Explicit Option Pricing Formula for a MeanReverting Asset in Energy Markets


 Brooke Richardson
 3 years ago
 Views:
Transcription
1 Explicit Option Pricing Formula for a MeanReverting Asset in Energy Markets Anatoliy Swishchuk Mathematical & Computational Finance Lab Dept of Math & Stat, University of Calgary, Calgary, AB, Canada QMF 2007 Conference Sydney, Australia December 1215, 2007 This research is supported by MITACS and NSERC
2 Outline MeanReverting Models (MRM): Deterministic vs. Stochastic MRM in Finance Markets: Variances or Volatilities (Not Asset Prices) MRM in Energy Markets: Asset Prices Change of Time Method (CTM) MeanReverting Model (MRM) Option Pricing Formula Drawback of OneFactor Models Future Work
3 Motivations for the Work Paper: Javaheri, Wilmott and Haug (2002) GARCH and Volatility Swaps, Wilmott Magazine, Jan Issue (they applied PDE approach to find a volatility swap for MRM and asked about the possible option pricing formula Paper: Bos, Ware and Pavlov (2002) On a SemiSpectral Method for Pricing an Option on a MeanReverting Asset, Quantit. Finance J. (PDE approach, semispectral method to calculate numerically the solution)
4 Wilmott, Javaheri & Haug (2002) Model Wilmott, Javaheri & Haug (GARCH and Volatility Swaps, Wilmott Magazine, 2002) volatility swap for continuoustime GARCH(1,1) model
5 M. Yor s Results M. Yor On some exponential functions of Brownian motion, Adv. In Applied Probab., v. 24, No. 3, (1992), started the research for the integral of an exponential Brownian motion H. Matsumoto, M. Yor Exponential Functionals of Brownian motion, I: Probability laws at fixed time, Probability Surveys, v. 2 (2005), there is still no closed form probability density function, while the best result is a function with a double integral.
6 MeanReversion Effect Guitar String Analogy: if we pluck the guitar string, the string will revert to its place of equilibrium To measure how quickly this reversion back to the equilibrium location would happen we had to pluck the string Similarly, the only way to measure mean reversion is when the variances of asset prices in financial markets and asset prices in energy markets get plucked away from their nonevent levels and we observe them go back to more or less the levels they started from
7 The MeanReverting Deterministic Process
8 MeanReverting Plot (a=4.6,l=2.5)
9 Meaning of MeanReverting Parameter The greater the meanreverting parameter value, a, the greater is the pull back to the equilibrium level For a daily variable change, the change in time, dt, in annualized terms is given by 1/365 If a=365, the mean reversion would act so quickly as to bring the variable back to its equilibrium within a single day The value of 365/a gives us an idea of how quickly the variable takes to get back to the equilibriumin days
10 MeanReverting Stochastic Process
11 MeanReverting Models in Financial Markets Stock (asset) Prices follow geometric Brownian motion The Variance of Stock Price follows MeanReverting Models Example: Heston Model
12 MeanReverting Models in Energy Markets Asset Prices follow MeanReverting Stochastic Processes Example: ContinuousTime GARCH Model (or Pilipovic OneFactor Model)
13 MeanReverting Models in Energy Markets
14 Change of Time: Definition and Examples Change of Timechange time from t to a nonnegative process T(t) with nondecreasing sample paths Example1 (Subordinator): X(t) and T(t)>0 are some processes, then X(T(t)) is subordinated to X(t); T(t) is change of time Example 2 (TimeChanged Brownian Motion): M(t)=B(T(t)), B(t)Brownian motion Example 3 (Product Process):
15 TimeChanged Brownian Motion by Bochner Bochner (1949) ( Diffusion Equation and Stochastic Process, Proc. N.A.S. USA, v. 35)introduced the notion of change of time (CT) (timechanged Brownian motion) Bochner (1955) ( Harmonic Analysis and the Theory of Probability, UCLA Press, 176)further development of CT
16 Change of Time: First Intro into Financial Economics Clark (1973) ( A Subordinated Stochastic Process Model with Fixed Variance for Speculative Prices, Econometrica, 41, )introduced Bochner s (1949) timechanged Brownian motion into financial economics: He wrote down a model for the logprice M as M(t)=B(T(t)), where B(t) is Brownian motion, T(t) is timechange (B and T are independent)
17 Change of Time: Short History. I. Feller (1966) ( An Introduction to Probability Theory, vol. II, NY: Wiley)introduced subordinated processes X(T(t)) with Markov process X(t) and T(t) as a process with independent increments (i.e., Poisson process); T(t) was called randomized operational time Johnson (1979) ( Option Pricing When the Variance Rate is Changing, working paper, UCLA) introduced timechanged SVM in continuous time Johnson & Shanno (1987) ( Option Pricing When the Variance is Changing, J. of Finan. & Quantit. Analysis, 22, )studied the pricing of options using timechanging SVM
18 Change of Time: Short History. II. Ikeda & Watanabe (1981) ( SDEs and Diffusion Processes, NorthHolland Publ. Co)introduced and studied CTM for the solution of SDEs BarndorffNielsen, Nicolato & Shephard (2003) ( Some recent development in stochastic volatility modelling )review and put in context some of their recent work on stochastic volatility (SV) modelling, including the relationship between subordination and SV (random timechronometer) Carr, Geman, Madan & Yor (2003) ( SV for Levy Processes, mathematical Finance, vol.13)used subordinated processes to construct SV for Levy Processes (T(t)business time)
19 CT and Embedding Problem Embedding Problem was first terated by Skorokhod (1965)sum of any sequence of i.r.v. with mean zero and finite variation could be embedded in Brownian motion (BM) using stopping time Dambis (1965) and Dubis and Schwartz (1965)every continuous martingale could be timechanged BM Huff (1969)every processes of pathwise bounded variation could be embedded in BM Monroe (1972)every right continuous martingale could be embedded in a BM Monroe (1978)local martingale can be embedded in BM
20 Change of Time: Simplest (Martingale) Case
21 Change of Time: Ito Integral s Case
22 Change of Time: SDE s Case
23 Geometric Brownian Motion SVM
24 Change of Time Method
25 Connection between phi_t and phi_t^(1)
26 Solution for GBM Equation Using Change of Time
27 Explicit Expression for
28 MeanReverting SV Model
29 Solution of MRM by CTM
30 Explicit Expression for
31 Explicit Expression for
32 Comparison: Solution of GBM & MRM GBM MRM
33 Explicit Expression for S(t) where
34 Properties of
35 Properties of
36 Properties of eta(t)
37 Properties of Eta(t). II.
38 Mean Value of MRM S(t)
39 Dependence of ES(t) on T
40 Dependence of ES(t) on S_0 and T
41 Variance for S(t)
42 Dependence of Variance of S(t) on S_0 and T
43 Dependence of Volatility of S(t) on S_0 and T
44 European Call Option for MRM.I.
45 European Call Option. II.
46 Expression for C_T in the case of MRM C_T=BS(T)+A(T)
47 Expression for C_T=BS(T)+A(T).II.
48 Expression for BS(T)
49 Expression for y_0 for MRM
50 Expression for A(T).I.
51 Moment generating) function of Eta(T)
52 Expression for A(T)
53 European Call Option for MRM (Explicit Formula)
54 European Call Option for MRM in RiskNeutral World
55
56
57 Dependence of C_T on T
58 Comparison of Three Solutions Heston Model MeanReverting Model BlackScholes Model
59 Comparison: Heston Model (1993)
60 Explicit Solution for CIR Process: CTM
61 Comparison: Solutions to the Three Models GBM MRM Heston model
62 Summary GBM Model 1. martingale MeanReverting Model 2. Heston Model sum of two martingales 3. martingale
63 Problem explicit expression? To calculate an option price for Heston model, for example We know all the moments at this moment, though
64 Drawback of OneFactor Mean Reverting Models The longterm mean L remains fixed over time: needs to be recalibrated on a continuous basis in order to ensure that the resulting curves are marked to market The biggest drawback is in option pricing: results in a modelimplied volatility term structure that has the volatilities going to zero as expiration time increases (spot volatilities have to be increased to nonintuitive levels so that the long term options do not lose all the volatility valueas in the marketplace they certainly do not)
65 Future Work Change of Time Method for Two Factor ContinuousTime GARCH Model
66 The End Thank You for Your Attention and Time!
Monte Carlo Methods and Models in Finance and Insurance
Chapman & Hall/CRC FINANCIAL MATHEMATICS SERIES Monte Carlo Methods and Models in Finance and Insurance Ralf Korn Elke Korn Gerald Kroisandt f r oc) CRC Press \ V^ J Taylor & Francis Croup ^^"^ Boca Raton
More 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 1718
More informationMonte Carlo Simulation of Stochastic Processes
Monte Carlo Simulation of Stochastic Processes Last update: January 10th, 2004. In this section are presented the steps to perform the simulation of the main stochastic processes used in real options applications,
More informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationMATHEMATICAL FINANCE and Derivatives (part II, PhD)
MATHEMATICAL FINANCE and Derivatives (part II, PhD) Lecturer: Prof. Dr. Marc CHESNEY Location: Time: Mon. 08.00 09.45 Uhr First lecture: 18.02.2008 Language: English Contents: Stochastic volatility models
More informationAnalytically Tractable Stochastic Stock Price Models
Archil Gulisashvili Analytically Tractable Stochastic Stock Price Models 4Q Springer Contents 1 Volatility Processes 1 1.1 Brownian Motion 1 1.2 s Geometric Brownian Motion 6 1.3 LongTime Behavior of
More informationARBITRAGEFREE OPTION PRICING MODELS. Denis Bell. University of North Florida
ARBITRAGEFREE 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 informationMaster of Mathematical Finance: Course Descriptions
Master of Mathematical Finance: Course Descriptions CS 522 Data Mining Computer Science This course provides continued exploration of data mining algorithms. More sophisticated algorithms such as support
More informationModeling the Implied Volatility Surface. Jim Gatheral Stanford Financial Mathematics Seminar February 28, 2003
Modeling the Implied Volatility Surface Jim Gatheral Stanford Financial Mathematics Seminar February 28, 2003 This presentation represents only the personal opinions of the author and not those of Merrill
More informationOnline Appendix. Supplemental Material for Insider Trading, Stochastic Liquidity and. Equilibrium Prices. by Pierre CollinDufresne and Vyacheslav Fos
Online Appendix Supplemental Material for Insider Trading, Stochastic Liquidity and Equilibrium Prices by Pierre CollinDufresne and Vyacheslav Fos 1. Deterministic growth rate of noise trader volatility
More informationOption Pricing. Prof. Dr. Svetlozar (Zari) Rachev
Option Pricing Prof. Dr. Svetlozar (Zari) Rachev Frey Family Foundation ChairProfessor, Applied Mathematics and Statistics, Stony Brook University Chief Scientific Officer, FinAnalytica Outline: Option
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 Exante (forwardlooking) approach based on Market Price
More informationOn BlackScholes Equation, Black Scholes Formula and Binary Option Price
On BlackScholes Equation, Black Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. BlackScholes Equation is derived using two methods: (1) riskneutral measure; (2)  hedge. II.
More informationStephane Crepey. Financial Modeling. A Backward Stochastic Differential Equations Perspective. 4y Springer
Stephane Crepey Financial Modeling A Backward Stochastic Differential Equations Perspective 4y Springer Part I An Introductory Course in Stochastic Processes 1 Some Classes of DiscreteTime Stochastic
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 informationMarkov modeling of Gas Futures
Markov modeling of Gas Futures p.1/31 Markov modeling of Gas Futures Leif Andersen Banc of America Securities February 2008 Agenda Markov modeling of Gas Futures p.2/31 This talk is based on a working
More informationThe BlackScholes pricing formulas
The BlackScholes pricing formulas Moty Katzman September 19, 2014 The BlackScholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock
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 informationLOGTYPE MODELS, HOMOGENEITY OF OPTION PRICES AND CONVEXITY. 1. Introduction
LOGTYPE MODELS, HOMOGENEITY OF OPTION PRICES AND CONVEXITY M. S. JOSHI Abstract. It is shown that the properties of convexity of call prices with respect to spot price and homogeneity of call prices as
More informationOn exponentially ane martingales. Johannes MuhleKarbe
On exponentially ane martingales AMAMEF 2007, Bedlewo Johannes MuhleKarbe Joint work with Jan Kallsen HVBInstitut für Finanzmathematik, Technische Universität München 1 Outline 1. Semimartingale characteristics
More informationHedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies
Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative
More informationGeneral price bounds for discrete and continuous arithmetic Asian options
General price bounds for discrete and continuous arithmetic Asian options 1 Ioannis.Kyriakou@city.ac.uk in collaboration with Gianluca Fusai 1,2 Gianluca.Fusai.1@city.ac.uk 1 Cass Business School, City
More informationQUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS
QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS L. M. Dieng ( Department of Physics, CUNY/BCC, New York, New York) Abstract: In this work, we expand the idea of Samuelson[3] and Shepp[,5,6] for
More informationThe Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models
780 w Interest Rate Models The Behavior of Bonds and Interest Rates Before discussing how a bond marketmaker would deltahedge, we first need to specify how bonds behave. Suppose we try to model a zerocoupon
More informationGeneration Asset Valuation with Operational Constraints A Trinomial Tree Approach
Generation Asset Valuation with Operational Constraints A Trinomial Tree Approach Andrew L. Liu ICF International September 17, 2008 1 Outline Power Plants Optionality  Intrinsic vs. Extrinsic Values
More informationFrom CFD to computational finance (and back again?)
computational finance p. 1/17 From CFD to computational finance (and back again?) Mike Giles mike.giles@maths.ox.ac.uk Oxford University Mathematical Institute OxfordMan Institute of Quantitative Finance
More informationPricing InterestRate Derivative Securities
Pricing InterestRate Derivative Securities John Hull Alan White University of Toronto This article shows that the onestatevariable interestrate models of Vasicek (1977) and Cox, Ingersoll, and Ross
More informationBlackScholes Option Pricing Model
BlackScholes 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 informationBROWNIAN MOTION DEVELOPMENT FOR MONTE CARLO METHOD APPLIED ON EUROPEAN STYLE OPTION PRICE FORECASTING
International journal of economics & law Vol. 1 (2011), No. 1 (1170) BROWNIAN MOTION DEVELOPMENT FOR MONTE CARLO METHOD APPLIED ON EUROPEAN STYLE OPTION PRICE FORECASTING Petar Koĉović, Fakultet za obrazovanje
More informationBinomial lattice model for stock prices
Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }
More informationPricing European and American bond option under the Hull White extended Vasicek model
1 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA Pricing European and American bond option under the Hull White extended Vasicek model Eva Maria Rapoo 1, Mukendi Mpanda 2
More informationA ClosedForm Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options
A ClosedForm Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options Steven L. Heston Yale University I use a new technique to derive a closedform solution for
More informationRolf Poulsen, Centre for Finance, University of Gothenburg, Box 640, SE40530 Gothenburg, Sweden. Email: rolf.poulsen@economics.gu.se.
The Margrabe Formula Rolf Poulsen, Centre for Finance, University of Gothenburg, Box 640, SE40530 Gothenburg, Sweden. Email: rolf.poulsen@economics.gu.se Abstract The Margrabe formula for valuation of
More informationLecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing
Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing Key concept: Ito s lemma Stock Options: A contract giving its holder the right, but not obligation, to trade shares of a common
More informationHandbook in. Monte Carlo Simulation. Applications in Financial Engineering, Risk Management, and Economics
Handbook in Monte Carlo Simulation Applications in Financial Engineering, Risk Management, and Economics PAOLO BRANDIMARTE Department of Mathematical Sciences Politecnico di Torino Torino, Italy WlLEY
More informationAsian Option Pricing Formula for Uncertain Financial Market
Sun and Chen Journal of Uncertainty Analysis and Applications (215) 3:11 DOI 1.1186/s446715357 RESEARCH Open Access Asian Option Pricing Formula for Uncertain Financial Market Jiajun Sun 1 and Xiaowei
More informationSimple Arbitrage. Motivated by and partly based on a joint work with T. Sottinen and E. Valkeila. Christian Bender. Saarland University
Simple Arbitrage Motivated by and partly based on a joint work with T. Sottinen and E. Valkeila Saarland University December, 8, 2011 Problem Setting Financial market with two assets (for simplicity) on
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, CergyPontoise, France hgeman@ems.bbk.ac.uk Summary.
More informationJungSoon Hyun and YoungHee Kim
J. Korean Math. Soc. 43 (2006), No. 4, pp. 845 858 TWO APPROACHES FOR STOCHASTIC INTEREST RATE OPTION MODEL JungSoon Hyun and YoungHee Kim Abstract. We present two approaches of the stochastic interest
More informationThe BlackScholes Formula
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 The BlackScholes Formula These notes examine the BlackScholes formula for European options. The BlackScholes formula are complex as they are based on the
More informationBINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract
BINOMIAL OPTIONS PRICING MODEL Mark Ioffe Abstract Binomial option pricing model is a widespread numerical method of calculating price of American options. In terms of applied mathematics this is simple
More informationProceedings of the 33rd Hawaii International Conference on System Sciences  2000
Pricing Electricity Derivatives Under Alternative Stochastic Spot Price Models Shijie Deng School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta, GA 30332, USA deng@isye.gatech.edu
More information2 The Term Structure of Interest Rates in a Hidden Markov Setting
2 The Term Structure of Interest Rates in a Hidden Markov Setting Robert J. Elliott 1 and Craig A. Wilson 2 1 Haskayne School of Business University of Calgary Calgary, Alberta, Canada relliott@ucalgary.ca
More informationOption Pricing with Time Varying Volatility
Corso di Laurea Specialistica in Economia, curriculum Models and Methods of Quantitative Economics per la Gestione dell Impresa Prova finale di Laurea Option Pricing with Time Varying Volatility Relatore
More information3 Results. σdx. df =[µ 1 2 σ 2 ]dt+ σdx. Integration both sides will form
Appl. Math. Inf. Sci. 8, No. 1, 107112 (2014) 107 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/080112 Forecasting Share Prices of Small Size Companies
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 informationVolatility Jumps. April 12, 2010
Volatility Jumps Viktor Todorov and George Tauchen April 12, 2010 Abstract The paper undertakes a nonparametric analysis of the high frequency movements in stock market volatility using very finely sampled
More informationDerivatives: Principles and Practice
Derivatives: Principles and Practice Rangarajan K. Sundaram Stern School of Business New York University New York, NY 10012 Sanjiv R. Das Leavey School of Business Santa Clara University Santa Clara, CA
More informationCalibration of Stock Betas from Skews of Implied Volatilities
Calibration of Stock Betas from Skews of Implied Volatilities JeanPierre Fouque University of California Santa Barbara Joint work with Eli Kollman (Ph.D. student at UCSB) Joint Seminar: Department of
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 EulerMaruyama scheme Implement in Excel&VBA 1.
More informationThe Effective Dimension of AssetLiability Management Problems in Life Insurance
The Effective Dimension of AssetLiability Management Problems in Life Insurance Thomas Gerstner, Michael Griebel, Markus Holtz Institute for Numerical Simulation, University of Bonn holtz@ins.unibonn.de
More informationRiskNeutral Valuation of Participating Life Insurance Contracts
RiskNeutral Valuation of Participating Life Insurance Contracts DANIEL BAUER with R. Kiesel, A. Kling, J. Russ, and K. Zaglauer ULM UNIVERSITY RTG 1100 AND INSTITUT FÜR FINANZ UND AKTUARWISSENSCHAFTEN
More informationSpikes. Shijie Deng 1. Georgia Institute of Technology. Email: deng@isye.gatech.edu. First draft: November 20, 1998
Stochastic Models of Energy ommodity Prices and Their Applications: Meanreversion with Jumps and Spikes Shijie Deng Industrial and Systems Engineering Georgia Institute of Technology Atlanta, GA 333225
More informationGrey Brownian motion and local times
Grey Brownian motion and local times José Luís da Silva 1,2 (Joint work with: M. Erraoui 3 ) 2 CCM  Centro de Ciências Matemáticas, University of Madeira, Portugal 3 University Cadi Ayyad, Faculty of
More informationReview of Basic Options Concepts and Terminology
Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some
More informationA new FeynmanKacformula for option pricing in Lévy models
A new FeynmanKacformula for option pricing in Lévy models Kathrin Glau Department of Mathematical Stochastics, Universtity of Freiburg (Joint work with E. Eberlein) 6th World Congress of the Bachelier
More informationThe interest volatility surface
The interest volatility surface David Kohlberg Kandidatuppsats i matematisk statistik Bachelor Thesis in Mathematical Statistics Kandidatuppsats 2011:7 Matematisk statistik Juni 2011 www.math.su.se Matematisk
More informationA nongaussian OrnsteinUhlenbeck process for electricity spot price modeling and derivatives pricing
A nongaussian OrnsteinUhlenbeck process for electricity spot price modeling and derivatives pricing Thilo MeyerBrandis Center of Mathematics for Applications / University of Oslo Based on joint work
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 informationValuation of Asian Options
Valuation of Asian Options  with Levy Approximation Master thesis in Economics Jan 2014 Author: Aleksandra Mraovic, Qian Zhang Supervisor: Frederik Lundtofte Department of Economics Abstract Asian options
More informationFrom Exotic Options to Exotic Underlyings: Electricity, Weather and Catastrophe Derivatives
From Exotic Options to Exotic Underlyings: Electricity, Weather and Catastrophe Derivatives Dr. Svetlana Borovkova Vrije Universiteit Amsterdam History of derivatives Derivative: a financial contract whose
More informationTwofactor capital structure models for equity and credit
Twofactor capital structure models for equity and credit T. R. Hurd and Zhuowei Zhou Dept. of Mathematics and Statistics McMaster University Hamilton ON L8S 4K1 Canada October 26, 2011 Abstract We extend
More informationFlash crashes and order avalanches
Flash crashes and order avalanches Friedrich Hubalek and Thorsten Rheinländer Vienna University of Technology November 29, 2014 Friedrich Hubalek and Thorsten Rheinländer (Vienna Flash University crashes
More informationBlackScholes Equation for Option Pricing
BlackScholes 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 informationFrom CFD to computational finance (and back again?)
computational finance p. 1/21 From CFD to computational finance (and back again?) Mike Giles mike.giles@maths.ox.ac.uk Oxford University Mathematical Institute OxfordMan Institute of Quantitative Finance
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 informationStochastic Modelling and Forecasting
Stochastic Modelling and Forecasting Department of Statistics and Modelling Science University of Strathclyde Glasgow, G1 1XH RSE/NNSFC Workshop on Management Science and Engineering and Public Policy
More informationStocks paying discrete dividends: modelling and option pricing
Stocks paying discrete dividends: modelling and option pricing Ralf Korn 1 and L. C. G. Rogers 2 Abstract In the BlackScholes model, any dividends on stocks are paid continuously, but in reality dividends
More informationLecture 1: Stochastic Volatility and Local Volatility
Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch Case Studies in Financial Modelling Course Notes, Courant Institute of Mathematical Sciences, Fall Term, 2002 Abstract
More informationSimulating Stochastic Differential Equations
Monte Carlo Simulation: IEOR E473 Fall 24 c 24 by Martin Haugh Simulating Stochastic Differential Equations 1 Brief Review of Stochastic Calculus and Itô s Lemma Let S t be the time t price of a particular
More informationOn the Existence of a Unique Optimal Threshold Value for the Early Exercise of Call Options
On the Existence of a Unique Optimal Threshold Value for the Early Exercise of Call Options Patrick Jaillet Ehud I. Ronn Stathis Tompaidis July 2003 Abstract In the case of early exercise of an Americanstyle
More informationMore Exotic Options. 1 Barrier Options. 2 Compound Options. 3 Gap Options
More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options Definition; Some types The payoff of a Barrier option is path
More informationThe BlackScholesMerton Approach to Pricing Options
he BlackScholesMerton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the BlackScholesMerton approach to determining
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 informationThe BlackScholes Model
The BlackScholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The BlackScholes Model Options Markets 1 / 19 The BlackScholesMerton
More informationInsider Trading, Stochastic Liquidity and Equilibrium Prices
Insider Trading, Stochastic Liquidity and Equilibrium Prices Pierre CollinDufresne Carson Family Professor of Finance, Columbia University, and EPFL & SFI and NBER Vyacheslav Fos University of Illinois
More information第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model
1 第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model Outline 有 关 股 价 的 假 设 The BS 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 informationIto Excursion Theory. Calum G. Turvey Cornell University
Ito Excursion Theory Calum G. Turvey Cornell University Problem Overview Times series and dynamics have been the mainstay of agricultural economic and agricultural finance for over 20 years. Much of the
More informationSTOCKS IN THE SHORT RUN
STOCKS IN THE SHORT RUN Bryan Ellickson, Benjamin Hood, Tin Shing Liu, Duke Whang and Peilan Zhou Department of Economics, UCLA November 11, 2011 Abstract This paper examines stockprice volatility in
More informationOption Pricing using Fourier Space Timestepping Framework. Vladimir Surkov
Option Pricing using Fourier Space Timestepping Framework by Vladimir Surkov A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Computer
More informationGraduate Programs in Statistics
Graduate Programs in Statistics Course Titles STAT 100 CALCULUS AND MATR IX ALGEBRA FOR STATISTICS. Differential and integral calculus; infinite series; matrix algebra STAT 195 INTRODUCTION TO MATHEMATICAL
More informationOPTIONS, FUTURES, & OTHER DERIVATI
Fifth Edition OPTIONS, FUTURES, & OTHER DERIVATI John C. Hull Maple Financial Group Professor of Derivatives and Risk Manage, Director, Bonham Center for Finance Joseph L. Rotinan School of Management
More informationMartingale Pricing Applied to Options, Forwards and Futures
IEOR E4706: Financial Engineering: DiscreteTime Asset Pricing Fall 2005 c 2005 by Martin Haugh Martingale Pricing Applied to Options, Forwards and Futures We now apply martingale pricing theory to the
More informationNotes on BlackScholes Option Pricing Formula
. Notes on BlackScholes Option Pricing Formula by DeXing Guan March 2006 These notes are a brief introduction to the BlackScholes formula, which prices the European call options. The essential reading
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 informationDoes BlackScholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem
Does BlackScholes 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 informationOption Valuation under Stochastic Volatility With Mathematica Code
Option Valuation under Stochastic Volatility With Mathematica Code Copyright µ 2000 by Alan L. Lewis All rights reserved. Except for the quotation of short passages for the purposes of criticism and review,
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 informationProperties of the SABR model
U.U.D.M. Project Report 2011:11 Properties of the SABR model Nan Zhang Examensarbete i matematik, 30 hp Handledare och examinator: Johan Tysk Juni 2011 Department of Mathematics Uppsala University ABSTRACT
More informationLogNormal stockprice models in Exams MFE/3 and C/4
Making sense of... LogNormal stockprice models in Exams MFE/3 and C/4 James W. Daniel Austin Actuarial Seminars http://www.actuarialseminars.com June 26, 2008 c Copyright 2007 by James W. Daniel; reproduction
More informationIN THE DEFERRED ANNUITIES MARKET, THE PORTION OF FIXED RATE ANNUITIES in annual
W ORKSHOP B Y H A N G S U C K L E E Pricing EquityIndexed Annuities Embedded with Exotic Options IN THE DEFERRED ANNUITIES MARKET, THE PORTION OF FIXED RATE ANNUITIES in annual sales has declined from
More informationSimple approximations for option pricing under mean reversion and stochastic volatility
Simple approximations for option pricing under mean reversion and stochastic volatility Christian M. Hafner Econometric Institute Report EI 2003 20 April 2003 Abstract This paper provides simple approximations
More informationIntroduction to ArbitrageFree Pricing: Fundamental Theorems
Introduction to ArbitrageFree Pricing: Fundamental Theorems Dmitry Kramkov Carnegie Mellon University Workshop on Interdisciplinary Mathematics, Penn State, May 810, 2015 1 / 24 Outline Financial market
More informationStochastic Skew Models for FX Options
Stochastic Skew Models for FX Options Peter Carr Bloomberg LP and Courant Institute, NYU Liuren Wu Zicklin School of Business, Baruch College Special thanks to Bruno Dupire, Harvey Stein, Arun Verma, and
More informationELECTRICITY REAL OPTIONS VALUATION
Vol. 37 (6) ACTA PHYSICA POLONICA B No 11 ELECTRICITY REAL OPTIONS VALUATION Ewa BroszkiewiczSuwaj Hugo Steinhaus Center, Institute of Mathematics and Computer Science Wrocław University of Technology
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 informationAN ACCESSIBLE TREATMENT OF MONTE CARLO METHODS, TECHNIQUES, AND APPLICATIONS IN THE FIELD OF FINANCE AND ECONOMICS
Brochure More information from http://www.researchandmarkets.com/reports/2638617/ Handbook in Monte Carlo Simulation. Applications in Financial Engineering, Risk Management, and Economics. Wiley Handbooks
More informationBarrier Options. Peter Carr
Barrier Options Peter Carr Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU March 14th, 2008 What are Barrier Options?
More informationCOMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS
COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS NICOLE BÄUERLE AND STEFANIE GRETHER Abstract. In this short note we prove a conjecture posed in Cui et al. 2012): Dynamic meanvariance problems in
More informationApplications of Stochastic Processes in Asset Price Modeling
Applications of Stochastic Processes in Asset Price Modeling TJHSST Computer Systems Lab Senior Research Project 20082009 Preetam D Souza November 11, 2008 Abstract Stock market forecasting and asset
More information