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

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

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

Implied Volatility Surface

Implied Volatility Surface Implied Volatility Surface Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 16) Liuren Wu Implied Volatility Surface Options Markets 1 / 18 Implied volatility Recall

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

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

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

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

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

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

Valuation of Long Term Equity Options and Guarantees under Stochastic Interest Rates. Bernard Wong, UNSW

Valuation of Long Term Equity Options and Guarantees under Stochastic Interest Rates. Bernard Wong, UNSW Valuation of Long Term Equity Options and Guarantees under Stochastic Interest Rates Bernard Wong, UNSW Outline 1. Long Term Guarantees and Interest Rate Variability 2. HJM framework and applicable 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

The Impact of Stochastic Volatility on Pricing, Hedging, and Hedge Efficiency of Variable Annuity Guarantees

The Impact of Stochastic Volatility on Pricing, Hedging, and Hedge Efficiency of Variable Annuity Guarantees The Impact of Stochastic Volatility on Pricing, Hedging, and Hedge Efficiency of Variable Annuity Guarantees Alexander Kling, Frederik Ruez, and Jochen Russ Frederik Ruez, Ulm University Research Purpose

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

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

The Feynman-Kac Theorem by Fabrice Douglas Rouah

The Feynman-Kac Theorem by Fabrice Douglas Rouah The Feynman-Kac Theorem by Fabrice Douglas Rouah wwwfrouahcom wwwvoloptacom In this Note we illustrate the Feynman-Kac theorem in one dimension, and in multiple dimensions We illustrate the use of the

More information

A Closed-form Exact Solution for Pricing Variance. Swaps with Stochastic Volatility

A Closed-form Exact Solution for Pricing Variance. Swaps with Stochastic Volatility A Closed-form Exact Solution for Pricing Variance Swaps with Stochastic Volatility Song-Ping Zhu, Guang-Hua Lian University of Wollongong, Australia Abstract In this paper, we present a highly efficient

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

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 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

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

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

MATH3075/3975 Financial Mathematics

MATH3075/3975 Financial Mathematics MATH3075/3975 Financial Mathematics Week 11: Solutions Exercise 1 We consider the Black-Scholes model M = B, S with the initial stock price S 0 = 9, the continuously compounded interest rate r = 0.01 per

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

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

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

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

Generating Random Numbers Variance Reduction Quasi-Monte Carlo. Simulation Methods. Leonid Kogan. MIT, Sloan. 15.450, Fall 2010

Generating Random Numbers Variance Reduction Quasi-Monte Carlo. Simulation Methods. Leonid Kogan. MIT, Sloan. 15.450, Fall 2010 Simulation Methods Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Simulation Methods 15.450, Fall 2010 1 / 35 Outline 1 Generating Random Numbers 2 Variance Reduction 3 Quasi-Monte

More information

The Black-Scholes Model

The Black-Scholes Model The Black-Scholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The Black-Scholes Model Options Markets 1 / 19 The Black-Scholes-Merton

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

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

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t.

Moreover, 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 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

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

Stochastic Processes Prof. Dr. S. Dharmaraja Department of Mathematics Indian Institute of Technology, Delhi

Stochastic Processes Prof. Dr. S. Dharmaraja Department of Mathematics Indian Institute of Technology, Delhi Stochastic Processes Prof. Dr. S. Dharmaraja Department of Mathematics Indian Institute of Technology, Delhi Module - 7 Brownian Motion and its Applications Lecture - 5 Ito Formula and its Variants This

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

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

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

金融隨機計算 : 第一章. Black-Scholes-Merton Theory of Derivative Pricing and Hedging. CH Han Dept of Quantitative Finance, Natl. Tsing-Hua Univ.

金融隨機計算 : 第一章. Black-Scholes-Merton Theory of Derivative Pricing and Hedging. CH Han Dept of Quantitative Finance, Natl. Tsing-Hua Univ. 金融隨機計算 : 第一章 Black-Scholes-Merton Theory of Derivative Pricing and Hedging CH Han Dept of Quantitative Finance, Natl. Tsing-Hua Univ. Derivative Contracts Derivatives, also called contingent claims, are

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

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

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

Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing

Lecture 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 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

Notes and exercises on Black-Scholes April 2010

Notes and exercises on Black-Scholes April 2010 Math 425 Dr. DeTurck Notes and exercises on Black-Scholes April 2010 On Thursday we talked in class about how to derive the Black-Scholes differential equation, which is used in mathematical finance to

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

Timer-Style Options Design, Pricing and Practice

Timer-Style Options Design, Pricing and Practice Timer-Style Options Design, Pricing and Practice RiO 2010 Carole Bernard (joint work with Zhenyu Cui) Carole Bernard Timer Options 1 Outline Realized volatility. What is a timer option? Model-free price

More information

Option hedging with stochastic volatility

Option hedging with stochastic volatility Option hedging with stochastic volatility Adam Kurpiel L.A.R.E. U.R.A. n 944, Université Montesquieu-Bordeaux IV, France Thierry Roncalli FERC, City University Business School, England December 8, 998

More information

Continuous time; continuous variable stochastic process. We assume that stock prices follow Markov processes. That is, the future movements in a

Continuous time; continuous variable stochastic process. We assume that stock prices follow Markov processes. That is, the future movements in a Continuous time; continuous variable stochastic process. We assume that stock prices follow Markov processes. That is, the future movements in a variable depend only on the present, and not the history

More information

Week 9 Stochastic differential equations

Week 9 Stochastic differential equations Week 9 Stochastic differential equations Jonathan Goodman November 19, 1 1 Introduction to the material for the week The material this week is all about the expression dx t = a t dt + b t dw t. (1 There

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

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

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

Derivation of Local Volatility by Fabrice Douglas Rouah www.frouah.com www.volopta.com

Derivation of Local Volatility by Fabrice Douglas Rouah www.frouah.com www.volopta.com Derivation of Local Volatility by Fabrice Douglas Rouah www.frouah.com www.volopta.com The derivation of local volatility is outlined in many papers and textbooks (such as the one by Jim Gatheral []),

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

The Three Methods of Pricing Derivatives by Fabrice Douglas Rouah

The Three Methods of Pricing Derivatives by Fabrice Douglas Rouah The Three Methods of Pricing Derivatives by Fabrice Douglas Rouah www.frouah.com www.volopta.com In this Note we illustrate the three methods for pricing derivatives: pricing by no arbitrage, pricing using

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

LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS

LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS Robert S. Pindyck Massachusetts Institute of Technology Cambridge, MA 02142 Robert Pindyck (MIT) LECTURES ON REAL OPTIONS PART II August, 2008 1 / 50

More information

Share Price Movements

Share Price Movements Share Price Movements Brian A. Eales April 2004 Share Price Movements ds = S µ dt + S σ dz ( 1) In continuous time or S = S µ t + S σ z ( 2) In discrete (measurable) time Page1 Where: ds or S represents

More information

Properties of the SABR model

Properties 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 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

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

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

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

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

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

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

Lecture 10. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 7

Lecture 10. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 7 Lecture 10 Sergei Fedotov 20912 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 7 Lecture 10 1 Binomial Model for Stock Price 2 Option Pricing on Binomial

More information

European Options Pricing Using Monte Carlo Simulation

European Options Pricing Using Monte Carlo Simulation European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical

More information

Time Series 6. Robert Almgren. Nov. 9, 2009

Time Series 6. Robert Almgren. Nov. 9, 2009 Time Series 6 Robert Almgren Nov. 9, 2009 This week we continue our discussion of state space models, focusing on the particle method approach for nonlinear models. Besides its practical application, this

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

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

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

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

Interest Rate Models: Paradigm shifts in recent years

Interest Rate Models: Paradigm shifts in recent years Interest Rate Models: Paradigm shifts in recent years Damiano Brigo Q-SCI, Managing Director and Global Head DerivativeFitch, 101 Finsbury Pavement, London Columbia University Seminar, New York, November

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

Options Vs. Futures: Which on Average Will Have the Greater Payoff?

Options Vs. Futures: Which on Average Will Have the Greater Payoff? Utah State University DigitalCommons@USU All Graduate Plan B and other Reports Graduate Studies 2012 Options Vs. Futures: Which on Average Will Have the Greater? Ryan Silvester Utah State University Follow

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

INTEREST RATES AND FX MODELS

INTEREST RATES AND FX MODELS INTEREST RATES AND FX MODELS 8. Portfolio greeks Andrew Lesniewski Courant Institute of Mathematical Sciences New York University New York March 27, 2013 2 Interest Rates & FX Models Contents 1 Introduction

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

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

Monte Carlo simulations in the case of several risk factors: Cholesky decomposition and copulas

Monte Carlo simulations in the case of several risk factors: Cholesky decomposition and copulas wwwijcsiorg 233 Monte Carlo simulations in the case of several risk factors: Cholesky decomposition and copulas Naima SOKHER 1, Boubker DAAFI 2, Jamal BOYAGHROMNI 1, Abdelwahed NAMIR 1 1 Department of

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

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

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

Stochastic Modelling and Forecasting

Stochastic 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 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

Numerical PDE methods for exotic options

Numerical PDE methods for exotic options Lecture 8 Numerical PDE methods for exotic options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Barrier options For barrier option part of the option contract is triggered if the asset

More information

EXACT SIMULATION OF OPTION GREEKS UNDER STOCHASTIC VOLATILITY AND JUMP DIFFUSION MODELS

EXACT SIMULATION OF OPTION GREEKS UNDER STOCHASTIC VOLATILITY AND JUMP DIFFUSION MODELS Proceedings of the 2004 Winter Simulation Conference R. G. Ingalls, M. D. Rossetti, J. S. Smith, and B. A. Peters, eds. EXACT SIMULATION OF OPTION GREEKS UNDER STOCHASTIC VOLATILITY AND JUMP DIFFUSION

More information

Option to contract An option to reduce the scale of a project s operation. Synonym of contraction option and option to scale back

Option to contract An option to reduce the scale of a project s operation. Synonym of contraction option and option to scale back Glossary @Risk Risk Analysis Software using Monte Carlo Simulation software developed by Palisade Abandonment option Option to sell or close down a project (a simple put option). Synonym of option to abandon

More information

Continuous-Time Derivative Pricing Models

Continuous-Time Derivative Pricing Models Continuous-Time Derivative Pricing Models Eric Zivot May 5, 2011 Outline 1. Derivative Pricing with Continuous-Time Models 2. Derivation of Black-Scholes (BS) SDE 3. BS Implied Volatility Reading APDVP,

More information

Financial Derivatives. An Introduction to the Black-Scholes PDE. The Pricing Problem. Example

Financial Derivatives. An Introduction to the Black-Scholes PDE. The Pricing Problem. Example Financial Derivatives April 23, 2009 Definition A derivative is a financial contract whose value is based on the value of an underlying asset. Typically, a derivative gives the holder the right to buy

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

Variance Reduction. Pricing American Options. Monte Carlo Option Pricing. Delta and Common Random Numbers

Variance Reduction. Pricing American Options. Monte Carlo Option Pricing. Delta and Common Random Numbers Variance Reduction The statistical efficiency of Monte Carlo simulation can be measured by the variance of its output If this variance can be lowered without changing the expected value, fewer replications

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

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