Implied Trinomial Trees of the Volatility Smile
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Trinomial Tree
Trinomial Tree • Set up a trinomial approximation to the geometric Brownian motion dS=S = r dt + σ dW .a • The three stock prices at time ∆t are S, Su, and Sd, where ud = 1. • Impose the matching of mean and that of variance: 1 = pu + pm + pd; SM = (puu + pm + (pd=u)) S; 2 2 2 2 S V = pu(Su − SM) + pm(S − SM) + pd(Sd − SM) : aBoyle (1988). ⃝c 2013 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 599 • Above, M ≡ er∆t; 2 V ≡ M 2(eσ ∆t − 1); by Eqs. (21) on p. 154. ⃝c 2013 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 600 * - pu* Su * * pm- - j- S S * pd j j j- Sd - j ∆t ⃝c 2013 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 601 Trinomial Tree (concluded) • Use linear algebra to verify that ( ) u V + M 2 − M − (M − 1) p = ; u (u − 1) (u2 − 1) ( ) u2 V + M 2 − M − u3(M − 1) p = : d (u − 1) (u2 − 1) { In practice, we must also make sure the probabilities lie between 0 and 1. • Countless variations. ⃝c 2013 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 602 A Trinomial Tree p • Use u = eλσ ∆t, where λ ≥ 1 is a tunable parameter. • Then ( ) p r + σ2 ∆t ! 1 pu 2 + ; 2λ ( 2λσ) p 1 r − 2σ2 ∆t p ! − : d 2λ2 2λσ p • A nice choice for λ is π=2 .a aOmberg (1988). ⃝c 2013 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 603 Barrier Options Revisited • BOPM introduces a specification error by replacing the barrier with a nonidentical effective barrier. -
A Fuzzy Real Option Model for Pricing Grid Compute Resources
A Fuzzy Real Option Model for Pricing Grid Compute Resources by David Allenotor A Thesis submitted to the Faculty of Graduate Studies of The University of Manitoba in partial fulfillment of the requirements of the degree of DOCTOR OF PHILOSOPHY Department of Computer Science University of Manitoba Winnipeg Copyright ⃝c 2010 by David Allenotor Abstract Many of the grid compute resources (CPU cycles, network bandwidths, computing power, processor times, and software) exist as non-storable commodities, which we call grid compute commodities (gcc) and are distributed geographically across organizations. These organizations have dissimilar resource compositions and usage policies, which makes pricing grid resources and guaranteeing their availability a challenge. Several initiatives (Globus, Legion, Nimrod/G) have developed various frameworks for grid resource management. However, there has been a very little effort in pricing the resources. In this thesis, we propose financial option based model for pricing grid resources by devising three research threads: pricing the gcc as a problem of real option, modeling gcc spot price using a discrete time approach, and addressing uncertainty constraints in the provision of Quality of Service (QoS) using fuzzy logic. We used GridSim, a simulation tool for resource usage in a Grid to experiment and test our model. To further consolidate our model and validate our results, we analyzed usage traces from six real grids from across the world for which we priced a set of resources. We designed a Price Variant Function (PVF) in our model, which is a fuzzy value and its application attracts more patronage to a grid that has more resources to offer and also redirect patronage from a grid that is very busy to another grid. -
Local Volatility Modelling
LOCAL VOLATILITY MODELLING Roel van der Kamp July 13, 2009 A DISSERTATION SUBMITTED FOR THE DEGREE OF Master of Science in Applied Mathematics (Financial Engineering) I have to understand the world, you see. - Richard Philips Feynman Foreword This report serves as a dissertation for the completion of the Master programme in Applied Math- ematics (Financial Engineering) from the University of Twente. The project was devised from the collaboration of the University of Twente with Saen Options BV (during the course of the project Saen Options BV was integrated into AllOptions BV) at whose facilities the project was performed over a period of six months. This research project could not have been performed without the help of others. Most notably I would like to extend my gratitude towards my supervisors: Michel Vellekoop of the University of Twente, Julien Gosme of AllOptions BV and Fran¸coisMyburg of AllOptions BV. They provided me with the theoretical and practical knowledge necessary to perform this research. Their constant guidance, involvement and availability were an essential part of this project. My thanks goes out to Irakli Khomasuridze, who worked beside me for six months on his own project for the same degree. The many discussions I had with him greatly facilitated my progress and made the whole experience much more enjoyable. Finally I would like to thank AllOptions and their staff for making use of their facilities, getting access to their data and assisting me with all practical issues. RvdK Abstract Many different models exist that describe the behaviour of stock prices and are used to value op- tions on such an underlying asset. -
MERTON JUMP-DIFFUSION MODEL VERSUS the BLACK and SCHOLES APPROACH for the LOG-RETURNS and VOLATILITY SMILE FITTING Nicola Gugole
International Journal of Pure and Applied Mathematics Volume 109 No. 3 2016, 719-736 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v109i3.19 ijpam.eu MERTON JUMP-DIFFUSION MODEL VERSUS THE BLACK AND SCHOLES APPROACH FOR THE LOG-RETURNS AND VOLATILITY SMILE FITTING Nicola Gugole Department of Computer Science University of Verona Strada le Grazie, 15-37134, Verona, ITALY Abstract: In the present paper we perform a comparison between the standard Black and Scholes model and the Merton jump-diffusion one, from the point of view of the study of the leptokurtic feature of log-returns and also concerning the volatility smile fitting. Provided results are obtained by calibrating on market data and by mean of numerical simulations which clearly show how the jump-diffusion model outperforms the classical geometric Brownian motion approach. AMS Subject Classification: 60H15, 60H35, 91B60, 91G20, 91G60 Key Words: Black and Scholes model, Merton model, stochastic differential equations, log-returns, volatility smile 1. Introduction In the early 1970’s the world of option pricing experienced a great contribu- tion given by the work of Fischer Black and Myron Scholes. They developed a new mathematical model to treat certain financial quantities publishing re- lated results in the article The Pricing of Options and Corporate Liabilities, see [1]. The latter work became soon a reference point in the financial scenario. Received: August 3, 2016 c 2016 Academic Publications, Ltd. Revised: September 16, 2016 url: www.acadpubl.eu Published: September 30, 2016 720 N. Gugole Nowadays, many traders still use the Black and Scholes (BS) model to price as well as to hedge various types of contingent claims. -
Pricing Options Using Trinomial Trees
Pricing Options Using Trinomial Trees Paul Clifford Yan Wang Oleg Zaboronski 30.12.2009 1 Introduction One of the first computational models used in the financial mathematics community was the binomial tree model. This model was popular for some time but in the last 15 years has become significantly outdated and is of little practical use. However it is still one of the first models students traditionally were taught. A more advanced model used for the project this semester, is the trinomial tree model. This improves upon the binomial model by allowing a stock price to move up, down or stay the same with certain probabilities, as shown in the diagram below. 2 Project description. The aim of the project is to apply the trinomial tree to the following problems: ² Pricing various European and American options ² Pricing barrier options ² Calculating the greeks More precisely, the students are asked to do the following: 1. Study the trinomial tree and its parameters, pu; pd; pm; u; d 2. Study the method to build the trinomial tree of share prices 3. Study the backward induction algorithms for option pricing on trees 4. Price various options such as European, American and barrier 1 5. Calculate the greeks using the tree Each of these topics will be explained very clearly in the following sections. Students are encouraged to ask questions during the lab sessions about certain terminology that they do not understand such as barrier options, hedging greeks and things of this nature. Answers to questions listed below should contain analysis of numerical results produced by the simulation. -
On Trinomial Trees for One-Factor Short Rate Models∗
On Trinomial Trees for One-Factor Short Rate Models∗ Markus Leippoldy Swiss Banking Institute, University of Zurich Zvi Wienerz School of Business Administration, Hebrew University of Jerusalem April 3, 2003 ∗Markus Leippold acknowledges the financial support of the Swiss National Science Foundation (NCCR FINRISK). Zvi Wiener acknowledges the financial support of the Krueger and Rosenberg funds at the He- brew University of Jerusalem. We welcome comments, including references to related papers we inadvertently overlooked. yCorrespondence Information: University of Zurich, ISB, Plattenstr. 14, 8032 Zurich, Switzerland; tel: +41 1-634-2951; fax: +41 1-634-4903; [email protected]. zCorrespondence Information: Hebrew University of Jerusalem, Mount Scopus, Jerusalem, 91905, Israel; tel: +972-2-588-3049; fax: +972-2-588-1341; [email protected]. On Trinomial Trees for One-Factor Short Rate Models ABSTRACT In this article we discuss the implementation of general one-factor short rate models with a trinomial tree. Taking the Hull-White model as a starting point, our contribution is threefold. First, we show how trees can be spanned using a set of general branching processes. Secondly, we improve Hull-White's procedure to calibrate the tree to bond prices by a much more efficient approach. This approach is applicable to a wide range of term structure models. Finally, we show how the tree can be adjusted to the volatility structure. The proposed approach leads to an efficient and flexible construction method for trinomial trees, which can be easily implemented and calibrated to both prices and volatilities. JEL Classification Codes: G13, C6. Key Words: Short Rate Models, Trinomial Trees, Forward Measure. -
The Equity Option Volatility Smile: an Implicit Finite-Difference Approach 5
The equity option volatility smile: an implicit finite-difference approach 5 The equity option volatility smile: an implicit ®nite-dierence approach Leif B. G. Andersen and Rupert Brotherton-Ratcliffe This paper illustrates how to construct an unconditionally stable finite-difference lattice consistent with the equity option volatility smile. In particular, the paper shows how to extend the method of forward induction on Arrow±Debreu securities to generate local instantaneous volatilities in implicit and semi-implicit (Crank±Nicholson) lattices. The technique developed in the paper provides a highly accurate fit to the entire volatility smile and offers excellent convergence properties and high flexibility of asset- and time-space partitioning. Contrary to standard algorithms based on binomial trees, our approach is well suited to price options with discontinuous payouts (e.g. knock-out and barrier options) and does not suffer from problems arising from negative branch- ing probabilities. 1. INTRODUCTION The Black±Scholes option pricing formula (Black and Scholes 1973, Merton 1973) expresses the value of a European call option on a stock in terms of seven parameters: current time t, current stock price St, option maturity T, strike K, interest rate r, dividend rate , and volatility1 . As the Black±Scholes formula is based on an assumption of stock prices following geometric Brownian motion with constant process parameters, the parameters r, , and are all considered constants independent of the particular terms of the option contract. Of the seven parameters in the Black±Scholes formula, all but the volatility are, in principle, directly observable in the ®nancial market. The volatility can be estimated from historical data or, as is more common, by numerically inverting the Black±Scholes formula to back out the level of Ðthe implied volatilityÐthat is consistent with observed market prices of European options. -
Volatility Curves of Incomplete Markets
Master's thesis Volatility Curves of Incomplete Markets The Trinomial Option Pricing Model KATERYNA CHECHELNYTSKA Department of Mathematical Sciences Chalmers University of Technology University of Gothenburg Gothenburg, Sweden 2019 Thesis for the Degree of Master Science Volatility Curves of Incomplete Markets The Trinomial Option Pricing Model KATERYNA CHECHELNYTSKA Department of Mathematical Sciences Chalmers University of Technology University of Gothenburg SE - 412 96 Gothenburg, Sweden Gothenburg, Sweden 2019 Volatility Curves of Incomplete Markets The Trinomial Asset Pricing Model KATERYNA CHECHELNYTSKA © KATERYNA CHECHELNYTSKA, 2019. Supervisor: Docent Simone Calogero, Department of Mathematical Sciences Examiner: Docent Patrik Albin, Department of Mathematical Sciences Master's Thesis 2019 Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg SE-412 96 Gothenburg Telephone +46 31 772 1000 Typeset in LATEX Printed by Chalmers Reproservice Gothenburg, Sweden 2019 4 Volatility Curves of Incomplete Markets The Trinomial Option Pricing Model KATERYNA CHECHELNYTSKA Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg Abstract The graph of the implied volatility of call options as a function of the strike price is called volatility curve. If the options market were perfectly described by the Black-Scholes model, the implied volatility would be independent of the strike price and thus the volatility curve would be a flat horizontal line. However the volatility curve of real markets is often found to have recurrent convex shapes called volatility smile and volatility skew. The common approach to explain this phenomena is by assuming that the volatility of the underlying stock is a stochastic process (while in Black-Scholes it is assumed to be a deterministic constant). -
Robust Option Pricing: the Uncertain Volatility Model
Imperial College London Department of Mathematics Robust option pricing: the uncertain volatility model Supervisor: Dr. Pietro SIORPAES Author: Hicham EL JERRARI (CID: 01792391) A thesis submitted for the degree of MSc in Mathematics and Finance, 2019-2020 El Jerrari Robust option pricing Declaration The work contained in this thesis is my own work unless otherwise stated. Signature and date: 06/09/2020 Hicham EL JERRARI 1 El Jerrari Robust option pricing Acknowledgements I would like to thank my supervisor: Dr. Pietro SIORPAES for giving me the opportunity to work on this exciting subject. I am very grateful to him for his help throughout my project. Thanks to his advice, his supervision, as well as the discussions I had with him, this experience was very enriching for me. I would also like to thank Imperial College and more particularly the \Msc Mathemat- ics and Finance" for this year rich in learning, for their support and their dedication especially in this particular year of pandemic. On a personal level, I want to thank my parents for giving me the opportunity to study this master's year at Imperial College and to be present and support me throughout my studies. I also thank my brother and my two sisters for their encouragement. Last but not least, I dedicate this work to my grandmother for her moral support and a tribute to my late grandfather. 2 El Jerrari Robust option pricing Abstract This study presents the uncertain volatility model (UVM) which proposes a new approach for the pricing and hedging of derivatives by considering a band of spot's volatility as input. -
The Hull-White Model • Hull and White (1987) Postulate the Following Model, Ds √ = R Dt + V Dw , S 1 Dv = Μvv Dt + Bv Dw2
The Hull-White Model • Hull and White (1987) postulate the following model, dS p = r dt + V dW ; S 1 dV = µvV dt + bV dW2: • Above, V is the instantaneous variance. • They assume µv depends on V and t (but not S). ⃝c 2014 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 599 The SABR Model • Hagan, Kumar, Lesniewski, and Woodward (2002) postulate the following model, dS = r dt + SθV dW ; S 1 dV = bV dW2; for 0 ≤ θ ≤ 1. • A nice feature of this model is that the implied volatility surface has a compact approximate closed form. ⃝c 2014 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 600 The Hilliard-Schwartz Model • Hilliard and Schwartz (1996) postulate the following general model, dS = r dt + f(S)V a dW ; S 1 dV = µ(V ) dt + bV dW2; for some well-behaved function f(S) and constant a. ⃝c 2014 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 601 The Blacher Model • Blacher (2002) postulates the following model, dS [ ] = r dt + σ 1 + α(S − S ) + β(S − S )2 dW ; S 0 0 1 dσ = κ(θ − σ) dt + ϵσ dW2: • So the volatility σ follows a mean-reverting process to level θ. ⃝c 2014 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 602 Heston's Stochastic-Volatility Model • Heston (1993) assumes the stock price follows dS p = (µ − q) dt + V dW1; (64) S p dV = κ(θ − V ) dt + σ V dW2: (65) { V is the instantaneous variance, which follows a square-root process. { dW1 and dW2 have correlation ρ. -
Local and Stochastic Volatility Models: an Investigation Into the Pricing of Exotic Equity Options
Local and Stochastic Volatility Models: An Investigation into the Pricing of Exotic Equity Options A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, South Africa, in ful¯llment of the requirements of the degree of Master of Science. Abstract The assumption of constant volatility as an input parameter into the Black-Scholes option pricing formula is deemed primitive and highly erroneous when one considers the terminal distribution of the log-returns of the underlying process. To account for the `fat tails' of the distribution, we consider both local and stochastic volatility option pricing models. Each class of models, the former being a special case of the latter, gives rise to a parametrization of the skew, which may or may not reflect the correct dynamics of the skew. We investigate a select few from each class and derive the results presented in the corresponding papers. We select one from each class, namely the implied trinomial tree (Derman, Kani & Chriss 1996) and the SABR model (Hagan, Kumar, Lesniewski & Woodward 2002), and calibrate to the implied skew for SAFEX futures. We also obtain prices for both vanilla and exotic equity index options and compare the two approaches. Lisa Majmin September 29, 2005 I declare that this is my own, unaided work. It is being submitted for the Degree of Master of Science to the University of the Witwatersrand, Johannesburg. It has not been submitted before for any degree or examination to any other University. (Signature) (Date) I would like to thank my supervisor, mentor and friend Dr Graeme West for his guidance, dedication and his persistent e®ort. -
Approximation and Calibration of Short-Term
APPROXIMATION AND CALIBRATION OF SHORT-TERM IMPLIED VOLATILITIES UNDER JUMP-DIFFUSION STOCHASTIC VOLATILITY Alexey MEDVEDEV and Olivier SCAILLETa 1 a HEC Genève and Swiss Finance Institute, Université de Genève, 102 Bd Carl Vogt, CH - 1211 Genève 4, Su- isse. [email protected], [email protected] Revised version: January 2006 Abstract We derive a closed-form asymptotic expansion formula for option implied volatility under a two-factor jump-diffusion stochastic volatility model when time-to-maturity is small. Based on numerical experiments we describe the range of time-to-maturity and moneyness for which the approximation is accurate. We further propose a simple calibration procedure of an arbitrary parametric model to short-term near-the-money implied volatilities. An important advantage of our approximation is that it is free of the unobserved spot volatility. Therefore, the model can be calibrated on option data pooled across different calendar dates in order to extract information from the dynamics of the implied volatility smile. An example of calibration to a sample of S&P500 option prices is provided. We find that jumps are significant. The evidence also supports an affine specification for the jump intensity and Constant-Elasticity-of-Variance for the dynamics of the return volatility. Key words: Option pricing, stochastic volatility, asymptotic approximation, jump-diffusion. JEL Classification: G12. 1 An early version of this paper has circulated under the title "A simple calibration procedure of stochastic volatility models with jumps by short term asymptotics". We would like to thank the Editor and the referee for constructive criticism and comments which have led to a substantial improvement over the previous version.