The stochastic modelling 5
|
|
- Mervyn Gordon Mathews
- 7 years ago
- Views:
Transcription
1 The stochastic modelling 5 Karol Dziedziul February 211 Gdansk
2 A schedule of the lecture. Stochastic differential equations Ito calculus, Ito process Ornstein - Uhlenbeck process Heston model
3 Stochastic differential equations Stochastic differential equations studied in this lecture are typically of the form dx t = a(t, X t )dt + b(t, X t )dw t, (1) where W t is the Wiener process. More explicitly the equation (1) has to be understood as X t = X + a(s, X s )ds + b(s, X s )dw s. Both integrals exist: the first in the sense of the Lebesgue integration, the second in the sense of the Ito integration when the functions a and b have to fulfill integrability conditions ( a(s, X s ) + b(s, X s ) 2 )ds, a.s., t.
4 Usually we assume that the functions a, b satisfy the Lipschitz condition in the second variable, i.e. there is constant K such that for all x, y R, t [, T ] a(t, x) a(t, y) + b(t, x) b(t, y) K x y. We also assume that there is constant K such for all x R, t [, T ] a(t, x) + b(t, x) K(1 + x ). In that cases the equation (1) has a unique solution X t, t T (so called a strong solution). In spite of the difficulties of the mathematical language there is a numerical procedure which can be helpful to solve (1) and to catch the ideas. This procedure is based on a discrete version of the equation (1). In discrete time = t < t 1 < t 2 < < t n = T we consider a recursive formula X tj+1 X tj = a tj (t j+1 t j ) + b tj (W tj+1 W tj ).
5 Geometric Brownian motion Symbolically a recursive formula is written as X t = a(t, X t ) t + b(t, X t ) W t. In this way we obtain the piecewise linear sample path of the process. If sup{ t j t j 1 : j n} then we obtain the solution of (1). We apply this method to the Geometric Brownian motion. Samuelson provided the arguments that the price of an asset S t follows the Geometric Brownian motion, i.e. ds t = S t µdt + S t σdw t. (2) Use Program 28 to find out that the price of the asstet S T /S (the solution of (2)) in the moment T has lognormal distribution, where S is an initial value of the asset.
6 Lognormal distribution. We say that a positive random variable X has the lognormal distribution with parameter m, s, if log X, (log X = log e X = ln X ) has the normal distribution N(m, s 2 ). Moreover, the mean and the variance are given by EX = e m+s2 /2, VarX = (e s2 1)e 2m+s2. Using the Itô calculus we will show that the exact solution of (2) is given by ln(s T /S ) = (µ σ 2 /2)T + σw T. Compare the estimated parameters of solution S T /S : s- a shape and m -a scale parameters in Program 28 with the exact value m = µ σ 2, s = σ.
7 Itô process. Let us define the Ito processes, the important class of stochastic processes, using the differential form: More explicitly dy t = a t dt + b t dw t. (3) Y t = Y + a s ds + b s dw s. We assume that the processes a, b satisfy the integrability conditions: T a s ds < a.s. and T b s 2 ds < a.s.
8 Itô calculus. Theorem (Itô formula) Let Y i t Let, i = 1, 2 be two Itô processes dy i t = a i tdt + b i tdw t. F : [, t] R 2 R be twice differentiable. Then ξ t = F (t, Y 1 t, Y 2 t ) is the Ito process such that dξ t = F t (t, Y t 1, Yt 2 )dt + F (t, Yt 1, Yt 2 )dyt 1 + F (t, Yt 1, Yt 2 )dyt 2 x 1 x F 2 x1 2 (t, Yt 1, Yt 2 )(bt 1 ) 2 dt F 2 x2 2 (t, Yt 1, Yt 2 )(bt 2 ) 2 dt + 2 F x 1 x 2 (t, Y 1 t, Y 2 t )b 1 t b 2 t dt.
9 Geometric Brownian motion We will use the Itô formula to check that the process ln(s t /S ) = (µ σ 2 /2)t + σw t = (µ σ 2 /2)ds + σdw s. or equivalently S t = S e (µ σ2 /2)ds+ σdws. is the solution of the Geometric Brownian motion, i.e. solution of the equation ds t = S t µdt + S t σdw t. (4)
10 We define Y t = (µ σ 2 /2)t + σw t = Note that Y t is the Ito process and S t = S exp(y t ). (µ σ 2 /2)ds + σdw s. The function F (x) = S e x is a smooth function, so twice differentiable and F (x) = S e x, F (x) = S e x. By the Itô formula and (4) we get ds t = S e Yt dy t S e Yt σ 2 dt = S e Yt ((µ σ 2 /2)dt + σdw t ) S e Yt σ 2 dt = S t µdt + S t σdw t.
11 Ornstein - Uhlenbeck process In the physics, a relaxation means the return of a perturbed system into equilibrium. Such processess are modeled by Ornstein - Uhlenbeck process given by dx t = θ(µ X t )dt + σdw t, (5) where θ, σ > and µ R. To solve this equation let us consider f (X t, t) = X t e θt, where X t is the solution of (5). From the Itô formula we obtain Since dx t is the solution of (5) then df (X t, t) = θx t e θt dt + e θt dx t. df (X t, t) = θx t e θt dt + e θt (θ(µ X t )dt + σdw t ) = θx t e θt dt + e θt θµdt e θt θx t dt + e θt σdw t Thus = e θt θµdt + e θt σdw t X t e θt = X + e θs θµds + e θs σdw s.
12 Consequently, the solution of (5), i.e. the Ornstein - Uhlenbeck process is given by X t = X e θt + µ(1 e θt ) + e θ(s t) σdw s. From the Itô integral theory we get that the stochastic process U t = e θ(s t) σdw s is the Gaussian process, U t N(, VarU t ), where Hence VarU t = ( σe θt) 2 VarU t = σ 2 e 2θt e2θt 1 2θ e 2θs ds. = σ 2 1 e 2θt. 2θ
13 Consequetly, EX t = X e θt + µ(1 e θt ), VarX t = VarU t. Since θ > we get that lim EX t = µ, (6) t lim VarX t = σ2 t 2θ. (7) Use Program 29 to create a sample path of the Ornstein - Uhlenbeck process for large n. Note that a trajectories according to (6) return to µ and their oscilations stabilize according to (7).
14 Heston model. One of the generalizations of the Samuelson model, is the Heston model. We assume that the price of an asset is given by where σ t is the Ornsteina-Uhlenbeck process ds t = µs t dt + σ t S t dz 1 t, (8) dσ t = βσ t dt + δdz 2 t. (9) We assume that the Wiener processes (Z 1 t, Z 2 t ) are correlated, i.e Cov(Z 1 t, Z 2 t ) = ρt. To obtain the equivalent formula of Heston model let us define From the Itô formula ν t = σ 2 t. dν t = 2σ t dσ t δ2 dt. Taking (9) and puting σ t = ν t we get dν t = 2 ν t ( β ν t dt + δdz 2 t ) δ2 dt.
15 Thus dν t = 2δ ν t dz 2 t + (δ 2 βν t )dt = 2δ ν t dz 2 t + β(δ 2 /β ν t )dt We obtain the system of equations equivalent to (8) i (9), ds t = µs t dt + ν t S t dzt 1 dν t = 2β(δ 2 /β ν t )dt + δ ν t dz 2 t. A solution of (9) is Ornstein Uhlenbeck process σ t = σ e βt + δ e β(s t) dz 2 s Using similar calculations as in the solution of the Geometric Brownian Motion we get { } S t = S exp (µ σ2 s 2 )ds + σ s dzs 1. Use Program 3 to create sample paths of S t and ν t in the Heston model.
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 informationNotes on Black-Scholes Option Pricing Formula
. Notes on Black-Scholes Option Pricing Formula by De-Xing Guan March 2006 These notes are a brief introduction to the Black-Scholes formula, which prices the European call options. The essential reading
More informationMoreover, under the risk neutral measure, it must be the case that (5) r t = µ t.
LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing
More informationThe Black-Scholes Formula
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Black-Scholes Formula These notes examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the
More information第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model
1 第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model Outline 有 关 股 价 的 假 设 The B-S Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American
More informationON THE RUIN PROBABILITY OF AN INSURANCE COMPANY DEALING IN A BS-MARKET
Teor Imov rtamatemstatist Theor Probability and Math Statist Vip 74, 6 No 74, 7, Pages 11 3 S 94-97)693-X Article electronically published on June 5, 7 ON THE RUIN PROBABILITY OF AN INSURANCE COMPANY DEALING
More informationARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida
ARBITRAGE-FREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic
More informationMonte 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 informationMerton-Black-Scholes model for option pricing. Peter Denteneer. 22 oktober 2009
Merton-Black-Scholes model for option pricing Instituut{Lorentz voor Theoretische Natuurkunde, LION, Universiteit Leiden 22 oktober 2009 With inspiration from: J. Tinbergen, T.C. Koopmans, E. Majorana,
More informationVolatility Index: VIX vs. GVIX
I. II. III. IV. Volatility Index: VIX vs. GVIX "Does VIX Truly Measure Return Volatility?" by Victor Chow, Wanjun Jiang, and Jingrui Li (214) An Ex-ante (forward-looking) approach based on Market Price
More informationDiusion processes. Olivier Scaillet. University of Geneva and Swiss Finance Institute
Diusion processes Olivier Scaillet University of Geneva and Swiss Finance Institute Outline 1 Brownian motion 2 Itô integral 3 Diusion processes 4 Black-Scholes 5 Equity linked life insurance 6 Merton
More informationTULLIE, TRACEY ANDREW. Variance Reduction for Monte Carlo Simulation of. for the evolution of an asset price under constant volatility.
Abstract TULLIE, TRACEY ANDREW. Variance Reduction for Monte Carlo Simulation of European, American or Barrier Options in a Stochastic Volatility Environment. (Under the direction of Jean-Pierre Fouque.)
More informationThe Black-Scholes pricing formulas
The Black-Scholes pricing formulas Moty Katzman September 19, 2014 The Black-Scholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock
More informationEXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL
EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL Exit Time problems and Escape from a Potential Well Escape From a Potential Well There are many systems in physics, chemistry and biology that exist
More informationIntroduction to Arbitrage-Free Pricing: Fundamental Theorems
Introduction to Arbitrage-Free Pricing: Fundamental Theorems Dmitry Kramkov Carnegie Mellon University Workshop on Interdisciplinary Mathematics, Penn State, May 8-10, 2015 1 / 24 Outline Financial market
More informationEssential Topic: Continuous cash flows
Essential Topic: Continuous cash flows Chapters 2 and 3 The Mathematics of Finance: A Deterministic Approach by S. J. Garrett CONTENTS PAGE MATERIAL Continuous payment streams Example Continuously paid
More informationBlack-Scholes Equation for Option Pricing
Black-Scholes Equation for Option Pricing By Ivan Karmazin, Jiacong Li 1. Introduction In early 1970s, Black, Scholes and Merton achieved a major breakthrough in pricing of European stock options and there
More informationAnalytic Approximations for Multi-Asset Option Pricing
Analytic Approximations for Multi-Asset Option Pricing Carol Alexander ICMA Centre, University of Reading Aanand Venkatramanan ICMA Centre, University of Reading First Version: March 2008 This Version:
More informationMathematical 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 informationExample 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 informationThe Black-Scholes-Merton Approach to Pricing Options
he Black-Scholes-Merton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the Black-Scholes-Merton approach to determining
More informationLECTURE 9: A MODEL FOR FOREIGN EXCHANGE
LECTURE 9: A MODEL FOR FOREIGN EXCHANGE 1. Foreign Exchange Contracts There was a time, not so long ago, when a U. S. dollar would buy you precisely.4 British pounds sterling 1, and a British pound sterling
More informationLecture 6 Black-Scholes PDE
Lecture 6 Black-Scholes PDE Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Pricing function Let the dynamics of underlining S t be given in the risk-neutral measure Q by If the contingent
More informationSensitivity analysis of European options in jump-diffusion models via the Malliavin calculus on the Wiener space
Sensitivity analysis of European options in jump-diffusion models via the Malliavin calculus on the Wiener space Virginie Debelley and Nicolas Privault Département de Mathématiques Université de La Rochelle
More informationA Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting Model
Applied Mathematical Sciences, vol 8, 14, no 143, 715-7135 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/11988/ams144644 A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting
More informationMicroeconomic Theory: Basic Math Concepts
Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts
More informationThe 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 informationTHE BLACK-SCHOLES MODEL AND EXTENSIONS
THE BLAC-SCHOLES MODEL AND EXTENSIONS EVAN TURNER Abstract. This paper will derive the Black-Scholes pricing model of a European option by calculating the expected value of the option. We will assume that
More informationMaximum 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 informationOverview of Monte Carlo Simulation, Probability Review and Introduction to Matlab
Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?
More informationOn 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 informationHedging 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 informationChapter 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= δx x + δy y. df ds = dx. ds y + xdy ds. Now multiply by ds to get the form of the equation in terms of differentials: df = y dx + x dy.
ERROR PROPAGATION For sums, differences, products, and quotients, propagation of errors is done as follows. (These formulas can easily be calculated using calculus, using the differential as the associated
More informationAsian 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 informationLecture. S t = S t δ[s t ].
Lecture In real life the vast majority of all traded options are written on stocks having at least one dividend left before the date of expiration of the option. Thus the study of dividends is important
More informationNumerical Methods for Pricing Exotic Options
Imperial College London Department of Computing Numerical Methods for Pricing Exotic Options by Hardik Dave - 00517958 Supervised by Dr. Daniel Kuhn Second Marker: Professor Berç Rustem Submitted in partial
More informationThe integrating factor method (Sect. 2.1).
The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable
More informationA spot price model feasible for electricity forward pricing Part II
A spot price model feasible for electricity forward pricing Part II Fred Espen Benth Centre of Mathematics for Applications (CMA) University of Oslo, Norway Wolfgang Pauli Institute, Wien January 17-18
More informationMath 526: Brownian Motion Notes
Math 526: Brownian Motion Notes Definition. Mike Ludkovski, 27, all rights reserved. A stochastic process (X t ) is called Brownian motion if:. The map t X t (ω) is continuous for every ω. 2. (X t X t
More informationOption Pricing. Stefan Ankirchner. January 20, 2014. 2 Brownian motion and Stochastic Calculus
Option Pricing Stefan Ankirchner January 2, 214 1 The Binomial Model 2 Brownian motion and Stochastic Calculus We next recall some basic results from Stochastic Calculus. We do not prove most of the results.
More information1 Error in Euler s Method
1 Error in Euler s Method Experience with Euler s 1 method raises some interesting questions about numerical approximations for the solutions of differential equations. 1. What determines the amount of
More informationCS 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 informationSensitivity analysis of utility based prices and risk-tolerance wealth processes
Sensitivity analysis of utility based prices and risk-tolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,
More informationDoes Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem
Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial
More informationLecture 8. Confidence intervals and the central limit theorem
Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of
More informationSingle period modelling of financial assets
Single period modelling of financial assets Pål Lillevold and Dag Svege 17. 10. 2002 Single period modelling of financial assets 1 1 Outline A possible - and common - approach to stochastic modelling of
More informationANALYZING INVESTMENT RETURN OF ASSET PORTFOLIOS WITH MULTIVARIATE ORNSTEIN-UHLENBECK PROCESSES
ANALYZING INVESTMENT RETURN OF ASSET PORTFOLIOS WITH MULTIVARIATE ORNSTEIN-UHLENBECK PROCESSES by Xiaofeng Qian Doctor of Philosophy, Boston University, 27 Bachelor of Science, Peking University, 2 a Project
More informationPricing catastrophe options in incomplete market
Pricing catastrophe options in incomplete market Arthur Charpentier arthur.charpentier@univ-rennes1.fr Actuarial and Financial Mathematics Conference Interplay between finance and insurance, February 2008
More informationModelling of Short Term Interest Rate Based on Fractional Relaxation Equation
Vol. 114 (28) ACTA PHYSICA POLONICA A No. 3 Proceedings of the 3rd Polish Symposium on Econo- and Sociophysics, Wroc law 27 Modelling of Short Term Interest Rate Based on Fractional Relaxation Equation
More informationOption Pricing. Chapter 4 Including dividends in the BS model. Stefan Ankirchner. University of Bonn. last update: 6th November 2013
Option Pricing Chapter 4 Including dividends in the BS model Stefan Ankirchner University of Bonn last update: 6th November 2013 Stefan Ankirchner Option Pricing 1 Dividend payments So far: we assumed
More informationCHAPTER IV - BROWNIAN MOTION
CHAPTER IV - BROWNIAN MOTION JOSEPH G. CONLON 1. Construction of Brownian Motion There are two ways in which the idea of a Markov chain on a discrete state space can be generalized: (1) The discrete time
More informationProbability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationIntroduction to Probability
Introduction to Probability EE 179, Lecture 15, Handout #24 Probability theory gives a mathematical characterization for experiments with random outcomes. coin toss life of lightbulb binary data sequence
More informationBROWNIAN MOTION DEVELOPMENT FOR MONTE CARLO METHOD APPLIED ON EUROPEAN STYLE OPTION PRICE FORECASTING
International journal of economics & law Vol. 1 (2011), No. 1 (1-170) BROWNIAN MOTION DEVELOPMENT FOR MONTE CARLO METHOD APPLIED ON EUROPEAN STYLE OPTION PRICE FORECASTING Petar Koĉović, Fakultet za obrazovanje
More informationLecture 17: Conformal Invariance
Lecture 17: Conformal Invariance Scribe: Yee Lok Wong Department of Mathematics, MIT November 7, 006 1 Eventual Hitting Probability In previous lectures, we studied the following PDE for ρ(x, t x 0 ) that
More informationHow To Price A Life Insurance Portfolio With A Mortality Risk
Indifference pricing of a life insurance portfolio with systematic mortality risk in a market with an asset driven by a Lévy process Łukasz Delong Institute of Econometrics, Division of Probabilistic Methods
More informationConstant Elasticity of Variance (CEV) Option Pricing Model:Integration and Detailed Derivation
Constant Elasticity of Variance (CEV) Option Pricing Model:Integration and Detailed Derivation Ying-Lin Hsu Department of Applied Mathematics National Chung Hsing University Co-authors: T. I. Lin and C.
More informationLecture 12: The Black-Scholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 12: The Black-Scholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The Black-Scholes-Merton Model
More informationMean Reversion versus Random Walk in Oil and Natural Gas Prices
Mean Reversion versus Random Walk in Oil and Natural Gas Prices Hélyette Geman Birkbeck, University of London, United Kingdom & ESSEC Business School, Cergy-Pontoise, France hgeman@ems.bbk.ac.uk Summary.
More informationAlternative Price Processes for Black-Scholes: Empirical Evidence and Theory
Alternative Price Processes for Black-Scholes: Empirical Evidence and Theory Samuel W. Malone April 19, 2002 This work is supported by NSF VIGRE grant number DMS-9983320. Page 1 of 44 1 Introduction This
More informationMarginal value approach to pricing temperature options and its empirical example on daily temperatures in Nagoya. Yoshiyuki Emoto and Tetsuya Misawa
Discussion Papers in Economics No. Marginal value approach to pricing temperature options and its empirical example on daily temperatures in Nagoya Yoshiyuki Emoto and Tetsuya Misawa August 2, 2006 Faculty
More informationSome remarks on two-asset options pricing and stochastic dependence of asset prices
Some remarks on two-asset options pricing and stochastic dependence of asset prices G. Rapuch & T. Roncalli Groupe de Recherche Opérationnelle, Crédit Lyonnais, France July 16, 001 Abstract In this short
More informationNumerical Solution of Differential Equations
Numerical Solution of Differential Equations Dr. Alvaro Islas Applications of Calculus I Spring 2008 We live in a world in constant change We live in a world in constant change We live in a world in constant
More information1 IEOR 4700: Introduction to stochastic integration
Copyright c 7 by Karl Sigman 1 IEOR 47: Introduction to stochastic integration 1.1 Riemann-Stieltjes integration Recall from calculus how the Riemann integral b a h(t)dt is defined for a continuous function
More informationCompensation of high interest rates by tax holidays: A real options approach
Compensation of high interest rates by tax holidays: A real options approach Vadim Arkin and Alexander Slastnikov There is an important problem how to attract investments to the real sector of economics
More informationRisk management of CPPI funds in switching regime markets.
Risk management of CPPI funds in switching regime markets. Donatien Hainaut October, 1 NSA-CRST. 945 Malako Cedex, France. mail: donatien.hainaut@ensae.fr Abstract The constant proportion portfolio insurance
More informationA note on exact moments of order statistics from exponentiated log-logistic distribution
ProbStat Forum, Volume 7, July 214, Pages 39 44 ISSN 974-3235 ProbStat Forum is an e-ournal. For details please visit www.probstat.org.in A note on exact moments of order statistics from exponentiated
More informationThe Fair Valuation of Life Insurance Participating Policies: The Mortality Risk Role
The Fair Valuation of Life Insurance Participating Policies: The Mortality Risk Role Massimiliano Politano Department of Mathematics and Statistics University of Naples Federico II Via Cinthia, Monte S.Angelo
More information3 Contour integrals and Cauchy s Theorem
3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of
More informationMonte 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 informationIntroduction to portfolio insurance. Introduction to portfolio insurance p.1/41
Introduction to portfolio insurance Introduction to portfolio insurance p.1/41 Portfolio insurance Maintain the portfolio value above a certain predetermined level (floor) while allowing some upside potential.
More information(1) Consider a European call option and a European put option on a nondividend-paying stock. You are given:
(1) Consider a European call option and a European put option on a nondividend-paying stock. You are given: (i) The current price of the stock is $60. (ii) The call option currently sells for $0.15 more
More informationDisability insurance: estimation and risk aggregation
Disability insurance: estimation and risk aggregation B. Löfdahl Department of Mathematics KTH, Royal Institute of Technology May 2015 Introduction New upcoming regulation for insurance industry: Solvency
More informationChapter 6: The Black Scholes Option Pricing Model
The Black Scholes Option Pricing Model 6-1 Chapter 6: The Black Scholes Option Pricing Model The Black Scholes Option Pricing Model 6-2 Differential Equation A common model for stock prices is the geometric
More informationBlack-Scholes Option Pricing Model
Black-Scholes Option Pricing Model Nathan Coelen June 6, 22 1 Introduction Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change,
More informationStochastic Processes LECTURE 5
128 LECTURE 5 Stochastic Processes We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that
More informationPayment streams and variable interest rates
Chapter 4 Payment streams and variable interest rates In this chapter we consider two extensions of the theory Firstly, we look at payment streams A payment stream is a payment that occurs continuously,
More informationHedging Options In The Incomplete Market With Stochastic Volatility. Rituparna Sen Sunday, Nov 15
Hedging Options In The Incomplete Market With Stochastic Volatility Rituparna Sen Sunday, Nov 15 1. Motivation This is a pure jump model and hence avoids the theoretical drawbacks of continuous path models.
More informationMark Howell Gonzaga High School, Washington, D.C.
Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,
More informationMarkovian 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 informationPricing participating policies with rate guarantees and bonuses
Pricing participating policies with rate guarantees and bonuses Chi Chiu Chu and Yue Kuen Kwok Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
More informationContents. 5 Numerical results 27 5.1 Single policies... 27 5.2 Portfolio of policies... 29
Abstract The capital requirements for insurance companies in the Solvency I framework are based on the premium and claim expenditure. This approach does not take the individual risk of the insurer into
More information1 The Brownian bridge construction
The Brownian bridge construction The Brownian bridge construction is a way to build a Brownian motion path by successively adding finer scale detail. This construction leads to a relatively easy proof
More informationELECTRICITY REAL OPTIONS VALUATION
Vol. 37 (6) ACTA PHYSICA POLONICA B No 11 ELECTRICITY REAL OPTIONS VALUATION Ewa Broszkiewicz-Suwaj Hugo Steinhaus Center, Institute of Mathematics and Computer Science Wrocław University of Technology
More informationOn exponentially ane martingales. Johannes Muhle-Karbe
On exponentially ane martingales AMAMEF 2007, Bedlewo Johannes Muhle-Karbe Joint work with Jan Kallsen HVB-Institut für Finanzmathematik, Technische Universität München 1 Outline 1. Semimartingale characteristics
More informationJung-Soon Hyun and Young-Hee Kim
J. Korean Math. Soc. 43 (2006), No. 4, pp. 845 858 TWO APPROACHES FOR STOCHASTIC INTEREST RATE OPTION MODEL Jung-Soon Hyun and Young-Hee Kim Abstract. We present two approaches of the stochastic interest
More informationThermodynamics: Lecture 2
Thermodynamics: Lecture 2 Chris Glosser February 11, 2001 1 OUTLINE I. Heat and Work. (A) Work, Heat and Energy: U = Q + W. (B) Methods of Heat Transport. (C) Infintesimal Work: Exact vs Inexact Differentials
More informationBuilding a Smooth Yield Curve. University of Chicago. Jeff Greco
Building a Smooth Yield Curve University of Chicago Jeff Greco email: jgreco@math.uchicago.edu Preliminaries As before, we will use continuously compounding Act/365 rates for both the zero coupon rates
More informationOscillatory Reduction in Option Pricing Formula Using Shifted Poisson and Linear Approximation
EPJ Web of Conferences 68, 0 00 06 (2014) DOI: 10.1051/ epjconf/ 20146800006 C Owned by the authors, published by EDP Sciences, 2014 Oscillatory Reduction in Option Pricing Formula Using Shifted Poisson
More informationOption Prices with Stochastic Interest Rates Black/Scholes and Ho/Lee unified. Jochen Wilhelm. Diskussionsbeitrag Nr. B 4 99
Herausgeber: Die Gruppe der betriebswirtschaftlichen Professoren der Wirtschaftswissenschaftlichen Fakultät der Universität Passau 943 Passau Option Prices with Stochastic Interest Rates Black/Scholes
More informationMath 461 Fall 2006 Test 2 Solutions
Math 461 Fall 2006 Test 2 Solutions Total points: 100. Do all questions. Explain all answers. No notes, books, or electronic devices. 1. [105+5 points] Assume X Exponential(λ). Justify the following two
More informationThe sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].
Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real
More information4. Complex integration: Cauchy integral theorem and Cauchy integral formulas. Definite integral of a complex-valued function of a real variable
4. Complex integration: Cauchy integral theorem and Cauchy integral formulas Definite integral of a complex-valued function of a real variable Consider a complex valued function f(t) of a real variable
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationPricing of an Exotic Forward Contract
Pricing of an Exotic Forward Contract Jirô Akahori, Yuji Hishida and Maho Nishida Dept. of Mathematical Sciences, Ritsumeikan University 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan E-mail: {akahori,
More informationThe Constant Elasticity of Variance Option Pricing Model
The Constant Elasticity of Variance Option Pricing Model John Randal A thesis submitted to the Victoria University of Wellington in partial fulfilment of the requirements for the degree of Master of Science
More informationPricing Frameworks for Securitization of Mortality Risk
1 Pricing Frameworks for Securitization of Mortality Risk Andrew Cairns Heriot-Watt University, Edinburgh Joint work with David Blake & Kevin Dowd Updated version at: http://www.ma.hw.ac.uk/ andrewc 2
More informationHow To Price A Call Option
Now by Itô s formula But Mu f and u g in Ū. Hence τ θ u(x) =E( Mu(X) ds + u(x(τ θ))) 0 τ θ u(x) E( f(x) ds + g(x(τ θ))) = J x (θ). 0 But since u(x) =J x (θ ), we consequently have u(x) =J x (θ ) = min
More information