1 Explicit Euler Scheme (or Euler Forward Scheme )
|
|
- Earl Williamson
- 6 years ago
- Views:
Transcription
1 Numerical methods for PDE in Finance - M2MO - Paris Diderot American options January 2018 Files: We look for a numerical approximation of the american put option v = v(t, s), t (0, T ) and s Ω := (S min, S max ), solution of the following Partial Differential Equation: with min( t v + Av, v ϕ) = 0, (t, s) (0, T ) Ω, (1a) v(t, S min ) = v l (t), t (0, T ), (1b) v(t, S max ) = v r (t) 0, t (0, T ), (1c) v(0, s) = ϕ(s), s Ω (1d) Av := σ2 2 s2 s,s v rs s v + rv, and σ, r, K are strictly positive constants. A logical choice for v l (t) is v l (t) : ϕ(s min ) = K S min. For the numerical applications we will chose the following financial parameters: K = 100, S max = 200, T = 1, σ = 0.2, r = 0.1, and S min = 0. We will consider two differents payoff functions ϕ 1 ou ϕ 2 for the tests: a classical payoff function (payoff=1): ϕ 1 (x) := (K x) +, with v l (t) := K S min, and a barrier payoff function (payoff=2): { K for K ϕ 2 (x) := x K otherwise }, with v l (t) = 0. 1 Explicit Euler Scheme (or Euler Forward Scheme ) Notations. We adopt the usual notations : mesh s j = S min + jh, j = 1,..., I, h = (S max S min )/(I + 1) (so that s 0 = S min and s I+1 = S max ), and t n = n,
2 0 n N, = T/N. We then look for Uj n, an approximation of v(t n, s j ). We choose to work with the unknown vector of R I : U n := U n 1. U n I Let us consider the explicit Euler Forward scheme (EE) with centered approximation: min ( U n+1 j Uj n U n 0 = v l (t n ), U n I+1 = v r(t n ),. + σ2 U 2 s2 j 1 n + UPj n Uj+1 n j ) h 2 U n+1 j ϕ(s j ) rs j U n j+1 U n j 1 2h + ru n j, = 0, 1 j I, for n = 0,..., N 1. The scheme is initialized with Uj 0 = ϕ(s j ). We denote by A the discretization matrix associated to the operator A, of size I, and q(t) R I, such that (AP + q(t)) j := + σ2 P j 1 + 2P j P j+1 P j+1 P j 1 2 s2 j rs h 2 j + rp j, 1 j I. 2h = (α j β j )P j 1 + (2α j + r)p j (α j + β j )P j+1, 1 j I. with α j = σ2 2 s 2 j h 2 and β j = rs j. We recall that A is the tridiagonal matrix 2h tridiag ( (α j β j ), 2α j + r, (α j + β j )), and q(t) := ( α 1 + β 1 )v l (t) 0. 0 ( α I β I )v r (t). This matrix A and vector q(t) are the same as the one used for European options. Let also g be the vector of R I with components g j := ϕ(s j ). We finaly obtain the following equivalent form of the scheme (EE) in R I : min( U n+1 U n U 0 = g. + AU n + q(t n ), U n+1 g) = 0, n = 0,..., N 1, (2)
3 (where the min must be understood component wise). We finaly check also that the main iteration can also be written or, in vector form, U n+1 i = max(u n i (AU n + q(t n )) i, g i ), 1 i I, U n+1 = max(u n (AU n + q(t n )), g). Download the program tp2.m (or tp2.sci): a matlab file (resp. scilab file) to complete. This is a similar program than the one for European options. For simplification, we will work with full matrices (TYPE=FULL). Modify the variable SCHEME to the value EE-AMER (this will correspond to the iterative scheme for the american option), and program the corresponding Euler Forward scheme case EE-AMER in the main loop. Check that the program does give a stable solution with the parameters I=20 and N=20. Check that there is an unstable behavior with other parameters (such as I=50 and N=20). 2 A first implicit scheme: the splitting scheme For stability reasons, we now turn on the implicit schemes. We propose a first implicit simplified scheme. 1 Although it might be less precise than exactly solving the implicit scheme (see next section), it is much simplier to program. The scheme is as follows: (i) compute U n+1,(1) s.t. U n+1,(1) U n + AU n+1,(1) + q(t n+1 ) = 0, (3) (ii) compute U n+1 s.t. U n+1 = max(u n+1,(1), g). (4) Program this method (SCHEME= EI-AMER-SPLITTING ). What is the advantage of this method with respect to Newton s method? (see for instance the scheme (5)) Propose a variant of Crank-Nicolson type (θ = 1 scheme). However we notice that 2 this Crank-Nicolson type method is not truly second order in time (as well as any method proposed here), because of the nonlinearity of the PDE. 2 3 Implicit Euler Scheme For stability reasons, we now turn on the time-implicit Euler Scheme for the american option, which naturally takes the following form: min( U n+1 U n + AU n+1 + q(t n+1 ), U n+1 g) = 0, n = 0,..., N 1, (5) U 0 = g. 1 For this method, we refer to Barles, Daher and Romano (1994). 2 A true second order method is proposed in Osterlee (2003), Bokanowski and Debrabant (2016).
4 (U n is known and we look for a solution U n+1 ). Let us define B := I d + A, and b := U n q(t n+1 ). For each n, one must solve a solution x R I of the following non-linear system min(bx b, x g) = 0, in R I. (6) Then, we will take U n+1 = x as the solution of the scheme (5). We now propose to test several algorithms for solving (6). 3.1 PSOR Algorithm (PSOR = Projected Successive Over Relaxation ). This is an iterative method based on the decomposition B = L + U where L is the lower triangular part of B and U is the strict upper triangular part. 3 Check that the solution x of min(lx b, x g) = 0 can be solved explicitly. Check that the solution x = x k+1 of min(lx (b Ux k ), x g) = 0 can be programmed using the following pseudo-algorithm: x <- x^k for i=1.. n x(i)=( b(i)- sum_{j<>i} (A(i,j) x(j)) ) / A(i,i) x(i)=max(x(i),g(i)) end Complete the iterative method in the file PSOR.m. (a solution is in PSORsol.m.) Branch on this method in the code : SCHEMA= EI-AMER-PSOR. Observe that the method slows down for larger I values (for instance, test with σ = 0.3, N = 10, I + 1 = 100, and observe a more important number of PSOR iterations at each time iteration). Observe that the method can be accelerated by using a relaxation method based on the following decomposition (instead of B = L + U): B = L w + U w where L w = ( 1 w 1)D + L, D = diag(a), U w = B L w. The parameter w can be tested with values in (1, 2) - for instance w = For a given starting vector x 0 R I, we define x k+1 as the solution of the system min(lx k+1 (b Ux k ), x k+1 g) = 0. Assuming that L i,i := B i,i > 0, the system can be solved explicitly, using that L is also lower triangular. By a fixed point argument the method can be shown to be convergent as soon as B is a strictly diagonal dominant matrix with B i,i > 0 i.
5 3.2 Brennan and Schwartz algorithm - or projected UL method There exists a direct method for solving min(bx b, x g) = 0, when the solution 4 x has a particular shape. The idea is to write a decomposition of the form B = UL (L: lower triangular matrix, and U: upper triangular matrix, with U ii = 1, i), and to use the equivalence, in some cases : min(ulx b, x g) = 0 min(lx U 1 b, x g) = 0. (7) Then, the right-hand-side of (7) has a simple explicit solution given by (i) solve c = U 1 b: upwind algorithm. (ii) solve min(lx c, x g) = 0: downwind algorithm. Set SCHEME== EI-AMER-UL in the main working file. Program the B = UL decomposition of a tridiagonal matrix B. In Matlab, one can fill the file uldecomp.m (to be completed). (In Scilab, one can work in the file uldecomp.sci.) Solution file : uldecomp_sol.m. First check that the decomposition B = UL is working on the specific matrix B := I d + A, in the case I = 10. To do so, one can introduce in the main loop a test that is only performed at iteration n=0, as follows: case SCHEME== EI-AMER-UL if n==0 // Here decompose B=UL and test the decomposition B=... [U,L]=uldecomp(B); // Here test that the norm of B-UL is zero : show norm of B-U*L end... //- here american option scheme Program the projected downwind algorithm (complete the function descente_p), in order to find the solution x of min(lx b, x g) = 0. Program the scheme by using the upwind algorithm (which is given) and the projected downwind algorithm. Test the method with N = 20, I + 1 = 50 (and 4 In the case of the american put with one asset, and for a finite element approach, see the reference of Jaillet, Lamberton and Lapeyere (1990). The algorithm has initially been introduced by Brennan and Schwartz.
6 with the classical payoff function). Check that we do solve correctly the equation min(bx b, x g) = 0 at each time iteration. To this end one can print the norm min(bx b, x g) after each new computation of the vector U in the main loop (example with Matlab): Pold=P; P=... % scheme definition err=norm(min(b*u-uold,u-payoff(s))); fprintf( Check: min(b x- b, x-g)= %f\n, err); Next, run the program again with the particular payoff ϕ 2 instead of ϕ 1. Check that in that case min(bx b, x g) 0 (as soon as n = 0). 3.3 Semi-smooth Newton s method The following proposed method will work whatever the form of the data and payoff functions. 5 We now want to apply a Newton type algorithm for solving F (x) = 0 with F (x) := min(bx b, x g). We consider the following algorithm: iterate over k 0 (for a given x 0 starting point of R I, to be choosen) x k+1 = x k F (x k ) 1 F (x k ), until F (x k ) = 0 (or, that x k+1 = x k ). We will take the following definition for F (x k ) (rwo by row derivative) { F (x k Bi,j if (Bx ) i,j := k b) i (x k g) i, otherwise. δ i,j (Note the specific choice F (x k ) i,j = B i,j when (Bx k b) i (x k g) i. The other choice F (x k ) i,j = δ i,j when (Bx k b) i (x k g) i works also but is less efficient : more iterations might be needed.) Fix the parameter SCHEME= EI-AMER-NEWTON. Program the algorithm by completing the function, in Matlab: newton.m (solution file : newton_sol.m). Test the method with N = 20, I = 50 and the classical payoff function ϕ 1. Test with the particular payoff function ϕ 2, and check that the method works now well, in the sense that there is no error when solving (6). Remark: there exist other schemes which are more or less equivalent. For instance, the Primal-Dual method, the policy iteration algorithm, or Howard s algorithm. 5 An analysis of the scheme can been found for instance in the reference Bokanowski, Maroso, Zidani (2009), although this type of algorithms goes back to Howard s algorithm, 1957.
7 Figure 1: American put option. Evaluated at time t = 0 for terminal time T = 1 (σ = 0.3, r = 0.1; N = 20, I = 20).
1 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 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 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 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 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 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 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 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 informationFINITE 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 informationLecture 4 - Finite differences methods for PDEs
Finite diff. Lecture 4 - Finite differences methods for PDEs Lina von Sydow Finite differences, Lina von Sydow, (1 : 18) Finite difference methods Finite diff. Black-Scholes equation @v @t + 1 2 2 s 2
More informationMartingale Pricing Theory in Discrete-Time and Discrete-Space Models
IEOR E4707: Foundations of Financial Engineering c 206 by Martin Haugh Martingale Pricing Theory in Discrete-Time and Discrete-Space Models These notes develop the theory of martingale pricing in a discrete-time,
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 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 informationWhat can we do with numerical optimization?
Optimization motivation and background Eddie Wadbro Introduction to PDE Constrained Optimization, 2016 February 15 16, 2016 Eddie Wadbro, Introduction to PDE Constrained Optimization, February 15 16, 2016
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 informationStrong Stability Preserving Time Discretizations
Strong Stability Preserving Time Discretizations Sigal Gottlieb University of Massachusetts Dartmouth AFOSR Computational Math Program Review August 2015 SSP time stepping August2015 1 / 33 Past and Current
More informationLecture 20 Heat Equation and Parabolic Problems
.539 Lecture 2 Heat Equation and Parabolic Problems Prof. Dean Wang. Classification of Differential Equations In most texts the classification is given for a linear second-order differential equation in
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 informationAN OPERATOR SPLITTING METHOD FOR PRICING THE ELS OPTION
J. KSIAM Vol.14, No.3, 175 187, 21 AN OPERATOR SPLITTING METHOD FOR PRICING THE ELS OPTION DARAE JEONG, IN-SUK WEE, AND JUNSEOK KIM DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY, SEOUL 136-71, KOREA E-mail
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 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 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 information32.4. Parabolic PDEs. Introduction. Prerequisites. Learning Outcomes
Parabolic PDEs 32.4 Introduction Second-order partial differential equations (PDEs) may be classified as parabolic, hyperbolic or elliptic. Parabolic and hyperbolic PDEs often model time dependent processes
More informationNumerical Methods For American Option Pricing. Peng Liu. June 2008
Numerical Methods For American Option Pricing Peng Liu June 2008 Abstract An analytic solution does not exist for evaluating the American put option. Usually, the value is obtained by applying numerical
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 informationPDE Project Course 1. Adaptive finite element methods
PDE Project Course 1. Adaptive finite element methods Anders Logg logg@math.chalmers.se Department of Computational Mathematics PDE Project Course 03/04 p. 1 Lecture plan Introduction to FEM FEM for Poisson
More informationGalerkin Least Square FEM for the European option price with CEV model
Galerkin Least Square FEM for the European option price with CEV model A Major Qualifying Project Submitted to the Faculty of Worcester Polytechnic Institute In partial fulfillment of requirements for
More informationTowards efficient option pricing in incomplete markets
Towards efficient option pricing in incomplete markets GPU TECHNOLOGY CONFERENCE 2016 Shih-Hau Tan 1 2 1 Marie Curie Research Project STRIKE 2 University of Greenwich Apr. 6, 2016 (University of Greenwich)
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 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 informationOutline. 1 Introduction. 2 Algorithms. 3 Examples. Algorithm 1 General coordinate minimization framework. 1: Choose x 0 R n and set k 0.
Outline Coordinate Minimization Daniel P. Robinson Department of Applied Mathematics and Statistics Johns Hopkins University November 27, 208 Introduction 2 Algorithms Cyclic order with exact minimization
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 informationOptimal Dam Management
Optimal Dam Management Michel De Lara et Vincent Leclère July 3, 2012 Contents 1 Problem statement 1 1.1 Dam dynamics.................................. 2 1.2 Intertemporal payoff criterion..........................
More informationEcon 582 Nonlinear Regression
Econ 582 Nonlinear Regression Eric Zivot June 3, 2013 Nonlinear Regression In linear regression models = x 0 β (1 )( 1) + [ x ]=0 [ x = x] =x 0 β = [ x = x] [ x = x] x = β it is assumed that the regression
More informationAppendix G: Numerical Solution to ODEs
Appendix G: Numerical Solution to ODEs The numerical solution to any transient problem begins with the derivation of the governing differential equation, which allows the calculation of the rate of change
More informationApplication of an Interval Backward Finite Difference Method for Solving the One-Dimensional Heat Conduction Problem
Application of an Interval Backward Finite Difference Method for Solving the One-Dimensional Heat Conduction Problem Malgorzata A. Jankowska 1, Andrzej Marciniak 2 and Tomasz Hoffmann 2 1 Poznan University
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 informationIteration. The Cake Eating Problem. Discount Factors
18 Value Function Iteration Lab Objective: Many questions have optimal answers that change over time. Sequential decision making problems are among this classification. In this lab you we learn how to
More informationOption Pricing Models for European Options
Chapter 2 Option Pricing Models for European Options 2.1 Continuous-time Model: Black-Scholes Model 2.1.1 Black-Scholes Assumptions We list the assumptions that we make for most of this notes. 1. The underlying
More informationHIGH ORDER DISCONTINUOUS GALERKIN METHODS FOR 1D PARABOLIC EQUATIONS. Ahmet İzmirlioğlu. BS, University of Pittsburgh, 2004
HIGH ORDER DISCONTINUOUS GALERKIN METHODS FOR D PARABOLIC EQUATIONS by Ahmet İzmirlioğlu BS, University of Pittsburgh, 24 Submitted to the Graduate Faculty of Art and Sciences in partial fulfillment of
More informationA potentially useful approach to model nonlinearities in time series is to assume different behavior (structural break) in different subsamples
1.3 Regime switching models A potentially useful approach to model nonlinearities in time series is to assume different behavior (structural break) in different subsamples (or regimes). If the dates, the
More information1 Residual life for gamma and Weibull distributions
Supplement to Tail Estimation for Window Censored Processes Residual life for gamma and Weibull distributions. Gamma distribution Let Γ(k, x = x yk e y dy be the upper incomplete gamma function, and let
More informationA Robust Option Pricing Problem
IMA 2003 Workshop, March 12-19, 2003 A Robust Option Pricing Problem Laurent El Ghaoui Department of EECS, UC Berkeley 3 Robust optimization standard form: min x sup u U f 0 (x, u) : u U, f i (x, u) 0,
More information13.3 A Stochastic Production Planning Model
13.3. A Stochastic Production Planning Model 347 From (13.9), we can formally write (dx t ) = f (dt) + G (dz t ) + fgdz t dt, (13.3) dx t dt = f(dt) + Gdz t dt. (13.33) The exact meaning of these expressions
More informationDepartment of Mathematics. Mathematics of Financial Derivatives
Department of Mathematics MA408 Mathematics of Financial Derivatives Thursday 15th January, 2009 2pm 4pm Duration: 2 hours Attempt THREE questions MA408 Page 1 of 5 1. (a) Suppose 0 < E 1 < E 3 and E 2
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 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 informationPh.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2017
Ph.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2017 The time limit for this exam is four hours. The exam has four sections. Each section includes two questions.
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 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 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 informationAn option-theoretic valuation model for residential mortgages with stochastic conditions and discount factors
Graduate Theses and Dissertations Iowa State University Capstones, Theses and Dissertations 2 An option-theoretic valuation model for residential mortgages with stochastic conditions and discount factors
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 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 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 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 informationMAT 4250: Lecture 1 Eric Chung
1 MAT 4250: Lecture 1 Eric Chung 2Chapter 1: Impartial Combinatorial Games 3 Combinatorial games Combinatorial games are two-person games with perfect information and no chance moves, and with a win-or-lose
More informationEquilibrium payoffs in finite games
Equilibrium payoffs in finite games Ehud Lehrer, Eilon Solan, Yannick Viossat To cite this version: Ehud Lehrer, Eilon Solan, Yannick Viossat. Equilibrium payoffs in finite games. Journal of Mathematical
More informationYao s Minimax Principle
Complexity of algorithms The complexity of an algorithm is usually measured with respect to the size of the input, where size may for example refer to the length of a binary word describing the input,
More informationAmerican Options; an American delayed- Exercise model and the free boundary. Business Analytics Paper. Nadra Abdalla
American Options; an American delayed- Exercise model and the free boundary Business Analytics Paper Nadra Abdalla [Geef tekst op] Pagina 1 Business Analytics Paper VU University Amsterdam Faculty of Sciences
More informationIntensity-based framework for optimal stopping
Intensity-based framework for optimal stopping problems Min Dai National University of Singapore Yue Kuen Kwok Hong Kong University of Science and Technology Hong You National University of Singapore Abstract
More informationExercise List: Proving convergence of the (Stochastic) Gradient Descent Method for the Least Squares Problem.
Exercise List: Proving convergence of the (Stochastic) Gradient Descent Method for the Least Squares Problem. Robert M. Gower. October 3, 07 Introduction This is an exercise in proving the convergence
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 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 informationc 2011 Society for Industrial and Applied Mathematics
SIAM J. SCI. COMPUT. Vol. 33, No. 5, pp. 244 268 c 20 Society for Industrial and Applied Mathematics METHODS FOR PRICING AMERICAN OPTIONS UNDER REGIME SWITCHING Y. HUANG, P. A. FORSYTH, AND G. LABAHN Abstract.
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 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 information5 Numerical Solution of Linear Systems
5 Numerical Solution of Linear Systems In this chapter we present several methods for solving linear systems of the form Ax = b. HereA is a (m m) matrix and both x and b are m-dimensional vectors. Our
More informationStock Loan Valuation Under Brownian-Motion Based and Markov Chain Stock Models
Stock Loan Valuation Under Brownian-Motion Based and Markov Chain Stock Models David Prager 1 1 Associate Professor of Mathematics Anderson University (SC) Based on joint work with Professor Qing Zhang,
More informationControl-theoretic framework for a quasi-newton local volatility surface inversion
Control-theoretic framework for a quasi-newton local volatility surface inversion Gabriel Turinici To cite this version: Gabriel Turinici. Control-theoretic framework for a quasi-newton local volatility
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 informationShort-time-to-expiry expansion for a digital European put option under the CEV model. November 1, 2017
Short-time-to-expiry expansion for a digital European put option under the CEV model November 1, 2017 Abstract In this paper I present a short-time-to-expiry asymptotic series expansion for a digital European
More informationApplied Mathematics Letters. On local regularization for an inverse problem of option pricing
Applied Mathematics Letters 24 (211) 1481 1485 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml On local regularization for an inverse
More informationBoundary conditions for options
Boundary conditions for options Boundary conditions for options can refer to the non-arbitrage conditions that option prices has to satisfy. If these conditions are broken, arbitrage can exist. to the
More informationMacroeconomics and finance
Macroeconomics and finance 1 1. Temporary equilibrium and the price level [Lectures 11 and 12] 2. Overlapping generations and learning [Lectures 13 and 14] 2.1 The overlapping generations model 2.2 Expectations
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 information16 MAKING SIMPLE DECISIONS
247 16 MAKING SIMPLE DECISIONS Let us associate each state S with a numeric utility U(S), which expresses the desirability of the state A nondeterministic action A will have possible outcome states Result
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 informationTEACHING NOTE 97-02: OPTION PRICING USING FINITE DIFFERENCE METHODS
TEACHING NOTE 970: OPTION PRICING USING FINITE DIFFERENCE METHODS Version date: August 1, 008 C:\Classes\Teaching Notes\TN970doc Under the appropriate assumptions, the price of an option is given by the
More informationReal Option Analysis for Adjacent Gas Producers to Choose Optimal Operating Strategy, such as Gas Plant Size, Leasing rate, and Entry Point
Real Option Analysis for Adjacent Gas Producers to Choose Optimal Operating Strategy, such as Gas Plant Size, Leasing rate, and Entry Point Gordon A. Sick and Yuanshun Li October 3, 4 Tuesday, October,
More informationAccelerated Stochastic Gradient Descent Praneeth Netrapalli MSR India
Accelerated Stochastic Gradient Descent Praneeth Netrapalli MSR India Presented at OSL workshop, Les Houches, France. Joint work with Prateek Jain, Sham M. Kakade, Rahul Kidambi and Aaron Sidford Linear
More informationDirect Methods for linear systems Ax = b basic point: easy to solve triangular systems
NLA p.1/13 Direct Methods for linear systems Ax = b basic point: easy to solve triangular systems... 0 0 0 etc. a n 1,n 1 x n 1 = b n 1 a n 1,n x n solve a n,n x n = b n then back substitution: takes n
More informationMonte Carlo Methods in Structuring and Derivatives Pricing
Monte Carlo Methods in Structuring and Derivatives Pricing Prof. Manuela Pedio (guest) 20263 Advanced Tools for Risk Management and Pricing Spring 2017 Outline and objectives The basic Monte Carlo algorithm
More informationMatrix-based numerical modelling of financial differential equations. Robert Piché and Juho Kanniainen
Matrix-based numerical modelling of financial differential equations Robert Piché and Juho Kanniainen Tampere University of Technology PO Box 553, FI-33101 Tampere, Finland E-mail: robert.piche@tut.fi
More informationChapter 20: An Introduction to ADI and Splitting Schemes
Chapter 20: An Introduction to ADI and Splitting Schemes 20.1INTRODUCTION AND OBJECTIVES In this chapter we discuss how to apply finite difference schemes to approximate the solution of multidimensional
More informationOptimization for Chemical Engineers, 4G3. Written midterm, 23 February 2015
Optimization for Chemical Engineers, 4G3 Written midterm, 23 February 2015 Kevin Dunn, kevin.dunn@mcmaster.ca McMaster University Note: No papers, other than this test and the answer booklet are allowed
More informationFinancial Optimization ISE 347/447. Lecture 15. Dr. Ted Ralphs
Financial Optimization ISE 347/447 Lecture 15 Dr. Ted Ralphs ISE 347/447 Lecture 15 1 Reading for This Lecture C&T Chapter 12 ISE 347/447 Lecture 15 2 Stock Market Indices A stock market index is a statistic
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 informationMTH6154 Financial Mathematics I Interest Rates and Present Value Analysis
16 MTH6154 Financial Mathematics I Interest Rates and Present Value Analysis Contents 2 Interest Rates 16 2.1 Definitions.................................... 16 2.1.1 Rate of Return..............................
More informationPOMDPs: Partially Observable Markov Decision Processes Advanced AI
POMDPs: Partially Observable Markov Decision Processes Advanced AI Wolfram Burgard Types of Planning Problems Classical Planning State observable Action Model Deterministic, accurate MDPs observable stochastic
More informationarxiv: v1 [q-fin.cp] 1 Nov 2016
Essentially high-order compact schemes with application to stochastic volatility models on non-uniform grids arxiv:1611.00316v1 [q-fin.cp] 1 Nov 016 Bertram Düring Christof Heuer November, 016 Abstract
More informationIs Greedy Coordinate Descent a Terrible Algorithm?
Is Greedy Coordinate Descent a Terrible Algorithm? Julie Nutini, Mark Schmidt, Issam Laradji, Michael Friedlander, Hoyt Koepke University of British Columbia Optimization and Big Data, 2015 Context: Random
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 informationOption Pricing. Chapter Discrete Time
Chapter 7 Option Pricing 7.1 Discrete Time In the next section we will discuss the Black Scholes formula. To prepare for that, we will consider the much simpler problem of pricing options when there are
More informationNo-Arbitrage Conditions for a Finite Options System
No-Arbitrage Conditions for a Finite Options System Fabio Mercurio Financial Models, Banca IMI Abstract In this document we derive necessary and sufficient conditions for a finite system of option prices
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 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 informationThe Correlation Smile Recovery
Fortis Bank Equity & Credit Derivatives Quantitative Research The Correlation Smile Recovery E. Vandenbrande, A. Vandendorpe, Y. Nesterov, P. Van Dooren draft version : March 2, 2009 1 Introduction Pricing
More informationExam Quantitative Finance (35V5A1)
Exam Quantitative Finance (35V5A1) Part I: Discrete-time finance Exercise 1 (20 points) a. Provide the definition of the pricing kernel k q. Relate this pricing kernel to the set of discount factors D
More informationA local RBF method based on a finite collocation approach
Boundary Elements and Other Mesh Reduction Methods XXXVIII 73 A local RBF method based on a finite collocation approach D. Stevens & H. Power Department of Mechanical Materials and Manufacturing Engineering,
More information