ ρ ρ 21 1 ρ 2N replace with ρ 1 ρ......, ρ N1 ρ N2 1 ρ ρ 1 then ρ = σ2 Basket N i=1 w i 2σ2 i 2 N N i=1 j=i+1 w (8) iw j σ i σ j is the average basket correlation. Nice property: for {1/(N 1)} < ρ < 1 - positive definite (see Mardia et al, 1979).
11 Estimating correlation matrix: implied correlation 4-1 Implied correlation Using (8) map the implied volatility surfaces of a basket σ Basket (κ, τ) and N constituents σ i (κ, τ) to ρ(τ, κ) the average implied correlation surface of a basket: ρ(κ, τ) = σ2 Basket (κ, τ) N i=1 w i 2 σ i 2 (κ, τ) 2 N N i=1 j=i+1 w iw j σ i (κ, τ) σ j (κ, τ) (9)
12 Estimating correlation matrix: implied correlation 4-2 Dynamic modeling of correlation surfaces Every t we observe (X t,j, Y t,j ), 1 j J t, 1 t T where Y t,j - implied correlation X t,j - two-dimensional vector of κ and τ T - number of observed time periods (days) J t - number of observations at day t
13 Estimating correlation matrix: implied correlation 4-3 Dynamic modeling of correlation surfaces Including explanatory variables X t,j influencing the factor loadings m l,j rewrite (10) Y t,j = L Z t,l m l (X t,j ) + ε t,j = Zt m(x t,j ) + ε t,j (10) l=1 where Z t = (Z t1,..., Z tl ) - unobservable L-dimensional process m - L-tuple (m 1,..., m L ) of unknown real-valued functions X t,j,..., X T,JT and ε t,j,..., ε T,JT are independent ε t,j are i.i.d. with zero mean and finite second moment In such setting the modelling of Y t can be simplified to modelling of Z t = (Z t,1,..., Z t,l ), which is more feasible for L << J.
14 Estimating correlation matrix: implied correlation 4-4 Dynamic modeling of correlation surfaces Y t,j = L K Z t,l l=1 k=1 a l,k ψ k (X t,j ) + ε t,j = Z t AΨ t + ε t (11) where A - L K coefficient matrix Ψ t = {ψ 1 (X t ),..., ψ R (X t )} - space basis, in Park et al. (2009) a tensor product of one dimensional B-spline basis.
15 Estimating correlation matrix: implied correlation 4-5 Choice of space basis Estimate basis functions in a FPCA framework, motivated by Hall et. al (2006): Find eigenfunctions corresponding to the K largest eigenvalues of the smoothed operator ψ(u, v) = φ(u, v) µ(u) µ(v)
16 Estimating correlation matrix: implied correlation 4-6 Choice of space basis 1. estimate µ(u)(µ(v)): T J {Y tj a t=1 j=1 2 ( ) b c (u c Xtj)} c 2 Xtj u K c=1 2. estimate φ(u, v): T 2 2 {Y tj Y tk a 0 b1(u c c Xtj) c b2(v c c Xtk c )}2 t=1 1 j k J t c=1 c=1 ( ) ( ) Xtj u Xtj v K K h φ 3. compute ψ(u, v) = φ(u, v) µ(u) µ(v) and take K eigenfunctions corresponding to the largest eigenvalues h φ h µ
17 Estimating correlation matrix: implied correlation 4-7 Basis functions 1st eigenfunction 2nd eigenfunction time to maturity moneyness time to maturity moneyness Figure 1: Eigenfunctions as basis functions estimated on 10x10 grid
18 Estimating correlation matrix: implied correlation 4-8 Estimated time series of factors Ẑt1, Ẑt2 Z
19 From equicorrelation to block correlation 5-1 From equicorrelation to block correlation Group assets in the basket into k blocks, then 1 ρ 1 ρ 1 ρ 1 1 ρ ρ k+1 ρ 1 ρ ρ k ρ k ρ k 1 ρ k ρ k ρ k ρ k 1
20 From equicorrelation to block correlation 5-2 Correlation matrix for 2 groups of assets (3 blocks) 1 ρ 1 ρ 1 ρ 1 1 ρ ρ 1 ρ 1 1 ρ 3 ρ 3 1 ρ 2 ρ 2 ρ 2 1 ρ ρ 2 ρ 2 1
21 From equicorrelation to block correlation 5-3 Block implied correlation, 3 blocks +2 σ 2 Basket (K, τ) = N M M i=1 j=i+1 N M i=1 M N M j=i+1 N i=1 j=m+1 i=1 w 2 i σ 2 i (K, τ)+ w i w j σ i (K, τ)σ j (K, τ)ρ 1 (K, τ)+ w i w j σ i (K, τ)σ j (K, τ)ρ 2 (K, τ)+ w i w j σ i (K, τ)σ j (K, τ)ρ 3 (K, τ) where M - number of assets in the 1-st block.
22 From equicorrelation to block correlation 5-4 Challenges Moving to high-dimensional portfolios (N ) with block structure of covariance matrix: need well-conditioned estimate of covariance matrix (Ledoit and Wolf (2003), Bickel and Levina (2008)) need to define the grouping procedure and way of finding the optimal block size (Hautsch, Kyj and Oomen (2009)) Improving correlation surface modeling: need to expand the time effect in a series model Z t as a sum of basis functions ( Song Härdle and Ritov (2010))
23 Genetics and/of basket options Wolfgang Karl Härdle Elena Silyakova Ladislaus von Bortkiewicz Chair of Statistics Humboldt-Universität zu Berlin
24 Bibliography 6-1 Bibliography Alexander, C. Market Models, A Guide to Financial Data Analysis John Wiley & Sons (2001) Bai, Z.D. Methodologies in Spectral Analysis of Large Dimensional Random Matrices, A Review Statistica Sinica, (1999) Efron, B. Bootstrap Methods: Another Look at the Jackknife Annals of Statistics, (1979)
25 Bibliography 6-2 Bibliography Fengler, M. R., Pilz K.F. and P. Schwendner Basket Volatility and Correlation Volatility As An Asset Class, Risk Publications (2007) Fengler, M. R. and P. Schwendner Quoting multiasset equity options in the presence of errors from estimating correlations Journal of Derivatives, (2004) Hall, P., Müller, H. G. and Wang J. Properties of principal component methods for functional and longitudinal data analysis Ann. Statist., 34(3): , (2006)
26 Bibliography 6-3 Bibliography Laloux, L., et al. Random Matrix Theory and Financial Correlations International Journal of Theoretical and Applied Finance, (2000) Ledoit, O., and M. Wolf Improved Estimation of the Covariance Matrix of Stock Returns with an Application to Portfolio Selection Journal of Empirical Finance 105, (2003) Mardia, K. V., Kent, J. T. and Bibby, J. M. Multivariate Analysis Academic Press,Duluth, London, (1979) Plerou, V., et al. Genetics Random and/ofmatrix basket options Approach - COMPSTAT to Cross 2010 Correlations in Financial Data
Volatility Investing with Variance Swaps
Volatility Investing with Variance Swaps Wolfgang Karl Härdle Elena Silyakova Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Centre for Applied Statistics and Economics School of Business and Economics
More informationEstimating Pricing Kernel via Series Methods
Estimating Pricing Kernel via Series Methods Maria Grith Wolfgang Karl Härdle Melanie Schienle Ladislaus von Bortkiewicz Chair of Statistics Chair of Econometrics C.A.S.E. Center for Applied Statistics
More informationQua de causa copulae me placent?
Barbara Choroś Wolfgang Härdle Institut für Statistik and Ökonometrie CASE - Center for Applied Statistics and Economics Humboldt-Universität zu Berlin Motivation - Dependence Matters! The normal world
More information2.1 Mathematical Basis: Risk-Neutral Pricing
Chapter Monte-Carlo Simulation.1 Mathematical Basis: Risk-Neutral Pricing Suppose that F T is the payoff at T for a European-type derivative f. Then the price at times t before T is given by f t = e r(t
More informationSkew Hedging. Szymon Borak Matthias R. Fengler Wolfgang K. Härdle. CASE-Center for Applied Statistics and Economics Humboldt-Universität zu Berlin
Szymon Borak Matthias R. Fengler Wolfgang K. Härdle CASE-Center for Applied Statistics and Economics Humboldt-Universität zu Berlin 6 4 2.22 Motivation 1-1 Barrier options Knock-out options are financial
More informationExtended Libor Models and Their Calibration
Extended Libor Models and Their Calibration Denis Belomestny Weierstraß Institute Berlin Vienna, 16 November 2007 Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Vienna, 16 November
More informationCDO Surfaces Dynamics
D Surfaces Dynamics Barbara Choroś-Tomczyk Wolfgang Karl Härdle stap khrin Ladislaus von Bortkiewicz Chair of Statistics C..S.E. - Center for pplied Statistics and Economics Humboldt-Universität zu Berlin
More informationSpatial Risk Premium on Weather and Hedging Weather Exposure in Electricity
and Hedging Weather Exposure in Electricity Wolfgang Karl Härdle Maria Osipenko Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Centre for Applied Statistics and Economics School of Business and
More informationKing s College London
King s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Simulating Stochastic Differential Equations Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationSTOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL
STOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL YOUNGGEUN YOO Abstract. Ito s lemma is often used in Ito calculus to find the differentials of a stochastic process that depends on time. This paper will introduce
More informationAdaptive Interest Rate Modelling
Modelling Mengmeng Guo Wolfgang Karl Härdle Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. - Center for Applied Statistics and Economics Humboldt-Universität zu Berlin http://lvb.wiwi.hu-berlin.de
More informationChapter 14. The Multi-Underlying Black-Scholes Model and Correlation
Chapter 4 The Multi-Underlying Black-Scholes Model and Correlation So far we have discussed single asset options, the payoff function depended only on one underlying. Now we want to allow multiple underlyings.
More informationComputational Efficiency and Accuracy in the Valuation of Basket Options. Pengguo Wang 1
Computational Efficiency and Accuracy in the Valuation of Basket Options Pengguo Wang 1 Abstract The complexity involved in the pricing of American style basket options requires careful consideration of
More informationWrite legibly. Unreadable answers are worthless.
MMF 2021 Final Exam 1 December 2016. This is a closed-book exam: no books, no notes, no calculators, no phones, no tablets, no computers (of any kind) allowed. Do NOT turn this page over until you are
More informationAdvanced Topics in Derivative Pricing Models. Topic 4 - Variance products and volatility derivatives
Advanced Topics in Derivative Pricing Models Topic 4 - Variance products and volatility derivatives 4.1 Volatility trading and replication of variance swaps 4.2 Volatility swaps 4.3 Pricing of discrete
More informationDynamic Relative Valuation
Dynamic Relative Valuation Liuren Wu, Baruch College Joint work with Peter Carr from Morgan Stanley October 15, 2013 Liuren Wu (Baruch) Dynamic Relative Valuation 10/15/2013 1 / 20 The standard approach
More informationImplementing the HJM model by Monte Carlo Simulation
Implementing the HJM model by Monte Carlo Simulation A CQF Project - 2010 June Cohort Bob Flagg Email: bob@calcworks.net January 14, 2011 Abstract We discuss an implementation of the Heath-Jarrow-Morton
More informationExtended Libor Models and Their Calibration
Extended Libor Models and Their Calibration Denis Belomestny Weierstraß Institute Berlin Haindorf, 7 Februar 2008 Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar
More informationKing s College London
King s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority
More informationComputational Finance. Computational Finance p. 1
Computational Finance Computational Finance p. 1 Outline Binomial model: option pricing and optimal investment Monte Carlo techniques for pricing of options pricing of non-standard options improving accuracy
More informationRisk Neutral Valuation
copyright 2012 Christian Fries 1 / 51 Risk Neutral Valuation Christian Fries Version 2.2 http://www.christian-fries.de/finmath April 19-20, 2012 copyright 2012 Christian Fries 2 / 51 Outline Notation Differential
More informationThe University of Chicago, Booth School of Business Business 41202, Spring Quarter 2011, Mr. Ruey S. Tsay. Solutions to Final Exam.
The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2011, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (32 pts) Answer briefly the following questions. 1. Suppose
More informationMulti-Asset Options. A Numerical Study VILHELM NIKLASSON FRIDA TIVEDAL. Master s thesis in Engineering Mathematics and Computational Science
Multi-Asset Options A Numerical Study Master s thesis in Engineering Mathematics and Computational Science VILHELM NIKLASSON FRIDA TIVEDAL Department of Mathematical Sciences Chalmers University of Technology
More informationTime Dependent Relative Risk Aversion
SFB 649 Discussion Paper 2006-020 Time Dependent Relative Risk Aversion Enzo Giacomini* Michael Handel** Wolfgang K. Härdle* * C.A.S.E. Center for Applied Statistics and Economics, Humboldt-Universität
More informationTheoretical Problems in Credit Portfolio Modeling 2
Theoretical Problems in Credit Portfolio Modeling 2 David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Jiaotong University(SJTU) November 3, 2017 Presented at the University of South California
More informationToward a coherent Monte Carlo simulation of CVA
Toward a coherent Monte Carlo simulation of CVA Lokman Abbas-Turki (Joint work with A. I. Bouselmi & M. A. Mikou) TU Berlin January 9, 2013 Lokman (TU Berlin) Advances in Mathematical Finance 1 / 16 Plan
More informationMASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS.
MASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS May/June 2006 Time allowed: 2 HOURS. Examiner: Dr N.P. Byott This is a CLOSED
More informationStochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models
Stochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models Eni Musta Università degli studi di Pisa San Miniato - 16 September 2016 Overview 1 Self-financing portfolio 2 Complete
More informationEuropean call option with inflation-linked strike
Mathematical Statistics Stockholm University European call option with inflation-linked strike Ola Hammarlid Research Report 2010:2 ISSN 1650-0377 Postal address: Mathematical Statistics Dept. of Mathematics
More informationIMPA Commodities Course : Forward Price Models
IMPA Commodities Course : Forward Price Models Sebastian Jaimungal sebastian.jaimungal@utoronto.ca Department of Statistics and Mathematical Finance Program, University of Toronto, Toronto, Canada http://www.utstat.utoronto.ca/sjaimung
More informationA Hybrid Commodity and Interest Rate Market Model
A Hybrid Commodity and Interest Rate Market Model University of Technology, Sydney June 1 Literature A Hybrid Market Model Recall: The basic LIBOR Market Model The cross currency LIBOR Market Model LIBOR
More informationMonte Carlo Simulations
Monte Carlo Simulations Lecture 1 December 7, 2014 Outline Monte Carlo Methods Monte Carlo methods simulate the random behavior underlying the financial models Remember: When pricing you must simulate
More informationOn VIX Futures in the rough Bergomi model
On VIX Futures in the rough Bergomi model Oberwolfach Research Institute for Mathematics, February 28, 2017 joint work with Antoine Jacquier and Claude Martini Contents VIX future dynamics under rbergomi
More informationMath 416/516: Stochastic Simulation
Math 416/516: Stochastic Simulation Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 13 Haijun Li Math 416/516: Stochastic Simulation Week 13 1 / 28 Outline 1 Simulation
More informationM.I.T Fall Practice Problems
M.I.T. 15.450-Fall 2010 Sloan School of Management Professor Leonid Kogan Practice Problems 1. Consider a 3-period model with t = 0, 1, 2, 3. There are a stock and a risk-free asset. The initial stock
More informationESTIMATION OF UTILITY FUNCTIONS: MARKET VS. REPRESENTATIVE AGENT THEORY
ESTIMATION OF UTILITY FUNCTIONS: MARKET VS. REPRESENTATIVE AGENT THEORY Kai Detlefsen Wolfgang K. Härdle Rouslan A. Moro, Deutsches Institut für Wirtschaftsforschung (DIW) Center for Applied Statistics
More informationStatistica Sinica Preprint No: SS R2
0 Statistica Sinica Preprint No: SS-2016-0434R2 Title Multi-asset empirical martingale price estimators derivatives Manuscript ID SS-2016-0434R2 URL http://www.stat.sinica.edu.tw/statistica/ DOI 10.5705/ss.202016.0434
More informationParametric Inference and Dynamic State Recovery from Option Panels. Nicola Fusari
Parametric Inference and Dynamic State Recovery from Option Panels Nicola Fusari Joint work with Torben G. Andersen and Viktor Todorov July 2012 Motivation Under realistic assumptions derivatives are nonredundant
More informationThe University of Chicago, Booth School of Business Business 41202, Spring Quarter 2017, Mr. Ruey S. Tsay. Solutions to Final Exam
The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2017, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (40 points) Answer briefly the following questions. 1. Describe
More informationMarket Risk Analysis Volume I
Market Risk Analysis Volume I Quantitative Methods in Finance Carol Alexander John Wiley & Sons, Ltd List of Figures List of Tables List of Examples Foreword Preface to Volume I xiii xvi xvii xix xxiii
More informationHEDGING RAINBOW OPTIONS IN DISCRETE TIME
Journal of the Chinese Statistical Association Vol. 50, (2012) 1 20 HEDGING RAINBOW OPTIONS IN DISCRETE TIME Shih-Feng Huang and Jia-Fang Yu Department of Applied Mathematics, National University of Kaohsiung
More informationInterest Rate Curves Calibration with Monte-Carlo Simulatio
Interest Rate Curves Calibration with Monte-Carlo Simulation 24 june 2008 Participants A. Baena (UCM) Y. Borhani (Univ. of Oxford) E. Leoncini (Univ. of Florence) R. Minguez (UCM) J.M. Nkhaso (UCM) A.
More informationComputer Exercise 2 Simulation
Lund University with Lund Institute of Technology Valuation of Derivative Assets Centre for Mathematical Sciences, Mathematical Statistics Fall 2017 Computer Exercise 2 Simulation This lab deals with pricing
More informationMonte Carlo Methods for Uncertainty Quantification
Monte Carlo Methods for Uncertainty Quantification Abdul-Lateef Haji-Ali Based on slides by: Mike Giles Mathematical Institute, University of Oxford Contemporary Numerical Techniques Haji-Ali (Oxford)
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets (Hull chapter: 12, 13, 14) Liuren Wu ( c ) The Black-Scholes Model colorhmoptions Markets 1 / 17 The Black-Scholes-Merton (BSM) model Black and Scholes
More informationLecture 3. Sergei Fedotov Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) / 6
Lecture 3 Sergei Fedotov 091 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 091 010 1 / 6 Lecture 3 1 Distribution for lns(t) Solution to Stochastic Differential Equation
More informationLIBOR models, multi-curve extensions, and the pricing of callable structured derivatives
Weierstrass Institute for Applied Analysis and Stochastics LIBOR models, multi-curve extensions, and the pricing of callable structured derivatives John Schoenmakers 9th Summer School in Mathematical Finance
More informationFinancial Risk Management
Financial Risk Management Professor: Thierry Roncalli Evry University Assistant: Enareta Kurtbegu Evry University Tutorial exercices #4 1 Correlation and copulas 1. The bivariate Gaussian copula is given
More informationRISK-NEUTRAL VALUATION AND STATE SPACE FRAMEWORK. JEL Codes: C51, C61, C63, and G13
RISK-NEUTRAL VALUATION AND STATE SPACE FRAMEWORK JEL Codes: C51, C61, C63, and G13 Dr. Ramaprasad Bhar School of Banking and Finance The University of New South Wales Sydney 2052, AUSTRALIA Fax. +61 2
More informationCDO Surfaces Dynamics
D Surfaces Dynamics Barbara Choroś-Tomczyk Wolfgang Karl Härdle stap khrin Ladislaus von Bortkiewicz Chair of Statistics C..S.E. - Center for pplied Statistics and Economics Humboldt-Universität zu Berlin
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets Liuren Wu ( c ) The Black-Merton-Scholes Model colorhmoptions Markets 1 / 18 The Black-Merton-Scholes-Merton (BMS) model Black and Scholes (1973) and Merton
More informationUtility Indifference Pricing and Dynamic Programming Algorithm
Chapter 8 Utility Indifference ricing and Dynamic rogramming Algorithm In the Black-Scholes framework, we can perfectly replicate an option s payoff. However, it may not be true beyond the Black-Scholes
More informationTangent Lévy Models. Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford.
Tangent Lévy Models Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford June 24, 2010 6th World Congress of the Bachelier Finance Society Sergey
More informationThe University of Chicago, Booth School of Business Business 41202, Spring Quarter 2009, Mr. Ruey S. Tsay. Solutions to Final Exam
The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2009, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (42 pts) Answer briefly the following questions. 1. Questions
More informationInterest rate models and Solvency II
www.nr.no Outline Desired properties of interest rate models in a Solvency II setting. A review of three well-known interest rate models A real example from a Norwegian insurance company 2 Interest rate
More informationEFFICIENT MONTE CARLO ALGORITHM FOR PRICING BARRIER OPTIONS
Commun. Korean Math. Soc. 23 (2008), No. 2, pp. 285 294 EFFICIENT MONTE CARLO ALGORITHM FOR PRICING BARRIER OPTIONS Kyoung-Sook Moon Reprinted from the Communications of the Korean Mathematical Society
More informationLecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing
Lecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing We shall go over this note quickly due to time constraints. Key concept: Ito s lemma Stock Options: A contract giving
More informationPolynomial processes in stochastic portofolio theory
Polynomial processes in stochastic portofolio theory Christa Cuchiero University of Vienna 9 th Bachelier World Congress July 15, 2016 Christa Cuchiero (University of Vienna) Polynomial processes in SPT
More informationLeveraged ETF options implied volatility paradox: a statistical study
SFB 649 Discussion Paper 216-4 Leveraged ETF options implied volatility paradox: a statistical study Wolfgang Karl Härdle* Sergey Nasekin* Zhiwu Hong*² * Humboldt-Universität zu Berlin, Germany *² Xiamen
More informationMOUNTAIN RANGE OPTIONS
MOUNTAIN RANGE OPTIONS Paolo Pirruccio Copyright Arkus Financial Services - 2014 Mountain Range options Page 1 Mountain Range options Introduction Originally marketed by Société Générale in 1998. Traded
More informationImplied Volatility String Dynamics
Szymon Borak Matthias R. Fengler Wolfgang K. Härdle Enno Mammen CASE-Center for Applied Statistics and Economics Humboldt-Universität zu Berlin and Universität Mannheim aims and generic challenges 1-1
More informationBasic Concepts and Examples in Finance
Basic Concepts and Examples in Finance Bernardo D Auria email: bernardo.dauria@uc3m.es web: www.est.uc3m.es/bdauria July 5, 2017 ICMAT / UC3M The Financial Market The Financial Market We assume there are
More informationRohini Kumar. Statistics and Applied Probability, UCSB (Joint work with J. Feng and J.-P. Fouque)
Small time asymptotics for fast mean-reverting stochastic volatility models Statistics and Applied Probability, UCSB (Joint work with J. Feng and J.-P. Fouque) March 11, 2011 Frontier Probability Days,
More informationValuation of performance-dependent options in a Black- Scholes framework
Valuation of performance-dependent options in a Black- Scholes framework Thomas Gerstner, Markus Holtz Institut für Numerische Simulation, Universität Bonn, Germany Ralf Korn Fachbereich Mathematik, TU
More informationLimit Theorems for the Empirical Distribution Function of Scaled Increments of Itô Semimartingales at high frequencies
Limit Theorems for the Empirical Distribution Function of Scaled Increments of Itô Semimartingales at high frequencies George Tauchen Duke University Viktor Todorov Northwestern University 2013 Motivation
More informationQuasi-Monte Carlo Methods in Financial Engineering: An Equivalence Principle and Dimension Reduction
Quasi-Monte Carlo Methods in Financial Engineering: An Equivalence Principle and Dimension Reduction Xiaoqun Wang,2, and Ian H. Sloan 2,3 Department of Mathematical Sciences, Tsinghua University, Beijing
More informationParametric Inference and Dynamic State Recovery from Option Panels. Torben G. Andersen
Parametric Inference and Dynamic State Recovery from Option Panels Torben G. Andersen Joint work with Nicola Fusari and Viktor Todorov The Third International Conference High-Frequency Data Analysis in
More informationROM Simulation with Exact Means, Covariances, and Multivariate Skewness
ROM Simulation with Exact Means, Covariances, and Multivariate Skewness Michael Hanke 1 Spiridon Penev 2 Wolfgang Schief 2 Alex Weissensteiner 3 1 Institute for Finance, University of Liechtenstein 2 School
More informationSimulating more interesting stochastic processes
Chapter 7 Simulating more interesting stochastic processes 7. Generating correlated random variables The lectures contained a lot of motivation and pictures. We'll boil everything down to pure algebra
More informationMultiname and Multiscale Default Modeling
Multiname and Multiscale Default Modeling Jean-Pierre Fouque University of California Santa Barbara Joint work with R. Sircar (Princeton) and K. Sølna (UC Irvine) Special Semester on Stochastics with Emphasis
More informationNon-semimartingales in finance
Non-semimartingales in finance Pricing and Hedging Options with Quadratic Variation Tommi Sottinen University of Vaasa 1st Northern Triangular Seminar 9-11 March 2009, Helsinki University of Technology
More informationThe University of Chicago, Booth School of Business Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay. Solutions to Final Exam
The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (40 points) Answer briefly the following questions. 1. Consider
More informationMonte Carlo Methods for Uncertainty Quantification
Monte Carlo Methods for Uncertainty Quantification Abdul-Lateef Haji-Ali Based on slides by: Mike Giles Mathematical Institute, University of Oxford Contemporary Numerical Techniques Haji-Ali (Oxford)
More informationModern Methods of Option Pricing
Modern Methods of Option Pricing Denis Belomestny Weierstraß Institute Berlin Motzen, 14 June 2007 Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 1 / 30 Overview 1 Introduction
More informationCopulas? What copulas? R. Chicheportiche & J.P. Bouchaud, CFM
Copulas? What copulas? R. Chicheportiche & J.P. Bouchaud, CFM Multivariate linear correlations Standard tool in risk management/portfolio optimisation: the covariance matrix R ij = r i r j Find the portfolio
More informationGraduate School of Business, University of Chicago Business 41202, Spring Quarter 2007, Mr. Ruey S. Tsay. Solutions to Final Exam
Graduate School of Business, University of Chicago Business 41202, Spring Quarter 2007, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (30 pts) Answer briefly the following questions. 1. Suppose that
More informationLattice (Binomial Trees) Version 1.2
Lattice (Binomial Trees) Version 1. 1 Introduction This plug-in implements different binomial trees approximations for pricing contingent claims and allows Fairmat to use some of the most popular binomial
More informationKØBENHAVNS UNIVERSITET (Blok 2, 2011/2012) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours
This question paper consists of 3 printed pages FinKont KØBENHAVNS UNIVERSITET (Blok 2, 211/212) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours This exam paper
More informationThe ruin probabilities of a multidimensional perturbed risk model
MATHEMATICAL COMMUNICATIONS 231 Math. Commun. 18(2013, 231 239 The ruin probabilities of a multidimensional perturbed risk model Tatjana Slijepčević-Manger 1, 1 Faculty of Civil Engineering, University
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Other Miscellaneous Topics and Applications of Monte-Carlo Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationDecomposition of life insurance liabilities into risk factors theory and application to annuity conversion options
Decomposition of life insurance liabilities into risk factors theory and application to annuity conversion options Joint work with Daniel Bauer, Marcus C. Christiansen, Alexander Kling Katja Schilling
More informationAsian Option Pricing: Monte Carlo Control Variate. A discrete arithmetic Asian call option has the payoff. S T i N N + 1
Asian Option Pricing: Monte Carlo Control Variate A discrete arithmetic Asian call option has the payoff ( 1 N N + 1 i=0 S T i N K ) + A discrete geometric Asian call option has the payoff [ N i=0 S T
More informationComputational Finance Improving Monte Carlo
Computational Finance Improving Monte Carlo School of Mathematics 2018 Monte Carlo so far... Simple to program and to understand Convergence is slow, extrapolation impossible. Forward looking method ideal
More informationNEWCASTLE UNIVERSITY SCHOOL OF MATHEMATICS, STATISTICS & PHYSICS SEMESTER 1 SPECIMEN 2 MAS3904. Stochastic Financial Modelling. Time allowed: 2 hours
NEWCASTLE UNIVERSITY SCHOOL OF MATHEMATICS, STATISTICS & PHYSICS SEMESTER 1 SPECIMEN 2 Stochastic Financial Modelling Time allowed: 2 hours Candidates should attempt all questions. Marks for each question
More informationHedging Under Jump Diffusions with Transaction Costs. Peter Forsyth, Shannon Kennedy, Ken Vetzal University of Waterloo
Hedging Under Jump Diffusions with Transaction Costs Peter Forsyth, Shannon Kennedy, Ken Vetzal University of Waterloo Computational Finance Workshop, Shanghai, July 4, 2008 Overview Overview Single factor
More informationPractical Hedging: From Theory to Practice. OSU Financial Mathematics Seminar May 5, 2008
Practical Hedging: From Theory to Practice OSU Financial Mathematics Seminar May 5, 008 Background Dynamic replication is a risk management technique used to mitigate market risk We hope to spend a certain
More informationMonte Carlo Methods in Finance
Monte Carlo Methods in Finance Peter Jackel JOHN WILEY & SONS, LTD Preface Acknowledgements Mathematical Notation xi xiii xv 1 Introduction 1 2 The Mathematics Behind Monte Carlo Methods 5 2.1 A Few Basic
More informationIlliquidity, Credit risk and Merton s model
Illiquidity, Credit risk and Merton s model (joint work with J. Dong and L. Korobenko) A. Deniz Sezer University of Calgary April 28, 2016 Merton s model of corporate debt A corporate bond is a contingent
More informationTHE MARTINGALE METHOD DEMYSTIFIED
THE MARTINGALE METHOD DEMYSTIFIED SIMON ELLERSGAARD NIELSEN Abstract. We consider the nitty gritty of the martingale approach to option pricing. These notes are largely based upon Björk s Arbitrage Theory
More information"Pricing Exotic Options using Strong Convergence Properties
Fourth Oxford / Princeton Workshop on Financial Mathematics "Pricing Exotic Options using Strong Convergence Properties Klaus E. Schmitz Abe schmitz@maths.ox.ac.uk www.maths.ox.ac.uk/~schmitz Prof. Mike
More informationSimulating Stochastic Differential Equations
IEOR E4603: Monte-Carlo Simulation c 2017 by Martin Haugh Columbia University Simulating Stochastic Differential Equations In these lecture notes we discuss the simulation of stochastic differential equations
More informationAsymptotic methods in risk management. Advances in Financial Mathematics
Asymptotic methods in risk management Peter Tankov Based on joint work with A. Gulisashvili Advances in Financial Mathematics Paris, January 7 10, 2014 Peter Tankov (Université Paris Diderot) Asymptotic
More informationQuasi-Monte Carlo for Finance
Quasi-Monte Carlo for Finance Peter Kritzer Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences Linz, Austria NCTS, Taipei, November 2016 Peter Kritzer
More informationSkewness and Kurtosis Trades
This is page 1 Printer: Opaque this Skewness and Kurtosis Trades Oliver J. Blaskowitz 1 Wolfgang K. Härdle 1 Peter Schmidt 2 1 Center for Applied Statistics and Economics (CASE), Humboldt Universität zu
More informationLecture 8: The Black-Scholes theory
Lecture 8: The Black-Scholes theory Dr. Roman V Belavkin MSO4112 Contents 1 Geometric Brownian motion 1 2 The Black-Scholes pricing 2 3 The Black-Scholes equation 3 References 5 1 Geometric Brownian motion
More informationSupplementary online material to Information tradeoffs in dynamic financial markets
Supplementary online material to Information tradeoffs in dynamic financial markets Efstathios Avdis University of Alberta, Canada 1. The value of information in continuous time In this document I address
More informationOptimal robust bounds for variance options and asymptotically extreme models
Optimal robust bounds for variance options and asymptotically extreme models Alexander Cox 1 Jiajie Wang 2 1 University of Bath 2 Università di Roma La Sapienza Advances in Financial Mathematics, 9th January,
More informationMODELING INVESTMENT RETURNS WITH A MULTIVARIATE ORNSTEIN-UHLENBECK PROCESS
MODELING INVESTMENT RETURNS WITH A MULTIVARIATE ORNSTEIN-UHLENBECK PROCESS by Zhong Wan B.Econ., Nankai University, 27 a Project submitted in partial fulfillment of the requirements for the degree of Master
More informationCredit Risk : Firm Value Model
Credit Risk : Firm Value Model Prof. Dr. Svetlozar Rachev Institute for Statistics and Mathematical Economics University of Karlsruhe and Karlsruhe Institute of Technology (KIT) Prof. Dr. Svetlozar Rachev
More information