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

Size: px
Start display at page:

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

## Transcription

1 Option Pricing Chapter 9 - Barrier Options - Stefan Ankirchner University of Bonn last update: 9th December 2013 Stefan Ankirchner Option Pricing 1

2 Standard barrier option Agenda What is a barrier option? Deriving pricing PDEs Valuation using the distribution of the maximum of a BM Double barrier options Further reading: S. Shreve: Stochastic Calculus for Finance II, Chapter 7.3. Stefan Ankirchner Option Pricing 2

3 Standard barrier option What is a barrier option? Definition: A barrier option is an option where the right to exercise depends on whether the underlying crosses a certain barrier level before expiration. Two cases: knock-out options: the right to exercise is lost if the barrier is crossed. The option becomes worthless. knock-in options: the right to exercise is obtained if the barrier is crossed. Why barrier options? smaller premiums: barrier options are cheaper! Stefan Ankirchner Option Pricing 3

4 Standard barrier option Example: Down and Out Call European call that is knocked out (you also say deactivated ) if the underlying crosses a barrier L before expiration. L is smaller than the present asset value S 0. Payoff of the D&O Call { CT D&O (ST K) = +, if S t > L for all t T, 0, if S t L for at least one t T. Stefan Ankirchner Option Pricing 4

5 Standard barrier option Example: Down and Out Call Special case K = L Asset evolution K L S 0 Payoff call Standard D&O S T K S T K S T K Time T Stefan Ankirchner Option Pricing 5

6 Standard barrier option Standard barrier options Categorizing barrier options standard option if active: call or put barrier level in relation to current asset price: down or up knock-in or knock-out 2 3 = 8 standard types Call Put Up Down Up Down In Out In Out In Out In Out Stefan Ankirchner Option Pricing 6

7 D&O call D&O call U&O call By following our 4 step recipe one can derive pricing PDEs for barrier options. We will do so first for D&O calls. Notation: S t = price of the underlying (think of a stock) m t = min 0 u t S u Payoff of the D&O call: C D&O T minimum price between 0 and t = (S T K) + 1 {mt >L}. The time t value of the D&O call depends on S t and m t. However, under the assumption that there has been no knock out prior to t, the value depends only on S t! Stefan Ankirchner Option Pricing 7

8 D&O call U&O call D&O call: Rolling the 4 steps 1) Assume that the D&O call is replicable. Denote by v(t, x) the time t option value / replicating portfolio value under the assumption that the barrier has not been attained before t and that S t = x. 2) v(t, S t ) is an Ito process. Ito s formula implies [ dv(t, S t) = v x(t, S t)σs tdw t + v x(t, S t)µs t + 1 ] 2 vxx(t, St)S t 2 σ 2 + v t(t, S t) dt. The self-financing condition yields dv(t, S t) = (t)ds t + (v(t, S t) (t)s t)rdt = (t)σs tdw t + (t)µs tdt + (v(t, S t) (t)s t)rdt. Stefan Ankirchner Option Pricing 8

9 D&O 4 steps cont d D&O call U&O call [ dv(t, S t) = v x(t, S t)σs tdw t + v x(t, S t)µs t + 1 ] 2 vxx(t, St)S t 2 σ 2 + v t(t, S t) dt, dv(t, S t) = (t)σs tdw t + (t)µs tdt + (v(t, S t) (t)s t)rdt. 3) Matching the coefficients: = (t) = v x (t, S t ) and v(t, x) has to satisfy the PDE v t (t, x) + rxv x (t, x) σ2 x 2 v xx (t, x) rv(t, x) = 0. Stefan Ankirchner Option Pricing 9

10 D&O 4 steps cont d D&O call U&O call Boundary conditions: at knock out the D&O call is worthless, i.e. v(t, L) = 0, 0 t T. payoff if no knock-out prior to expiration: (S T K) +. Thus v(t, x) = (x K) +, x > L. Stefan Ankirchner Option Pricing 10

11 D&O 4 steps cont d D&O call U&O call 4) Solving the PDE: Feynman-Kac (applies only up to knock-out), numerical solution: straightforward Boundary conditions for solving the PDE with a finite difference scheme: v(t, L) = 0, 0 t T, v(t, x) = (x K) +, x > L, v(t, S max ) S max e r(t t) K, 0 t T. Stefan Ankirchner Option Pricing 11

12 D&O call D&O call U&O call Stefan Ankirchner Option Pricing 12

13 D&O call: summary D&O call U&O call Theorem Let v(t, x) be the time t D&O call value under the assumption that it has not been knocked out before t and that S t = x. Then v(t, x) satisfies the Black-Scholes PDE v t (t, x) + rxv x (t, x) σ2 x 2 v xx (t, x) rv(t, x) = 0, with boundary conditions v(t, L) = 0, 0 t T, v(t, x) = (x K) +, x > L. Remark: formulation similar to Thm in Shreve: Stochastic Calculus for Finance II Stefan Ankirchner Option Pricing 13

14 U&O call D&O call U&O call Similarly we can derive pricing PDEs for U&O calls. Assumptions and notation: upper barrier U > K S t = price of the underlying (think of a stock) M t = max 0 u t S u Payoff of the U&O call: C U&O T maximal price between 0 and t = (S T K) + 1 {MT <U}. Stefan Ankirchner Option Pricing 14

15 D&O call U&O call Pricing PDE for U&O calls Theorem (see Thm in Shreve) Let v(t, x) be the time t U&O call value under the assumption that it has not been knocked out before t and that S t = x. Then v(t, x) satisfies the Black-Scholes PDE v t (t, x) + rxv x (t, x) σ2 x 2 v xx (t, x) rv(t, x) = 0, with boundary conditions v(t, U) = 0, 0 t T, v(t, 0) = 0, 0 t T, v(t, x) = (x K) +, x < U. Caution: v(t, x) is not continuous in (T, U)! Stefan Ankirchner Option Pricing 15

16 Maximum of a Brownian motion with drift Explicit pricing formula for U&O calls U&O call revisited Payoff of an U&O call: where U > K, C U&O T M t = max 0 u t S u. The value at time t = 0: = (S T K) + 1 {MT <U}, C U&O 0 = e rt E Q [(S T K) + 1 {MT <U}], where Q is the risk neutral measure. Next: the joint Q-distribution of (M T, S T ) is known. Therefore the value of the U&O call can be calculated explicitly. Stefan Ankirchner Option Pricing 16

17 Maximum of a Brownian motion with drift Explicit pricing formula for U&O calls The maximum of a Brownian motion with drift Payoff of an U&O call: Dynamics of S: C U&O T = (S T K) + 1 {MT <U}. ds t = rs t dt + σs t dw Q t, where W Q is a Brownian motion under the risk neutral measure Q. Definition: Let α = 1 σ2 σ (r 2 ) and Note that Ŵ t = αt + W Q t. S T = S 0 e σw Q T = S 0 e σŵt, and S T K iff ŴT k := 1 σ log ( K S 0 ). +(r σ2 2 )T Stefan Ankirchner Option Pricing 17

18 Maximum of a Brownian motion with drift Explicit pricing formula for U&O calls The maximum of a Brownian motion with drift Definition: Let M T denote the maximum Then M T = max Ŵ u. 0 u T max S u = max S 0e σŵu = S 0 e σ M T, 0 u T 0 u T and the barrier U is hit iff M T b := 1 σ log ( U S 0 ). The time t = 0 value of an U&O call is given by C U&O 0 = e rt E Q [(S 0 e σŵt K) + 1 { MT <b} ] = e rt E Q [(S 0 e σŵt K) 1 { MT <b,ŵt k} ]. Stefan Ankirchner Option Pricing 18

19 Maximum of a Brownian motion with drift Explicit pricing formula for U&O calls Distribution of the maximum of a BM with drift Theorem (see Thm in Shreve) The density of the joint distribution of ( M T, ŴT ) under Q is given by f (m, w) = 2(2m w) T 2πT eαw 1 2 α2 T 1 2T (2m w)2, w m, m 0, and f (m, w) = 0 for other values of m and w. Stefan Ankirchner Option Pricing 19

20 Maximum of a Brownian motion with drift Explicit pricing formula for U&O calls Price of an U&O call Recall: The time t = 0 value of an U&O call is given by C U&O 0 = e rt E Q [(S 0 e σŵt K) 1 { MT <b,ŵt k} ]. Using the joint density of ( M T, ŴT ) we obtain that the price of an U&O call with barrier U and strike K is given by C U&O 0 = e rt b k b (S 0 e σw 2(2m w) K) w + T 2πT eαw 1 2 α2 T 1 2T (2m w)2 dmdw Stefan Ankirchner Option Pricing 20

21 Maximum of a Brownian motion with drift Explicit pricing formula for U&O calls Price of an U&O call cont d b b C U&O 0 = e rt (S 0e σw K) k w + 2(2m w) T 2πT 1 eαw α 2 T 1 (2m w) 2 2 2T dmdw. With some tedious calculations one can show that ( C0 U&O = S 0 [Φ (δ + T, S0 K e rt K [ Φ )) ( Φ (δ + T, S0 U ( (δ T, S0 K )) Φ ( ) 2r [ S0 σ U 2 Φ (δ + (T, U2 U KS 0 ( ) 1 2r [ +e rt S0 σ K 2 Φ (δ (T, U2 U KS 0 )) ] ( (δ T, S0 U )) Φ ))] (δ + (T, US0 ))] )) ))] Φ (δ (T, US0, where δ ±(T, s) = 1 [ σ log s + (r ± 12 ) ] T σ2 T. Stefan Ankirchner Option Pricing 21

22 Pricing PDEs In-Out Parity U&I call and D&I call Assumptions and notation: L = lower barrier and U = upper barrier M t = max 0 u t S u maximum price between 0 and t m t = min 0 u t S u minimum price between 0 and t Payoff of an U&I call with strike K: C U&I T = (S T K) + 1 {MT U}. Payoff of an D&I call with strike K: C D&I T = (S T K) + 1 {mt L}. Stefan Ankirchner Option Pricing 22

23 Pricing PDEs In-Out Parity Pricing PDE for U&I calls Recall that BS call(s, K, T t, σ, r) denotes the price of a Plain Vanilla Call, where S is the current price of the underlying,... Theorem Let v(t, x) be the time t U&I call value under the assumption that it has not been activated before t and that S t = x. Then v(t, x) satisfies the Black-Scholes PDE v t (t, x) + rxv x (t, x) σ2 x 2 v xx (t, x) rv(t, x) = 0, (1) with boundary conditions v(t, U) = BS call(u, K, T t, σ, r) 0 t T, v(t, 0) = 0 0 t T, v(t, x) = 0 x < U. Stefan Ankirchner Option Pricing 23

24 Pricing PDEs In-Out Parity Pricing PDE for D&I calls Theorem Let v(t, x) be the time t D&I call value under the assumption that it has not been activated before t and that S t = x. Then v(t, x) satisfies the Black-Scholes PDE (1) with boundary conditions v(t, L) = BS call(l, K, T t, σ, r) 0 t T, v(t, x) = 0 x > L. Stefan Ankirchner Option Pricing 24

25 In-Out Parity Pricing PDEs In-Out Parity Let C T = (S T K) + be the payoff of a Plain Vanilla Call. Let CT U&O and CT U&I be the payoff of barrier calls with strike K and barrier U > S 0. Notice that C T = C U&O T + C U&I T. Thus the price of the U&I call at time 0 satisfies C U&I 0 = C 0 C U&O 0. Similarly, we have an In-Out parity for Down-options: C D&I 0 = C 0 C D&O 0. Stefan Ankirchner Option Pricing 25

26 Double barrier options Parisian barrier options Second generation barrier options First Generation Barrier Options: one barrier knock-in resp. knock-out if barrier is crossed once Second Generation Barrier Options: Double barrier options: additional barrier Parisian barrier options: knock-in resp. knock-out only if a certain amount of time is spent beyond the barrier Stefan Ankirchner Option Pricing 26

27 Double barrier options Parisian barrier options Double knock out barrier options Barriers: lower barrier L upper barrier U Double knock out call and put: Type DKOC DKOP Payoff (S T K) + 1 {min0 t T S t>l, max 0 t T S t<u} (K S T ) + 1 {min0 t T S t>l, max 0 t T S t<u} Stefan Ankirchner Option Pricing 27

28 Double barrier options Parisian barrier options Double knock out barrier options Asset evolution B u S 0 B L Time T Stefan Ankirchner Option Pricing 28

29 Double barrier options Parisian barrier options Double knock out call Theorem Let v(t, x) be the time t DKOC value under the assumption that it has not been knocked out before t and that S t = x. Then v(t, x) satisfies the Black-Scholes PDE v t (t, x) + rxv x (t, x) σ2 x 2 v xx (t, x) rv(t, x) = 0, with boundary conditions v(t, L) = 0, 0 t T, v(t, U) = 0, 0 t T, v(t, x) = (x K) +, L < x < U. Stefan Ankirchner Option Pricing 29

30 Parisian Option Double barrier options Parisian barrier options Standard barrier options: the option trigger only depends on a single touching of the barrier by the underlying price process. The counterparty may manipulate the underlying for a short time such that the barrier option is knocked-out resp. knocked in Parisian barrier options require the knock-out / knock-in condition to be satisfied for a certain time, and thus prevent the counterparty to influence prices. Stefan Ankirchner Option Pricing 30

31 Double barrier options Parisian barrier options Standard and cumulative Parisian options 2 types of Parisian barrier options: Standard Parisian barrier option: option is knocked out if the underlying asset value stays consecutively below the barrier for a time longer than some pre specified time window d before the maturity date. Cumulative Parisian barrier option: option is knocked out if the underlying asset value spends until maturity in total d units of time below the barrier. Pricing of Parisian barrier options: Monte-Carlo method (see next Chapter) Stefan Ankirchner Option Pricing 31

### 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

### 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

### 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

### 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

### 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

### THE BLACK-SCHOLES MODEL AND EXTENSIONS

THE BLAC-SCHOLES MODEL AND EXTENSIONS EVAN TURNER Abstract. This paper will derive the Black-Scholes pricing model of a European option by calculating the expected value of the option. We will assume that

### 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

### 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

### Pricing Barrier Options under Local Volatility

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

### Option Pricing. Stefan Ankirchner. January 20, 2014. 2 Brownian motion and Stochastic Calculus

Option Pricing Stefan Ankirchner January 2, 214 1 The Binomial Model 2 Brownian motion and Stochastic Calculus We next recall some basic results from Stochastic Calculus. We do not prove most of the results.

### Barrier Options. Peter Carr

Barrier Options Peter Carr Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU March 14th, 2008 What are Barrier Options?

### 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

### 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

### A Vega-Gamma Relationship for European-Style or Barrier Options in the Black-Scholes Model

A Vega-Gamma Relationship for European-Style or Barrier Options in the Black-Scholes Model Fabio Mercurio Financial Models, Banca IMI Abstract In this document we derive some fundamental relationships

### 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

### 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

### 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

### Monte Carlo Methods in Finance

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

### Path-dependent options

Chapter 5 Path-dependent options The contracts we have seen so far are the most basic and important derivative products. In this chapter, we shall discuss some complex contracts, including barrier 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

### 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

### 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

### 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.

### TABLE OF CONTENTS. A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13

TABLE OF CONTENTS 1. McDonald 9: "Parity and Other Option Relationships" A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 2. McDonald 10: "Binomial Option Pricing:

### 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

### Introduction to Arbitrage-Free Pricing: Fundamental Theorems

Introduction to Arbitrage-Free Pricing: Fundamental Theorems Dmitry Kramkov Carnegie Mellon University Workshop on Interdisciplinary Mathematics, Penn State, May 8-10, 2015 1 / 24 Outline Financial market

### 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

### Schonbucher Chapter 9: Firm Value and Share Priced-Based Models Updated 07-30-2007

Schonbucher Chapter 9: Firm alue and Share Priced-Based Models Updated 07-30-2007 (References sited are listed in the book s bibliography, except Miller 1988) For Intensity and spread-based models of default

### 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

### 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

### 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

### Black-Scholes and the Volatility Surface

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

### 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

### 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

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

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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Pricing European Barrier Options with Partial Differential Equations

Pricing European Barrier Options with Partial Differential Equations Akinyemi David Supervised by: Dr. Alili Larbi Erasmus Mundus Masters in Complexity Science, Complex Systems Science, University of Warwick

### arxiv:1108.4393v2 [q-fin.pr] 25 Aug 2011

arxiv:1108.4393v2 [q-fin.pr] 25 Aug 2011 Pricing Variable Annuity Contracts with High-Water Mark Feature V.M. Belyaev Allianz Investment Management, Allianz Life Minneapolis, MN, USA August 26, 2011 Abstract

### Characterizing Option Prices by Linear Programs

Contemporary Mathematics Characterizing Option Prices by Linear Programs Richard H. Stockbridge Abstract. The price of various options on a risky asset are characterized via a linear program involving

### Introduction to Options. Derivatives

Introduction to Options Econ 422: Investment, Capital & Finance University of Washington Summer 2010 August 18, 2010 Derivatives A derivative is a security whose payoff or value depends on (is derived

### 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

### 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

### A new Feynman-Kac-formula for option pricing in Lévy models

A new Feynman-Kac-formula for option pricing in Lévy models Kathrin Glau Department of Mathematical Stochastics, Universtity of Freiburg (Joint work with E. Eberlein) 6th World Congress of the Bachelier

### 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,

### 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

### 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

### 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 This Version:

### ENGINEERING AND HEDGING OF CORRIDOR PRODUCTS - with focus on FX linked instruments -

AARHUS SCHOOL OF BUSINESS AARHUS UNIVERSITY MASTER THESIS ENGINEERING AND HEDGING OF CORRIDOR PRODUCTS - with focus on FX linked instruments - AUTHORS: DANIELA ZABRE GEORGE RARES RADU SIMIAN SUPERVISOR:

### Computational Finance Options

1 Options 1 1 Options Computational Finance Options An option gives the holder of the option the right, but not the obligation to do something. Conversely, if you sell an option, you may be obliged to

### 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

### Diusion processes. Olivier Scaillet. University of Geneva and Swiss Finance Institute

Diusion processes Olivier Scaillet University of Geneva and Swiss Finance Institute Outline 1 Brownian motion 2 Itô integral 3 Diusion processes 4 Black-Scholes 5 Equity linked life insurance 6 Merton

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

Risk-Neutral Valuation of Participating Life Insurance Contracts DANIEL BAUER with R. Kiesel, A. Kling, J. Russ, and K. Zaglauer ULM UNIVERSITY RTG 1100 AND INSTITUT FÜR FINANZ- UND AKTUARWISSENSCHAFTEN

### 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

### 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

### 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,

### The Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models

780 w Interest Rate Models The Behavior of Bonds and Interest Rates Before discussing how a bond market-maker would delta-hedge, we first need to specify how bonds behave. Suppose we try to model a zero-coupon

### 1 The Black-Scholes model: extensions and hedging

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

### Some Practical Issues in FX and Equity Derivatives

Some Practical Issues in FX and Equity Derivatives Phenomenology of the Volatility Surface The volatility matrix is the map of the implied volatilities quoted by the market for options of different strikes

### Calibration of Stock Betas from Skews of Implied Volatilities

Calibration of Stock Betas from Skews of Implied Volatilities Jean-Pierre Fouque University of California Santa Barbara Joint work with Eli Kollman (Ph.D. student at UCSB) Joint Seminar: Department of

### Knock Out Power Options in Foreign Exchange Markets

U.U.D.M. Project Report 04:0 Knock Out Power Options in Foreign Echange Markets omé Eduardo Sicuaio Eamensarbete i matematik, 30 hp Handledare och eaminator: Johan ysk Maj 04 Department of Mathematics

### Introduction to portfolio insurance. Introduction to portfolio insurance p.1/41

Introduction to portfolio insurance Introduction to portfolio insurance p.1/41 Portfolio insurance Maintain the portfolio value above a certain predetermined level (floor) while allowing some upside potential.

### Valuation of Equity-Linked Insurance Products and Practical Issues in Equity Modeling. March 12, 2013

Valuation of Equity-Linked Insurance Products and Practical Issues in Equity Modeling March 12, 2013 The University of Hong Kong (A SOA Center of Actuarial Excellence) Session 2 Valuation of Equity-Linked

### Options/1. Prof. Ian Giddy

Options/1 New York University Stern School of Business Options Prof. Ian Giddy New York University Options Puts and Calls Put-Call Parity Combinations and Trading Strategies Valuation Hedging Options2

### 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,

### 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

### 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

### An Introduction to Exotic Options

An Introduction to Exotic Options Jeff Casey Jeff Casey is entering his final semester of undergraduate studies at Ball State University. He is majoring in Financial Mathematics and has been a math tutor

### 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

### Pricing Formulae for Foreign Exchange Options 1

Pricing Formulae for Foreign Exchange Options Andreas Weber and Uwe Wystup MathFinance AG Waldems, Germany www.mathfinance.com 22 December 2009 We would like to thank Peter Pong who pointed out an error

### CHAPTER 3 Pricing Models for One-Asset European Options

CHAPTER 3 Pricing Models for One-Asset European Options The revolution on trading and pricing derivative securities in financial markets and academic communities began in early 1970 s. In 1973, the Chicago

### 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

### LECTURE 10.1 Default risk in Merton s model

LECTURE 10.1 Default risk in Merton s model Adriana Breccia March 12, 2012 1 1 MERTON S MODEL 1.1 Introduction Credit risk is the risk of suffering a financial loss due to the decline in the creditworthiness

### Buy Low and Sell High

Buy Low and Sell High Min Dai Hanqing Jin Yifei Zhong Xun Yu Zhou This version: Sep 009 Abstract In trading stocks investors naturally aspire to buy low and sell high (BLSH). This paper formalizes the

### 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 on Market-Making and Delta-Hedging

More on Market-Making and Delta-Hedging What do market makers do to delta-hedge? Recall that the delta-hedging strategy consists of selling one option, and buying a certain number shares An example of

### The interest volatility surface

The interest volatility surface David Kohlberg Kandidatuppsats i matematisk statistik Bachelor Thesis in Mathematical Statistics Kandidatuppsats 2011:7 Matematisk statistik Juni 2011 www.math.su.se Matematisk

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

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

### 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

### Valuation of American Options

Valuation of American Options Among the seminal contributions to the mathematics of finance is the paper F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political

### The Effect of Management Discretion on Hedging and Fair Valuation of Participating Policies with Maturity Guarantees

The Effect of Management Discretion on Hedging and Fair Valuation of Participating Policies with Maturity Guarantees Torsten Kleinow Heriot-Watt University, Edinburgh (joint work with Mark Willder) Market-consistent

### IN THE DEFERRED ANNUITIES MARKET, THE PORTION OF FIXED- RATE ANNUITIES in annual

W ORKSHOP B Y H A N G S U C K L E E Pricing Equity-Indexed Annuities Embedded with Exotic Options IN THE DEFERRED ANNUITIES MARKET, THE PORTION OF FIXED- RATE ANNUITIES in annual sales has declined from

### 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

### INTEREST RATES AND FX MODELS

INTEREST RATES AND FX MODELS 4. Convexity and CMS Andrew Lesniewski Courant Institute of Mathematical Sciences New York University New York February 20, 2013 2 Interest Rates & FX Models Contents 1 Introduction

### 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

### Accurate Approximation Formulae for Evaluating Barrier Stock Options with Discrete Dividends and the Application in Credit Risk Valuation

Accurate Approximation Formulae for Evaluating Barrier Stock Options with Discrete Dividends and the Application in Credit Risk Valuation Tian-Shyr Dai Chun-Yuan Chiu Abstract To price the stock options

### 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

### 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

### 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

### Pricing Onion Options: A Probabilistic Approach

Pricing Onion Options: A Probabilistic Approach Thorsten Upmann 1 1 Mercator School of Management, University Duisburg-Essen, Duisburg, Germany Correspondence: Thorsten Upmann, Mercator School of Management,

### ON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACK-SCHOLES MODEL

ON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACK-SCHOLES MODEL A. B. M. Shahadat Hossain, Sharif Mozumder ABSTRACT This paper investigates determinant-wise effect of option prices when