Pricing and calibration in local volatility models via fast quantization

Size: px
Start display at page:

Download "Pricing and calibration in local volatility models via fast quantization"

Transcription

1 Pricing and calibration in local volatility models via fast quantization Parma, 29 th January Joint work with Giorgia Callegaro and Martino Grasselli

2 Quantization: a brief history Birth: back to the 50 s, due to the necessity to optimize signal transmission, by appropriate discretization procedures; Applications: information theory, cluster analysis, pattern and speech recognition, numerical integration and numerical probability (90 s); Idea: approximating a signal admitting a continuum of possible values, by a signal that takes values in a discrete set; Two types (probability): Vector quantization random variables; Functional quantization stochastic processes; How: numerical procedures mostly based on stochastic optimization algorithms very time consuming.

3 Quantization: technical introduction Given an R d -valued random variable X on (Ω, A, P), X L r, quantizing X on a grid Γ = (x 1,..., x N ) consists in projecting X on Γ, following the closest neighbor rule. A grid Γ minimizing the L r mean quantization error X Proj Γ (X ) r = min 1 i N X x i r over all the grids with size at most N is the L r -optimal quantizer. The projection of X on Γ, Proj Γ (X ) is called the quantization of X : Proj Γ (X ) = N x i 1 Ci (Γ )(X ) i=1 where C i (Γ ) {ξ R d : ξ x i = min 1 j N ξ x j }, is called the Voronoi partition, or tessellation induced by Γ.

4 Figure: Voronoi diagram for the optimal grid of N (0, I 2 ). The L r mean quantization error goes to zero as the grid size N + and the convergence rate is rules by the so-called Zador Theorem. From a numerical point of view, finding an optimal quantizer may be a very challenging and time consuming task. This motivates the introduction of sub-optimal criteria: stationary quantizers.

5 Stationary quantizers Definition: An N-quantizer Γ = {x 1,, x N } inducing the quantization Proj Γ (X ) of X is said to be stationary if E [X Proj Γ (X )] = Proj Γ (X ). Optimal quantizers are stationary; Stationary quantizers Γ are critical points of the distortion function associated with Γ: N D(Γ) := z x i 2 dp X (z). i=1 C i (Γ) Stationary quantizers are interesting insofar they can be found through zero search recursive procedures like Newton s algorithm or fixed point procedures.

6 Step by step marginal quantization : ideas A new quantization approach recently introduced by Pagès and Sagna, Consider a continuous-time Markov process Y dy t = b(t, Y t )dt + a(t, Y t )dw t, Y 0 = y 0 > 0, where W is a standard Brownian motion and a and b satisfy the usual conditions ensuring the existence of a (strong) solution to the SDE; Given T > 0 and {t 0, t 1,..., t M }, with t 0 = 0 and t M = T, = t k t k 1, k 1, the Euler scheme for the process Y is Ỹ tk = Ỹt k 1 + b(t k 1, Ỹt k 1 ) + a(t k 1, Ỹt k 1 ) W Ỹ t0 = Ỹ0 = y 0 where t k = k and W := (W tk W tk 1 ) N (0, );

7 KEY REMARK: for every k = 1,..., M where ) L (Ỹtk Ỹt = x k 1 N ( m k 1 (x), σk 1 2 (x)) (1) m k 1 (x) = x + b(t k 1, x) σk 1 2 (x) = [a(t k 1, x)] 2. IDEA: quantize recursively, using vector quantization, every marginal random variable Ỹt k, given its (Gaussian) conditional distribution given Ỹt k 1 ; It can be seen (see Pagès and Sagna, 2014) that the error made by quantizing the Euler scheme can be controlled, under some mild regularity assumptions on the process.

8 Step by step marginal quantization : stationary quantizers The distortion function at time t k+1, relative to Ỹt k+1, is D k+1 (x) = N i=1 C i (x) (y k+1 x i ) 2 P (Ỹtk+1 dy k+1 ) (2) where N is the (fixed) size of the quantizer x = {x 1, x 2,..., x N }. We are looking for x R N such that D k+1 (x) = 0. QUESTION: Is it possible to apply Newton-Raphson now? ANSWER: NO! We do NOT know the distribution of Ỹt k+1!

9 Using the conditional distribution in (1) we have P(Ỹt k+1 dy k+1 ) = φ mk (y k ),σ k (y k )(y k+1 ) P(Ỹt k dy k )dy k+1 R where φ m,σ is the density function of a N (m, σ 2 ). Due to the discrete nature of the quantizer, the integral in (2) becomes a finite sum; We deduce a recursive procedure to obtain the stationary (here optimal) quantizer at time t k+1, based on the quantizer at time t k, k {1,..., M} (Y 0 = y 0 is not random); The distorsion is continuously differentiable: it is possible to compute the gradient and the Hessian matrix of the distortion function Newton-Raphson faster computations wrt stochastic algorithms.

10 The Quadratic Normal Volatility model Much attention in the financial industry due to its analytic tractability and flexibility (Blacher, 2001; Ingersoll, 1997; Lipton, 2002; Andersen, 2011): dy t = (e 1 Y 2 t + e 2 Y t + e 3 )dw t, Y 0 = y 0 > 0, (3) for some e 1, e 2, e 3 R, where W is taken under the risk neutral measure. Includes, as special cases, the Brownian motion, the geometric Brownian motion and the inverse of a three-dimensional Bessel process. We refer to (Andersen, 2011) and (Carr, Fisher and Ruf, 2013) for technical properties of the model. Mimicking a quadratic spot volatility gives some chances to get an implied volatility curve that reproduces the smile and skew effects using a parsimonious number of parameters.

11 Vanilla options in the QNV model RE-PARAMETRIZATION : dy t = σ (qy t + (1 q)x s (Y t x 0 ) 2 ) dw (t). PRICING: Closed-form solutions for vanilla derivatives available (see Andersen, 2011); Solutions depend on the roots of the polynomial in (3); Even the implementations of closed form solutions requires some care; CALIBRATION: must allow for the possibility to switch from the real roots case to the complex roots case without constraints; x 0

12 Numerical results: pricing of Vanilla options Given stationary quantization grids, pricing is immediate: the price (with r = 0) of an European Vanilla Put option on Y with maturity T and strike K, given an N-quantization grid y = (y 1,..., y N ) at t = T and associated optimal quantizer ŶT, is E[(K Y T ) + ] = N ) (K y i ) + P (ŶT = y i. i=1 The dimension of every grid is 20 and we have 10 time steps; Parameters as in Andersen, 2011; Error measure: sum of the squared differences between implied volatilities ( Res-norm ) on 7 strikes, from 85% to 115% of the spot initial value (y 0 = 100) and 6 maturities, from 2 months to 2 years.

13 Results obtained using Matlab on a CPU 2.4 GHz and 8 Gb memory laptop. Real roots Complex roots Res. Norm e e 04 Comp. Time (closed form) sec sec Comp. Time (quantization) sec sec Figure: Quantization grids, an example.

14 Numerical results: calibration We work on European Vanilla Call-Put option on the Dax Index, as of 19 June 2014; Calibration is done via a standard non-linear least-squares optimizer that minimizes ( 2 σ imp n, market n,model) σimp. n Using closed form formulas, it turns out that the implied volatility smile produced by the market is fitted better when the two roots are complex. Closed form formulas do not perform well for short maturities. Short maturities from 2 to 5 months, long maturities from 1 to 3 years, 7 strikes.

15 Short maturities Long maturities Closed formulas Res. Norm e 04 Comp. Time sec Quantization Res. Norm e e 04 Comp. Time sec sec Figure: Calibration via quantization: squared errors of the implied volatilities. On the left long maturities, on the right short maturities.

16 Numerical results: pricing of barrier options Same model data as before, we fix T = 1 3 and K = 100. We compare the price of up-and-out put options obtained via the quantization method, with 100-dimensional quantizers, with Monte Carlo, with simulations. The time step equal to The benchmark price is computed via Monte Carlo, with 10 3 discretization points in the time grid and 10 6 simulations. Benchmark price Quantization price Monte Carlo price L = L = L = L = L = Computational time sec sec

17 Future developments Increasing dimensionality: fast quantization in stochastic volatility models, like Heston, SABR, multi-heston, Wishart; Increasing derivatives s complexity: pricing of (more) exotic derivatives; Application to more general discretization schemes?

18 Thank you for your attention

Stephane Crepey. Financial Modeling. A Backward Stochastic Differential Equations Perspective. 4y Springer

Stephane Crepey. Financial Modeling. A Backward Stochastic Differential Equations Perspective. 4y Springer Stephane Crepey Financial Modeling A Backward Stochastic Differential Equations Perspective 4y Springer Part I An Introductory Course in Stochastic Processes 1 Some Classes of Discrete-Time Stochastic

More information

Applied Computational Economics and Finance

Applied Computational Economics and Finance Applied Computational Economics and Finance Mario J. Miranda and Paul L. Fackler The MIT Press Cambridge, Massachusetts London, England Preface xv 1 Introduction 1 1.1 Some Apparently Simple Questions

More information

Using the SABR Model

Using the SABR Model Definitions Ameriprise Workshop 2012 Overview Definitions The Black-76 model has been the standard model for European options on currency, interest rates, and stock indices with it s main drawback being

More information

Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies

Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative

More information

Hedging Exotic Options

Hedging Exotic Options Kai Detlefsen Wolfgang Härdle Center for Applied Statistics and Economics Humboldt-Universität zu Berlin Germany introduction 1-1 Models The Black Scholes model has some shortcomings: - volatility is not

More information

Markovian projection for volatility calibration

Markovian projection for volatility calibration cutting edge. calibration Markovian projection for volatility calibration Vladimir Piterbarg looks at the Markovian projection method, a way of obtaining closed-form approximations of European-style option

More information

Pricing Barrier Options under Local Volatility

Pricing Barrier Options under Local Volatility Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly

More information

PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION

PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION Introduction In the previous chapter, we explored a class of regression models having particularly simple analytical

More information

STA 4273H: Statistical Machine Learning

STA 4273H: Statistical Machine Learning STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.cs.toronto.edu/~rsalakhu/ Lecture 6 Three Approaches to Classification Construct

More information

ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB. Sohail A. Dianat. Rochester Institute of Technology, New York, U.S.A. Eli S.

ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB. Sohail A. Dianat. Rochester Institute of Technology, New York, U.S.A. Eli S. ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB Sohail A. Dianat Rochester Institute of Technology, New York, U.S.A. Eli S. Saber Rochester Institute of Technology, New York, U.S.A. (g) CRC Press Taylor

More information

Simulating Stochastic Differential Equations

Simulating Stochastic Differential Equations Monte Carlo Simulation: IEOR E473 Fall 24 c 24 by Martin Haugh Simulating Stochastic Differential Equations 1 Brief Review of Stochastic Calculus and Itô s Lemma Let S t be the time t price of a particular

More information

Stochastic Modelling and Forecasting

Stochastic Modelling and Forecasting Stochastic Modelling and Forecasting Department of Statistics and Modelling Science University of Strathclyde Glasgow, G1 1XH RSE/NNSFC Workshop on Management Science and Engineering and Public Policy

More information

(Quasi-)Newton methods

(Quasi-)Newton methods (Quasi-)Newton methods 1 Introduction 1.1 Newton method Newton method is a method to find the zeros of a differentiable non-linear function g, x such that g(x) = 0, where g : R n R n. Given a starting

More information

Mathematical Finance

Mathematical Finance Mathematical Finance Option Pricing under the Risk-Neutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European

More information

Monte Carlo Methods and Models in Finance and Insurance

Monte Carlo Methods and Models in Finance and Insurance Chapman & Hall/CRC FINANCIAL MATHEMATICS SERIES Monte Carlo Methods and Models in Finance and Insurance Ralf Korn Elke Korn Gerald Kroisandt f r oc) CRC Press \ V^ J Taylor & Francis Croup ^^"^ Boca Raton

More information

CCNY. BME I5100: Biomedical Signal Processing. Linear Discrimination. Lucas C. Parra Biomedical Engineering Department City College of New York

CCNY. BME I5100: Biomedical Signal Processing. Linear Discrimination. Lucas C. Parra Biomedical Engineering Department City College of New York BME I5100: Biomedical Signal Processing Linear Discrimination Lucas C. Parra Biomedical Engineering Department CCNY 1 Schedule Week 1: Introduction Linear, stationary, normal - the stuff biology is not

More information

An Option Pricing Formula for the GARCH Diffusion Model

An Option Pricing Formula for the GARCH Diffusion Model An Option Pricing Formula for the GARCH Diffusion Model Giovanni Barone-Adesi a, Henrik Rasmussen b and Claudia Ravanelli a First Version: January 3 Revised: September 3 Abstract We derive analytically

More information

Asian Option Pricing Formula for Uncertain Financial Market

Asian Option Pricing Formula for Uncertain Financial Market Sun and Chen Journal of Uncertainty Analysis and Applications (215) 3:11 DOI 1.1186/s4467-15-35-7 RESEARCH Open Access Asian Option Pricing Formula for Uncertain Financial Market Jiajun Sun 1 and Xiaowei

More information

IL GOES OCAL A TWO-FACTOR LOCAL VOLATILITY MODEL FOR OIL AND OTHER COMMODITIES 15 // MAY // 2014

IL GOES OCAL A TWO-FACTOR LOCAL VOLATILITY MODEL FOR OIL AND OTHER COMMODITIES 15 // MAY // 2014 IL GOES OCAL A TWO-FACTOR LOCAL VOLATILITY MODEL FOR OIL AND OTHER COMMODITIES 15 MAY 2014 2 Marie-Lan Nguyen / Wikimedia Commons Introduction 3 Most commodities trade as futures/forwards Cash+carry arbitrage

More information

Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering

Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering Department of Industrial Engineering and Management Sciences Northwestern University September 15th, 2014

More information

Master of Mathematical Finance: Course Descriptions

Master of Mathematical Finance: Course Descriptions Master of Mathematical Finance: Course Descriptions CS 522 Data Mining Computer Science This course provides continued exploration of data mining algorithms. More sophisticated algorithms such as support

More information

Pricing Variable Annuity Guarantees in a Local Volatility framework

Pricing Variable Annuity Guarantees in a Local Volatility framework Pricing Variable Annuity Guarantees in a Local Volatility framework Griselda Deelstra and Grégory Rayée Department of Mathematics, Université Libre de Bruxelles, Boulevard du Triomphe, CP 210, Brussels

More information

Models Used in Variance Swap Pricing

Models Used in Variance Swap Pricing Models Used in Variance Swap Pricing Final Analysis Report Jason Vinar, Xu Li, Bowen Sun, Jingnan Zhang Qi Zhang, Tianyi Luo, Wensheng Sun, Yiming Wang Financial Modelling Workshop 2011 Presentation Jan

More information

OpenGamma Quantitative Research Adjoint Algorithmic Differentiation: Calibration and Implicit Function Theorem

OpenGamma Quantitative Research Adjoint Algorithmic Differentiation: Calibration and Implicit Function Theorem OpenGamma Quantitative Research Adjoint Algorithmic Differentiation: Calibration and Implicit Function Theorem Marc Henrard marc@opengamma.com OpenGamma Quantitative Research n. 1 November 2011 Abstract

More information

ELEC-E8104 Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems

ELEC-E8104 Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems Minimum Mean Square Error (MMSE) MMSE estimation of Gaussian random vectors Linear MMSE estimator for arbitrarily distributed

More information

Markov modeling of Gas Futures

Markov modeling of Gas Futures Markov modeling of Gas Futures p.1/31 Markov modeling of Gas Futures Leif Andersen Banc of America Securities February 2008 Agenda Markov modeling of Gas Futures p.2/31 This talk is based on a working

More information

Finite Differences Schemes for Pricing of European and American Options

Finite Differences Schemes for Pricing of European and American Options Finite Differences Schemes for Pricing of European and American Options Margarida Mirador Fernandes IST Technical University of Lisbon Lisbon, Portugal November 009 Abstract Starting with the Black-Scholes

More information

Maximum likelihood estimation of mean reverting processes

Maximum likelihood estimation of mean reverting processes Maximum likelihood estimation of mean reverting processes José Carlos García Franco Onward, Inc. jcpollo@onwardinc.com Abstract Mean reverting processes are frequently used models in real options. For

More information

Parallel Computing for Option Pricing Based on the Backward Stochastic Differential Equation

Parallel Computing for Option Pricing Based on the Backward Stochastic Differential Equation Parallel Computing for Option Pricing Based on the Backward Stochastic Differential Equation Ying Peng, Bin Gong, Hui Liu, and Yanxin Zhang School of Computer Science and Technology, Shandong University,

More information

Analytically Tractable Stochastic Stock Price Models

Analytically Tractable Stochastic Stock Price Models Archil Gulisashvili Analytically Tractable Stochastic Stock Price Models 4Q Springer Contents 1 Volatility Processes 1 1.1 Brownian Motion 1 1.2 s Geometric Brownian Motion 6 1.3 Long-Time Behavior of

More information

CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options

CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common

More information

Equity-Based Insurance Guarantees Conference November 1-2, 2010. New York, NY. Operational Risks

Equity-Based Insurance Guarantees Conference November 1-2, 2010. New York, NY. Operational Risks Equity-Based Insurance Guarantees Conference November -, 00 New York, NY Operational Risks Peter Phillips Operational Risk Associated with Running a VA Hedging Program Annuity Solutions Group Aon Benfield

More information

Bias in the Estimation of Mean Reversion in Continuous-Time Lévy Processes

Bias in the Estimation of Mean Reversion in Continuous-Time Lévy Processes Bias in the Estimation of Mean Reversion in Continuous-Time Lévy Processes Yong Bao a, Aman Ullah b, Yun Wang c, and Jun Yu d a Purdue University, IN, USA b University of California, Riverside, CA, USA

More information

Lecture 5 Least-squares

Lecture 5 Least-squares EE263 Autumn 2007-08 Stephen Boyd Lecture 5 Least-squares least-squares (approximate) solution of overdetermined equations projection and orthogonality principle least-squares estimation BLUE property

More information

Quanto Adjustments in the Presence of Stochastic Volatility

Quanto Adjustments in the Presence of Stochastic Volatility Quanto Adjustments in the Presence of tochastic Volatility Alexander Giese March 14, 01 Abstract This paper considers the pricing of quanto options in the presence of stochastic volatility. While it is

More information

Example of High Dimensional Contract

Example of High Dimensional Contract Example of High Dimensional Contract An exotic high dimensional option is the ING-Coconote option (Conditional Coupon Note), whose lifetime is 8 years (24-212). The interest rate paid is flexible. In the

More information

Open issues in equity derivatives modelling

Open issues in equity derivatives modelling Open issues in equity derivatives modelling Lorenzo Bergomi Equity Derivatives Quantitative Research ociété Générale lorenzo.bergomi@sgcib.com al Outline Equity derivatives at G A brief history of equity

More information

More Exotic Options. 1 Barrier Options. 2 Compound Options. 3 Gap Options

More Exotic Options. 1 Barrier Options. 2 Compound Options. 3 Gap Options More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options Definition; Some types The payoff of a Barrier option is path

More information

Optimization under uncertainty: modeling and solution methods

Optimization under uncertainty: modeling and solution methods Optimization under uncertainty: modeling and solution methods Paolo Brandimarte Dipartimento di Scienze Matematiche Politecnico di Torino e-mail: paolo.brandimarte@polito.it URL: http://staff.polito.it/paolo.brandimarte

More information

金融隨機計算 : 第一章. Black-Scholes-Merton Theory of Derivative Pricing and Hedging. CH Han Dept of Quantitative Finance, Natl. Tsing-Hua Univ.

金融隨機計算 : 第一章. Black-Scholes-Merton Theory of Derivative Pricing and Hedging. CH Han Dept of Quantitative Finance, Natl. Tsing-Hua Univ. 金融隨機計算 : 第一章 Black-Scholes-Merton Theory of Derivative Pricing and Hedging CH Han Dept of Quantitative Finance, Natl. Tsing-Hua Univ. Derivative Contracts Derivatives, also called contingent claims, are

More information

NEURAL NETWORK FUNDAMENTALS WITH GRAPHS, ALGORITHMS, AND APPLICATIONS

NEURAL NETWORK FUNDAMENTALS WITH GRAPHS, ALGORITHMS, AND APPLICATIONS NEURAL NETWORK FUNDAMENTALS WITH GRAPHS, ALGORITHMS, AND APPLICATIONS N. K. Bose HRB-Systems Professor of Electrical Engineering The Pennsylvania State University, University Park P. Liang Associate Professor

More information

AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS

AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEY-INTERSCIENCE A John Wiley & Sons, Inc.,

More information

Introduction to Engineering System Dynamics

Introduction to Engineering System Dynamics CHAPTER 0 Introduction to Engineering System Dynamics 0.1 INTRODUCTION The objective of an engineering analysis of a dynamic system is prediction of its behaviour or performance. Real dynamic systems are

More information

On the Valuation of Power-Reverse Duals and Equity-Rates Hybrids

On the Valuation of Power-Reverse Duals and Equity-Rates Hybrids On the Valuation of Power-Reverse Duals and Equity-Rates Hybrids Oliver Caps oliver.caps@dkib.com RMT Model Validation Rates Dresdner Bank Examples of Hybrid Products Pricing of Hybrid Products using a

More information

A Novel Fourier Transform B-spline Method for Option Pricing*

A Novel Fourier Transform B-spline Method for Option Pricing* A Novel Fourier Transform B-spline Method for Option Pricing* Paper available from SSRN: http://ssrn.com/abstract=2269370 Gareth G. Haslip, FIA PhD Cass Business School, City University London October

More information

Stochastic Gradient Method: Applications

Stochastic Gradient Method: Applications Stochastic Gradient Method: Applications February 03, 2015 P. Carpentier Master MMMEF Cours MNOS 2014-2015 114 / 267 Lecture Outline 1 Two Elementary Exercices on the Stochastic Gradient Two-Stage Recourse

More information

Vanna-Volga Method for Foreign Exchange Implied Volatility Smile. Copyright Changwei Xiong 2011. January 2011. last update: Nov 27, 2013

Vanna-Volga Method for Foreign Exchange Implied Volatility Smile. Copyright Changwei Xiong 2011. January 2011. last update: Nov 27, 2013 Vanna-Volga Method for Foreign Exchange Implied Volatility Smile Copyright Changwei Xiong 011 January 011 last update: Nov 7, 01 TABLE OF CONTENTS TABLE OF CONTENTS...1 1. Trading Strategies of Vanilla

More information

Hedging Barriers. Liuren Wu. Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/)

Hedging Barriers. Liuren Wu. Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/) Hedging Barriers Liuren Wu Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/) Based on joint work with Peter Carr (Bloomberg) Modeling and Hedging Using FX Options, March

More information

Implementing CCR and CVA in a Primary International Bank

Implementing CCR and CVA in a Primary International Bank Implementing CCR and CVA in a Primary International Bank www.iasonltd.com 2011 Index Introduction 1 Introduction 2 3 Index Introduction 1 Introduction 2 3 Counterparty Credit Risk and Credit Value Adjustment

More information

Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics

Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics Dimitri Van De Ville Ecole Polytechnique Fédérale de Lausanne Biomedical Imaging Group dimitri.vandeville@epfl.ch

More information

Curve Fitting. Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization.

Curve Fitting. Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization. Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization Subsections Least-Squares Regression Linear Regression General Linear Least-Squares Nonlinear

More information

Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-Normal Distribution

Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-Normal Distribution Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-ormal Distribution October 4, 200 Limiting Distribution of the Scaled Random Walk Recall that we defined a scaled simple random walk last

More information

Math 241, Exam 1 Information.

Math 241, Exam 1 Information. Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

More information

Monte Carlo Methods in Finance

Monte Carlo Methods in Finance Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction

More information

Probability and Statistics

Probability and Statistics Probability and Statistics Syllabus for the TEMPUS SEE PhD Course (Podgorica, April 4 29, 2011) Franz Kappel 1 Institute for Mathematics and Scientific Computing University of Graz Žaneta Popeska 2 Faculty

More information

Pricing European and American bond option under the Hull White extended Vasicek model

Pricing European and American bond option under the Hull White extended Vasicek model 1 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA Pricing European and American bond option under the Hull White extended Vasicek model Eva Maria Rapoo 1, Mukendi Mpanda 2

More information

Pricing complex options using a simple Monte Carlo Simulation

Pricing complex options using a simple Monte Carlo Simulation A subsidiary of Sumitomo Mitsui Banking Corporation Pricing complex options using a simple Monte Carlo Simulation Peter Fink Among the different numerical procedures for valuing options, the Monte Carlo

More information

HPCFinance: New Thinking in Finance. Calculating Variable Annuity Liability Greeks Using Monte Carlo Simulation

HPCFinance: New Thinking in Finance. Calculating Variable Annuity Liability Greeks Using Monte Carlo Simulation HPCFinance: New Thinking in Finance Calculating Variable Annuity Liability Greeks Using Monte Carlo Simulation Dr. Mark Cathcart, Standard Life February 14, 2014 0 / 58 Outline Outline of Presentation

More information

THE TI-NSPIRE PROGRAMS

THE TI-NSPIRE PROGRAMS THE TI-NSPIRE PROGRAMS JAMES KEESLING The purpose of this document is to list and document the programs that will be used in this class. For each program there is a screen shot containing an example and

More information

4. Factor polynomials over complex numbers, describe geometrically, and apply to real-world situations. 5. Determine and apply relationships among syn

4. Factor polynomials over complex numbers, describe geometrically, and apply to real-world situations. 5. Determine and apply relationships among syn I The Real and Complex Number Systems 1. Identify subsets of complex numbers, and compare their structural characteristics. 2. Compare and contrast the properties of real numbers with the properties of

More information

Applications of Stochastic Processes in Asset Price Modeling

Applications of Stochastic Processes in Asset Price Modeling Applications of Stochastic Processes in Asset Price Modeling TJHSST Computer Systems Lab Senior Research Project 2008-2009 Preetam D Souza November 11, 2008 Abstract Stock market forecasting and asset

More information

AN ACCESSIBLE TREATMENT OF MONTE CARLO METHODS, TECHNIQUES, AND APPLICATIONS IN THE FIELD OF FINANCE AND ECONOMICS

AN ACCESSIBLE TREATMENT OF MONTE CARLO METHODS, TECHNIQUES, AND APPLICATIONS IN THE FIELD OF FINANCE AND ECONOMICS Brochure More information from http://www.researchandmarkets.com/reports/2638617/ Handbook in Monte Carlo Simulation. Applications in Financial Engineering, Risk Management, and Economics. Wiley Handbooks

More information

No-Arbitrage Condition of Option Implied Volatility and Bandwidth Selection

No-Arbitrage Condition of Option Implied Volatility and Bandwidth Selection Kamla-Raj 2014 Anthropologist, 17(3): 751-755 (2014) No-Arbitrage Condition of Option Implied Volatility and Bandwidth Selection Milos Kopa 1 and Tomas Tichy 2 1 Institute of Information Theory and Automation

More information

Option Pricing. Chapter 12 - Local volatility models - Stefan Ankirchner. University of Bonn. last update: 13th January 2014

Option Pricing. Chapter 12 - Local volatility models - Stefan Ankirchner. University of Bonn. last update: 13th January 2014 Option Pricing Chapter 12 - Local volatility models - Stefan Ankirchner University of Bonn last update: 13th January 2014 Stefan Ankirchner Option Pricing 1 Agenda The volatility surface Local volatility

More information

Properties of the SABR model

Properties of the SABR model U.U.D.M. Project Report 2011:11 Properties of the SABR model Nan Zhang Examensarbete i matematik, 30 hp Handledare och examinator: Johan Tysk Juni 2011 Department of Mathematics Uppsala University ABSTRACT

More information

American Option Valuation with Particle Filters

American Option Valuation with Particle Filters American Option Valuation with Particle Filters Bhojnarine R. Rambharat Abstract A method to price American style option contracts in a limited information framework is introduced. The pricing methodology

More information

Consistent pricing and hedging of an FX options book

Consistent pricing and hedging of an FX options book Consistent pricing and hedging of an FX options book L. Bisesti, A. Castagna and F. Mercurio 1 Introduction In the foreign exchange (FX) options market away-from-the-money options are quite actively traded,

More information

A non-gaussian Ornstein-Uhlenbeck process for electricity spot price modeling and derivatives pricing

A non-gaussian Ornstein-Uhlenbeck process for electricity spot price modeling and derivatives pricing A non-gaussian Ornstein-Uhlenbeck process for electricity spot price modeling and derivatives pricing Thilo Meyer-Brandis Center of Mathematics for Applications / University of Oslo Based on joint work

More information

Pricing of cross-currency interest rate derivatives on Graphics Processing Units

Pricing of cross-currency interest rate derivatives on Graphics Processing Units Pricing of cross-currency interest rate derivatives on Graphics Processing Units Duy Minh Dang Department of Computer Science University of Toronto Toronto, Canada dmdang@cs.toronto.edu Joint work with

More information

Efficient visual search of local features. Cordelia Schmid

Efficient visual search of local features. Cordelia Schmid Efficient visual search of local features Cordelia Schmid Visual search change in viewing angle Matches 22 correct matches Image search system for large datasets Large image dataset (one million images

More information

Lecture 1 Newton s method.

Lecture 1 Newton s method. Lecture 1 Newton s method. 1 Square roots 2 Newton s method. The guts of the method. A vector version. Implementation. The existence theorem. Basins of attraction. The Babylonian algorithm for finding

More information

1 Simulating Brownian motion (BM) and geometric Brownian motion (GBM)

1 Simulating Brownian motion (BM) and geometric Brownian motion (GBM) Copyright c 2013 by Karl Sigman 1 Simulating Brownian motion (BM) and geometric Brownian motion (GBM) For an introduction to how one can construct BM, see the Appendix at the end of these notes A stochastic

More information

Hedging Variable Annuity Guarantees

Hedging Variable Annuity Guarantees p. 1/4 Hedging Variable Annuity Guarantees Actuarial Society of Hong Kong Hong Kong, July 30 Phelim P Boyle Wilfrid Laurier University Thanks to Yan Liu and Adam Kolkiewicz for useful discussions. p. 2/4

More information

Pricing Formula for 3-Period Discrete Barrier Options

Pricing Formula for 3-Period Discrete Barrier Options Pricing Formula for 3-Period Discrete Barrier Options Chun-Yuan Chiu Down-and-Out Call Options We first give the pricing formula as an integral and then simplify the integral to obtain a formula similar

More information

Affine-structure models and the pricing of energy commodity derivatives

Affine-structure models and the pricing of energy commodity derivatives Affine-structure models and the pricing of energy commodity derivatives Nikos K Nomikos n.nomikos@city.ac.uk Cass Business School, City University London Joint work with: Ioannis Kyriakou, Panos Pouliasis

More information

Estimation of Fractal Dimension: Numerical Experiments and Software

Estimation of Fractal Dimension: Numerical Experiments and Software Institute of Biomathematics and Biometry Helmholtz Center Münhen (IBB HMGU) Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of Russian Academy of Sciences, Novosibirsk

More information

Barrier Options. Peter Carr

Barrier Options. Peter Carr Barrier Options Peter Carr Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU March 14th, 2008 What are Barrier Options?

More information

COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS

COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS NICOLE BÄUERLE AND STEFANIE GRETHER Abstract. In this short note we prove a conjecture posed in Cui et al. 2012): Dynamic mean-variance problems in

More information

Rolf Poulsen, Centre for Finance, University of Gothenburg, Box 640, SE-40530 Gothenburg, Sweden. E-mail: rolf.poulsen@economics.gu.se.

Rolf Poulsen, Centre for Finance, University of Gothenburg, Box 640, SE-40530 Gothenburg, Sweden. E-mail: rolf.poulsen@economics.gu.se. The Margrabe Formula Rolf Poulsen, Centre for Finance, University of Gothenburg, Box 640, SE-40530 Gothenburg, Sweden. E-mail: rolf.poulsen@economics.gu.se Abstract The Margrabe formula for valuation of

More information

Linear Threshold Units

Linear Threshold Units Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear

More information

From CFD to computational finance (and back again?)

From CFD to computational finance (and back again?) computational finance p. 1/17 From CFD to computational finance (and back again?) Mike Giles mike.giles@maths.ox.ac.uk Oxford University Mathematical Institute Oxford-Man Institute of Quantitative Finance

More information

BIG DATA PROBLEMS AND LARGE-SCALE OPTIMIZATION: A DISTRIBUTED ALGORITHM FOR MATRIX FACTORIZATION

BIG DATA PROBLEMS AND LARGE-SCALE OPTIMIZATION: A DISTRIBUTED ALGORITHM FOR MATRIX FACTORIZATION BIG DATA PROBLEMS AND LARGE-SCALE OPTIMIZATION: A DISTRIBUTED ALGORITHM FOR MATRIX FACTORIZATION Ş. İlker Birbil Sabancı University Ali Taylan Cemgil 1, Hazal Koptagel 1, Figen Öztoprak 2, Umut Şimşekli

More information

Introduction to machine learning and pattern recognition Lecture 1 Coryn Bailer-Jones

Introduction to machine learning and pattern recognition Lecture 1 Coryn Bailer-Jones Introduction to machine learning and pattern recognition Lecture 1 Coryn Bailer-Jones http://www.mpia.de/homes/calj/mlpr_mpia2008.html 1 1 What is machine learning? Data description and interpretation

More information

Essential Mathematics for Computer Graphics fast

Essential Mathematics for Computer Graphics fast John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made

More information

Diploma Plus in Certificate in Advanced Engineering

Diploma Plus in Certificate in Advanced Engineering Diploma Plus in Certificate in Advanced Engineering Mathematics New Syllabus from April 2011 Ngee Ann Polytechnic / School of Interdisciplinary Studies 1 I. SYNOPSIS APPENDIX A This course of advanced

More information

Numerical Methods For Derivative Pricing with Applications to Barrier Options

Numerical Methods For Derivative Pricing with Applications to Barrier Options Numerical Methods For Derivative Pricing with Applications to Barrier Options by Kavin Sin Supervisor: Professor Lilia Krivodonova A thesis presented to the University of Waterloo in fulfillment of the

More information

Some Practical Issues in FX and Equity Derivatives

Some Practical Issues in FX and Equity Derivatives Some Practical Issues in FX and Equity Derivatives Phenomenology of the Volatility Surface The volatility matrix is the map of the implied volatilities quoted by the market for options of different strikes

More information

PURE MATHEMATICS AM 27

PURE MATHEMATICS AM 27 AM SYLLABUS (013) PURE MATHEMATICS AM 7 SYLLABUS 1 Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics and

More information

PURE MATHEMATICS AM 27

PURE MATHEMATICS AM 27 AM Syllabus (015): Pure Mathematics AM SYLLABUS (015) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (015): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)

More information

USING MONTE CARLO SIMULATION AND IMPORTANCE SAMPLING TO RAPIDLY OBTAIN JUMP-DIFFUSION PRICES OF CONTINUOUS BARRIER OPTIONS

USING MONTE CARLO SIMULATION AND IMPORTANCE SAMPLING TO RAPIDLY OBTAIN JUMP-DIFFUSION PRICES OF CONTINUOUS BARRIER OPTIONS USING MONTE CARLO SIMULATION AND IMPORTANCE SAMPLING TO RAPIDLY OBTAIN JUMP-DIFFUSION PRICES OF CONTINUOUS BARRIER OPTIONS MARK S. JOSHI AND TERENCE LEUNG Abstract. The problem of pricing a continuous

More information

arxiv:1204.0453v2 [q-fin.pr] 3 Apr 2012

arxiv:1204.0453v2 [q-fin.pr] 3 Apr 2012 Pricing Variable Annuity Guarantees in a Local Volatility framework Griselda Deelstra and Grégory Rayée arxiv:1204.0453v2 [q-fin.pr] 3 Apr 2012 Department of Mathematics, Université Libre de Bruxelles,

More information

The Heston Model. Hui Gong, UCL http://www.homepages.ucl.ac.uk/ ucahgon/ May 6, 2014

The Heston Model. Hui Gong, UCL http://www.homepages.ucl.ac.uk/ ucahgon/ May 6, 2014 Hui Gong, UCL http://www.homepages.ucl.ac.uk/ ucahgon/ May 6, 2014 Generalized SV models Vanilla Call Option via Heston Itô s lemma for variance process Euler-Maruyama scheme Implement in Excel&VBA 1.

More information

Modeling Counterparty Credit Exposure

Modeling Counterparty Credit Exposure Modeling Counterparty Credit Exposure Michael Pykhtin Federal Reserve Board PRMIA Global Risk Seminar Counterparty Credit Risk New York, NY May 14, 2012 The opinions expressed here are my own, and do not

More information

Chapter 2: Binomial Methods and the Black-Scholes Formula

Chapter 2: Binomial Methods and the Black-Scholes Formula Chapter 2: Binomial Methods and the Black-Scholes Formula 2.1 Binomial Trees We consider a financial market consisting of a bond B t = B(t), a stock S t = S(t), and a call-option C t = C(t), where the

More information

Stock Option Pricing Using Bayes Filters

Stock Option Pricing Using Bayes Filters Stock Option Pricing Using Bayes Filters Lin Liao liaolin@cs.washington.edu Abstract When using Black-Scholes formula to price options, the key is the estimation of the stochastic return variance. In this

More information

Bildverarbeitung und Mustererkennung Image Processing and Pattern Recognition

Bildverarbeitung und Mustererkennung Image Processing and Pattern Recognition Bildverarbeitung und Mustererkennung Image Processing and Pattern Recognition 1. Image Pre-Processing - Pixel Brightness Transformation - Geometric Transformation - Image Denoising 1 1. Image Pre-Processing

More information

Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

More information

Lisa Borland. A multi-timescale statistical feedback model of volatility: Stylized facts and implications for option pricing

Lisa Borland. A multi-timescale statistical feedback model of volatility: Stylized facts and implications for option pricing Evnine-Vaughan Associates, Inc. A multi-timescale statistical feedback model of volatility: Stylized facts and implications for option pricing Lisa Borland October, 2005 Acknowledgements: Jeremy Evnine

More information

Numerical Methods for Pricing Exotic Options

Numerical Methods for Pricing Exotic Options Numerical Methods for Pricing Exotic Options Dimitra Bampou Supervisor: Dr. Daniel Kuhn Second Marker: Professor Berç Rustem 18 June 2008 2 Numerical Methods for Pricing Exotic Options 0BAbstract 3 Abstract

More information

European Options Pricing Using Monte Carlo Simulation

European Options Pricing Using Monte Carlo Simulation European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical

More information