Static Replication of Exotic Options Andrew Chou JUL 241997 Eng
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Pricing and Hedging of Lookback Options in Hyper-Exponential Jump Diffusion Models
Pricing and hedging of lookback options in hyper-exponential jump diffusion models Markus Hofer∗ Philipp Mayer y Abstract In this article we consider the problem of pricing lookback options in certain exponential Lévy market models. While in the classic Black-Scholes models the price of such options can be calculated in closed form, for more general asset price model one typically has to rely on (rather time-intense) Monte- Carlo or P(I)DE methods. However, for Lévy processes with double exponentially distributed jumps the lookback option price can be expressed as one-dimensional Laplace transform (cf. Kou [Kou et al., 2005]). The key ingredient to derive this representation is the explicit availability of the first passage time distribution for this particular Lévy process, which is well-known also for the more general class of hyper-exponential jump diffusions (HEJD). In fact, Jeannin and Pistorius [Jeannin and Pistorius, 2010] were able to derive formulae for the Laplace transformed price of certain barrier options in market models described by HEJD processes. Here, we similarly derive the Laplace transforms of floating and fixed strike lookback option prices and propose a numerical inversion scheme, which allows, like Fourier inversion methods for European vanilla options, the calculation of lookback options with different strikes in one shot. Additionally, we give semi-analytical formulae for several Greeks of the option price and discuss a method of extending the proposed method to generalised hyper- exponential (as e.g. NIG or CGMY) models by fitting a suitable HEJD process. Finally, we illustrate the theoretical findings by some numerical experiments. -
A Discrete-Time Approach to Evaluate Path-Dependent Derivatives in a Regime-Switching Risk Model
risks Article A Discrete-Time Approach to Evaluate Path-Dependent Derivatives in a Regime-Switching Risk Model Emilio Russo Department of Economics, Statistics and Finance, University of Calabria, Ponte Bucci cubo 1C, 87036 Rende (CS), Italy; [email protected] Received: 29 November 2019; Accepted: 25 January 2020 ; Published: 29 January 2020 Abstract: This paper provides a discrete-time approach for evaluating financial and actuarial products characterized by path-dependent features in a regime-switching risk model. In each regime, a binomial discretization of the asset value is obtained by modifying the parameters used to generate the lattice in the highest-volatility regime, thus allowing a simultaneous asset description in all the regimes. The path-dependent feature is treated by computing representative values of the path-dependent function on a fixed number of effective trajectories reaching each lattice node. The prices of the analyzed products are calculated as the expected values of their payoffs registered over the lattice branches, invoking a quadratic interpolation technique if the regime changes, and capturing the switches among regimes by using a transition probability matrix. Some numerical applications are provided to support the model, which is also useful to accurately capture the market risk concerning path-dependent financial and actuarial instruments. Keywords: regime-switching risk; market risk; path-dependent derivatives; insurance policies; binomial lattices; discrete-time models 1. Introduction With the aim of providing an accurate evaluation of the risks affecting financial markets, a wide range of empirical research evidences that asset returns show stochastic volatility patterns and fatter tails with respect to the standard normal model. -
Dixit-Pindyck and Arrow-Fisher-Hanemann-Henry Option Concepts in a Finance Framework
Dixit-Pindyck and Arrow-Fisher-Hanemann-Henry Option Concepts in a Finance Framework January 12, 2015 Abstract We look at the debate on the equivalence of the Dixit-Pindyck (DP) and Arrow-Fisher- Hanemann-Henry (AFHH) option values. Casting the problem into a financial framework allows to disentangle the discussion without unnecessarily introducing new definitions. Instead, the option values can be easily translated to meaningful terms of financial option pricing. We find that the DP option value can easily be described as time-value of an American plain-vanilla option, while the AFHH option value is an exotic chooser option. Although the option values can be numerically equal, they differ for interesting, i.e. non-trivial investment decisions and benefit-cost analyses. We find that for applied work, compared to the Dixit-Pindyck value, the AFHH concept has only limited use. Keywords: Benefit cost analysis, irreversibility, option, quasi-option value, real option, uncertainty. Abbreviations: i.e.: id est; e.g.: exemplum gratia. 1 Introduction In the recent decades, the real options1 concept gained a large foothold in the strat- egy and investment literature, even more so with Dixit & Pindyck (1994)'s (DP) seminal volume on investment under uncertainty. In environmental economics, the importance of irreversibility has already been known since the contributions of Arrow & Fisher (1974), Henry (1974), and Hanemann (1989) (AFHH) on quasi-options. As Fisher (2000) notes, a unification of the two option concepts would have the benefit of applying the vast results of real option pricing in environmental issues. In his 1The term real option has been coined by Myers (1977). -
Analysis of Employee Stock Options and Guaranteed Withdrawal Benefits for Life by Premal Shah B
Analysis of Employee Stock Options and Guaranteed Withdrawal Benefits for Life by Premal Shah B. Tech. & M. Tech., Electrical Engineering (2003), Indian Institute of Technology (IIT) Bombay, Mumbai Submitted to the Sloan School of Management in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Operations Research at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2008 c Massachusetts Institute of Technology 2008. All rights reserved. Author.............................................................. Sloan School of Management July 03, 2008 Certified by. Dimitris J. Bertsimas Boeing Professor of Operations Research Thesis Supervisor Accepted by . Cynthia Barnhart Professor Co-director, Operations Research Center 2 Analysis of Employee Stock Options and Guaranteed Withdrawal Benefits for Life by Premal Shah B. Tech. & M. Tech., Electrical Engineering (2003), Indian Institute of Technology (IIT) Bombay, Mumbai Submitted to the Sloan School of Management on July 03, 2008, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Operations Research Abstract In this thesis we study three problems related to financial modeling. First, we study the problem of pricing Employee Stock Options (ESOs) from the point of view of the issuing company. Since an employee cannot trade or effectively hedge ESOs, she exercises them to maximize a subjective criterion of value. Modeling this exercise behavior is key to pricing ESOs. We argue that ESO exercises should not be modeled on a one by one basis, as is commonly done, but at a portfolio level because exercises related to different ESOs that an employee holds would be coupled. Using utility based models we also show that such coupled exercise behavior leads to lower average ESO costs for the commonly used utility functions such as power and exponential utilities. -
Sequential Compound Options and Investments Valuation
Sequential compound options and investments valuation Luigi Sereno Dottorato di Ricerca in Economia - XIX Ciclo - Alma Mater Studiorum - Università di Bologna Marzo 2007 Relatore: Prof. ssa Elettra Agliardi Coordinatore: Prof. Luca Lambertini Settore scienti…co-disciplinare: SECS-P/01 Economia Politica ii Contents I Sequential compound options and investments valua- tion 1 1 An overview 3 1.1 Introduction . 3 1.2 Literature review . 6 1.2.1 R&D as real options . 11 1.2.2 Exotic Options . 12 1.3 An example . 17 1.3.1 Value of expansion opportunities . 18 1.3.2 Value with abandonment option . 23 1.3.3 Value with temporary suspension . 26 1.4 Real option modelling with jump processes . 31 1.4.1 Introduction . 31 1.4.2 Merton’sapproach . 33 1.4.3 Further reading . 36 1.5 Real option and game theory . 41 1.5.1 Introduction . 41 1.5.2 Grenadier’smodel . 42 iii iv CONTENTS 1.5.3 Further reading . 45 1.6 Final remark . 48 II The valuation of new ventures 59 2 61 2.1 Introduction . 61 2.2 Literature Review . 63 2.2.1 Flexibility of Multiple Compound Real Options . 65 2.3 Model and Assumptions . 68 2.3.1 Value of the Option to Continuously Shut - Down . 69 2.4 An extension . 74 2.4.1 The mathematical problem and solution . 75 2.5 Implementation of the approach . 80 2.5.1 Numerical results . 82 2.6 Final remarks . 86 III Valuing R&D investments with a jump-di¤usion process 93 3 95 3.1 Introduction . -
Calibration Risk for Exotic Options
Forschungsgemeinschaft through through Forschungsgemeinschaft SFB 649DiscussionPaper2006-001 * CASE - Center for Applied Statistics and Economics, Statisticsand Center forApplied - * CASE Calibration Riskfor This research was supported by the Deutsche the Deutsche by was supported This research Wolfgang K.Härdle** Humboldt-Universität zuBerlin,Germany SFB 649, Humboldt-Universität zu Berlin zu SFB 649,Humboldt-Universität Exotic Options Spandauer Straße 1,D-10178 Berlin Spandauer http://sfb649.wiwi.hu-berlin.de http://sfb649.wiwi.hu-berlin.de Kai Detlefsen* ISSN 1860-5664 the SFB 649 "Economic Risk". "Economic the SFB649 SFB 6 4 9 E C O N O M I C R I S K B E R L I N Calibration Risk for Exotic Options K. Detlefsen and W. K. H¨ardle CASE - Center for Applied Statistics and Economics Humboldt-Universit¨atzu Berlin Wirtschaftswissenschaftliche Fakult¨at Spandauer Strasse 1, 10178 Berlin, Germany Abstract Option pricing models are calibrated to market data of plain vanil- las by minimization of an error functional. From the economic view- point, there are several possibilities to measure the error between the market and the model. These different specifications of the error give rise to different sets of calibrated model parameters and the resulting prices of exotic options vary significantly. These price differences often exceed the usual profit margin of exotic options. We provide evidence for this calibration risk in a time series of DAX implied volatility surfaces from April 2003 to March 2004. We analyze in the Heston and in the Bates model factors influencing these price differences of exotic options and finally recommend an error func- tional. -
Model Risk in the Pricing of Exotic Options
Model Risk in the Pricing of Exotic Options Jacinto Marabel Romo∗ José Luis Crespo Espert† [email protected] [email protected] Abstract The growth experimented in recent years in both the variety and volume of structured products implies that banks and other financial institutions have become increasingly exposed to model risk. In this article we focus on the model risk associated with the local volatility (LV) model and with the Vari- ance Gamma (VG) model. The results show that the LV model performs better than the VG model in terms of its ability to match the market prices of European options. Nevertheless, both models are subject to significant pricing errors when compared with the stochastic volatility framework. Keywords: Model risk, exotic options, local volatility, stochastic volatil- ity, Variance Gamma process, path dependence. JEL: G12, G13. ∗BBVA and University Institute for Economic and Social Analysis, University of Alcalá (UAH). Mailing address: Vía de los Poblados s/n, 28033, Madrid, Spain. The content of this paper represents the author’s personal opinion and does not reflect the views of BBVA. †University of Alcalá (UAH) and University Institute for Economic and Social Analysis. Mail- ing address: Plaza de la Victoria, 2, 28802, Alcalá de Henares, Madrid, Spain. 1 Introduction In recent years there has been a remarkable growth of structured products with embedded exotic options. In this sense, the European Commission1 stated that the use of derivatives has grown exponentially over the last decade, with over-the- counter transactions being the main contributor to this growth. At the end of December 2009, the size of the over-the-counter derivatives market by notional value equaled approximately $615 trillion, a 12% increase with respect to the end of 2008. -
Quanto Lookback Options
Quanto lookback options Min Dai Institute of Mathematics and Department of Financial Mathematics Peking University, Beijing 100871, China (e-mails: [email protected]) Hoi Ying Wong Department of Statistics, Chinese University of HongKong, Shatin, Hong Kong, China (e-mail: [email protected]) Yue Kuen Kwoky Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China (e-mail: [email protected]) Date of submission: 1 December, 2001 Abstract. The lookback feature in a quanto option refers to the payoff structure where the terminal payoff of the quanto option depends on the realized extreme value of either the stock price or the exchange rate. In this paper, we study the pricing models of European and American lookback option with the quanto feature. The analytic price formulas for two types of European style quanto lookback options are derived. The success of the analytic tractability of these quanto lookback options depends on the availability of a succinct analytic representation of the joint density function of the extreme value and terminal value of the stock price and exchange rate. We also analyze the early exercise policies and pricing behaviors of the quanto lookback option with the American feature. The early exercise boundaries of these American quanto lookback options exhibit properties that are distinctive from other two-state American option models. Key words: Lookback options, quanto feature, early exercise policies JEL classification number: G130 Mathematics Subject Classification (1991): 90A09, 60G44 y Corresponding author 1 1. Introduction Lookback options are contingent claims whose payoff depends on the extreme value of the underlying asset price process realized over a specified period of time within the life of the option. -
Discretely Monitored Look-Back Option Prices and Their Sensitivities in Levy´ Models
Discretely Monitored Look-Back Option Prices and their Sensitivities in Levy´ Models Farid AitSahlia1 Gudbjort Gylfadottir2 (Preliminary Draft) May 22, 2017 Abstract We present an efficient method to price discretely monitored lookback options when the underlying asset price follows an exponential Levy´ process. Our approach extends the random walk duality results of AitSahlia and Lai (1998) originally developed in the Black- Scholes set-up and exploits the very fast numerical scheme recently developed by Linetsky and Feng (2008, 2009) to compute and invert Hilbert transforms. Though Linetsky and Feng (2009) do apply these transforms to price lookback options, they require an explicit transition probability density of the Levy´ process and impose a condition that excludes the pure jump variance gamma process, among others. In contrast, our approach is much simpler and makes use of only the characteristic function of the log-increment, which is central to Levy´ processes. Furthermore, by focusing our approach on determining the distribution function of the maximum of the Levy´ process we can also determine price sensitivities with minimal additional computational effort. 1: Department of Finance, Warrington College of Business Administration, University of Florida, Gainesville, Florida, 32611. Tel: 352.392.5058. Fax: 352.392.0301. email: [email protected] . (Corresponding author) 2: Bloomberg L.P., 731 Lexington Ave., New York, 10022. Email: [email protected] 1 1 Introduction Lookback options provide the largest payoff potential because their holders can choose (in hindsight) the exercise date with the advantage of having full path information. Lookback options were initially devised mainly for speculative purposes but starting with currency mar- kets, their adoption has been increasing significantly, especially in insurance and structured products during the past decade. -
Lecture 2: Options and Investments
Binnenlandse Franqui leerstoel –VUB December, 2004 Opties André Farber Lecture 2: Options and investments 1. Introduction binomial option pricing – Review 1-period binomial option pricing formulas: σ ∆t u = e d = 1/u -r∆t f = [ p fu + (1-p) fd ] e p = (er∆t – d)/(u - d) 2. Black Scholes formula (European option on non dividend paying stock) -rT European call: C = S N(d1) – K e N(d2) European put: P = K N(-d2) – S N(-d1) S ln( ) Ke −rT d1 = + .5σ T σ T S ln( ) Ke −rT d 2 = − .5σ T = d1 −σ T σ T Note: Using put call parity: P = C – S + K e-rT -rT -rT = S N(d1) – K e N(d2) – S + K e -rT = K e [1 – N(d2)] – S [1 – N(d1)] -rT = K e N(-d2)] – S N(-d1) If stock pays continuous dividend yield q: replace S by S e-qT Illustration: Excel Lecture 2 worksheet B&S formula 3. American options Non dividend paying stock Call: Black Scholes formula (no early exercise) Put: No closed form solution – use binomial model -r∆t f = Max(K-S, [ p fu + (1-p) fd ] e ) Dividend paying stock (assume constant dividend yield q) No closed form solution – Use binomial model Put-Call parity: C + PV(K) = P + Se-qT Risk neutral probability of up: p = (e(r-q)∆t – d)/(u - d) Illustration: Excel Lecture 2 Binomial model 3. The Greek letters ∂f Slope Delta : δ = = f ' ∂S S -qT Black Scholes: Delta Call = N(d1) e -q∆t Binomial model: Delta = (fu – fd) / [(u-d)Se ] 1 ∂δ ∂²f Convexity Gamma: Γ = = = f " ∂S ∂S ² SS ∂f Time Theta: Θ = = f ' ∂T T ∂f Volatility Vega: = ∂σ ∂f Interest rate Rho: = ∂r Illustration: Excel Lecture 2 worksheet B&S Formula 4. -
Pricing Lookback Option Under Stochastic Volatility
PRICING LOOKBACK OPTION UNDER STOCHASTIC VOLATILITY TEFERI DEREJE WIRTU MASTER OF SCIENCE (Mathematics - Financial Option) PAN AFRICAN UNIVERSITY INSTITUTE OF BASIC SCIENCES, TECHNOLOGY AND INNOVATION 2018 Pricing Lookback Option under Stochastic Volatility Teferi Dereje Wirtu MF300-0001/2016 A Thesis submitted to Pan African University Institute of Basic Sciences, Technology and Innovation in partial fulfillment of the requirements for the award of the Degree of Master of Science in Mathematics (Financial Option) 2018 DECLARATION I, Teferi Dereje Wirtu, declare that this thesis entitled: "Pricing lookback option under stochastic volatility" is my original work and has not been presented for a degree in any University. I also confirm that this work was done wholly or mainly while in candidature for a research degree at this noble University. The work was done under the guidence of both Dr. Philip Ngare and Dr. Ananda Kube. Signature .............................................. date.......................................... Name: Teferi Dereje Wirtu (Registration number MF 300-0001/16) This thesis has been submitted for final thesis with my approval as University Supervisors. Signature ........................................... Date ............................................ Name: Dr. Philip Ngare (Supervisor) This thesis has been submitted for final thesis with my approval as University Supervisors Signature ............................................ date ......................................... Name: Dr. Ananda Kube (Supervisor) i DEDICATION To my mother Yashi Mokonon, my father Dereje Wirtu, my brother Sekata Asefa, Fikire Dereje and Tesfaye Dereje my sister Aberash Dereje my friends Megersa Tadesse, Solomon Teshome, Abriham Mengesha and Yosef Hamba. ii Acknowledgements First, I would like to express my special thanks and gratitude to God. Next, I would like to express my gratitude to my supervisors Dr.Philip Ngare and Dr. -
Barrier Options in the Black-Scholes Framework
Barrier Options This note is several years old and very preliminary. It has no references to the literature. Do not trust its accuracy! Note that there is a lot of more recent literature, especially on static hedging. 0.1 Introduction In this note we discuss various kinds of barrier options. The four basic forms of these path- dependent options are down-and-out, down-and-in, up-and-out and up-and-in. That is, the right to exercise either appears (‘in’) or disappears (‘out’) on some barrier in (S, t) space, usually of the form S = constant. The barrier is set above (‘up’) or below (‘down’) the asset price at the time the option is created. They are also often called knock-out, or knock-in options. An example of a knock-out contract is a European-style option which immediately expires worthless if, at any time before expiry, the asset price falls to a lower barrier S = B−, set below S(0). If the barrier is not reached, the holder receives the payoff at expiry. When the payoff is the same as that for a vanilla call, the barrier option is termed a European down-and-out call. Figure 1 shows two realisations of the random walk, of which one ends in knock-out, while the other does not. The second walk in the figure leads to a payout of S(T ) − E at expiry, but if it had finished between B− and E, the payout would have been zero. An up-and-out call has similar characteristics except that it becomes worthless if the asset price ever rises to an upper barrier S = B+.