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


 Flora Dawson
 3 years ago
 Views:
Transcription
1 Applied Mathematical Sciences, vol 8, 14, no 143, HIKARI Ltd, wwwmhikaricom A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting Model Joseph AckoraPrah Department of Mathematics Kwame Nkrumah University of Science and Technology, Kumasi, Ghana Perpetual Saah Andam Department of Mathematics Kwame Nkrumah University of Science and Technology, Kumasi, Ghana Samuel Asante Gyamerah Department of Mathematics Kwame Nkrumah University of Science and Technology, Kumasi, Ghana Daniel Gyamfi Department of Mathematics Kwame Nkrumah University of Science and Technology, Kumasi, Ghana Copyright 14 Joseph AckoraPrah et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Abstract Evolutionary computation have been used in different areas of research in finance The more the perfect price of option we obtain the more attractive it becomes to the investors Investors have developed much interest in option investment but when the option is exercised at a wrong time, it can lead to massive loss for the investor This paper is mainly focused on pricing a European put option when the underlying security price is geometric mean reverting with the assumption that the Girsanov change of measure has already been implemented and it has a constant interest rate We provide a Genetic Algorithm which gives a
2 716 J AckoraPrah, P S Andam, S A Gyamera and D Gyamfi perfect option price needed to be redeemed by the option buyer so as the option seller gets some profit rather than the asset expiring worthless Keywords: European put option, Geometric mean reverting model, Genetic Algorithm 1 Introduction An option investment can turn into great massive gains for the investor This is because an option allows an investor to control the profit potential of an investment many times the size of the actual amount the investor has at risk in the market 4 When an investor invests in options, the investor protects himself or herself from total loss by taking positions on the option market that minimize risk through hedging We use the Geometric mean reverting model to simulate the underlying asset price It determines the proper valuation of an option and sets accurate prices for the options using available information obtained In this paper we consider European option style A European option is an option that gives the right to the holder to trade an underlying asset S for prescribed price K at the expiry date without being obliged to do so There are two types of European options namely; European put option and European call option European call option provides the holder the right to buy an underlying asset at the expiry date for the strike price without being under obligation to do so European put option provides the holder the right to sell an underlying asset at the expiry date for the strike price without being under obligation to do so Genetic Algorithms evolved from both natural and artificial genetics John Henry Holland was the key brain behind Genetic Algorithms It has brought up many insight in using Genetic Algorithms to solve practical problems He published a book known as Adaptation in natural and artificial systems in AckoraPrah et al (14 4 presented a Genetic Algorithm to price a fixed term American put option when the underlying asset price is Geometric Brownian Motion They used Genetic Algorithm and Black Scholes model to calculate the option price and the optimal stopping time They compared the performances of the Genetic Algorithm and the Black Scholes model and found a perfect price for the American put option using Genetic Algorithm which was lower than that of Black Scholes model under the same condition They concluded that the Genetic Algorithm approach performed better than the Black Scholes model
3 A genetic algorithm to price an European put option 717 ShuCheng and Lee ( presented the use of Genetic Algorithm in option pricing in which they concentrated on the European call option They found the price of European call options whose exact solution was known from the Black Scholes option pricing theory using Genetic Algorithm They use GENESIS 5 software and they noticed the boundary conditions using the Genetic Algorithm was arbitrarily imposed and it only satisfied the case when the stock price was greater than the exercise price The solutions that they found using the basic Genetic Algorithm were compared to the exact solution and their results showed that Genetic Algorithm was a powerful tool for option pricing Investing in an asset that follows Geometric mean reverting model is a difficult decision to take This is because the price of the asset is always going down What if the asset follows the Geometric Mean Reverting model, will Genetic Algorithm still perform better? This leaves decision to the investors as to whether to invest in it or not Preliminaries Definition 1 Itô Integral An Itô integral is defined as, T S t d B n 1 t = S i ( B ti+1 B ti, i= where B t is a standard Brownian motion adapted to the filtration F t 8 Lemma Itô Lemma Let S t be a stochastic process and f(x, t be a measurable function with continuous partial derivatives up to the second order then, df(t, S t = f t (t, S tdt + f x (t, S tds t + 1 f x (t, S t(ds t Let B :, T Ω R be a Wiener process defined up to T > and let S :, T Ω R be a stochastic process that is adapted to the natural filtration F of a Wiener process Then ( T T E S t d B t = E St dt
4 718 J AckoraPrah, P S Andam, S A Gyamera and D Gyamfi 1 Geometric Mean Reverting Model Models predict the way prices of asset behave and the movement of the asset in the market The geometric mean reverting Brownian motion is also known as the BlackKarasinski model Suppose the price of the underlying asset S t which follows the geometric mean reverting model is given as, ds t = k(ρ ln S t S t dt + σs t d B t, (1 from the model we determine the underlying asset price using the Itô Lemma The assumption made is that the Girsanov change of measure has already been implemented and that B(t is a Brownian motion with respect to the risk neutral measure Q and ρ being the constant force of interest The degree of Mean Reverting, k and the volatility rate, σ are constants Bt is a onedimensional standard Brownian motion and d B t N, dt S t is the underlying asset price at time t Let B t, t T be a Brownian motion on a probability space (Ω, F, Q and F(t, t T be a filtration for this Brownian motion where T is a fixed final time We want to solve equation (1 explicitly, We let Y t = ln S t Then we have, Y t t =, From the Itô Lemma we obtain, Y t S t = 1 S t, Y t S t = 1 St dy t = Y t t dt + Y t ds t + 1 Y t (ds S t St t, ( 1 (ds t, = + 1 ds t + 1 S t St = 1 ds t 1 (ds S t St t We use the fact that (d B t = d B t d B t = dt 1 But we have, (ds t = ( k(ρ ln S t S t dt + σs t d B t, taking the term having (d B t only we obtain, σ S t (d B t = σ S t (d B t = σ S t dt
5 A genetic algorithm to price an European put option 719 Then, dy t = 1 k(ρ ln S t S t dt + σs t d S B t 1 σ S t St t dt, dy t = k(ρ ln S t dt + σd B t σ dt dy t = k(ρ Y t dt + σd B t σ dt, dy t = (kρ σ dt ky t dt + σd B t, dy t + ky t dt = (kρ σ dt + σd B t Using the integrating factor e kdt = e kt we have, ( (dy t + ky t dte kt = e kt kρ σ d(y t e kt = e kt ( kρ σ dt + σe kt d B t, dt + σe kt d B t Then finding the underlying asset price S t at time t, have, d(y s e ks = Ys e ks t = e ks k ( e ks kρ σ (kρ σ ds + t + σ σe ks d B s e ks d B s < t T we Substituting the limits and solving it further we obtain, Y t = Y e + (ρ σ (1 + σe e ks d B s, but Y t = ln S t then we have, ln S t = e ln S + { S t = exp e ln S + (ρ σ (1 + σe e ks d B s (ρ σ (1 } + σe e ks d B s (
6 713 J AckoraPrah, P S Andam, S A Gyamera and D Gyamfi Then, S t = S e exp (ρ σ (1 + σe e ks d B s (3 e e ks d B s is an Itô integral (1 with respect to Q, then, E e e ks d B s = The formula for finding the variance is given by, ( ( E e e ks d B s E e e ks d B s = Var(S t but E e e ks d B s = = then from ( we have, ( E e e ks d B s = E ( E e e ks d B s = e t e s ds Then integrating and substituting the limits gives, ( E e e ks d B s = 1 ( 1 t Then it implies that, E Hence, e e ks d B s = and V ar e e ks d B s = 1 ( 1 t e e ks d B s N Therefore, we can rewrite S t from (3 as, S t = S e exp, 1 ( 1 t {(ρ σ (1 } + σy, where Y N, 1 ( 1 t (4
7 A genetic algorithm to price an European put option 7131 At maturity we have, S T = S e kt exp {(ρ σ (1 } kt + σy, where Y N, 1 ( 1 T (5 Finding the Expectation and Variance of Underlying asset Price From (4 we have, e S t = S exp {(ρ σ (1 } + σy, let Y = V Z, V = 1 ( 1 t, Z follows the normal distribution that is Z N, 1 then it follows that, S t = S e exp {(ρ σ (1 + σv Z} (6 Then finding the expectation of underlying asset price it follows that, e ES t = E S exp (ρ σ (1 + σv Z, e ES t = S exp (ρ σ (1 E e σv Z, but, E e σv Z = e σ V, then the expectation of the underlying asset price is, e ES t = S exp (ρ σ (1 + σ V (7 Finding variance of the underlying asset price it follows that, ES t = E V ars t = ES T (ES t, { e S exp (ρ σ (1 + σv Z},
8 713 J AckoraPrah, P S Andam, S A Gyamera and D Gyamfi but, ESt e = S exp (ρ σ (1 E e σv Z, E e σv Z = e σ V, ESt e = S exp (ρ σ (1 + σ V Also, from (7 we obtain, then we get, (ES t e = S exp (ρ σ (1 + V σ, (ES t = Then the variance of S t is, e S exp (ρ σ (1 + V σ e V ars t = S exp (ρ σ (1 ( e V σ e σ V 1 Using Numerical Values to Simulate the Underlying Asset price Fixed numerical values were used to simulate the underlying asset price (S t over the period of, T using equation (6 The following values were used; S = $1, σ = 35, k = 1, ρ = 1% and T = 1 S is the initial underlying asset price, σ is the volatility rate, ρ is the interest rate, k is the degree of Mean Reverting and T is the maturity time
9 A genetic algorithm to price an European put option 7133 Simulated underlying asset price Figure 1: Simulated Geometric Mean Reverting Underlying Asset The graph in figure 1 displays a simulated underlying asset price over the period of, T It can be observed from the graph that the underlying asset price is decreasing when there is an increase in time 3 The Geometric Mean Reverting Model using Genetic Algorithm In European call option the holder expects the price of the underlying asset to rise at the expiry date Let S T be the underlying asset at the expiry date and K be the strike price then the holder of this European call option expects the payoff of S T K for S T > K when the right is exercised and if the right is not exercised then it is zero ( For European put option, the holder expects the price of the underlying asset to fall at the expiry date Then we expect the holder of this European put option s payoff to be K S T for S T < K when the right is exercised and if the right is not exercised then it is zero (
10 7134 J AckoraPrah, P S Andam, S A Gyamera and D Gyamfi We denote the payoff of a European put option as P T = (K S T + Let OP T = OP (S T be the price of the option at time T, then at maturity the payoff is OP T = P T We use this brief procedure to price a European put option when the underlying asset is geometric mean reverting We first generate random asset price using equation (5 and because we are writing a European put, we use a fitness function of max{k S T, } We use Roulette wheel selection for the candidates to be drawn independently We use onepoint crossover and flip bit mutation because we decoded it into binary form 4 Results and Discussion We assign numerical values to the parameters to find the option price of a European put when the underlying asset price is geometric mean reverting We let the initial underlying asset price, S = $1, the strike price, K = $1, the volatility rate σ = 35, the speed of reverting, k = 1, the interest rate, ρ = 1% and the maturity time T = 1 year We used python software to price the European put option when the underlying asset price follows the Black Karasinski model as given in equation (5 We obtain the option price as $ 14 at maturity time An option price of $14 means that the holder of the option should pay $14 The seller of the option buys assets and bonds at the initial time with the $14 received to make the same profit as the option buyer This is because the movement of the prices is always going down and an investor will be at risk if the investors exercises a European call on this asset A European put is supposed to be exercised on this asset and the investor is suppose to sell out the asset to make profit or else the asset will expire worthless 5 Conclusion We have used Geometric mean reverting model to simulate the underlying asset with Python software for the programming Our Genetic Algorithm was used to calculate for the option price under a European put We found a perfect option price for exercising a European put option that follows a Geometric Mean Reverting asset The result obtained from exercising a European put option on a mean reverting asset was supportive and this will benefit the
11 A genetic algorithm to price an European put option 7135 option seller because if this asset is not sold off it will expire worthless Acknowledgements We express our gratitude to the Almighty God and the Department of Mathematics, Kwame Nkrumah University of Science and Technology for providing us resources to help complete this research successfully References 1 B Oksendal, Stochastic Differential Equations: An introduction with applications SpringerVerlag, New York, 6 (3 DA Coley, An Introduction to Genetic Algorithms for Scientists and Engineers, World Scientific Publishing, ( E V Mvanda, Pricing of a Perpetual American put option when stock prices are mean reverting University of Dar es Salaam, (1 4 J AckoraPrah, S K Amponsah, P S Andam and S A Gyamerah, A Genetic Algorithm for Option Pricing: The American Put Option, Applied Mathematical Sciences, Hikari ltd, 8 (14, J C Hull, Options, Futures, and other Derivatives, PrinticeHall international limited, ( J H Holland, Adaptation in natural and artificial systems, University of Michigan Press, ( K A Sidarto, On the Calculation of Implied Volatility using a Genetic Algorithm, Seminar Nasional Aplikasi Teknologi Informasi, (6 8 S E Shreve, Stochastic Calculus for finance, pringer Science and Business Media New York, 1 (4 9 SH Cheng and WC Lee, Option Pricing with Genetic Algorithm: The case of European  Style Options Proceedings of the seventh International Conference on Genetic Algorithms, ( Y Chen, S Chang, C Wu, A Dynamic Hybrid Option Pricing Model by Genetic Algorithm and BlackScholes Model, World Academy of Science,7 (1, Received: June 1, 14
The 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 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 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 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 informationQUANTIZED 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 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 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 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 informationLECTURE 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 informationOscillatory 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 informationAmerican and European. Put Option
American and European Put Option Analytical Finance I Kinda Sumlaji 1 Table of Contents: 1. Introduction... 3 2. Option Style... 4 3. Put Option 4 3.1 Definition 4 3.2 Payoff at Maturity... 4 3.3 Example
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 informationApplication of options in hedging of crude oil price risk
AMERICAN JOURNAL OF SOCIAL AND MANAGEMEN SCIENCES ISSN rint: 156154, ISSN Online: 1511559 doi:1.551/ajsms.1.1.1.67.74 1, ScienceHuβ, http://www.scihub.org/ajsms Application of options in hedging of crude
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 informationAsian Option Pricing Formula for Uncertain Financial Market
Sun and Chen Journal of Uncertainty Analysis and Applications (215) 3:11 DOI 1.1186/s446715357 RESEARCH Open Access Asian Option Pricing Formula for Uncertain Financial Market Jiajun Sun 1 and Xiaowei
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 informationAssessing Credit Risk for a Ghanaian Bank Using the Black Scholes Model
Assessing Credit Risk for a Ghanaian Bank Using the Black Scholes Model VK Dedu 1, FT Oduro 2 1,2 Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana. Abstract
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 informationSOLVING PARTIAL DIFFERENTIAL EQUATIONS RELATED TO OPTION PRICING WITH NUMERICAL METHOD. KENNEDY HAYFORD, (B.Sc. Mathematics)
SOLVING PARTIAL DIFFERENTIAL EQUATIONS RELATED TO OPTION PRICING WITH NUMERICAL METHOD BY KENNEDY HAYFORD, (B.Sc. Mathematics) A Thesis submitted to the Department of Mathematics, Kwame Nkrumah University
More informationFinancial Modeling. An introduction to financial modelling and financial options. Conall O Sullivan
Financial Modeling An introduction to financial modelling and financial options Conall O Sullivan Banking and Finance UCD Smurfit School of Business 31 May / UCD Maths Summer School Outline Introduction
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 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 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 informationExam MFE Spring 2007 FINAL ANSWER KEY 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D
Exam MFE Spring 2007 FINAL ANSWER KEY Question # Answer 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D **BEGINNING OF EXAMINATION** ACTUARIAL MODELS FINANCIAL ECONOMICS
More 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 informationLECTURE 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 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 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 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 informationPRICING DIGITAL CALL OPTION IN THE HESTON STOCHASTIC VOLATILITY MODEL
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVIII, Number 3, September 003 PRICING DIGITAL CALL OPTION IN THE HESTON STOCHASTIC VOLATILITY MODEL VASILE L. LAZAR Dedicated to Professor Gheorghe Micula
More informationFINANCIAL 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 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 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 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 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 informationCall Price as a Function of the Stock Price
Call Price as a Function of the Stock Price Intuitively, the call price should be an increasing function of the stock price. This relationship allows one to develop a theory of option pricing, derived
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 informationLecture 4: Derivatives
Lecture 4: Derivatives School of Mathematics Introduction to Financial Mathematics, 2015 Lecture 4 1 Financial Derivatives 2 uropean Call and Put Options 3 Payoff Diagrams, Short Selling and Profit Derivatives
More informationDerivation and Comparative Statics of the BlackScholes Call and Put Option Pricing Formulas
Derivation and Comparative Statics of the BlackScholes Call and Put Option Pricing Formulas James R. Garven Latest Revision: 27 April, 2015 Abstract This paper provides an alternative derivation of the
More informationBlackScholes Option Pricing Model
BlackScholes 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 informationJorge Cruz Lopez  Bus 316: Derivative Securities. Week 11. The BlackScholes Model: Hull, Ch. 13.
Week 11 The BlackScholes Model: Hull, Ch. 13. 1 The BlackScholes Model Objective: To show how the BlackScholes formula is derived and how it can be used to value options. 2 The BlackScholes Model 1.
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 informationBINOMIAL 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 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 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 BlackScholes Formula
ECO30004 OPTIONS AND FUTURES SPRING 2008 The BlackScholes Formula The BlackScholes Formula We next examine the BlackScholes formula for European options. The BlackScholes formula are complex as they
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 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 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 informationAmerican 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 informationValuing Stock Options: The BlackScholesMerton Model. Chapter 13
Valuing Stock Options: The BlackScholesMerton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The BlackScholesMerton Random Walk Assumption
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 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 informationBlackScholesMerton approach merits and shortcomings
BlackScholesMerton approach merits and shortcomings Emilia Matei 1005056 EC372 Term Paper. Topic 3 1. Introduction The BlackScholes and Merton method of modelling derivatives prices was first introduced
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 informationMonte 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 informationBlack Scholes Merton Approach To Modelling Financial Derivatives Prices Tomas Sinkariovas 0802869. Words: 3441
Black Scholes Merton Approach To Modelling Financial Derivatives Prices Tomas Sinkariovas 0802869 Words: 3441 1 1. Introduction In this paper I present Black, Scholes (1973) and Merton (1973) (BSM) general
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 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 informationOPTIONS CALCULATOR QUICK GUIDE. Reshaping Canada s Equities Trading Landscape
OPTIONS CALCULATOR QUICK GUIDE Reshaping Canada s Equities Trading Landscape OCTOBER 2014 Table of Contents Introduction 3 Valuing options 4 Examples 6 Valuing an American style nondividend paying stock
More informationDerivatives: Options
Derivatives: Options Call Option: The right, but not the obligation, to buy an asset at a specified exercise (or, strike) price on or before a specified date. Put Option: The right, but not the obligation,
More informationA 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 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 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 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 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 informationBefore we discuss a Call Option in detail we give some Option Terminology:
Call and Put Options As you possibly have learned, the holder of a forward contract is obliged to trade at maturity. Unless the position is closed before maturity the holder must take possession of the
More informationBond 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 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 information2. 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 : Options1 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. BlackScholes
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 Research Problems in Uncertainty Theory
Journal of Uncertain Systems Vol.3, No.1, pp.310, 2009 Online at: www.jus.org.uk Some Research Problems in Uncertainty Theory aoding Liu Uncertainty Theory Laboratory, Department of Mathematical Sciences
More informationOption 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 informationOption Pricing Basics
Option Pricing Basics Aswath Damodaran Aswath Damodaran 1 What is an option? An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called
More informationUnderstanding Options and Their Role in Hedging via the Greeks
Understanding Options and Their Role in Hedging via the Greeks Bradley J. Wogsland Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 379961200 Options are priced assuming that
More informationEXP 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 Blackcholes model PutCall Parity Implied Volatility Options: Definitions A call option gives the buyer the
More informationResearch on Option Trading Strategies
Research on Option Trading Strategies An Interactive Qualifying Project Report: Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE In partial fulfillment of the requirements for the Degree
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 informationUCLA Anderson School of Management Daniel Andrei, Derivative Markets 237D, Winter 2014. MFE Midterm. February 2014. Date:
UCLA Anderson School of Management Daniel Andrei, Derivative Markets 237D, Winter 2014 MFE Midterm February 2014 Date: Your Name: Your Equiz.me email address: Your Signature: 1 This exam is open book,
More informationChapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO ENDOFCHAPTER QUESTIONS
Chapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO ENDOFCHAPTER QUESTIONS 81 a. An option is a contract which gives its holder the right to buy or sell an asset at some predetermined
More information3 Results. σdx. df =[µ 1 2 σ 2 ]dt+ σdx. Integration both sides will form
Appl. Math. Inf. Sci. 8, No. 1, 107112 (2014) 107 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/080112 Forecasting Share Prices of Small Size Companies
More informationOPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION
OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION NITESH AIDASANI KHYAMI Abstract. Option contracts are used by all major financial institutions and investors, either to speculate on stock market
More informationBINOMIAL OPTION PRICING
Darden Graduate School of Business Administration University of Virginia BINOMIAL OPTION PRICING Binomial option pricing is a simple but powerful technique that can be used to solve many complex optionpricing
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 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 informationThe Valuation of Currency Options
The Valuation of Currency Options Nahum Biger and John Hull Both Nahum Biger and John Hull are Associate Professors of Finance in the Faculty of Administrative Studies, York University, Canada. Introduction
More informationWeek 1: Futures, Forwards and Options derivative three Hedge: Speculation: Futures Contract: buy or sell
Week 1: Futures, Forwards and Options  A derivative is a financial instrument which has a value which is determined by the price of something else (or an underlying instrument) E.g. energy like coal/electricity
More informationFIN 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 Blackcholes model PutCall Parity Implied Volatility Options: Definitions A call option gives the buyer the right, but
More informationStochastic Processes and Advanced Mathematical Finance. Multiperiod Binomial Tree Models
Steven R. Dunbar Department of Mathematics 203 Avery Hall University of NebraskaLincoln Lincoln, NE 685880130 http://www.math.unl.edu Voice: 4024723731 Fax: 4024728466 Stochastic Processes and Advanced
More informationThe Intuition Behind Option Valuation: A Teaching Note
The Intuition Behind Option Valuation: A Teaching Note Thomas Grossman Haskayne School of Business University of Calgary Steve Powell Tuck School of Business Dartmouth College Kent L Womack Tuck School
More informationHedging of Currency Option in Trading Market
International Journal of Economic and Management Strategy. ISSN 22783636 Volume 3, Number 1 (2013), pp. 16 Research India Publications http://www.ripublication.com/jems.htm Hedging of Currency Option
More informationWhen 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 informationPricing European and American bond option under the Hull White extended Vasicek model
1 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA Pricing European and American bond option under the Hull White extended Vasicek model Eva Maria Rapoo 1, Mukendi Mpanda 2
More informationNew Pricing Formula for Arithmetic Asian Options. Using PDE Approach
Applied Mathematical Sciences, Vol. 5, 0, no. 77, 3803809 New Pricing Formula for Arithmetic Asian Options Using PDE Approach Zieneb Ali Elshegmani, Rokiah Rozita Ahmad and 3 Roza Hazli Zakaria, School
More informationConsider a European call option maturing at time T
Lecture 10: Multiperiod Model Options BlackScholesMerton model Prof. Markus K. Brunnermeier 1 Binomial Option Pricing Consider a European call option maturing at time T with ihstrike K: C T =max(s T
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 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 informationELECTRICITY REAL OPTIONS VALUATION
Vol. 37 (6) ACTA PHYSICA POLONICA B No 11 ELECTRICITY REAL OPTIONS VALUATION Ewa BroszkiewiczSuwaj Hugo Steinhaus Center, Institute of Mathematics and Computer Science Wrocław University of Technology
More information