Monte Carlo Methods in Finance

Size: px
Start display at page:

Download "Monte Carlo Methods in Finance"

Transcription

1 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

2 Outline Introduction 1 Introduction 2 Generating Random Numbers Generating Random Variables Generating Sample Path 3 Variance Reduction Techniques Quasi Monte Carlo Method 4 Black-Scholes Equations Monte Carlo Simulations for Option Pricing

3 Introduction Background History: John von Neumann, Stainslaw Ulam and Nicholas Metroplis Manhattan Project in Los Alamous National Laboratory Monte Carlo Casino, Monaco Monte Carlo methods: experimental mathematics large number of random variable simulation strong law of large numbers the sample average converges almost surely to the expected value X1 + + Xn a.s. X n = µ n i.e. n ( ) P lim X n = µ = 1. n

4 Introduction (Cont.) Two broad classes of Monte Carlo methods: Direct simulation of a naturally random system Operations research (inventory control, hospital management) Statistics: properties of complicated distribution Finance: models for stock prices, credit risk Physical, biology and social science: models with complex nondeterministic time evolution Adding artificial randomness to a system, then simulating the new system Solving some partial differential equations Markov chain Monte Carlo methods: for problems in statistical physics and in Bayesian statistics Optimization: travelling salesman, genetic algorithms

5 Example Introduction Objective: integral ˆ 1 α = f (x) dx = E [f (U)] 0 U uniformly distributed between 0 and 1 Get points U 1, U 2, independently and uniformly from [0, 1] The Monte carlo estimate ˆα n = 1 n f (U i ) n i=1 If f is integrable over [0, 1], by strong law of large numbers ˆα n α with probability 1 as n The error α n α is approximately normally distributed with mean 0 and standard deviation σ f/ n, where σf 2 = 1 0 (f (x) α)2 dx and can be approximated by smaple standard deviation s f = n i=1 (f (U i ) ˆα n) 2 1 n 1

6 Principles of Derivative Pricing Principles of theory for Monte Carlo If a derivative security can be perfectly replicated through trading in other assets, then the price of the derivative security is the cost of the replicating trading strategy. Discounted asset prices are martingales under a probability measure associated with the choice of discount factor. Prices are expectation of discounted payoffs under such martingale measure. In a complete market, any payoff can be realized through a trading strategy and the martingale measure associated with the discount rate is unique.

7 Random Number Generator Generating Random Numbers Generating Random Variables Generating Sample Path A generator of genuinely random numbers has the mechanism for producing random variables U 1, U 2, such that each U i is uniformly distirbuted between 0 and 1 the U i are mutually indepedent A random number generator produces a finite sequence of numbers u 1, u 2,, u K in the unit interval pseudorandom number generator not real random number, only mimics randomness

8 Linear Congruential Gnerators Generating Random Numbers Generating Random Variables Generating Sample Path Definition The general linear congruential generator proposed by Lehmer takes the form x i+1 = (ax i + c) mod m, u i+1 = x i+1 m. a, c, m are integer constants that determine the value generated. Initial value x 0 is called seed.

9 Inverse Transform Method Generating Random Numbers Generating Random Variables Generating Sample Path Definition In order to sample from a cumulative distribution function F, i.e. generate a random variable X with the property that P (X x) = F (x) for all x. The inverse transform method sets X = F 1 (U), U Unif [0, 1] where F 1 is the inverse of F and Unif [0, 1] denotes the uniform distribution on [0, 1]. Proof sketch: P (X x) = P ( F 1 (U) x ) = P (U F (x)) = F (x) Example: The exponential distribution with mean θ has distribution F (x) = 1 e x/θ, x 0. Inverting the exponential distirbution yields X = θ log (1 U) and can be implemented as X = θ log (U).

10 Acceptance-Rejection Method Definition Generating Random Numbers Generating Random Variables Generating Sample Path In order to generate random variable X with density function f (x), we first generate X from distribution g (x). Then generate U from Unif [0, 1]. If U f (X ) /cg(x ), this is the expected X ; else, repeat above steps. Proof sketch: Let Y be a sample returned by the algorithm and observe that Y has the distribution of X conditional on U f (X ) /cg(x ). For any A R P (Y A) = P (X A U f (X )/cg(x ) ) = P (X A, U f (X )/cg(x )) P (U f (X )/cg(x )) P (X A, U f (X ) /cg(x )) = A f (x) g (x) dx = 1 f (x) dx cg(x) c A P (U f (X ) /cg(x )) = R f (x) g (x) dx = 1 cg(x) c P (Y A) = f (x) dx conclusion proved. A

11 Brownian Motion Introduction Generating Random Numbers Generating Random Variables Generating Sample Path Definition A standard one-dimensional Brownian motion on [0, T ] is a stochastic process {W (t), 0 t T } with following properties: W (0) = 0; The mapping t W (t) is a continuous function; The increments W (t 1) W (t 0), W (t 2) W (t 1),, W (t k ) W (t k 1 ) are independent for any k and any 0 < t 0 < t 1 < < t k T ; W (t) W (s) N (0, t s) for any 0 < s < t < T. Simulation of Brownian Motion Based on the stationary and independent increment properties, generate n independent and identically distributed random variable B 1,, B n such that B i N ( ) 0, T n, i = 1,, n. Define Ŵ ( ) kt n = k i=1 B i, then as n, Ŵ is an appropriate of W.

12 Brownian Motion (Cont.) Generating Random Numbers Generating Random Variables Generating Sample Path

13 Geometric Brownian Motion Generating Random Numbers Generating Random Variables Generating Sample Path Definition A stochastic process S (t) is a geometric Brownian motion if log S (t) is a Brownian motion with initial value log S (0). Geometric Brownian motion is the most fundamental model of the value of a financial asset. Suppose W is a standard Brownian motion and a geometric Brownian motion process is often specified by an SDE By Itô formula, we have ds (t) S (t) d (log S (t)) = = µdt + σdw (t) (µ 12 σ2 ) dt + σdw (t) and if S has initial value S (0) then S (t) = S (0) exp [(µ 12 ) ] σ2 t + σw (t).

14 Geometric Brownian Motion (Cont.) Generating Random Numbers Generating Random Variables Generating Sample Path

15 The Stratified Sampling Variance Reduction Techniques Quasi Monte Carlo Method Definition Stratified sampling refers broadly to any sampling mechanism that constrains the fraction of observations drawn from specific subsets of the sample space. Let A 1,, A k be disjoint subsets of the real line for which P (Y i A i ) = 1, then K K E [Y ] = P (Y A i ) E [Y Y A i ] = p i E [Y Y A i ] i=1 Decide in advance what fraction of the sample should be drawn from each stratum A i and the theoretical probability p i = P (Y A i ) An unbiased estimator of E [Y ] is provided by i=1 Ŷ = K i=1 p i 1 n i n i j=1 Y ij = 1 n n K i i=1 j=1 Y ij

16 Stratified Sampling (Cont.) Variance Reduction Techniques Quasi Monte Carlo Method comparison of stratified sample (left) and random sample (right)

17 Antithetic Variates Introduction Variance Reduction Techniques Quasi Monte Carlo Method Definition The method of antithetic variates attempts to reduce variance by introducing negative dependence between pairs of replications. U and 1 U are both uniformly distributed over [0, 1] F 1 (U) and F 1 (1 U) both have distribution F and are mutually antithetic. Implement of the antithetic variates method: ) ) ) the pairs (Y 1, Ỹ 1, (Y 2, Ỹ 2,, (Y n, Ỹ n are i.i.d. ) each Y i and Ỹ i have same distribution and Cov (Y i, Ỹ i < 0 ) Monte Carlo estimate is Ŷ = 1 n 2n i=1 (Y i + Ỹ i ) Var (Ŷ = 1 n ( ) n Var Yi + Ỹ i = 1 ( )) Var (Y 1) + Cov (Y 1, Ỹ i=1

18 Importance Sampling Variance Reduction Techniques Quasi Monte Carlo Method Definition Compute an expectation under a given probability measure Q of a random variable X, there is another measure Q equivalent to Q such that [ E Q [X ] = E Q X dq ]. d Q define Radon-Nikodym derivative L = dq d Q and measure Q is called an importance measure which give more weight to important outcomes. Theorem Let Q be define by dq dq = X E X the importance sampling estimator ZL under Q has a smaller variance than the estimator ZL under any other Q.

19 Quasi Monte Carlo Method Variance Reduction Techniques Quasi Monte Carlo Method Definition Quasi Monte Carlo method is a method numerical integration and solving some problems using low-discrepancy sequences (or quasi-random sequence or sub-random sequences) Properties of Quasi Monte Carlo Method Quasi Monte Carlo method make no attempt to mimic randomness, but to generating points evenly distributed. ( ) 1 Accelerate convergence of ordinary Monte Carlo method from O n to Quasi Monte Carlo method O ( 1 n ). Example: Suppose the objective is to calculate ˆ E [f (U 1,, U d )] = f (x) dx 1 [0,1) d n n f (x i ) for carefully and deterministically chosen points x 1,, x n in [0, 1) d. i=1

20 European Option Introduction Black-Scholes Equations Monte Carlo Simulations for Option Pricing Definition An option is a derivative financial instrument that specifies a contract between two parties for future transaction on an asset at a reference price (the strike). The buyer of the option gains the right, but not the obligation, to engage in the transaction, while the seller incurs the corresponding obligation to fulfill the transaction. An option conveys the right to buy something is called a call option; an option conveys the right to sell something is called a put option. The reference price at which the underlying asset may be traded is called strike price or exercise price. Most options have an expiration date and if it is not exercised by the expiration date, it becomes worthless. A European option may be exercised only at the expiration date of the option.

21 Black-Scholes Equations Black-Scholes Equations Monte Carlo Simulations for Option Pricing Black-Scholes model of the market follows these assumptions: There is no arbitrage opportunity. The stock price follows a geometric Brownian motion with constant drift and volatility. It is possible to borrow and lend cash at a known constant risk free interest rate. It is possible to but and sell any amount, even fractional of stock. The transactions do not incur any fees or costs and underlying security does not pay a dividend. Definition The Black-Scholes equation is a partial differential equation which describes the price of the option over time: V t σ2 S 2 2 V S 2 V + rs S rv = 0.

22 Black-Scholes Solution Black-Scholes Equations Monte Carlo Simulations for Option Pricing The value of a call option for a non-dividend paying underlying stock in terms of Black-Scholes parameter is ( ln S K d 2 = r(t t) C (S, t) = N (d 1 ) S N (d 2 ) Ke ( ln S K d 1 = ) + ) + (r σ2 2 σ T t (r + σ2 2 σ T t ) (T t) ) (T t) = d 1 σ T t. The price of a corresponding put option based on put-call parity is P (S, t) = Ke r(t t) S + C (S, t) = N ( d 2 ) Ke r(t t) N ( d 1 ) S. N ( ) is the cumulative distribution function of the standard normal distribution. T t is the time to maturity. S is spot price of the underlying asset and K is strike price. r is the risk free rate and σ is the volatility of the returns.

23 Numerical Scheme Introduction Black-Scholes Equations Monte Carlo Simulations for Option Pricing With boundary conditions C (0, t) = 0 for all t, C (S, t) S as S, C (S, T ) = max {S K, 0} corresponding numerical schemes can be developed.

24 No-Arbitrage Pricing Formula Black-Scholes Equations Monte Carlo Simulations for Option Pricing Non-Arbitrage Pricing Formula Price of a European option can be obtained by the expectation of the present value of the payoff for the options under the equivalent martingale measure Q. That is, at time t < T, the non-arbitrage price of a European option V t with the payoff Π(T ) and the maturity T is obtained by V t = e r(t t) E Q [Π(T ) F t ]. Consider European call option, then Π(T ) = (S T K) +. Then we have C (S t, t) = e r(t t) E Q [ (ST K) + F t ] and the stock price follows geometric Brownian motion ds t S t = µdt + σdw t.

25 Monte Carlo Simulation Black-Scholes Equations Monte Carlo Simulations for Option Pricing Simulate N (about ten thousands scale) paths of S i, i = 1,, N Every path S i is genererated from time t to T step by step S k = S k 1 exp [(µ 12 ) ] σ2 t + σw t, W t N (0, t) Start from Si 0 T t t = S t, i = 1,, N we have N paths ended with Si, then the simulated Monte Carlo result of call option can be approximated as C (S t, t) 1 N N ( S i=1 i T t t ) + K

26 Monte Carlo Simulation (Cont.) Black-Scholes Equations Monte Carlo Simulations for Option Pricing

27 References Introduction Black-Scholes Equations Monte Carlo Simulations for Option Pricing Paul Glasserman, Monte Carlo Methods in Financial Engineering, Springer, ISBN-10: , ISBN-13: , August 2003 Phelim P. Boyle, Options: A Monte Carlo Approach, Journal of Financial Economics 4(1977) Phelim Boyle, Mark Broadie, Paul Glasserman, Monte Carlo Methods for Security Pricing, Journal of Economic Dynamics and Control 21(1997)

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

Mathematical Finance

Mathematical Finance Mathematical Finance Option Pricing under the Risk-Neutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European

More information

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. Black-Scholes Equation is derived using two methods: (1) risk-neutral measure; (2) - hedge. II.

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

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

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

The Black-Scholes Formula

The Black-Scholes Formula FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Black-Scholes Formula These notes examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the

More information

Option Valuation. Chapter 21

Option Valuation. Chapter 21 Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of in-the-money options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price

More information

American Options. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan American Options

American Options. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan American Options American Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Early Exercise Since American style options give the holder the same rights as European style options plus

More information

CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options

CS 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 information

Chapter 1: Financial Markets and Financial Derivatives

Chapter 1: Financial Markets and Financial Derivatives Chapter 1: Financial Markets and Financial Derivatives 1.1 Financial Markets Financial markets are markets for financial instruments, in which buyers and sellers find each other and create or exchange

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

Stephane Crepey. Financial Modeling. A Backward Stochastic Differential Equations Perspective. 4y Springer

Stephane 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 Discrete-Time Stochastic

More information

Monte Carlo Methods and Models in Finance and Insurance

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 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

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

A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting Model

A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting Model Applied Mathematical Sciences, vol 8, 14, no 143, 715-7135 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/11988/ams144644 A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting

More information

Binomial lattice model for stock prices

Binomial 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 information

Option Pricing. Chapter 11 Options on Futures. Stefan Ankirchner. University of Bonn. last update: 13/01/2014 at 14:25

Option Pricing. Chapter 11 Options on Futures. Stefan Ankirchner. University of Bonn. last update: 13/01/2014 at 14:25 Option Pricing Chapter 11 Options on Futures Stefan Ankirchner University of Bonn last update: 13/01/2014 at 14:25 Stefan Ankirchner Option Pricing 1 Agenda Forward contracts Definition Determining forward

More information

Review of Basic Options Concepts and Terminology

Review 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 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

Lecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena Lecture 7: Bounds on Options Prices Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Option Price Quotes Reading the

More information

The Black-Scholes-Merton Approach to Pricing Options

The Black-Scholes-Merton Approach to Pricing Options he Black-Scholes-Merton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the Black-Scholes-Merton approach to determining

More information

Oscillatory Reduction in Option Pricing Formula Using Shifted Poisson and Linear Approximation

Oscillatory 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 information

Lecture. S t = S t δ[s t ].

Lecture. S t = S t δ[s t ]. Lecture In real life the vast majority of all traded options are written on stocks having at least one dividend left before the date of expiration of the option. Thus the study of dividends is important

More 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

Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-Normal Distribution

Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-Normal Distribution Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-ormal Distribution October 4, 200 Limiting Distribution of the Scaled Random Walk Recall that we defined a scaled simple random walk last

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

Option pricing. Vinod Kothari

Option pricing. Vinod Kothari Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate

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

Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)

Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald) Copyright 2003 Pearson Education, Inc. Slide 08-1 Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared

More information

Assignment 2: Option Pricing and the Black-Scholes formula The University of British Columbia Science One CS 2015-2016 Instructor: Michael Gelbart

Assignment 2: Option Pricing and the Black-Scholes formula The University of British Columbia Science One CS 2015-2016 Instructor: Michael Gelbart Assignment 2: Option Pricing and the Black-Scholes formula The University of British Columbia Science One CS 2015-2016 Instructor: Michael Gelbart Overview Due Thursday, November 12th at 11:59pm Last updated

More information

BINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract

BINOMIAL 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 information

Chapter 21: Options and Corporate Finance

Chapter 21: Options and Corporate Finance Chapter 21: Options and Corporate Finance 21.1 a. An option is a contract which gives its owner the right to buy or sell an underlying asset at a fixed price on or before a given date. b. Exercise is the

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

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

Put-Call Parity. chris bemis

Put-Call Parity. chris bemis Put-Call Parity chris bemis May 22, 2006 Recall that a replicating portfolio of a contingent claim determines the claim s price. This was justified by the no arbitrage principle. Using this idea, we obtain

More information

Sensitivity analysis of utility based prices and risk-tolerance wealth processes

Sensitivity analysis of utility based prices and risk-tolerance wealth processes Sensitivity analysis of utility based prices and risk-tolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,

More information

Option Pricing. Chapter 9 - Barrier Options - Stefan Ankirchner. University of Bonn. last update: 9th December 2013

Option Pricing. Chapter 9 - Barrier Options - Stefan Ankirchner. University of Bonn. last update: 9th December 2013 Option Pricing Chapter 9 - Barrier Options - Stefan Ankirchner University of Bonn last update: 9th December 2013 Stefan Ankirchner Option Pricing 1 Standard barrier option Agenda What is a barrier option?

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

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

On Market-Making and Delta-Hedging

On Market-Making and Delta-Hedging On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing What to market makers do? Provide

More information

Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab

Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?

More 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

Option Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration

Option Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration CHAPTER 16 Option Valuation 16.1 OPTION VALUATION: INTRODUCTION Option Values Intrinsic value - profit that could be made if the option was immediately exercised Call: stock price - exercise price Put:

More information

Pricing of an Exotic Forward Contract

Pricing of an Exotic Forward Contract Pricing of an Exotic Forward Contract Jirô Akahori, Yuji Hishida and Maho Nishida Dept. of Mathematical Sciences, Ritsumeikan University 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan E-mail: {akahori,

More information

Asian Option Pricing Formula for Uncertain Financial Market

Asian Option Pricing Formula for Uncertain Financial Market Sun and Chen Journal of Uncertainty Analysis and Applications (215) 3:11 DOI 1.1186/s4467-15-35-7 RESEARCH Open Access Asian Option Pricing Formula for Uncertain Financial Market Jiajun Sun 1 and Xiaowei

More 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

A Comparison of Option Pricing Models

A Comparison of Option Pricing Models A Comparison of Option Pricing Models Ekrem Kilic 11.01.2005 Abstract Modeling a nonlinear pay o generating instrument is a challenging work. The models that are commonly used for pricing derivative might

More information

EXP 481 -- Capital Markets Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices 1) C > 0

EXP 481 -- Capital Markets Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices 1) C > 0 EXP 481 -- Capital Markets Option Pricing imple arbitrage relations Payoffs to call options Black-choles model Put-Call Parity Implied Volatility Options: Definitions A call option gives the buyer the

More information

ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida

ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida ARBITRAGE-FREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic

More 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

Bond Options, Caps and the Black Model

Bond Options, Caps and the Black Model Bond Options, Caps and the Black Model Black formula Recall the Black formula for pricing options on futures: C(F, K, σ, r, T, r) = Fe rt N(d 1 ) Ke rt N(d 2 ) where d 1 = 1 [ σ ln( F T K ) + 1 ] 2 σ2

More information

QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS

QUANTIZED 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 information

τ θ What is the proper price at time t =0of this option?

τ θ What is the proper price at time t =0of this option? Now by Itô s formula But Mu f and u g in Ū. Hence τ θ u(x) =E( Mu(X) ds + u(x(τ θ))) 0 τ θ u(x) E( f(x) ds + g(x(τ θ))) = J x (θ). 0 But since u(x) =J x (θ ), we consequently have u(x) =J x (θ ) = min

More information

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

1 Geometric Brownian motion

1 Geometric Brownian motion Copyright c 006 by Karl Sigman Geometric Brownian motion Note that since BM can take on negative values, using it directly for modeling stock prices is questionable. There are other reasons too why BM

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

UNIVERSITY OF CALIFORNIA Los Angeles. A Monte Carlo Approach to Pricing An Exotic Currency Derivative Structure

UNIVERSITY OF CALIFORNIA Los Angeles. A Monte Carlo Approach to Pricing An Exotic Currency Derivative Structure UNIVERSITY OF CALIFORNIA Los Angeles A Monte Carlo Approach to Pricing An Exotic Currency Derivative Structure A thesis submitted in partial satisfaction of the requirements for the degree of Master of

More information

Martingale Pricing Applied to Options, Forwards and Futures

Martingale Pricing Applied to Options, Forwards and Futures IEOR E4706: Financial Engineering: Discrete-Time 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 information

COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS

COMPLETE 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 mean-variance problems in

More information

When to Refinance Mortgage Loans in a Stochastic Interest Rate Environment

When to Refinance Mortgage Loans in a Stochastic Interest Rate Environment When to Refinance Mortgage Loans in a Stochastic Interest Rate Environment Siwei Gan, Jin Zheng, Xiaoxia Feng, and Dejun Xie Abstract Refinancing refers to the replacement of an existing debt obligation

More information

Lecture 21 Options Pricing

Lecture 21 Options Pricing Lecture 21 Options Pricing Readings BM, chapter 20 Reader, Lecture 21 M. Spiegel and R. Stanton, 2000 1 Outline Last lecture: Examples of options Derivatives and risk (mis)management Replication and Put-call

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

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

The Black-Scholes Formula

The Black-Scholes Formula ECO-30004 OPTIONS AND FUTURES SPRING 2008 The Black-Scholes Formula The Black-Scholes Formula We next examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they

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

Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies

Hedging 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 information

Chapter 11 Properties of Stock Options. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull

Chapter 11 Properties of Stock Options. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull Chapter 11 Properties of Stock Options 1 Notation c: European call option price p: European put option price S 0 : Stock price today K: Strike price T: Life of option σ: Volatility of stock price C: American

More information

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes consider the way put and call options and the underlying can be combined to create hedges, spreads and combinations. We will consider the

More information

Chapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options.

Chapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options. Chapter 11 Options Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of risk-adjusted discount rate. Part D Introduction to derivatives. Forwards

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

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

Call and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options

Call and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options Call and Put Options A call option gives its holder the right to purchase an asset for a specified price, called the strike price, on or before some specified expiration date. A put option gives its holder

More information

Option Properties. Liuren Wu. Zicklin School of Business, Baruch College. Options Markets. (Hull chapter: 9)

Option Properties. Liuren Wu. Zicklin School of Business, Baruch College. Options Markets. (Hull chapter: 9) Option Properties Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 9) Liuren Wu (Baruch) Option Properties Options Markets 1 / 17 Notation c: European call option price.

More information

Notes on Black-Scholes Option Pricing Formula

Notes on Black-Scholes Option Pricing Formula . Notes on Black-Scholes Option Pricing Formula by De-Xing Guan March 2006 These notes are a brief introduction to the Black-Scholes formula, which prices the European call options. The essential reading

More information

Factors Affecting Option Prices

Factors Affecting Option Prices Factors Affecting Option Prices 1. The current stock price S 0. 2. The option strike price K. 3. The time to expiration T. 4. The volatility of the stock price σ. 5. The risk-free interest rate r. 6. The

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

Pricing Discrete Barrier Options

Pricing Discrete Barrier Options Pricing Discrete Barrier Options Barrier options whose barrier is monitored only at discrete times are called discrete barrier options. They are more common than the continuously monitored versions. The

More information

where N is the standard normal distribution function,

where N is the standard normal distribution function, The Black-Scholes-Merton formula (Hull 13.5 13.8) Assume S t is a geometric Brownian motion w/drift. Want market value at t = 0 of call option. European call option with expiration at time T. Payout at

More information

Financial Modeling. Class #06B. Financial Modeling MSS 2012 1

Financial Modeling. Class #06B. Financial Modeling MSS 2012 1 Financial Modeling Class #06B Financial Modeling MSS 2012 1 Class Overview Equity options We will cover three methods of determining an option s price 1. Black-Scholes-Merton formula 2. Binomial trees

More information

Properties of Stock Options. Chapter 10

Properties of Stock Options. Chapter 10 Properties of Stock Options Chapter 10 1 Notation c : European call option price C : American Call option price p : European put option price P : American Put option price S 0 : Stock price today K : Strike

More information

Merton-Black-Scholes model for option pricing. Peter Denteneer. 22 oktober 2009

Merton-Black-Scholes model for option pricing. Peter Denteneer. 22 oktober 2009 Merton-Black-Scholes model for option pricing Instituut{Lorentz voor Theoretische Natuurkunde, LION, Universiteit Leiden 22 oktober 2009 With inspiration from: J. Tinbergen, T.C. Koopmans, E. Majorana,

More information

FIN 411 -- Investments Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices

FIN 411 -- Investments Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices FIN 411 -- Investments Option Pricing imple arbitrage relations s to call options Black-choles model Put-Call Parity Implied Volatility Options: Definitions A call option gives the buyer the right, but

More information

Forward Price. The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow.

Forward Price. The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow. Forward Price The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow. The forward price is the delivery price which makes the forward contract zero

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

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

FINANCIAL ECONOMICS OPTION PRICING

FINANCIAL ECONOMICS OPTION PRICING OPTION PRICING Options are contingency contracts that specify payoffs if stock prices reach specified levels. A call option is the right to buy a stock at a specified price, X, called the strike price.

More information

n(n + 1) 2 1 + 2 + + n = 1 r (iii) infinite geometric series: if r < 1 then 1 + 2r + 3r 2 1 e x = 1 + x + x2 3! + for x < 1 ln(1 + x) = x x2 2 + x3 3

n(n + 1) 2 1 + 2 + + n = 1 r (iii) infinite geometric series: if r < 1 then 1 + 2r + 3r 2 1 e x = 1 + x + x2 3! + for x < 1 ln(1 + x) = x x2 2 + x3 3 ACTS 4308 FORMULA SUMMARY Section 1: Calculus review and effective rates of interest and discount 1 Some useful finite and infinite series: (i) sum of the first n positive integers: (ii) finite geometric

More information

LECTURE 9: A MODEL FOR FOREIGN EXCHANGE

LECTURE 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 information

A spot price model feasible for electricity forward pricing Part II

A spot price model feasible for electricity forward pricing Part II A spot price model feasible for electricity forward pricing Part II Fred Espen Benth Centre of Mathematics for Applications (CMA) University of Oslo, Norway Wolfgang Pauli Institute, Wien January 17-18

More information

Financial Options: Pricing and Hedging

Financial Options: Pricing and Hedging Financial Options: Pricing and Hedging Diagrams Debt Equity Value of Firm s Assets T Value of Firm s Assets T Valuation of distressed debt and equity-linked securities requires an understanding of financial

More information

2. Exercising the option - buying or selling asset by using option. 3. Strike (or exercise) price - price at which asset may be bought or sold

2. Exercising the option - buying or selling asset by using option. 3. Strike (or exercise) price - price at which asset may be bought or sold Chapter 21 : Options-1 CHAPTER 21. OPTIONS Contents I. INTRODUCTION BASIC TERMS II. VALUATION OF OPTIONS A. Minimum Values of Options B. Maximum Values of Options C. Determinants of Call Value D. Black-Scholes

More information

Chapter 21 Valuing Options

Chapter 21 Valuing Options Chapter 21 Valuing Options Multiple Choice Questions 1. Relative to the underlying stock, a call option always has: A) A higher beta and a higher standard deviation of return B) A lower beta and a higher

More information

LECTURE 15: AMERICAN OPTIONS

LECTURE 15: AMERICAN OPTIONS LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These

More information

Master of Mathematical Finance: Course Descriptions

Master 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 information

Lecture 6: Option Pricing Using a One-step Binomial Tree. Friday, September 14, 12

Lecture 6: Option Pricing Using a One-step Binomial Tree. Friday, September 14, 12 Lecture 6: Option Pricing Using a One-step Binomial Tree An over-simplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond

More information

Options 1 OPTIONS. Introduction

Options 1 OPTIONS. Introduction Options 1 OPTIONS Introduction A derivative is a financial instrument whose value is derived from the value of some underlying asset. A call option gives one the right to buy an asset at the exercise or

More information

Finance 400 A. Penati - G. Pennacchi. Option Pricing

Finance 400 A. Penati - G. Pennacchi. Option Pricing Finance 400 A. Penati - G. Pennacchi Option Pricing Earlier we derived general pricing relationships for contingent claims in terms of an equilibrium stochastic discount factor or in terms of elementary

More information