Price an Asian option by PDE approach 5/24 2007. PDE & the pricing of an option The advantages of the PDE approach are that it is generally faster than.

Slides:

Advertisements

Similar presentations

A Multi-Phase, Flexible, and Accurate Lattice for Pricing Complex Derivatives with Multiple Market Variables.

Advertisements

Multi-period BTM Dr. DAI Min.

指導教授：戴天時學生：王薇婷 7.3 Knock-out Barrier Option. There are several types of barrier options. Some “Knock out” when the underlying asset price crosses a.

Futures Options Chapter 16 1 Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008.

Who is Afraid of Black Scholes A Gentle Introduction to Quantitative Finance Day 2 July 12 th 13 th and 15 th 2013 UNIVERSIDAD NACIONAL MAYOR DE SAN MARCOS.

Options, Futures, and Other Derivatives, 6 th Edition, Copyright © John C. Hull The Black-Scholes- Merton Model Chapter 13.

Chapter 14 The Black-Scholes-Merton Model

Valuing Stock Options: The Black-Scholes-Merton Model.

By: Piet Nova The Binomial Tree Model.  Important problem in financial markets today  Computation of a particular integral  Methods of valuation 

Basic Numerical Procedures Chapter 19 1 資管所柯婷瑱 2009/07/17.

Stochastic Volatility Modelling Bruno Dupire Nice 14/02/03.

L7: Stochastic Process 1 Lecture 7: Stochastic Process The following topics are covered: –Markov Property and Markov Stochastic Process –Wiener Process.

Pricing Derivative Financial Products: Linear Programming (LP) Formulation Donald C. Williams Doctoral Candidate Department of Computational and Applied.

Derivatives Inside Black Scholes

Mathematics in Finance Binomial model of options pricing.

Options and Speculative Markets Introduction to option pricing André Farber Solvay Business School University of Brussels.

Numerical Methods for Option Pricing

Derivatives Financial products that depend on another, generally more basic, product such as a stock.

Options and Speculative Markets Inside Black Scholes Professor André Farber Solvay Business School Université Libre de Bruxelles.

Week 5 Options: Pricing. Pricing a call or a put (1/3) To price a call or a put, we will use a similar methodology as we used to price the portfolio of.

Derivatives Options on Bonds and Interest Rates Professor André Farber Solvay Business School Université Libre de Bruxelles.

Pricing an Option Monte Carlo Simulation. We will explore a technique, called Monte Carlo simulation, to numerically derive the price of an option or.

An Idiot’s Guide to Option Pricing

Asset Pricing Theory Option A right to buy (or sell) an underlying asset. Strike price: K Maturity date: T. Price of the underlying asset: S(t)

Option Pricing and Dynamic Modeling of Stock Prices Investments 2004.

Binnenlandse Francqui Leerstoel VUB Black Scholes and beyond André Farber Solvay Business School University of Brussels.

7.5 Asian Options 指導老師：戴天時演講者：鄭凱允. 序 An Asian option is one whose payoff includes a time average of the underlying asset price. The average may be over.

9.4 Forward Measure Forward Price Zero-Coupon Bond as Numeraire Theorem

Derivatives Introduction to option pricing André Farber Solvay Business School University of Brussels.

Lecturer :Jan Röman Students:Daria Novoderejkina,Arad Tahmidi,Dmytro Sheludchenko.

Valuing Stock Options:The Black-Scholes Model

11.1 Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull The Black-Scholes Model Chapter 11.

1 The Black-Scholes-Merton Model MGT 821/ECON 873 The Black-Scholes-Merton Model.

Ewa Lukasik - Jakub Lawik - Juan Mojica - Xiaodong Xu.

Advanced Risk Management I Lecture 6 Non-linear portfolios.

The Pricing of Stock Options Using Black- Scholes Chapter 12.

Exercises of computational methods in finance

Simulating the value of Asian Options Vladimir Kozak.

Chapter 13 Wiener Processes and Itô’s Lemma

Chapter 26 More on Models and Numerical Procedures Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull

Valuing Stock Options: The Black- Scholes Model Chapter 11.

The Math and Magic of Financial Derivatives Klaus Volpert Villanova University March 31, 2008.

Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009.

Lecture 1: Introduction to QF4102 Financial Modeling

Final Review. Basic Derivatives Options –Non-linear Payoffs Futures and Forward Contracts –Linear Payoffs.

Option Pricing Models: The Black-Scholes-Merton Model aka Black – Scholes Option Pricing Model (BSOPM)

Functional Itô Calculus and Volatility Risk Management

© K.Cuthbertson, D. Nitzsche FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche Lecture Asset Price.

Non-Uniform Adaptive Meshing for One-Asset Problems in Finance

Functional Itô Calculus and PDEs for Path-Dependent Options Bruno Dupire Bloomberg L.P. Rutgers University Conference New Brunswick, December 4, 2009.

Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.2004.Jun.29 1 Financial Derivatives The Mathematics Fang-Bo Yeh Mathematics Department System and Control.

Computational Finance

The Black-Scholes-Merton Model Chapter 13 Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull

Monte-Carlo Simulations Seminar Project. Task  To Build an application in Excel/VBA to solve option prices.  Use a stochastic volatility in the model.

1 Advanced Finite Difference Methods for Financial Instrument Pricing Core Processes PDE theory in general Applications to financial engineering State.

Valuing Stock Options:The Black-Scholes Model

Small Dimension PDE for Discrete Asian Options Eric BenHamou (LSE, UK) & Alexandre Duguet (LSE, UK) CEF2000 Conference (Barcelona)

The Black- Scholes Equation

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 18.1 Exotic Options Chapter 18.

Lecture 8: Finance: GBM model for share prices, futures and options, hedging, and the Black-Scholes equation Outline: GBM as a model of share prices futures.

Numerical Methods for derivatives pricing. 2 American Monte Carlo It’s difficult to price an option with early exercise using Monte Carlo But some options.

Chapter 13 Wiener Processes and Itô’s Lemma 1. Stochastic Processes Describes the way in which a variable such as a stock price, exchange rate or interest.

Primbs, MS&E Applications of the Linear Functional Form: Pricing Exotics.

Wiener Processes and Itô’s Lemma

Financial Risk Management of Insurance Enterprises

Pricing Asian Basket Multi-Digital Options

Mathematical Finance An Introduction

High-accuracy PDE Method for Financial Derivative Pricing Shan Zhao and G. W. Wei Department of Computational Science National University of Singapore,

Chapter 14 Wiener Processes and Itô’s Lemma

Numerical Methods in Finance

Presentation transcript:

Price an Asian option by PDE approach 5/

PDE & the pricing of an option The advantages of the PDE approach are that it is generally faster than Monte Carlo methods and that it gives the results for all initial prices (and even for all strikes or all maturities T in some cases). The drawback is that the numerical methods are usually more complicated to implement for PDE. -- Francois Dubois & Tony Lelievre

B.S. PDE Suppose an option has payoff function the value of the option at time t is By Black-Scholes assumption, where dz is a standard Brownian motion. By Ito lemma,

B.S. PDE Long: 1 share option Short: V s share underlying asset the value of this portfolio is riskless This is a PDE for vanilla option

Pricing an Asian option by PDE approach Suppose an option has payoff function the value of the option at time t is where By Black-Scholes assumption, where dz is a standard Brownian motion. By multi-dimension Ito lemma,

Pricing an Asian option by PDE approach Long: 1 share Asian option Short: V s share underlying asset the value of this portfolio is riskless This is a PDE for Asian option

PDE & the pricing of an option the PDE approach gives the results for all the time, all initial prices, all running average.

Pricing an Asian option by PDE approach Solving this PDE using finite difference take time for V is a function with 3 variables.

Change of variable Francois Dubois & Tony Lelievre (2005) Change of variable

PDE (1) with boundary condition is reduced to: Solving this PDE by finite difference method take time Note that this PDE has been obtained by Rogers & Shi (1995) by using some different approach. Change of variable Francois Dubois & Tony Lelievre (2005)

Numerical results for Rogers & Shi PDE

Change of variable Francois Dubois & Tony Lelievre (2005) Rogers & Shi’s PDE gives poor results, especially when the volatility is small. These poor results are due to the fact when x is close to zero, the advective term is larger than the diffusion term. Change of variable This PDE has been obtained by Vecer (2001) by using some financial arguments

The reason why we chose this change of variable The PDE (2) :

The reason why we chose this change of variable In order to fit the term we solve the ODE So that we chose

Change of variable This approach can be generalized. For example, in order to completely get rid of the advective term, we solve the ODE: Change of variable