Hyperbolic Normal Stochastic Volatility Model
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A Tree-Based Method to Price American Options in the Heston Model
The Journal of Computational Finance (1–21) Volume 13/Number 1, Fall 2009 A tree-based method to price American options in the Heston model Michel Vellekoop Financial Engineering Laboratory, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands; email: [email protected] Hans Nieuwenhuis University of Groningen, Faculty of Economics, PO Box 800, 9700 AV Groningen, The Netherlands; email: [email protected] We develop an algorithm to price American options on assets that follow the stochastic volatility model defined by Heston. We use an approach which is based on a modification of a combined tree for stock prices and volatilities, where the number of nodes grows quadratically in the number of time steps. We show in a number of numerical tests that we get accurate results in a fast manner, and that features which are essential for the practical use of stock option pricing algorithms, such as the incorporation of cash dividends and a term structure of interest rates, can easily be incorporated. 1 INTRODUCTION One of the most popular models for equity option pricing under stochastic volatility is the one defined by Heston (1993): 1 dSt = µSt dt + Vt St dW (1.1) t = − + 2 dVt κ(θ Vt ) dt ω Vt dWt (1.2) In this model for the stock price process S and squared volatility process V the processes W 1 and W 2 are standard Brownian motions that may have a non-zero correlation coefficient ρ, while µ, κ, θ and ω are known strictly positive parameters. -
Black-Scholes Model
Chapter 5 Black-Scholes Model Copyright c 2008{2012 Hyeong In Choi, All rights reserved. 5.1 Modeling the Stock Price Process One of the basic building blocks of the Black-Scholes model is the stock price process. In particular, in the Black-Scholes world, the stock price process, denoted by St, is modeled as a geometric Brow- nian motion satisfying the following stochastic differential equation: dSt = St(µdt + σdWt); where µ and σ are constants called the drift and volatility, respec- tively. Also, σ is always assumed to be a positive constant. In order to see if such model is \reasonable," let us look at the price data of KOSPI 200 index. In particular Figure 5.1 depicts the KOSPI 200 index from January 2, 2001 till May 18, 2011. If it is reasonable to model it as a geometric Brownian motion, we must have ∆St ≈ µ∆t + σ∆Wt: St In particular if we let ∆t = h (so that ∆St = St+h − St and ∆Wt = Wt+h − Wt), the above approximate equality becomes St+h − St ≈ µh + σ(Wt+h − Wt): St 2 Since σ(Wt+h − Wt) ∼ N(0; σ h), the histogram of (St+h − St)p=St should resemble a normal distribution with standard deviation σ h. (It is called the h-day volatility.) To see it is indeed the case we have plotted the histograms for h = 1; 2; 3; 5; 7 (days) in Figures 5.2 up to 5.1. MODELING THE STOCK PRICE PROCESS 156 Figure 5.1: KOSPI 200 from 2001. -
THE BLACK-SCHOLES EQUATION in STOCHASTIC VOLATILITY MODELS 1. Introduction in Financial Mathematics There Are Two Main Approache
THE BLACK-SCHOLES EQUATION IN STOCHASTIC VOLATILITY MODELS ERIK EKSTROM¨ 1,2 AND JOHAN TYSK2 Abstract. We study the Black-Scholes equation in stochastic volatility models. In particular, we show that the option price is the unique classi- cal solution to a parabolic differential equation with a certain boundary behaviour for vanishing values of the volatility. If the boundary is at- tainable, then this boundary behaviour serves as a boundary condition and guarantees uniqueness in appropriate function spaces. On the other hand, if the boundary is non-attainable, then the boundary behaviour is not needed to guarantee uniqueness, but is nevertheless very useful for instance from a numerical perspective. 1. Introduction In financial mathematics there are two main approaches to the calculation of option prices. Either the price of an option is viewed as a risk-neutral expected value, or it is obtained by solving the Black-Scholes partial differ- ential equation. The connection between these approaches is furnished by the classical Feynman-Kac theorem, which states that a classical solution to a linear parabolic PDE has a stochastic representation in terms of an ex- pected value. In the standard Black-Scholes model, a standard logarithmic change of variables transforms the Black-Scholes equation into an equation with constant coefficients. Since such an equation is covered by standard PDE theory, the existence of a unique classical solution is guaranteed. Con- sequently, the option price given by the risk-neutral expected value is the unique classical solution to the Black-Scholes equation. However, in many situations outside the standard Black-Scholes setting, the pricing equation has degenerate, or too fast growing, coefficients and standard PDE theory does not apply. -
Lecture 4: Risk Neutral Pricing 1 Part I: the Girsanov Theorem
SEEM 5670 { Advanced Models in Financial Engineering Professor: Nan Chen Oct. 17 &18, 2017 Lecture 4: Risk Neutral Pricing 1 Part I: The Girsanov Theorem 1.1 Change of Measure and Girsanov Theorem • Change of measure for a single random variable: Theorem 1. Let (Ω; F;P ) be a sample space and Z be an almost surely nonnegative random variable with E[Z] = 1. For A 2 F, define Z P~(A) = Z(!)dP (!): A Then P~ is a probability measure. Furthermore, if X is a nonnegative random variable, then E~[X] = E[XZ]: If Z is almost surely strictly positive, we also have Y E[Y ] = E~ : Z Definition 1. Let (Ω; F) be a sample space. Two probability measures P and P~ on (Ω; F) are said to be equivalent if P (A) = 0 , P~(A) = 0 for all such A. Theorem 2 (Radon-Nikodym). Let P and P~ be equivalent probability measures defined on (Ω; F). Then there exists an almost surely positive random variable Z such that E[Z] = 1 and Z P~(A) = Z(!)dP (!) A for A 2 F. We say Z is the Radon-Nikodym derivative of P~ with respect to P , and we write dP~ Z = : dP Example 1. Change of measure on Ω = [0; 1]. Example 2. Change of measure for normal random variables. • We can also perform change of measure for a whole process rather than for a single random variable. Suppose that there is a probability space (Ω; F) and a filtration fFt; 0 ≤ t ≤ T g. Suppose further that FT -measurable Z is an almost surely positive random variable satisfying E[Z] = 1. -
Empirical Performance of Alternative Option Pricing Models For
Empirical Performance of Alternative Option Pricing Models for Commodity Futures Options (Very draft: Please do not quote) Gang Chen, Matthew C. Roberts, and Brian Roe¤ Department of Agricultural, Environmental, and Development Economics The Ohio State University 2120 Fy®e Road Columbus, Ohio 43210 Contact: [email protected] Selected Paper prepared for presentation at the American Agricultural Economics Association Annual Meeting, Providence, Rhode Island, July 24-27, 2005 Copyright 2005 by Gang Chen, Matthew C. Roberts, and Brian Boe. All rights reserved. Readers may make verbatim copies of this document for non-commercial purposes by any means, provided that this copyright notice appears on all such copies. ¤Graduate Research Associate, Assistant Professor, and Associate Professor, Department of Agri- cultural, Environmental, and Development Economics, The Ohio State University, Columbus, OH 43210. Empirical Performance of Alternative Option Pricing Models for Commodity Futures Options Abstract The central part of pricing agricultural commodity futures options is to ¯nd appro- priate stochastic process of the underlying assets. The Black's (1976) futures option pricing model laid the foundation for a new era of futures option valuation theory. The geometric Brownian motion assumption girding the Black's model, however, has been regarded as unrealistic in numerous empirical studies. Option pricing models incor- porating discrete jumps and stochastic volatility have been studied extensively in the literature. This study tests the performance of major alternative option pricing models and attempts to ¯nd the appropriate model for pricing commodity futures options. Keywords: futures options, jump-di®usion, option pricing, stochastic volatility, sea- sonality Introduction Proper model for pricing agricultural commodity futures options is crucial to estimating implied volatility and e®ectively hedging in agricultural ¯nancial markets. -
Risk Neutral Measures
We will denote expectations and conditional expectations with respect to the new measure by E˜. That is, given a random variable X, Z E˜X = EZ(T )X = Z(T )X dP . CHAPTER 4 Also, given a σ-algebra F, E˜(X | F) is the unique F-measurable random variable such that Risk Neutral Measures Z Z (1.2) E˜(X | F) dP˜ = X dP˜ , F F Our aim in this section is to show how risk neutral measures can be used to holds for all F measurable events F . Of course, equation (1.2) is equivalent to price derivative securities. The key advantage is that under a risk neutral measure requiring the discounted hedging portfolio becomes a martingale. Thus the price of any Z Z 0 ˜ ˜ derivative security can be computed by conditioning the payoff at maturity. We will (1.2 ) Z(T )E(X | F) dP = Z(T )X dP , F F use this to provide an elegant derivation of the Black-Scholes formula, and discuss for all F measurable events F . the fundamental theorems of asset pricing. The main goal of this section is to prove the Girsanov theorem. 1. The Girsanov Theorem. Theorem 1.6 (Cameron, Martin, Girsanov). Let b(t) = (b1(t), b2(t), . , bd(t)) be a d-dimensional adapted process, W be a d-dimensional Brownian motion, and ˜ Definition 1.1. Two probability measures P and P are said to be equivalent define if for every event A, P (A) = 0 if and only if P˜(A) = 0. Z t W˜ (t) = W (t) + b(s) ds . -
Stochastic Volatility and Black – Scholes Model Evidence of Amman Stock Exchange
R M B www.irmbrjournal.com June 2017 R International Review of Management and Business Research Vol. 6 Issue.2 I Stochastic Volatility and Black – Scholes Model Evidence of Amman Stock Exchange MOHAMMAD. M. ALALAYA Associate Prof in Economic Methods Al-Hussein Bin Talal University, Collage of Administrative Management and Economics, Ma‟an, Jordan. Email: [email protected] SULIMAN ALKHATAB Professer in Marketing, Al-Hussein Bin Talal University, Collage of Administrative Management and Economics, Ma‟an, Jordan. AHMAD ALMUHTASEB Assistant Prof in Marketing, Al-Hussein Bin Talal University, Collage of Administrative Management and Economics, Ma‟an, Jordan. JIHAD ALAFARJAT Assistant Prof in Management, Al-Hussein Bin Talal University, Collage of Administrative Management and Economics, Ma‟an, Jordan. Abstract This paper makes an attempt to decompose the Black – Scholes into components in Garch option model, and to examine the path of dependence in the terminal stock price distribution of Amman Stock Exchange (ASE), such as Black – Scholes’, the leverage effect in this paper which represents the result of analysis is important to determine the direction of the model bias, a time varying risk, may give fruitful help in explaining the under pricing of traded stock sheers and traded options in ASE. Generally, this study considers various pricing bias related to warrant of strike prices, and observes time to time maturity. The Garch option price does not seem overly sensitive to (a, B1) parameters, or to the time risk premium variance persistence parameter, Ω = a1+B1, heaving on the magnitude of Black –Scholes’ bias of the result of analysis, where the conditional variance bias does not improve in accuracy to justify the model to ASE data. -
The Girsanov Theorem Without (So Much) Stochastic Analysis Antoine Lejay
The Girsanov theorem without (so much) stochastic analysis Antoine Lejay To cite this version: Antoine Lejay. The Girsanov theorem without (so much) stochastic analysis. 2017. hal-01498129v1 HAL Id: hal-01498129 https://hal.inria.fr/hal-01498129v1 Preprint submitted on 29 Mar 2017 (v1), last revised 22 Sep 2018 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Girsanov theorem without (so much) stochastic analysis Antoine Lejay March 29, 2017 Abstract In this pedagogical note, we construct the semi-group associated to a stochastic differential equation with a constant diffusion and a Lipschitz drift by composing over small times the semi-groups generated respectively by the Brownian motion and the drift part. Similarly to the interpreta- tion of the Feynman-Kac formula through the Trotter-Kato-Lie formula in which the exponential term appears naturally, we construct by doing so an approximation of the exponential weight of the Girsanov theorem. As this approach only relies on the basic properties of the Gaussian distribution, it provides an alternative explanation of the form of the Girsanov weights without referring to a change of measure nor on stochastic calculus. -
Comparison of Option Pricing Between ARMA-GARCH and GARCH-M Models
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Scholarship@Western Western University Scholarship@Western Electronic Thesis and Dissertation Repository 4-22-2013 12:00 AM Comparison of option pricing between ARMA-GARCH and GARCH-M models Yi Xi The University of Western Ontario Supervisor Dr. Reginald J. Kulperger The University of Western Ontario Graduate Program in Statistics and Actuarial Sciences A thesis submitted in partial fulfillment of the equirr ements for the degree in Master of Science © Yi Xi 2013 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Other Statistics and Probability Commons Recommended Citation Xi, Yi, "Comparison of option pricing between ARMA-GARCH and GARCH-M models" (2013). Electronic Thesis and Dissertation Repository. 1215. https://ir.lib.uwo.ca/etd/1215 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. COMPARISON OF OPTION PRICING BETWEEN ARMA-GARCH AND GARCH-M MODELS (Thesis format: Monograph) by Yi Xi Graduate Program in Statistics and Actuarial Science A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science The School of Graduate and Postdoctoral Studies The University of Western Ontario London, Ontario, Canada ⃝c Yi Xi 2013 Abstract Option pricing is a major area in financial modeling. Option pricing is sometimes based on normal GARCH models. Normal GARCH models fail to capture the skewness and the leptokurtosis in financial data. -
Equivalent and Absolutely Continuous Measure Changes for Jump-Diffusion
The Annals of Applied Probability 2005, Vol. 15, No. 3, 1713–1732 DOI: 10.1214/105051605000000197 c Institute of Mathematical Statistics, 2005 EQUIVALENT AND ABSOLUTELY CONTINUOUS MEASURE CHANGES FOR JUMP-DIFFUSION PROCESSES By Patrick Cheridito1, Damir Filipovic´ and Marc Yor Princeton University, University of Munich and Universit´e Pierre et Marie Curie, Paris VI We provide explicit sufficient conditions for absolute continuity and equivalence between the distributions of two jump-diffusion pro- cesses that can explode and be killed by a potential. 1. Introduction. The purpose of this paper is to give explicit, easy-to- check sufficient conditions for the distributions of two jump-diffusion pro- cesses to be equivalent or absolutely continuous. We consider jump-diffusions that can explode and be killed by a potential. These processes are, in general, not semimartingales. We characterize them by their infinitesimal generators. The structure of the paper is as follows. In Section 2 we introduce the notation and state the paper’s main result, which gives sufficient conditions for the distributions of two jump-diffusions to be equivalent or absolutely continuous. The conditions consist of local bounds on the transformation of one generator into the other one and the assumption that the martingale problem for the second generator has for all initial distributions a unique solution. The formulation of the main theorem involves two sequences of stopping times. Stopping times of the first sequence stop the process before it explodes. The second sequence consists of exit times of the process from regions in the state space where the transformation of the first generator into the second one can be controlled. -
The Impact of Implied Volatility Fluctuations on Vertical Spread Option Strategies: the Case of WTI Crude Oil Market
energies Article The Impact of Implied Volatility Fluctuations on Vertical Spread Option Strategies: The Case of WTI Crude Oil Market Bartosz Łamasz * and Natalia Iwaszczuk Faculty of Management, AGH University of Science and Technology, 30-059 Cracow, Poland; [email protected] * Correspondence: [email protected]; Tel.: +48-696-668-417 Received: 31 July 2020; Accepted: 7 October 2020; Published: 13 October 2020 Abstract: This paper aims to analyze the impact of implied volatility on the costs, break-even points (BEPs), and the final results of the vertical spread option strategies (vertical spreads). We considered two main groups of vertical spreads: with limited and unlimited profits. The strategy with limited profits was divided into net credit spread and net debit spread. The analysis takes into account West Texas Intermediate (WTI) crude oil options listed on New York Mercantile Exchange (NYMEX) from 17 November 2008 to 15 April 2020. Our findings suggest that the unlimited vertical spreads were executed with profits less frequently than the limited vertical spreads in each of the considered categories of implied volatility. Nonetheless, the advantage of unlimited strategies was observed for substantial oil price movements (above 10%) when the rates of return on these strategies were higher than for limited strategies. With small price movements (lower than 5%), the net credit spread strategies were by far the best choice and generated profits in the widest price ranges in each category of implied volatility. This study bridges the gap between option strategies trading, implied volatility and WTI crude oil market. The obtained results may be a source of information in hedging against oil price fluctuations. -
Optimal Importance Sampling for Diffusion Processes
U.U.D.M. Project Report 2019:38 Optimal Importance Sampling for Diffusion Processes Malvina Fröberg Examensarbete i matematik, 30 hp Handledare: Erik Ekström Examinator: Denis Gaidashev Juni 2019 Department of Mathematics Uppsala University Acknowledgements I would like to express my sincere gratitude to my supervisor, Professor Erik Ekström, for all his help and support. I am thankful for his encouragement and for introducing me to the topic, as well as for the hours spent guiding me. 1 Abstract Variance reduction techniques are used to increase the precision of estimates in numerical calculations and more specifically in Monte Carlo simulations. This thesis focuses on a particular variance reduction technique, namely importance sampling, and applies it to diffusion processes with applications within the field of financial mathematics. Importance sampling attempts to reduce variance by changing the probability measure. The Girsanov theorem is used when chang- ing measure for stochastic processes. However, a change of the probability measure gives a new drift coefficient for the underlying diffusion, which may lead to an increased computational cost. This issue is discussed and formu- lated as a stochastic optimal control problem and studied further by using the Hamilton-Jacobi-Bellman equation with a penalty term to account for compu- tational costs. The objective of this thesis is to examine whether there is an optimal change of measure or an optimal new drift for a diffusion process. This thesis provides examples of optimal measure changes in cases where the set of possible measure changes is restricted but not penalized, as well as examples for unrestricted measure changes but with penalization.