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


 Megan Charles
 2 years ago
 Views:
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: knockout options: the right to exercise is lost if the barrier is crossed. The option becomes worthless. knockin 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 knockin or knockout 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 selffinancing 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 knockout 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: FeynmanKac (applies only up to knockout), 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 BlackScholes 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 BlackScholes 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 Qdistribution 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 InOut 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 InOut 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 BlackScholes 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 InOut 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 BlackScholes 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 InOut Parity Pricing PDEs InOut 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 InOut parity for Downoptions: 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 knockin resp. knockout if barrier is crossed once Second Generation Barrier Options: Double barrier options: additional barrier Parisian barrier options: knockin resp. knockout 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 BlackScholes 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 knockedout resp. knocked in Parisian barrier options require the knockout / knockin 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: MonteCarlo 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
More informationOption 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 informationOption Pricing. Chapter 12  Local volatility models  Stefan Ankirchner. University of Bonn. last update: 13th January 2014
Option Pricing Chapter 12  Local volatility models  Stefan Ankirchner University of Bonn last update: 13th January 2014 Stefan Ankirchner Option Pricing 1 Agenda The volatility surface Local volatility
More informationLecture. 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 informationThe BlackScholes pricing formulas
The BlackScholes pricing formulas Moty Katzman September 19, 2014 The BlackScholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock
More informationTHE BLACKSCHOLES MODEL AND EXTENSIONS
THE BLACSCHOLES MODEL AND EXTENSIONS EVAN TURNER Abstract. This paper will derive the BlackScholes pricing model of a European option by calculating the expected value of the option. We will assume that
More informationMore 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 informationNotes on BlackScholes Option Pricing Formula
. Notes on BlackScholes Option Pricing Formula by DeXing Guan March 2006 These notes are a brief introduction to the BlackScholes formula, which prices the European call options. The essential reading
More informationPricing Barrier Options under Local Volatility
Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly
More informationOption 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.
More informationBarrier 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?
More informationNumerical PDE methods for exotic options
Lecture 8 Numerical PDE methods for exotic options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Barrier options For barrier option part of the option contract is triggered if the asset
More informationThe BlackScholes Formula
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 The BlackScholes Formula These notes examine the BlackScholes formula for European options. The BlackScholes formula are complex as they are based on the
More informationA VegaGamma Relationship for EuropeanStyle or Barrier Options in the BlackScholes Model
A VegaGamma Relationship for EuropeanStyle or Barrier Options in the BlackScholes Model Fabio Mercurio Financial Models, Banca IMI Abstract In this document we derive some fundamental relationships
More informationARBITRAGEFREE OPTION PRICING MODELS. Denis Bell. University of North Florida
ARBITRAGEFREE 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 informationCS 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 informationNumerical 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 informationMonte Carlo Methods in Finance
Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction
More informationPathdependent options
Chapter 5 Pathdependent 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,
More informationFinite 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 BlackScholes
More informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationThe BlackScholesMerton Approach to Pricing Options
he BlackScholesMerton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the BlackScholesMerton approach to determining
More informationOn BlackScholes Equation, Black Scholes Formula and Binary Option Price
On BlackScholes Equation, Black Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. BlackScholes Equation is derived using two methods: (1) riskneutral measure; (2)  hedge. II.
More informationTABLE OF CONTENTS. A. PutCall 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. PutCall Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 2. McDonald 10: "Binomial Option Pricing:
More informationτ 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 informationIntroduction to ArbitrageFree Pricing: Fundamental Theorems
Introduction to ArbitrageFree Pricing: Fundamental Theorems Dmitry Kramkov Carnegie Mellon University Workshop on Interdisciplinary Mathematics, Penn State, May 810, 2015 1 / 24 Outline Financial market
More informationReview 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 informationSchonbucher Chapter 9: Firm Value and Share PricedBased Models Updated 07302007
Schonbucher Chapter 9: Firm alue and Share PricedBased Models Updated 07302007 (References sited are listed in the book s bibliography, except Miller 1988) For Intensity and spreadbased models of default
More informationChapter 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 informationOption Valuation. Chapter 21
Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of inthemoney options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price
More informationLecture 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 BlackScholes Equation and Replicating Portfolio 2 Static
More informationBlackScholes and the Volatility Surface
IEOR E4707: Financial Engineering: ContinuousTime Models Fall 2009 c 2009 by Martin Haugh BlackScholes and the Volatility Surface When we studied discretetime models we used martingale pricing to derive
More informationDoes BlackScholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem
Does BlackScholes 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 informationThe BlackScholes Model
The BlackScholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The BlackScholes Model Options Markets 1 / 19 The BlackScholesMerton
More informationFour Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com
Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com In this Note we derive the Black Scholes PDE for an option V, given by @t + 1 + rs @S2 @S We derive the
More information第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model
1 第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model Outline 有 关 股 价 的 假 设 The BS 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 informationOption Pricing. 1 Introduction. Mrinal K. Ghosh
Option Pricing Mrinal K. Ghosh 1 Introduction We first introduce the basic terminology in option pricing. Option: An option is the right, but not the obligation to buy (or sell) an asset under specified
More informationLecture 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 informationCall 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 informationLecture 12: The BlackScholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 12: The BlackScholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The BlackScholesMerton Model
More informationPutCall Parity. chris bemis
PutCall 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 informationBlackScholes Equation for Option Pricing
BlackScholes 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 informationLecture 6 BlackScholes PDE
Lecture 6 BlackScholes PDE Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Pricing function Let the dynamics of underlining S t be given in the riskneutral measure Q by If the contingent
More informationPricing 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
More informationarxiv:1108.4393v2 [qfin.pr] 25 Aug 2011
arxiv:1108.4393v2 [qfin.pr] 25 Aug 2011 Pricing Variable Annuity Contracts with HighWater Mark Feature V.M. Belyaev Allianz Investment Management, Allianz Life Minneapolis, MN, USA August 26, 2011 Abstract
More informationCharacterizing 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
More informationIntroduction 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
More informationEuropean 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 informationA Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting Model
Applied Mathematical Sciences, vol 8, 14, no 143, 7157135 HIKARI Ltd, wwwmhikaricom http://dxdoiorg/11988/ams144644 A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting
More informationA new FeynmanKacformula for option pricing in Lévy models
A new FeynmanKacformula 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
More informationPricing of an Exotic Forward Contract
Pricing of an Exotic Forward Contract Jirô Akahori, Yuji Hishida and Maho Nishida Dept. of Mathematical Sciences, Ritsumeikan University 111 Nojihigashi, Kusatsu, Shiga 5258577, Japan Email: {akahori,
More informationOption 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 informationA 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 Email address: gshim@ajou.ac.kr ABSTRACT. I introduce
More informationAnalytic Approximations for MultiAsset Option Pricing
Analytic Approximations for MultiAsset Option Pricing Carol Alexander ICMA Centre, University of Reading Aanand Venkatramanan ICMA Centre, University of Reading First Version: March 2008 This Version:
More informationENGINEERING 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:
More informationComputational 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
More informationOn MarketMaking and DeltaHedging
On MarketMaking and DeltaHedging 1 Market Makers 2 MarketMaking and BondPricing On MarketMaking and DeltaHedging 1 Market Makers 2 MarketMaking and BondPricing What to market makers do? Provide
More informationDiusion 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 BlackScholes 5 Equity linked life insurance 6 Merton
More informationRiskNeutral Valuation of Participating Life Insurance Contracts
RiskNeutral Valuation of Participating Life Insurance Contracts DANIEL BAUER with R. Kiesel, A. Kling, J. Russ, and K. Zaglauer ULM UNIVERSITY RTG 1100 AND INSTITUT FÜR FINANZ UND AKTUARWISSENSCHAFTEN
More informationHedging Barriers. Liuren Wu. Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/)
Hedging Barriers Liuren Wu Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/) Based on joint work with Peter Carr (Bloomberg) Modeling and Hedging Using FX Options, March
More informationwhere N is the standard normal distribution function,
The BlackScholesMerton 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 informationPricing 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 informationVannaVolga Method for Foreign Exchange Implied Volatility Smile. Copyright Changwei Xiong 2011. January 2011. last update: Nov 27, 2013
VannaVolga 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 informationThe 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 marketmaker would deltahedge, we first need to specify how bonds behave. Suppose we try to model a zerocoupon
More information1 The BlackScholes model: extensions and hedging
1 The BlackScholes model: extensions and hedging 1.1 Dividends Since we are now in a continuous time framework the dividend paid out at time t (or t ) is given by dd t = D t D t, where as before D denotes
More informationSome 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
More informationCalibration of Stock Betas from Skews of Implied Volatilities
Calibration of Stock Betas from Skews of Implied Volatilities JeanPierre Fouque University of California Santa Barbara Joint work with Eli Kollman (Ph.D. student at UCSB) Joint Seminar: Department of
More informationKnock 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
More informationIntroduction 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.
More informationValuation of EquityLinked Insurance Products and Practical Issues in Equity Modeling. March 12, 2013
Valuation of EquityLinked 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 EquityLinked
More informationOptions/1. Prof. Ian Giddy
Options/1 New York University Stern School of Business Options Prof. Ian Giddy New York University Options Puts and Calls PutCall Parity Combinations and Trading Strategies Valuation Hedging Options2
More informationMertonBlackScholes model for option pricing. Peter Denteneer. 22 oktober 2009
MertonBlackScholes 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 informationHedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging
Hedging An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Definition Hedging is the practice of making a portfolio of investments less sensitive to changes in
More informationHedging 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 informationAn 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
More informationJungSoon Hyun and YoungHee Kim
J. Korean Math. Soc. 43 (2006), No. 4, pp. 845 858 TWO APPROACHES FOR STOCHASTIC INTEREST RATE OPTION MODEL JungSoon Hyun and YoungHee Kim Abstract. We present two approaches of the stochastic interest
More informationPricing 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
More informationCHAPTER 3 Pricing Models for OneAsset European Options
CHAPTER 3 Pricing Models for OneAsset 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
More informationMoreover, 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 informationLECTURE 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
More informationBuy 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
More informationFIN40008 FINANCIAL INSTRUMENTS SPRING 2008
FIN40008 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 informationMore on MarketMaking and DeltaHedging
More on MarketMaking and DeltaHedging What do market makers do to deltahedge? Recall that the deltahedging strategy consists of selling one option, and buying a certain number shares An example of
More informationThe 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
More informationOn the Valuation of PowerReverse Duals and EquityRates Hybrids
On the Valuation of PowerReverse Duals and EquityRates Hybrids Oliver Caps oliver.caps@dkib.com RMT Model Validation Rates Dresdner Bank Examples of Hybrid Products Pricing of Hybrid Products using a
More informationLecture 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 Putcall
More informationValuation 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
More informationThe 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 HeriotWatt University, Edinburgh (joint work with Mark Willder) Marketconsistent
More informationIN 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 EquityIndexed Annuities Embedded with Exotic Options IN THE DEFERRED ANNUITIES MARKET, THE PORTION OF FIXED RATE ANNUITIES in annual sales has declined from
More informationNumerical Methods for Pricing Exotic Options
Imperial College London Department of Computing Numerical Methods for Pricing Exotic Options by Hardik Dave  00517958 Supervised by Dr. Daniel Kuhn Second Marker: Professor Berç Rustem Submitted in partial
More informationINTEREST 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
More informationPrivate Equity Fund Valuation and Systematic Risk
An Equilibrium Approach and Empirical Evidence Axel Buchner 1, Christoph Kaserer 2, Niklas Wagner 3 Santa Clara University, March 3th 29 1 Munich University of Technology 2 Munich University of Technology
More informationAccurate 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 TianShyr Dai ChunYuan Chiu Abstract To price the stock options
More informationLectures. 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 informationChapter 2: Binomial Methods and the BlackScholes Formula
Chapter 2: Binomial Methods and the BlackScholes 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 calloption C t = C(t), where the
More informationMartingale Pricing Applied to Options, Forwards and Futures
IEOR E4706: Financial Engineering: DiscreteTime 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 informationPricing Onion Options: A Probabilistic Approach
Pricing Onion Options: A Probabilistic Approach Thorsten Upmann 1 1 Mercator School of Management, University DuisburgEssen, Duisburg, Germany Correspondence: Thorsten Upmann, Mercator School of Management,
More informationON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACKSCHOLES MODEL
ON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACKSCHOLES MODEL A. B. M. Shahadat Hossain, Sharif Mozumder ABSTRACT This paper investigates determinantwise effect of option prices when
More informationPricing Parisian Options
Pricing Parisian Options Richard J. Haber Mathematical Institute, 2429 St Giles, Oxford OX1 3LB Philipp J. Schönbucher University of Bonn, Statistical Department, Adenaueralle 2442, 53113 Bonn, Germany
More informationAdditional questions for chapter 4
Additional questions for chapter 4 1. A stock price is currently $ 1. Over the next two sixmonth periods it is expected to go up by 1% or go down by 1%. The riskfree interest rate is 8% per annum with
More information