The Heston Model. Hui Gong, UCL ucahgon/ May 6, 2014

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 Hui Gong, UCL ucahgon/ May 6, 2014

2 Generalized SV models Vanilla Call Option via Heston Itô s lemma for variance process Euler-Maruyama scheme Implement in Excel&VBA

3 1. Why the Black-Scholes model is not popular in the industry? 2. What is the stochastic volatility models? Stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed.

4 Generalized SV models Vanilla Call Option via Heston A general expression for non-dividend stock with stochastic volatility is as below: with ds t = µ t S t dt + v t S t dw 1 t, (1) dv t = α(s t, v t, t)dt + β(s t, v t, t)dw 2 t, (2) dw 1 t dw 2 t = ρdt, where S t denotes the stock price and v t denotes its variance. Examples: Heston model SABR volatility model GARCH model 3/2 model Chen model

5 Generalized SV models Vanilla Call Option via Heston The Heston model is a typical model which takes α(s t, v t, t) = κ(θ v t ) and β(s t, v t, t) = σ v t, i.e. ds t = µs t dt + v t S t dw 1,t, (3) dv t = κ(θ v t )dt + σ v t dw 2,t, (4) with dw 1,t dw 2,t = ρdt, (5) where θ is the long term mean of v t, κ denotes the speed of reversion and σ is the volatility of volatility. The instantaneous variance v t here is a CIR process (square root process).

6 Generalized SV models Vanilla Call Option via Heston Let x t = ln S t, the risk-neutral dynamics of Heston model is ( dx t = r 1 ) 2 v t dt + v t dw1,t, (6) with dv t = κ (θ v t )dt + σ v t dw 2,t, (7) dw 1,tdW 2,t = ρdt. (8) where κ = κ + λ and θ = κθ κ+λ. Using these dynamics, the probability of the call option expires in-the-money, conditional on the log of the stock price, can be interpreted as risk-adjusted or risk-neutral probabilities. Hence, F j (x, v, T ; ln K) = Pr(x(T ) ln K x t = x, v t = v).

7 Generalized SV models Vanilla Call Option via Heston The price of vanilla call option is: C(S, v, t) = SF 1 e r(t t) KF 2, (9) where F 1 and F 2 should satisfy the PDE (for j = 1, 2) 1 2 v 2 F j x 2 + ρσv 2 F j x v σ2 v 2 F j v 2 +(r + u j v) F j x + (a j b j v) F j v + F j t = 0. (10) The parameter in Equation (10) is as follows u 1 = 1 2, u 2 = 1 2, a = κθ, b 1 = κ+λ ρσ, b 2 = κ+λ.

8 Itô s lemma for variance process Euler-Maruyama scheme Implement in Excel&VBA The simulated variance can be inspected to check whether it is negative (v < 0). In this case, the variance can be set to zero (v = 0), or its sign can be inverted so that v becomes v. Alternatively, the variance process can be modified in the same way as the stock process, by defining a process for natural log variances by using Itô s lemma d ln v t = 1 ( κ (θ v t ) 1 ) v t 2 σ2 dt + σ 1 dw2,t. (11) vt

9 Itô s lemma for variance process Euler-Maruyama scheme Implement in Excel&VBA The Heston model can be discretized as following ( ln S t+ t = ln S t + r 1 ) 2 v t t + v t tɛs,t+1, ln v t+ t = ln v t + 1 v t ( κ (θ v t ) 1 2 σ2 ) t + σ 1 vt tɛv,t+1. Shocks to the volatility, ɛ v,t+1, are correlated with the shocks to the stock price process, ɛ S,t+1. This correlation is denoted ρ, so that ρ = Corr(ɛ S,t+1, ɛ v,t+1 ) and the relationship between the shocks can be written as ɛ v,t+1 = ρɛ S,t ρ 2 ɛ t+1 where ɛ t+1 are independently with ɛ S,t+1.

10 Itô s lemma for variance process Euler-Maruyama scheme Implement in Excel&VBA Figure: Heston (1993) Call Price by Monte Carlo

11 Itô s lemma for variance process Euler-Maruyama scheme Implement in Excel&VBA Figure: VBA code for Heston (1993) Call Price by Monte Carlo

12 Use the Closed-Form Approach to implement Heston Call & Put.

The Evaluation of Barrier Option Prices Under Stochastic Volatility. BFS 2010 Hilton, Toronto June 24, 2010

The Evaluation of Barrier Option Prices Under Stochastic Volatility. BFS 2010 Hilton, Toronto June 24, 2010 The Evaluation of Barrier Option Prices Under Stochastic Volatility Carl Chiarella, Boda Kang and Gunter H. Meyer School of Finance and Economics University of Technology, Sydney School of Mathematics

More information

Simulating Stochastic Differential Equations

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

More information

Private Equity Fund Valuation and Systematic Risk

Private Equity Fund Valuation and Systematic Risk An Equilibrium Approach and Empirical Evidence Axel Buchner 1, Christoph Kaserer 2, Niklas Wagner 3 Santa Clara University, March 3th 29 1 Munich University of Technology 2 Munich University of Technology

More information

Using the SABR Model

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

More information

Lecture 1: Stochastic Volatility and Local Volatility

Lecture 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 information

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

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

More information

Stock Price Dynamics, Dividends and Option Prices with Volatility Feedback

Stock Price Dynamics, Dividends and Option Prices with Volatility Feedback Stock Price Dynamics, Dividends and Option Prices with Volatility Feedback Juho Kanniainen Tampere University of Technology New Thinking in Finance 12 Feb. 2014, London Based on J. Kanniainen and R. Piche,

More information

Pricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation

Pricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation Pricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation Yoon W. Kwon CIMS 1, Math. Finance Suzanne A. Lewis CIMS, Math. Finance May 9, 000 1 Courant Institue of Mathematical Science,

More information

Jung-Soon Hyun and Young-Hee Kim

Jung-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 information

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

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

More information

The Black-Scholes pricing formulas

The 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 information

Numerical methods for American options

Numerical methods for American options Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment

More information

Numerical Methods for Option Pricing

Numerical Methods for Option Pricing Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly

More information

Hedging 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 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 information

Exam MFE Spring 2007 FINAL ANSWER KEY 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D

Exam MFE Spring 2007 FINAL ANSWER KEY 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D Exam MFE Spring 2007 FINAL ANSWER KEY Question # Answer 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D **BEGINNING OF EXAMINATION** ACTUARIAL MODELS FINANCIAL ECONOMICS

More information

Finite Differences Schemes for Pricing of European and American Options

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

More information

Analytic Approximations for Multi-Asset Option Pricing

Analytic 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 June 23, 2009

More information

Lecture 6 Black-Scholes PDE

Lecture 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 information

Estimating the Degree of Activity of jumps in High Frequency Financial Data. joint with Yacine Aït-Sahalia

Estimating the Degree of Activity of jumps in High Frequency Financial Data. joint with Yacine Aït-Sahalia Estimating the Degree of Activity of jumps in High Frequency Financial Data joint with Yacine Aït-Sahalia Aim and setting An underlying process X = (X t ) t 0, observed at equally spaced discrete times

More information

Lecture 12: The Black-Scholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 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 information

Quanto Adjustments in the Presence of Stochastic Volatility

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

More information

Lecture 15. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 6

Lecture 15. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 6 Lecture 15 Sergei Fedotov 20912 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 6 Lecture 15 1 Black-Scholes Equation and Replicating Portfolio 2 Static

More information

Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 11. The Black-Scholes Model: Hull, Ch. 13.

Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 11. The Black-Scholes Model: Hull, Ch. 13. Week 11 The Black-Scholes Model: Hull, Ch. 13. 1 The Black-Scholes Model Objective: To show how the Black-Scholes formula is derived and how it can be used to value options. 2 The Black-Scholes Model 1.

More information

From CFD to computational finance (and back again?)

From 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 Oxford-Man Institute of Quantitative Finance

More information

Convenience Yield-Based Pricing of Commodity Futures

Convenience Yield-Based Pricing of Commodity Futures Convenience Yield-Based Pricing of Commodity Futures Takashi Kanamura, J-POWER BFS2010 6th World Congress in Toronto, Canada June 26th, 2010 1 Agenda 1. The objectives and results 2. The convenience yield-based

More information

Hedging Exotic Options

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

More information

Online Appendix. Supplemental Material for Insider Trading, Stochastic Liquidity and. Equilibrium Prices. by Pierre Collin-Dufresne and Vyacheslav Fos

Online Appendix. Supplemental Material for Insider Trading, Stochastic Liquidity and. Equilibrium Prices. by Pierre Collin-Dufresne and Vyacheslav Fos Online Appendix Supplemental Material for Insider Trading, Stochastic Liquidity and Equilibrium Prices by Pierre Collin-Dufresne and Vyacheslav Fos 1. Deterministic growth rate of noise trader volatility

More information

Option Pricing. 1 Introduction. Mrinal K. Ghosh

Option Pricing. 1 Introduction. Mrinal K. Ghosh Option Pricing Mrinal K. Ghosh 1 Introduction We first introduce the basic terminology in option pricing. Option: An option is the right, but not the obligation to buy (or sell) an asset under specified

More information

Likewise, the payoff of the better-of-two note may be decomposed as follows: Price of gold (US$/oz) 375 400 425 450 475 500 525 550 575 600 Oil price

Likewise, the payoff of the better-of-two note may be decomposed as follows: Price of gold (US$/oz) 375 400 425 450 475 500 525 550 575 600 Oil price Exchange Options Consider the Double Index Bull (DIB) note, which is suited to investors who believe that two indices will rally over a given term. The note typically pays no coupons and has a redemption

More information

Markovian projection for volatility calibration

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

More information

Stochastic Skew Models for FX Options

Stochastic 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 information

Pricing Currency Options Under Stochastic Volatility

Pricing Currency Options Under Stochastic Volatility Pricing Currency Options Under Stochastic Volatility Ming-Hsien Chen Department of Finance National Cheng Chi University Yin-Feng Gau * Department of International Business Studies National Chi Nan University

More information

Some 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 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 information

Hedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging

Hedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging Hedging An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Definition Hedging is the practice of making a portfolio of investments less sensitive to changes in

More information

Estimation of Stochastic Volatility Models with Implied Volatility Indices and Pricing of

Estimation of Stochastic Volatility Models with Implied Volatility Indices and Pricing of Estimation of Stochastic Volatility Models with Implied Volatility Indices and Pricing of Straddle Option Yue Peng and Steven C. J. Simon University of Essex Centre for Computational Finance and Economic

More information

Implied Volatility of Leveraged ETF Options: Consistency and Scaling

Implied Volatility of Leveraged ETF Options: Consistency and Scaling Implied Volatility of Leveraged ETF Options: Consistency and Scaling Industrial Engineering & Operations Research Dept Columbia University Finance and Stochastics (FAST) Seminar University of Sussex March

More information

Option Pricing under Heston and 3/2 Stochastic Volatility Models: an Approximation to the Fast Fourier Transform

Option Pricing under Heston and 3/2 Stochastic Volatility Models: an Approximation to the Fast Fourier Transform Aarhus University Master s thesis Option Pricing under Heston and 3/2 Stochastic Volatility Models: an Approximation to the Fast Fourier Transform Author: Dessislava Koleva Supervisor: Elisa Nicolato July,

More information

Static Hedging and Model Risk for Barrier Options

Static Hedging and Model Risk for Barrier Options Static Hedging and Model Risk for Barrier Options Morten Nalholm Rolf Poulsen Abstract We investigate how sensitive different dynamic and static hedge strategies for barrier options are to model risk.

More information

Black-Scholes Equation for Option Pricing

Black-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 information

Stochastic Skew in Currency Options

Stochastic Skew in Currency Options Stochastic Skew in Currency Options PETER CARR Bloomberg LP and Courant Institute, NYU LIUREN WU Zicklin School of Business, Baruch College Citigroup Wednesday, September 22, 2004 Overview There is a huge

More information

Implied Volatility of Leveraged ETF Options: Consistency and Scaling

Implied Volatility of Leveraged ETF Options: Consistency and Scaling Implied Volatility of Leveraged ETF Options: Consistency and Scaling Tim Leung Industrial Engineering & Operations Research Dept Columbia University http://www.columbia.edu/ tl2497 Risk USA Post-Conference

More information

Modeling 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 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 information

International Stock Market Integration: A Dynamic General Equilibrium Approach

International Stock Market Integration: A Dynamic General Equilibrium Approach International Stock Market Integration: A Dynamic General Equilibrium Approach Harjoat S. Bhamra London Business School 2003 Outline of talk 1 Introduction......................... 1 2 Economy...........................

More information

Chapter 2: Binomial Methods and the Black-Scholes Formula

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

More information

Variance Reduction for Monte Carlo Methods to Evaluate Option Prices under Multi-factor Stochastic Volatility Models

Variance Reduction for Monte Carlo Methods to Evaluate Option Prices under Multi-factor Stochastic Volatility Models Variance Reduction for Monte Carlo Methods to Evaluate Option Prices under Multi-factor Stochastic Volatility Models Jean-Pierre Fouque and Chuan-Hsiang Han Submitted April 24, Accepted October 24 Abstract

More information

Lecture 11: The Greeks and Risk Management

Lecture 11: The Greeks and Risk Management Lecture 11: The Greeks and Risk Management This lecture studies market risk management from the perspective of an options trader. First, we show how to describe the risk characteristics of derivatives.

More information

Case Studies in Acceleration of Heston s Stochastic Volatility Financial Engineering Model: GPU, Cloud and FPGA Implementations

Case Studies in Acceleration of Heston s Stochastic Volatility Financial Engineering Model: GPU, Cloud and FPGA Implementations Case Studies in Acceleration of Heston s Stochastic Volatility Financial Engineering Model: GPU, Cloud and FPGA Implementations by Christos Delivorias Supervised by Dr. Peter Richtárik and Martin Takáč

More information

3. Monte Carlo Simulations. Math6911 S08, HM Zhu

3. Monte Carlo Simulations. Math6911 S08, HM Zhu 3. Monte Carlo Simulations Math6911 S08, HM Zhu References 1. Chapters 4 and 8, Numerical Methods in Finance. Chapters 17.6-17.7, Options, Futures and Other Derivatives 3. George S. Fishman, Monte Carlo:

More information

Generation Asset Valuation with Operational Constraints A Trinomial Tree Approach

Generation 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 information

第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model

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 information

Risk-Neutral Valuation of Participating Life Insurance Contracts

Risk-Neutral Valuation of Participating Life Insurance Contracts Risk-Neutral 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 information

Does 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 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 information

An Analytical Pricing Formula for VIX Futures and Its Empirical Applications

An Analytical Pricing Formula for VIX Futures and Its Empirical Applications Faculty of Informatics, University of Wollongong An Analytical Pricing Formula for VIX Futures and Its Empirical Applications Song-Ping Zhu and Guang-Hua Lian School of Mathematics and Applied Statistics

More information

The real P&L in Black-Scholes and Dupire Delta hedging

The real P&L in Black-Scholes and Dupire Delta hedging International Journal of Theoretical and Applied Finance c World Scientific Publishing Company The real P&L in Black-Scholes and Dupire Delta hedging MARTIN FORDE University of Bristol, Department of Mathematics,

More information

Valuing double barrier options with time-dependent parameters by Fourier series expansion

Valuing double barrier options with time-dependent parameters by Fourier series expansion IAENG International Journal of Applied Mathematics, 36:1, IJAM_36_1_1 Valuing double barrier options with time-dependent parameters by Fourier series ansion C.F. Lo Institute of Theoretical Physics and

More information

ARMA, GARCH and Related Option Pricing Method

ARMA, GARCH and Related Option Pricing Method ARMA, GARCH and Related Option Pricing Method Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook September

More information

A SNOWBALL CURRENCY OPTION

A SNOWBALL CURRENCY OPTION J. KSIAM Vol.15, No.1, 31 41, 011 A SNOWBALL CURRENCY OPTION GYOOCHEOL SHIM 1 1 GRADUATE DEPARTMENT OF FINANCIAL ENGINEERING, AJOU UNIVERSITY, SOUTH KOREA E-mail address: gshim@ajou.ac.kr ABSTRACT. I introduce

More information

Numerical Methods for Pricing Exotic Options

Numerical 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 information

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

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

More information

Estimating Option Prices with Heston s Stochastic Volatility Model

Estimating Option Prices with Heston s Stochastic Volatility Model Estimating Option Prices with Heston s Stochastic Volatility odel Robin Dunn, Paloma Hauser, Tom Seibold 3, Hugh Gong 4. Department of athematics and Statistics, Kenyon College, Gambier, OH 430. Department

More information

Simple approximations for option pricing under mean reversion and stochastic volatility

Simple 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 information

Arbitrage-Free Pricing Models

Arbitrage-Free Pricing Models Arbitrage-Free Pricing Models Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Arbitrage-Free Pricing Models 15.450, Fall 2010 1 / 48 Outline 1 Introduction 2 Arbitrage and SPD 3

More information

α α λ α = = λ λ α ψ = = α α α λ λ ψ α = + β = > θ θ β > β β θ θ θ β θ β γ θ β = γ θ > β > γ θ β γ = θ β = θ β = θ β = β θ = β β θ = = = β β θ = + α α α α α = = λ λ λ λ λ λ λ = λ λ α α α α λ ψ + α =

More information

Markov modeling of Gas Futures

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

More information

DRAFT. Geng Deng, PhD, CFA, FRM Tim Dulaney, PhD, FRM Craig McCann, PhD, CFA Mike Yan, PhD, FRM. January 7, 2014

DRAFT. Geng Deng, PhD, CFA, FRM Tim Dulaney, PhD, FRM Craig McCann, PhD, CFA Mike Yan, PhD, FRM. January 7, 2014 Crooked Volatility Smiles: Evidence from Leveraged and Inverse ETF Options Geng Deng, PhD, CFA, FRM Tim Dulaney, PhD, FRM Craig McCann, PhD, CFA Mike Yan, PhD, FRM January 7, 214 Abstract We find that

More information

Risk-Neutral Valuation of Participating Life Insurance Contracts in a Stochastic Interest Rate Environment

Risk-Neutral Valuation of Participating Life Insurance Contracts in a Stochastic Interest Rate Environment Risk-Neutral Valuation of Participating Life Insurance Contracts in a Stochastic Interest Rate Environment Katharina Zaglauer a, Daniel Bauer b, a University of Wisconsin-Milwaukee, Department of Mathematical

More information

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

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

More information

Valuation of commodity derivatives when spot prices revert to a cyclical mean

Valuation of commodity derivatives when spot prices revert to a cyclical mean Valuation of commodity derivatives when spot prices revert to a cyclical mean April, 24 Abstract This paper introduces a new continuous-time model based on the logarithm of the commodity spot price assuming

More information

A Study on Heston-Nandi GARCH Option Pricing Model

A Study on Heston-Nandi GARCH Option Pricing Model 2011 3rd International Conference on Information and Financial Engineering IPEDR vol.12 (2011) (2011) IACSIT Press, Singapore A Study on Heston-Nandi GARCH Option Pricing Model Suk Joon Byun KAIST Business

More information

Black-Scholes Option Pricing Model

Black-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 information

Master's thesis Calibration of FX options and pricing of barrier options. Anders Persson June 4, 2013

Master's thesis Calibration of FX options and pricing of barrier options. Anders Persson June 4, 2013 Master's thesis Calibration of FX options and pricing of barrier options Anders Persson June 4, 213 1 Abstract This paper examines the calibration of foreign exchange options during one year using the

More information

Valuation, Pricing of Options / Use of MATLAB

Valuation, Pricing of Options / Use of MATLAB CS-5 Computational Tools and Methods in Finance Tom Coleman Valuation, Pricing of Options / Use of MATLAB 1.0 Put-Call Parity (review) Given a European option with no dividends, let t current time T exercise

More information

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

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

More information

Option 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 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 information

Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com

Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com In this Note we derive the Black Scholes PDE for an option V, given by @t + 1 + rs @S2 @S We derive the

More information

S 1 S 2. Options and Other Derivatives

S 1 S 2. Options and Other Derivatives Options and Other Derivatives The One-Period Model The previous chapter introduced the following two methods: Replicate the option payoffs with known securities, and calculate the price of the replicating

More information

From CFD to computational finance (and back again?)

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

More information

1 The Black-Scholes model: extensions and hedging

1 The Black-Scholes model: extensions and hedging 1 The Black-Scholes model: extensions and hedging 1.1 Dividends Since we are now in a continuous time framework the dividend paid out at time t (or t ) is given by dd t = D t D t, where as before D denotes

More information

Pricing Stock Options with Stochastic Interest Rate

Pricing Stock Options with Stochastic Interest Rate Pricing Stock Options with Stochastic Interest Rate Menachem Abudy* and Yehuda Izhakian** Abstract This paper constructs a closed-form generalization of the Black-Scholes model for the case where the short-term

More information

Pricing Options with Discrete Dividends by High Order Finite Differences and Grid Stretching

Pricing Options with Discrete Dividends by High Order Finite Differences and Grid Stretching Pricing Options with Discrete Dividends by High Order Finite Differences and Grid Stretching Kees Oosterlee Numerical analysis group, Delft University of Technology Joint work with Coen Leentvaar, Ariel

More information

Pricing Barrier Options under Local Volatility

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

More information

Is Infrastructure Capital Productive? A Dynamic Heterogeneous Approach.

Is Infrastructure Capital Productive? A Dynamic Heterogeneous Approach. Is Infrastructure Capital Productive? A Dynamic Heterogeneous Approach. César Calderón a, Enrique Moral-Benito b, Luis Servén a a The World Bank b CEMFI International conference on Infrastructure Economics

More information

Introduction to Stochastic Differential Equations (SDEs) for Finance

Introduction to Stochastic Differential Equations (SDEs) for Finance Introduction to Stochastic Differential Equations (SDEs) for Finance Andrew Papanicolaou January, 013 Contents 1 Financial Introduction 3 1.1 A Market in Discrete Time and Space..................... 3

More information

European Call Option Pricing using the Adomian Decomposition Method

European Call Option Pricing using the Adomian Decomposition Method Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 9, Number 1, pp. 75 85 (2014) http://campus.mst.edu/adsa European Call Option Pricing using the Adomian Decomposition Method Martin

More information

Numerical Methods For Derivative Pricing with Applications to Barrier Options

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

More information

Black-Scholes and the Volatility Surface

Black-Scholes and the Volatility Surface IEOR E4707: Financial Engineering: Continuous-Time Models Fall 2009 c 2009 by Martin Haugh Black-Scholes and the Volatility Surface When we studied discrete-time models we used martingale pricing to derive

More information

Realized Volatility and Variance: Options via Swaps

Realized Volatility and Variance: Options via Swaps Realized Volatility and Variance: Options via Swaps Peter Carr and Roger Lee This version: October 26, 2007 In this paper we develop strategies for pricing and hedging options on realized variance and

More information

Monte Carlo Methods in Finance

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

More information

Recent Developments of Statistical Application in. Finance. Ruey S. Tsay. Graduate School of Business. The University of Chicago

Recent Developments of Statistical Application in. Finance. Ruey S. Tsay. Graduate School of Business. The University of Chicago Recent Developments of Statistical Application in Finance Ruey S. Tsay Graduate School of Business The University of Chicago Guanghua Conference, June 2004 Summary Focus on two parts: Applications in Finance:

More information

Open issues in equity derivatives modelling

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

More information

ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 10, 11, 12, 18. October 21, 2010 (Thurs)

ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 10, 11, 12, 18. October 21, 2010 (Thurs) Problem ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 0,, 2, 8. October 2, 200 (Thurs) (i) The current exchange rate is 0.0$/. (ii) A four-year dollar-denominated European put option

More information

Monte Carlo simulations and option pricing

Monte Carlo simulations and option pricing Monte Carlo simulations and option pricing by Bingqian Lu Undergraduate Mathematics Department Pennsylvania State University University Park, PA 16802 Project Supervisor: Professor Anna Mazzucato July,

More information

Estimating Volatility

Estimating Volatility Estimating Volatility Daniel Abrams Managing Partner FAS123 Solutions, LLC Copyright 2005 FAS123 Solutions, LLC Definition of Volatility Historical Volatility as a Forecast of the Future Definition of

More information

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

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

More information

Lectures. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. No tutorials in the first week

Lectures. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. No tutorials in the first week Lectures Sergei Fedotov 20912 - Introduction to Financial Mathematics No tutorials in the first week Sergei Fedotov (University of Manchester) 20912 2010 1 / 1 Lecture 1 1 Introduction Elementary economics

More information

Tutorial: Structural Models of the Firm

Tutorial: Structural Models of the Firm Tutorial: Structural Models of the Firm Peter Ritchken Case Western Reserve University February 16, 2015 Peter Ritchken, Case Western Reserve University Tutorial: Structural Models of the Firm 1/61 Tutorial:

More information

Life-insurance-specific optimal investment: the impact of stochastic interest rate and shortfall constraint

Life-insurance-specific optimal investment: the impact of stochastic interest rate and shortfall constraint Life-insurance-specific optimal investment: the impact of stochastic interest rate and shortfall constraint An Radon Workshop on Financial and Actuarial Mathematics for Young Researchers May 30-31 2007,

More information

Valuation of Asian Options

Valuation 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 information

Pricing American Options without Expiry Date

Pricing American Options without Expiry Date Pricing American Options without Expiry Date Carisa K. W. Yu Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom, Hong Kong E-mail: carisa.yu@polyu.edu.hk Abstract This paper

More information