Lecture 4 - Finite differences methods for PDEs
|
|
- Loren Pierce
- 6 years ago
- Views:
Transcription
1 Finite diff. Lecture 4 - Finite differences methods for PDEs Lina von Sydow Finite differences, Lina von Sydow, (1 : 18)
2 Finite difference methods Finite diff s v rv = 0 v(s T ; T ) = Φ(S) Grid in space and time: (1) Finite differences, Lina von Sydow, (2 : 18)
3 Finite diff. We know v(t ; s) = Φ(s). Want to compute v(0; s) using (1). Equation (1) holds for all s 0, for practical reasons we have to decide on computational domain 0 s s max ; 0 t T : As a rule of thumb we use s max = 4K. Finite differences, Lina von Sydow, (3 : 18)
4 Finite diff. Introduce grid-points (s ; t n ) ) s = s ; = 0; : : : ; M; t n = n ; n = 0; : : : ; N; s = smax M ; = T N : Finite differences, Lina von Sydow, (4 : 18)
5 Approximate derivatives (v n v(s ; t n )). (s ; t n ) = v n v n+1 (s ; t n ) = v +1 n v n 1 2 s + O(( s) 2 ) = = D 0 v n + O(( s) (s ; t n ) = v n +1 v n s + O( s) = D + v n + (s ; t n ) = v n v n 1 s + O( s) = D v n + O( s) Finite differences, Lina von Sydow, (5 : 18)
6 Finite 2 2 (s ; t n ) = D + D v n + O(( s) 2 ) = v n = D + = v +1 n v n s 2 v n 1 s v n + O(( s) 2 ) = v n 1 s 2 + O(( s) 2 ) = = v n +1 2v n +v n 1 s 2 + O(( s) 2 ) Finite differences, Lina von Sydow, (6 : 18)
7 Finite diff. Call option: Put option: v(0; t) = 0; v(s max ; t) = s max Ke r(t t) : (2) v(s max ; t) = 0; v(0; t) = Ke r(t t) : Other possible boundary 2 s = 0 and s = s max. 0 at Finite differences, Lina von Sydow, (7 : 18)
8 Finite diff. Example: Consider European call option. Use boundary conditions (2): = 1: v+1 n v n 1 2 s v+1 n 2v n + v n 1 s 2 = M 1: = v n +1 2 s ; = v +1 n 2v n s 2 : v n +1 v n 1 2 s v+1 n 2v n + v n 1 s 2 = s max K e r(t tn) v n 2 s 1 ; = s max K e r(t tn) 2v n + v n 1 s 2 : Finite differences, Lina von Sydow, (8 : 18)
9 Finite diff. Finite differences, Lina von Sydow, (9 : 18)
10 Finite diff. Finite differences - (European options) for = 0; : : : ; M v N = Φ(s ) (final condition) end for [ General formula: v n v n 1 rv n = 0: v+1 + rs n v n 1 2 s s2 Multiply with, move v n 1 ) v n +1 2v n +v n 1 s 2 (3) v n 1 = v n s2 + rs 2 s (v +1 n v n 1 )+ (v n s v n Finite differences, Lina von Sydow, (10 : 18) + v n 1 ) rv n : ]
11 for n = N; : : : ; 1 Finite diff. v n 1 0 = 0 v n 1 M n = s max K e r(t t n 1) for = 1; : : : ; M 1 v n s2 = v n + rs 2 s (v n +1 v n (v n s v n 1 )+ + v n 1 ) rv n end for end for Finite differences, Lina von Sydow, (11 : 18)
12 Finite diff. Taylor expanding ) local truncation error '(; s). Introduce notation v = v(s ; t n ). v(s ; t n 1 ) = v v t v 3 tt 6 v ttt + + O( 4 ) v(s 1 ; t n ) = v s v s + s2 s 2 v 3 ss 6 v sss + + s4 24 v ssss + O( s 5 ) v(s +1 ; t n ) = v + s v s + s2 2 v ss + s3 6 v sss + + s4 24 v ssss + O( s 5 ) Finite differences, Lina von Sydow, (12 : 18)
13 Use in approximation of PDE (3): Finite diff. v (v v t+ 2 2 vtt 3 6 vttt+o(4 )) + 2 s v s+ +rs s 3 3 vsss+o( s5 ) 2 s s2 v t s 2 v ss++ s4 12 vssss+o( s6 ) s 2 rv = 2 v tt + O( 2 ) + rs v s + O( s 2 )+ 2 2 s 2 v ss + O( s 2 ) rv = [Use(1)] = (4) O() + O( s 2 ) = ': Finite differences, Lina von Sydow, (13 : 18)
14 Finite diff. From Fourier-analysis it is possible to derive stability condition for periodic problem. These conditions also say something about stability for problem with boundary conditions. Finite differences, Lina von Sydow, (14 : 18)
15 Finite diff. For parabolic problems explicit methods have conditions like s 2 : Implicit methods do not have same type of stability conditions. Finite differences, Lina von Sydow, (15 : 18)
16 Euler backwards in time: Finite diff. v n v n 1 + rs v n 1 rv n 1 Crank-Nicholson: +1 v n s s2 = v n v n 1 + Lv n 1 = 0; ' = O() + O( s 2 ): v n v n v n v n 1 +v n 1 1 s 2 Lv n + Lv n 1 = 0; ' = O( 2 ) + O( s 2 ): Finite differences, Lina von Sydow, (16 : 18)
17 Finite diff. Introduce notation v n = v1 n v2 n : : : vm n 1 Explicit method: v n 1 = Av n + b. Implicit method: Av n 1 = b. A matrix of order M 1 M 1. T : Finite differences, Lina von Sydow, (17 : 18)
18 Transformations Finite diff. Possible to transform Black-Scholes PDE to @x 2 : Finite differences, Lina von Sydow, (18 : 18)
FINITE DIFFERENCE METHODS
FINITE DIFFERENCE METHODS School of Mathematics 2013 OUTLINE Review 1 REVIEW Last time Today s Lecture OUTLINE Review 1 REVIEW Last time Today s Lecture 2 DISCRETISING THE PROBLEM Finite-difference approximations
More informationComputational Finance Finite Difference Methods
Explicit finite difference method Computational Finance Finite Difference Methods School of Mathematics 2018 Today s Lecture We now introduce the final numerical scheme which is related to the PDE solution.
More informationChapter 5 Finite Difference Methods. Math6911 W07, HM Zhu
Chapter 5 Finite Difference Methods Math69 W07, HM Zhu References. Chapters 5 and 9, Brandimarte. Section 7.8, Hull 3. Chapter 7, Numerical analysis, Burden and Faires Outline Finite difference (FD) approximation
More informationFinite difference method for the Black and Scholes PDE (TP-1)
Numerical metods for PDE in Finance - ENSTA - S1-1/MMMEF Finite difference metod for te Black and Scoles PDE (TP-1) November 2015 1 Te Euler Forward sceme We look for a numerical approximation of te European
More informationCS476/676 Mar 6, Today s Topics. American Option: early exercise curve. PDE overview. Discretizations. Finite difference approximations
CS476/676 Mar 6, 2019 1 Today s Topics American Option: early exercise curve PDE overview Discretizations Finite difference approximations CS476/676 Mar 6, 2019 2 American Option American Option: PDE Complementarity
More information4. Black-Scholes Models and PDEs. Math6911 S08, HM Zhu
4. Black-Scholes Models and PDEs Math6911 S08, HM Zhu References 1. Chapter 13, J. Hull. Section.6, P. Brandimarte Outline Derivation of Black-Scholes equation Black-Scholes models for options Implied
More informationAmerican Equity Option Valuation Practical Guide
Valuation Practical Guide John Smith FinPricing Summary American Equity Option Introduction The Use of American Equity Options Valuation Practical Guide A Real World Example American Option Introduction
More informationLecture 4. Finite difference and finite element methods
Finite difference and finite element methods Lecture 4 Outline Black-Scholes equation From expectation to PDE Goal: compute the value of European option with payoff g which is the conditional expectation
More informationLecture 11: Ito Calculus. Tuesday, October 23, 12
Lecture 11: Ito Calculus Continuous time models We start with the model from Chapter 3 log S j log S j 1 = µ t + p tz j Sum it over j: log S N log S 0 = NX µ t + NX p tzj j=1 j=1 Can we take the limit
More informationA Study on Numerical Solution of Black-Scholes Model
Journal of Mathematical Finance, 8, 8, 37-38 http://www.scirp.org/journal/jmf ISSN Online: 6-44 ISSN Print: 6-434 A Study on Numerical Solution of Black-Scholes Model Md. Nurul Anwar,*, Laek Sazzad Andallah
More information1 Explicit Euler Scheme (or Euler Forward Scheme )
Numerical methods for PDE in Finance - M2MO - Paris Diderot American options January 2017 Files: https://ljll.math.upmc.fr/bokanowski/enseignement/2016/m2mo/m2mo.html We look for a numerical approximation
More informationPDE Methods for the Maximum Drawdown
PDE Methods for the Maximum Drawdown Libor Pospisil, Jan Vecer Columbia University, Department of Statistics, New York, NY 127, USA April 1, 28 Abstract Maximum drawdown is a risk measure that plays an
More informationUsing radial basis functions for option pricing
Using radial basis functions for option pricing Elisabeth Larsson Division of Scientific Computing Department of Information Technology Uppsala University Actuarial Mathematics Workshop, March 19, 2013,
More informationMAFS Computational Methods for Pricing Structured Products
MAFS550 - Computational Methods for Pricing Structured Products Solution to Homework Two Course instructor: Prof YK Kwok 1 Expand f(x 0 ) and f(x 0 x) at x 0 into Taylor series, where f(x 0 ) = f(x 0 )
More informationQueens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane.
Queens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 217 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 217 13 Lecture 13 November 15, 217 Derivation of the Black-Scholes-Merton
More informationNumerical Methods in Option Pricing (Part III)
Numerical Methods in Option Pricing (Part III) E. Explicit Finite Differences. Use of the Forward, Central, and Symmetric Central a. In order to obtain an explicit solution for the price of the derivative,
More informationEvaluation of Asian option by using RBF approximation
Boundary Elements and Other Mesh Reduction Methods XXVIII 33 Evaluation of Asian option by using RBF approximation E. Kita, Y. Goto, F. Zhai & K. Shen Graduate School of Information Sciences, Nagoya University,
More informationAs an example, we consider the following PDE with one variable; Finite difference method is one of numerical method for the PDE.
7. Introduction to the numerical integration of PDE. As an example, we consider the following PDE with one variable; Finite difference method is one of numerical method for the PDE. Accuracy requirements
More information1 Explicit Euler Scheme (or Euler Forward Scheme )
Numerical methods for PDE in Finance - M2MO - Paris Diderot American options January 2018 Files: https://ljll.math.upmc.fr/bokanowski/enseignement/2017/m2mo/m2mo.html We look for a numerical approximation
More informationFE610 Stochastic Calculus for Financial Engineers. Stevens Institute of Technology
FE610 Stochastic Calculus for Financial Engineers Lecture 13. The Black-Scholes PDE Steve Yang Stevens Institute of Technology 04/25/2013 Outline 1 The Black-Scholes PDE 2 PDEs in Asset Pricing 3 Exotic
More informationResearch Article Exponential Time Integration and Second-Order Difference Scheme for a Generalized Black-Scholes Equation
Applied Mathematics Volume 1, Article ID 796814, 1 pages doi:11155/1/796814 Research Article Exponential Time Integration and Second-Order Difference Scheme for a Generalized Black-Scholes Equation Zhongdi
More informationMATH60082 Example Sheet 6 Explicit Finite Difference
MATH68 Example Sheet 6 Explicit Finite Difference Dr P Johnson Initial Setup For the explicit method we shall need: All parameters for the option, such as X and S etc. The number of divisions in stock,
More informationProject 1: Double Pendulum
Final Projects Introduction to Numerical Analysis II http://www.math.ucsb.edu/ atzberg/winter2009numericalanalysis/index.html Professor: Paul J. Atzberger Due: Friday, March 20th Turn in to TA s Mailbox:
More informationFinancial derivatives exam Winter term 2014/2015
Financial derivatives exam Winter term 2014/2015 Problem 1: [max. 13 points] Determine whether the following assertions are true or false. Write your answers, without explanations. Grading: correct answer
More informationFrom Discrete Time to Continuous Time Modeling
From Discrete Time to Continuous Time Modeling Prof. S. Jaimungal, Department of Statistics, University of Toronto 2004 Arrow-Debreu Securities 2004 Prof. S. Jaimungal 2 Consider a simple one-period economy
More informationFinal Exam Key, JDEP 384H, Spring 2006
Final Exam Key, JDEP 384H, Spring 2006 Due Date for Exam: Thursday, May 4, 12:00 noon. Instructions: Show your work and give reasons for your answers. Write out your solutions neatly and completely. There
More informationAn IMEX-method for pricing options under Bates model using adaptive finite differences Rapport i Teknisk-vetenskapliga datorberäkningar
PROJEKTRAPPORT An IMEX-method for pricing options under Bates model using adaptive finite differences Arvid Westlund Rapport i Teknisk-vetenskapliga datorberäkningar Jan 2014 INSTITUTIONEN FÖR INFORMATIONSTEKNOLOGI
More informationMath 623 (IOE 623), Winter 2008: Final exam
Math 623 (IOE 623), Winter 2008: Final exam Name: Student ID: This is a closed book exam. You may bring up to ten one sided A4 pages of notes to the exam. You may also use a calculator but not its memory
More informationOption Valuation with Sinusoidal Heteroskedasticity
Option Valuation with Sinusoidal Heteroskedasticity Caleb Magruder June 26, 2009 1 Black-Scholes-Merton Option Pricing Ito drift-diffusion process (1) can be used to derive the Black Scholes formula (2).
More informationA distributed Laplace transform algorithm for European options
A distributed Laplace transform algorithm for European options 1 1 A. J. Davies, M. E. Honnor, C.-H. Lai, A. K. Parrott & S. Rout 1 Department of Physics, Astronomy and Mathematics, University of Hertfordshire,
More informationSpace time adaptive finite difference method for European multi-asset options
Computers and Mathematics with Applications 53 (2007) 1159 1180 www.elsevier.com/locate/camwa Space time adaptive finite difference method for European multi-asset options Per Lötstedt a,, Jonas Persson
More informationFinite Difference Methods for Option Pricing
Finite Difference Methods for Option Pricing Muhammad Usman, Ph.D. University of Dayton CASM Workshop - Black Scholes and Beyond: Pricing Equity Derivatives LUMS, Lahore, Pakistan, May 16 18, 2014 Outline
More informationCalibration Lecture 4: LSV and Model Uncertainty
Calibration Lecture 4: LSV and Model Uncertainty March 2017 Recap: Heston model Recall the Heston stochastic volatility model ds t = rs t dt + Y t S t dw 1 t, dy t = κ(θ Y t ) dt + ξ Y t dw 2 t, where
More informationNumerical Solution of BSM Equation Using Some Payoff Functions
Mathematics Today Vol.33 (June & December 017) 44-51 ISSN 0976-38, E-ISSN 455-9601 Numerical Solution of BSM Equation Using Some Payoff Functions Dhruti B. Joshi 1, Prof.(Dr.) A. K. Desai 1 Lecturer in
More informationThe Use of Numerical Methods in Solving Pricing Problems for Exotic Financial Derivatives with a Stochastic Volatility
The Use of Numerical Methods in Solving Pricing Problems for Exotic Financial Derivatives with a Stochastic Volatility Rachael England September 6, 2006 1 Rachael England 2 Declaration I confirm that this
More informationNUMERICAL AND SIMULATION TECHNIQUES IN FINANCE
NUMERICAL AND SIMULATION TECHNIQUES IN FINANCE Edward D. Weinberger, Ph.D., F.R.M Adjunct Assoc. Professor Dept. of Finance and Risk Engineering edw2026@nyu.edu Office Hours by appointment This half-semester
More informationMAS452/MAS6052. MAS452/MAS Turn Over SCHOOL OF MATHEMATICS AND STATISTICS. Stochastic Processes and Financial Mathematics
t r t r2 r t SCHOOL OF MATHEMATICS AND STATISTICS Stochastic Processes and Financial Mathematics Spring Semester 2017 2018 3 hours t s s tt t q st s 1 r s r t r s rts t q st s r t r r t Please leave this
More informationValuation of derivative assets Lecture 6
Valuation of derivative assets Lecture 6 Magnus Wiktorsson September 14, 2017 Magnus Wiktorsson L6 September 14, 2017 1 / 13 Feynman-Kac representation This is the link between a class of Partial Differential
More informationHomework Set 6 Solutions
MATH 667-010 Introduction to Mathematical Finance Prof. D. A. Edwards Due: Apr. 11, 018 P Homework Set 6 Solutions K z K + z S 1. The payoff diagram shown is for a strangle. Denote its option value by
More informationRisk Neutral Pricing Black-Scholes Formula Lecture 19. Dr. Vasily Strela (Morgan Stanley and MIT)
Risk Neutral Pricing Black-Scholes Formula Lecture 19 Dr. Vasily Strela (Morgan Stanley and MIT) Risk Neutral Valuation: Two-Horse Race Example One horse has 20% chance to win another has 80% chance $10000
More informationFROM NAVIER-STOKES TO BLACK-SCHOLES: NUMERICAL METHODS IN COMPUTATIONAL FINANCE
Irish Math. Soc. Bulletin Number 75, Summer 2015, 7 19 ISSN 0791-5578 FROM NAVIER-STOKES TO BLACK-SCHOLES: NUMERICAL METHODS IN COMPUTATIONAL FINANCE DANIEL J. DUFFY Abstract. In this article we give a
More informationIn chapter 5, we approximated the Black-Scholes model
Chapter 7 The Black-Scholes Equation In chapter 5, we approximated the Black-Scholes model ds t /S t = µ dt + σ dx t 7.1) with a suitable Binomial model and were able to derive a pricing formula for option
More informationLab12_sol. November 21, 2017
Lab12_sol November 21, 2017 1 Sample solutions of exercises of Lab 12 Suppose we want to find the current prices of an European option so that the error at the current stock price S(0) = S 0 was less that
More informationOptions. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Options
Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2014 Definitions and Terminology Definition An option is the right, but not the obligation, to buy or sell a security such
More informationJean-Paul Murara, Västeras, 26-April Mälardalen University, Sweden. Pricing EO under 2-dim. B S PDE by. using the Crank-Nicolson Method
Prcng EO under Mälardalen Unversty, Sweden Västeras, 26-Aprl-2017 1 / 15 Outlne 1 2 3 2 / 15 Optons - contracts that gve to the holder the rght but not the oblgaton to buy/sell an asset sometmes n the
More informationBinomial model: numerical algorithm
Binomial model: numerical algorithm S / 0 C \ 0 S0 u / C \ 1,1 S0 d / S u 0 /, S u 3 0 / 3,3 C \ S0 u d /,1 S u 5 0 4 0 / C 5 5,5 max X S0 u,0 S u C \ 4 4,4 C \ 3 S u d / 0 3, C \ S u d 0 S u d 0 / C 4
More informationChapter 3: Black-Scholes Equation and Its Numerical Evaluation
Chapter 3: Black-Scholes Equation and Its Numerical Evaluation 3.1 Itô Integral 3.1.1 Convergence in the Mean and Stieltjes Integral Definition 3.1 (Convergence in the Mean) A sequence {X n } n ln of random
More informationValuing the Guaranteed Minimum Death Benefit Clause with Partial Withdrawals
1 2 3 4 Valuing the Guaranteed Minimum Death Benefit Clause with Partial Withdrawals A. C. Bélanger, P. A. Forsyth and G. Labahn January 30, 2009 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Abstract In
More informationExtensions to the Black Scholes Model
Lecture 16 Extensions to the Black Scholes Model 16.1 Dividends Dividend is a sum of money paid regularly (typically annually) by a company to its shareholders out of its profits (or reserves). In this
More informationDerivative Securities Fall 2012 Final Exam Guidance Extended version includes full semester
Derivative Securities Fall 2012 Final Exam Guidance Extended version includes full semester Our exam is Wednesday, December 19, at the normal class place and time. You may bring two sheets of notes (8.5
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 informationPricing 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 informationThe Black-Scholes Equation
The Black-Scholes Equation MATH 472 Financial Mathematics J. Robert Buchanan 2018 Objectives In this lesson we will: derive the Black-Scholes partial differential equation using Itô s Lemma and no-arbitrage
More information(RP13) Efficient numerical methods on high-performance computing platforms for the underlying financial models: Series Solution and Option Pricing
(RP13) Efficient numerical methods on high-performance computing platforms for the underlying financial models: Series Solution and Option Pricing Jun Hu Tampere University of Technology Final conference
More informationPricing American Options Using a Space-time Adaptive Finite Difference Method
Pricing American Options Using a Space-time Adaptive Finite Difference Method Jonas Persson Abstract American options are priced numerically using a space- and timeadaptive finite difference method. The
More informationBlack-Scholes-Merton Model
Black-Scholes-Merton Model Weerachart Kilenthong University of the Thai Chamber of Commerce c Kilenthong 2017 Weerachart Kilenthong University of the Thai Chamber Black-Scholes-Merton of Commerce Model
More informationSpace-time adaptive finite difference method for European multi-asset options
Space-time adaptive finite difference method for European multi-asset options Per Lötstedt 1, Jonas Persson 1, Lina von Sydow 1 Ý, Johan Tysk 2 Þ 1 Division of Scientific Computing, Department of Information
More informationPLEASE SCROLL DOWN FOR ARTICLE
This article was downloaded by: [Swets Content Distribution] On: 1 October 2009 Access details: Access Details: [subscription number 912280237] Publisher Routledge Informa Ltd Registered in England and
More informationOptions, American Style. Comparison of American Options and European Options
Options, American Style Comparison of American Options and European Options Background on Stocks On time domain [0, T], an asset (such as a stock) changes in value from S 0 to S T At each period n, the
More informationNUMERICAL METHODS OF PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS FOR OPTION PRICE
Trends in Mathematics - New Series Information Center for Mathematical Sciences Volume 13, Number 1, 011, pages 1 5 NUMERICAL METHODS OF PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS FOR OPTION PRICE YONGHOON
More informationDerivative Securities Section 9 Fall 2004 Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences.
Derivative Securities Section 9 Fall 2004 Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. Futures, and options on futures. Martingales and their role in option pricing. A brief introduction
More informationLecture 15. Concepts of Black-Scholes options model. I. Intuition of Black-Scholes Pricing formulas
Lecture 15 Concepts of Black-Scholes options model Agenda: I. Intuition of Black-Scholes Pricing formulas II. III. he impact of stock dilution: an example of stock warrant pricing model he impact of Dividends:
More informationPricing Algorithms for financial derivatives on baskets modeled by Lévy copulas
Pricing Algorithms for financial derivatives on baskets modeled by Lévy copulas Christoph Winter, ETH Zurich, Seminar for Applied Mathematics École Polytechnique, Paris, September 6 8, 26 Introduction
More informationHIGHER ORDER BINARY OPTIONS AND MULTIPLE-EXPIRY EXOTICS
Electronic Journal of Mathematical Analysis and Applications Vol. (2) July 203, pp. 247-259. ISSN: 2090-792X (online) http://ejmaa.6te.net/ HIGHER ORDER BINARY OPTIONS AND MULTIPLE-EXPIRY EXOTICS HYONG-CHOL
More informationIn this lecture we will solve the final-value problem derived in the previous lecture 4, V (1) + rs = rv (t < T )
MSC FINANCIAL ENGINEERING PRICING I, AUTUMN 2010-2011 LECTURE 5: THE BLACK AND SCHOLES FORMULA AND ITS GREEKS RAYMOND BRUMMELHUIS DEPARTMENT EMS BIRKBECK In this lecture we will solve the final-value problem
More informationPDE Methods for Option Pricing under Jump Diffusion Processes
PDE Methods for Option Pricing under Jump Diffusion Processes Prof Kevin Parrott University of Greenwich November 2009 Typeset by FoilTEX Summary Merton jump diffusion American options Levy Processes -
More informationInfinite Reload Options: Pricing and Analysis
Infinite Reload Options: Pricing and Analysis A. C. Bélanger P. A. Forsyth April 27, 2006 Abstract Infinite reload options allow the user to exercise his reload right as often as he chooses during the
More informationBlack-Scholes Option Pricing
Black-Scholes Option Pricing The pricing kernel furnishes an alternate derivation of the Black-Scholes formula for the price of a call option. Arbitrage is again the foundation for the theory. 1 Risk-Free
More informationFinite Element Method
In Finite Difference Methods: the solution domain is divided into a grid of discrete points or nodes the PDE is then written for each node and its derivatives replaced by finite-divided differences In
More informationComputational Methods in Finance
Chapman & Hall/CRC FINANCIAL MATHEMATICS SERIES Computational Methods in Finance AM Hirsa Ltfi) CRC Press VV^ J Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor &
More informationStock loan valuation under a stochastic interest rate model
University of Wollongong Research Online University of Wollongong Thesis Collection 1954-2016 University of Wollongong Thesis Collections 2015 Stock loan valuation under a stochastic interest rate model
More informationPROJECT REPORT. Dimension Reduction for the Black-Scholes Equation. Alleviating the Curse of Dimensionality
Dimension Reduction for the Black-Scholes Equation Alleviating the Curse of Dimensionality Erik Ekedahl, Eric Hansander and Erik Lehto Report in Scientic Computing, Advanced Course June 2007 PROJECT REPORT
More informationFourier Space Time-stepping Method for Option Pricing with Lévy Processes
FST method Extensions Indifference pricing Fourier Space Time-stepping Method for Option Pricing with Lévy Processes Vladimir Surkov University of Toronto Computational Methods in Finance Conference University
More informationA Simple Numerical Approach for Solving American Option Problems
Proceedings of the World Congress on Engineering 013 Vol I, WCE 013, July 3-5, 013, London, U.K. A Simple Numerical Approach for Solving American Option Problems Tzyy-Leng Horng and Chih-Yuan Tien Abstract
More informationSystems of Ordinary Differential Equations. Lectures INF2320 p. 1/48
Systems of Ordinary Differential Equations Lectures INF2320 p. 1/48 Lectures INF2320 p. 2/48 ystems of ordinary differential equations Last two lectures we have studied models of the form y (t) = F(y),
More informationPAijpam.eu ANALYTIC SOLUTION OF A NONLINEAR BLACK-SCHOLES EQUATION
International Journal of Pure and Applied Mathematics Volume 8 No. 4 013, 547-555 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.173/ijpam.v8i4.4
More informationImplementing Models in Quantitative Finance: Methods and Cases
Gianluca Fusai Andrea Roncoroni Implementing Models in Quantitative Finance: Methods and Cases vl Springer Contents Introduction xv Parti Methods 1 Static Monte Carlo 3 1.1 Motivation and Issues 3 1.1.1
More informationOption pricing using TR-BDF2 time stepping method
Option pricing using TR-BDF time stepping method by Ming Ma Aresearchpaper presented to the University of Waterloo in partial fulfilment of the requirement for the degree of Master of Mathematics in Computational
More informationModule 10:Application of stochastic processes in areas like finance Lecture 36:Black-Scholes Model. Stochastic Differential Equation.
Stochastic Differential Equation Consider. Moreover partition the interval into and define, where. Now by Rieman Integral we know that, where. Moreover. Using the fundamentals mentioned above we can easily
More informationA Worst-Case Approach to Option Pricing in Crash-Threatened Markets
A Worst-Case Approach to Option Pricing in Crash-Threatened Markets Christoph Belak School of Mathematical Sciences Dublin City University Ireland Department of Mathematics University of Kaiserslautern
More informationCRANK-NICOLSON SCHEME FOR ASIAN OPTION
CRANK-NICOLSON SCHEME FOR ASIAN OPTION By LEE TSE YUENG A thesis submitted to the Department of Mathematical and Actuarial Sciences, Faculty of Engineering and Science, Universiti Tunku Abdul Rahman, in
More informationConservative and Finite Volume Methods for the Pricing Problem
Conservative and Finite Volume Methods for the Pricing Problem Master Thesis M.Sc. Computer Simulation in Science Germán I. Ramírez-Espinoza Faculty of Mathematics and Natural Science Bergische Universität
More informationCompleteness and Hedging. Tomas Björk
IV Completeness and Hedging Tomas Björk 1 Problems around Standard Black-Scholes We assumed that the derivative was traded. How do we price OTC products? Why is the option price independent of the expected
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 informationA High-order Front-tracking Finite Difference Method for Pricing American Options under Jump-Diffusion Models
A High-order Front-tracking Finite Difference Method for Pricing American Options under Jump-Diffusion Models Jari Toivanen Abstract A free boundary formulation is considered for the price of American
More informationLecture 6: Option Pricing Using a One-step Binomial Tree. Thursday, September 12, 13
Lecture 6: Option Pricing Using a One-step Binomial Tree An over-simplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
More information6. Numerical methods for option pricing
6. Numerical methods for option pricing Binomial model revisited Under the risk neutral measure, ln S t+ t ( ) S t becomes normally distributed with mean r σ2 t and variance σ 2 t, where r is 2 the riskless
More informationRadial Basis Function Methods for Pricing Multi-Asset Options
IT Licentiate theses 2016-001 Radial Basis Function Methods for Pricing Multi-Asset Options VICTOR SHCHERBAKOV UPPSALA UNIVERSITY Department of Information Technology Radial Basis Function Methods for
More informationEquations of Mathematical Finance. Fall 2007
Equations of Mathematical Finance Fall 007 Introduction In the early 1970s, Fisher Black and Myron Scholes made a major breakthrough by deriving a differential equation that must be satisfied by the price
More informationContents Critique 26. portfolio optimization 32
Contents Preface vii 1 Financial problems and numerical methods 3 1.1 MATLAB environment 4 1.1.1 Why MATLAB? 5 1.2 Fixed-income securities: analysis and portfolio immunization 6 1.2.1 Basic valuation of
More informationMATH6911: Numerical Methods in Finance. Final exam Time: 2:00pm - 5:00pm, April 11, Student Name (print): Student Signature: Student ID:
MATH6911 Page 1 of 16 Winter 2007 MATH6911: Numerical Methods in Finance Final exam Time: 2:00pm - 5:00pm, April 11, 2007 Student Name (print): Student Signature: Student ID: Question Full Mark Mark 1
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 informationAEM Computational Fluid Dynamics Instructor: Dr. M. A. R. Sharif
AEM 620 - Computational Fluid Dynamics Instructor: Dr. M. A. R. Sharif Numerical Solution Techniques for 1-D Parabolic Partial Differential Equations: Transient Flow Problem by Parshant Dhand September
More informationPackage multiassetoptions
Package multiassetoptions February 20, 2015 Type Package Title Finite Difference Method for Multi-Asset Option Valuation Version 0.1-1 Date 2015-01-31 Author Maintainer Michael Eichenberger
More information1 The Hull-White Interest Rate Model
Abstract Numerical Implementation of Hull-White Interest Rate Model: Hull-White Tree vs Finite Differences Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 30 April 2002 We implement the
More informationDefinition Pricing Risk management Second generation barrier options. Barrier Options. Arfima Financial Solutions
Arfima Financial Solutions Contents Definition 1 Definition 2 3 4 Contenido Definition 1 Definition 2 3 4 Definition Definition: A barrier option is an option on the underlying asset that is activated
More informationApplied Stochastic Processes and Control for Jump-Diffusions
Applied Stochastic Processes and Control for Jump-Diffusions Modeling, Analysis, and Computation Floyd B. Hanson University of Illinois at Chicago Chicago, Illinois siam.. Society for Industrial and Applied
More informationNumerical schemes for SDEs
Lecture 5 Numerical schemes for SDEs Lecture Notes by Jan Palczewski Computational Finance p. 1 A Stochastic Differential Equation (SDE) is an object of the following type dx t = a(t,x t )dt + b(t,x t
More informationAdvanced Numerical Methods for Financial Problems
Advanced Numerical Methods for Financial Problems Pricing of Derivatives Krasimir Milanov krasimir.milanov@finanalytica.com Department of Research and Development FinAnalytica Ltd. Seminar: Signal Analysis
More informationNo ANALYTIC AMERICAN OPTION PRICING AND APPLICATIONS. By A. Sbuelz. July 2003 ISSN
No. 23 64 ANALYTIC AMERICAN OPTION PRICING AND APPLICATIONS By A. Sbuelz July 23 ISSN 924-781 Analytic American Option Pricing and Applications Alessandro Sbuelz First Version: June 3, 23 This Version:
More information