DOCSLIB.ORG

Computing the Implied Volatility in Stochastic Volatility Models

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Computing the Implied Volatility in Stochastic Volatility Models Computing the Implied Volatility in Stochastic Volatility Models HENRI BERESTYCKI École des Hautes Études en Sciences Sociales JÉRÔME BUSCA CNRS and Université Paris Dauphine AND IGOR FLORENT HSBC CCF 1 Introduction and Main Results The Black-Scholes model [6, 23] has gained wide recognition on financial mar- kets. One of its shortcomings, however, is that it is inconsistent with most observed option prices. Although the model can still be used very efficiently, it has been pro- posed to relax its assumptions, and, for instance, to consider that the volatility of the underlying asset S is no longer a constant but rather a stochastic process. There are two well-known approaches to achieve this goal. In the first class of models, the volatility is assumed to depend on the variables t (time) and S, giving rise to the so-called local volatility models. The second one, conceptually more ambitious, considers that the volatility has a stochastic component of its own. In the latter, the number of factors is increased by the amount of stochastic factors entering the volatility modeling. Both models are of practical interest. In these contexts, it is relevant to express the resulting prices in terms of implied volatilities. Given a price, the Black-Scholes implied volatility is determined, for each given product (that is for each given strike and expiry date defining, say, the call option) as the unique value of the volatility parameter for which the Black- Scholes pricing formula agrees with that given price. Actually, it is common prac- tice on trading floors to quote and to observe prices in this way. A great advantage of having prices expressed in such dimensionless units is to provide easy compari- son between products with different characteristics. In principle, the implied volatility can be inferred from computed options prices by inverting the Black-Scholes formula. It is more convenient, however, to directly analyze the implied volatility. Indeed, this approach allows us to shed light on qualitative properties that would otherwise be more difficult to establish. In partic- ular, we derive here several asymptotic formulae that are of practical interest, for example, in the calibration problem. The latter—an inverse problem that consists Communications on Pure and Applied Mathematics, Vol. LVII, 1352–1373 (2004) c 2004 Wiley Periodicals, Inc. STOCHASTIC VOLATILITY MODELS 1353 in determining model parameters from the observation of market instruments—is typically computationally intensive. This paper addresses precisely these questions for stochastic volatility models. In an earlier paper, we had carried out a similar program in the framework of lo- cal volatility models (see [4, 5]). There, the asymptotic behavior near expiry was given explicitly by an ordinary differential equation. The situation here is more involved. We introduce a new method to determine the implied volatility in the framework of stochastic volatility models for European call or put options. It is described in Section 1.2, where we show that the implied volatility is obtained by solving a quasi-linear parabolic partial differential equation, the initial condition of which is given by the solution of an eikonal (first-order) Hamilton-Jacobi equation. We establish a uniqueness result for this equation that is new and of independent interest. The proof takes up Section 3 and involves the auxiliary notion of “effective volatility.” Its definition and some useful results about the effective volatility are given in Section 2. This notion is due to Derman and Kani [12]. Using a result of Varadhan and a large-deviation approach, the behavior of the effective volatility near expiry has been obtained by Avellaneda et al. [2, 3] in the context of basket options, and we apply the same methodology here. This is described in Section 2. We give original proofs of all these various results related to the effective volatility introducing a PDE type approach to this topic. These take up Sections 4 and 5. Furthermore, we establish here a new characterization of the Varadhan geodesic distance that is involved in the limiting theorem near expiry as a viscosity solution to a Hamilton-Jacobi equation. Lastly, in Section 6 we show how our general approach applies to a variety of specific popular stochastic volatility models. There we derive closed-form approx- imate formulae. Finally, numerical computations illustrate the high-order accuracy that is achieved through the approximation with only one- or two-term expansions. 1.1 Stochastic Volatility Models We assume that the volatility of the underlying asset St is given as a function − = ( 1,..., n−1) of n 1 stochastic factors yt yt yt that follow diffusion processes. Specifically, we consider the following S.D.E. dSt = rdt + σ(St , yt , t)dWt (1.1) St dyt = θ(yt , t)dt + ν(yt , t)dZt ≡ 0 = ( 1,..., n−1) where Wt Zt , Zt Zt Zt , are standard Wiener processes. We define = (ρ ) i , j =ρ θ = the correlation matrix ij 0≤i, j≤n−1 by Zt Zt ijdt. In (1.1), t (θ 1,...,θn−1) ν( , ) = (ν ) t t are drift coefficients and yt t ij 1≤i, j≤n−1 is a diffusion matrix. Precise regularity and growth conditions on these coefficients will be stated below. We assume that St ∈ (0, +∞) and that yt belongs to a domain O ⊂ Rn−1 a.s. In applications the domain O will typically be the whole space or a 1354 H. BERESTYCKI, J. BUSCA, AND I. FLORENT half-space. This framework includes such popular stochastic volatility models as the Heston model and the log-normal volatility model. We assume for simplicity in the sequel that O = Rn−1 (an example with O a half-space is discussed later on in Section 6). As is classical, we will assume that the fair value of any option is the expec- tation of its discounted payoff at maturity T under the probability for which (1.1) holds, i.e., −r(T −t) (1.2) C(S, y, t; K, T ) = E e (ST − K )+ | Ft , where Ft is the natural filtration. A discussion of this property as well as the choice of the probability measure are outside the scope of this paper. Equivalently, from the Feynman-Kac relation, C can be obtained as the solution of the linear parabolic partial differential equation in n space dimensions C + LC = 0 (1.3) t C(S, yt , t = T ; K, T ) = (S − K )+ in the domain {S > 0, y ∈ Rn−1}, where the pricing operator L is defined by 1 ∂2ϕ Lϕ = σ 2(S, y, t)S2 2 ∂ S2 ∂2ϕ + σ(S, y, t)S ρ ν , (y, t) 0 j k j ∂ S∂y 1≤ j,k≤n−1 k (1.4) 1 ∂2ϕ + ρ ν (y, t)ν (y, t) 2 ij ik jl ∂y ∂y 1≤i, j,k,l≤n−1 k l ∂ϕ ∂ϕ + rS + θ − rϕ. ∂ S i ∂y 1≤i≤n−1 i The implied volatility function (S, y, t; K, T ) is uniquely defined by the re- lation (1.5) C(S, y, t; K, T ) = CBS(S, t; K, T ; (S, y, t; K, T )) , where CBS(S, t; K, T ; ) is the price of call options in the Black-Scholes model [6] for a given volatility (constant) parameter >0. Recall that it is given by BS −r(T −t) (1.6) C (S, t; K, T ; ) = SN(d1) − Ke N(d2) with ln(Ser(T −t)/K ) 1 √ √ d1 = √ + T − t , d2 = d1 − T − t . T − t 2 STOCHASTIC VOLATILITY MODELS 1355 1.2 Main Results Henceforth we use the following notation corresponding to indices being shifted by one digit. The correlation matrix is now = (ωij) with ωij = ρi−1, j−1, 1 ≤ i, j ≤ n, the diffusion matrix is given by M = (mij) with m11 = σ , m1k = mk1 = 0ifk = 1, mij = νi−1, j−1 if 2 ≤ i, j ≤ n. The drift terms are de- 1 2 = − σ ( , , ) ( ,τ)= θ − ( , ) = ,..., fined by q1 rS 2 S y t , qi x i 1 y t , i 2 n. For later pur- ˜ ˜ poses we also introduce a modified drift coefficient θ1 = 0 and θi = qi + ω1i σνi−1 for i = 2,...,n. Let us introduce the reduced variables S τ = T − t , x = ln + r(T − t), x = y − for i ≥ 2. 1 K i i 1 From now on, the “space” variables are x = (x1,...,xn). With a slight abuse, we keep the notation σ(x,τ) = σ(S, y, t) and likewise for other functions. Last, we introduce a normalized price C(S, y, t; K, T ) u(x,τ)≡ u(x,τ; K, T ) = erτ . K Throughout the paper, we shall make the following technical assumptions on the diffusion coefficients: q ∈ Cα,α/2 (H) MMT ∈ Cα,α/2 C(1 +|x|)−2|ξ|2 ≤MMT(x,τ)ξ,ξ≤C(1 +|x|)2|ξ|2 for all x,ξ ∈ Rn, τ ∈ (0, T ). Here Cα,α/2 denotes the space of functions having uniformly bounded partial Hölder differential quotients with exponent α (respec- tively, α/2) in the space (respectively, time) variables; see [15]. A straightforward computation shows that the pricing PDE in the reduced vari- ables is 1 T 2 τ = + ( ,τ)· u 2 Tr M M D u q x Du (1.7) x u(x, 0) = (e 1 − 1)+, the equation being satisfied in Rn × (0, T ). Note that the matrix M = M(x,τ) depends on the variables x and τ. Throughout the paper, D denotes the gradient vector (∂/∂x1,...,∂/∂xn), D2 the Hessian matrix , MT the transpose of the matrix M, and Tr the trace. The following classical result asserts the solvability of the pricing equation.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • CORRECTION to BLACK-SCHOLES FORMULA DUE to FRACTIONAL STOCHASTIC VOLATILITY 1. Introduction. Our Aim in This Paper Is to Provide
    CORRECTION TO BLACK-SCHOLES FORMULA DUE TO FRACTIONAL STOCHASTIC VOLATILITY JOSSELIN GARNIER∗AND KNUT SØLNAy Abstract. Empirical studies show that the volatility may exhibit correlations that decay as a fractional power of the time offset. The paper presents a rigorous analysis for the case when the stationary stochastic volatility model is constructed in terms of a fractional Ornstein Uhlenbeck process to have such correlations. It is shown how the associated implied volatility has a term structure that is a function of maturity to a fractional power. Key words. Stochastic volatility, implied volatility, fractional Brownian motion, long-range dependence. AMS subject classifications. 91G80, 60H10, 60G22, 60K37. 1. Introduction. Our aim in this paper is to provide a framework for analysis of stochastic volatility problems in the context when the volatility process possesses long-range correlations. Replacing the constant volatility of the Black-Scholes model with a random process gives price modifications in financial contracts. It is impor- tant to understand the qualitative behavior of such price modifications for a (class of) stochastic volatility models since this can be used for calibration purposes. Typ- ically the price modifications are parameterized by the implied volatility relative to the Black-Scholes model [25, 41]. For illustration we consider here European option pricing and then the implied volatility depends on the moneyness, the ratio between the strike price and the current price, moreover, the time to maturity. The term and moneyness structure of the implied volatility can be calibrated with respect to liquid contracts and then used for pricing of related but less liquid contracts.
    [Show full text]
  • Complete Models with Stochastic Volatility
    Complete Mo dels with Sto chastic Volatility 1 David G. Hobson Scho ol of Mathematical Sciences, University of Bath and 2 L.C.G. Rogers Scho ol of Mathematical Sciences, UniversityofBath Abstract The pap er prop oses an original class of mo dels for the continuous time price pro cess of a nancial security with non-constantvolatility. The idea is to de ne instantaneous volatility in terms of exp onentially-weighted moments of historic log-price. The instanta- neous volatility is therefore driven by the same sto chastic factors as the price pro cess, so that unlike many other mo dels of non-constantvolatility, it is not necessary to intro duce additional sources of randomness. Thus the market is complete and there are unique, preference-indep endent options prices. We nd a partial di erential equation for the price of a Europ ean Call Option. Smiles and skews are found in the resulting plots of implied volatility. Keywords: Option pricing, sto chastic volatility, complete markets, smiles. Acknow ledgemen t. It is a pleasure to thank the referees of an earlier draft of this pap er whose p erceptive comments have resulted in manyimprovements. 1 Research supp orted in part byRecordTreasury Management 2 Research supp orted in part by SERCgrant GR/K00448 1 Sto chastic Volatility The work on option pricing of BlackandScholes (1973) represents one of the most striking developments in nancial economics. In practice b oth the pricing and hedging of derivative securities is to daygoverned by Black-Scholes, to the extent that prices are often quoted in terms of the volatility parameters implied by the mo del.
    [Show full text]
  • Local Volatility, Stochastic Volatility and Jump-Diffusion Models
    IEOR E4707: Financial Engineering: Continuous-Time Models Fall 2013 ⃝c 2013 by Martin Haugh Local Volatility, Stochastic Volatility and Jump-Diffusion Models These notes provide a brief introduction to local and stochastic volatility models as well as jump-diffusion models. These models extend the geometric Brownian motion model and are often used in practice to price exotic derivative securities. It is worth emphasizing that the prices of exotics and other non-liquid securities are generally not available in the market-place and so models are needed in order to both price them and calculate their Greeks. This is in contrast to vanilla options where prices are available and easily seen in the market. For these more liquid options, we only need a model, i.e. Black-Scholes, and the volatility surface to calculate the Greeks and perform other risk-management tasks. In addition to describing some of these models, we will also provide an introduction to a commonly used fourier transform method for pricing vanilla options when analytic solutions are not available. This transform method is quite general and can also be used in any model where the characteristic function of the log-stock price is available. Transform methods now play a key role in the numerical pricing of derivative securities. 1 Local Volatility Models The GBM model for stock prices states that dSt = µSt dt + σSt dWt where µ and σ are constants. Moreover, when pricing derivative securities with the cash account as numeraire, we know that µ = r − q where r is the risk-free interest rate and q is the dividend yield.
    [Show full text]
  • Derivatives Pricing Using Quantlib: an Introduction
    INDIAN INSTITUTE OF MANAGEMENT AHMEDABAD • INDIA Research and Publications Derivatives Pricing using QuantLib: An Introduction Jayanth R. Varma Vineet Virmani W.P. No. 2015-03-16 April 2015 The main objective of the Working Paper series of IIMA is to help faculty members, research staff, and doctoral students to speedily share their research findings with professional colleagues and to test out their research findings at the pre-publication stage. INDIAN INSTITUTE OF MANAGEMENT AHMEDABAD – 380015 INDIA W.P. No. 2015-03-16 Page No. 1 IIMA • INDIA Research and Publications Derivatives Pricing using QuantLib: An Introduction Jayanth R. Varma Vineet Virmani Abstract Given the complexity of over-the-counter derivatives and structured products, al- most all of derivatives pricing today is based on numerical methods. While large fi- nancial institutions typically have their own team of developers who maintain state- of-the-art financial libraries, till a few years ago none of that sophistication was avail- able for use in teaching and research. For the last decade„ there is now a reliable C++ open-source library available called QuantLib. This note introduces QuantLib for pricing derivatives and documents our experience using QuantLib in our course on Computational Finance at the Indian Institute of Management Ahmedabad. The fact that it is also available (and extendable) in Python has allowed us to harness the power of C++ with the ease of iPython notebooks in the classroom as well as for stu- dent’s projects. Keywords: Derivatives pricing, Financial engineering, Open-source computing, Python, QuantLib 1 Introduction Financial engineering and algorithmic trading are perhaps two of the most computation- ally intensive parts of all of finance.
    [Show full text]
  • FX Derivatives: Stochastic-Local-Volatility Model
    Review Vanna Volga Stochastic-Local-Volatility Mixture Local Volatility Model Summary FX Derivatives: Stochastic-Local-Volatility Model Uwe Wystup Fr´ed´ericBossens, Andreas Weber MathFinance AG uwe.wystup@mathfinance.com July 2020 FX Derivatives: Stochastic-Local-Volatility Model c by MathFinance AG uwe.wystup@mathfinance.com 1 / 37 Review Vanna Volga Stochastic-Local-Volatility Mixture Local Volatility Model Summary Agenda 1 Review Vanna Volga Early Days and Drawbacks Versions of Vanna-Volga Design and Consistency Issues 2 Stochastic-Local-Volatility LV and SV vanilla smile fit SLV Step by Step SLV Pricing / Validation 3 Mixture Local Volatility Model MLV Main Features Vol Process Comparison MLV vs. SLV Granular Model Marking 4 Summary Product/Model Matrix Key Take-Aways FX Derivatives: Stochastic-Local-Volatility Model c by MathFinance AG uwe.wystup@mathfinance.com 2 / 37 Review Vanna Volga Stochastic-Local-Volatility Mixture Local Volatility Model Summary Agenda 1 Review Vanna Volga Early Days and Drawbacks Versions of Vanna-Volga Design and Consistency Issues 2 Stochastic-Local-Volatility LV and SV vanilla smile fit SLV Step by Step SLV Pricing / Validation 3 Mixture Local Volatility Model MLV Main Features Vol Process Comparison MLV vs. SLV Granular Model Marking 4 Summary Product/Model Matrix Key Take-Aways FX Derivatives: Stochastic-Local-Volatility Model c by MathFinance AG uwe.wystup@mathfinance.com 3 / 37 2 For a long time it matched market prices for barrier options and touch contracts quite good. 3 Market fit at least better than the alternatively available Local Volatility (LV) or Stochastic Volatility (SV) models. Review Vanna Volga Early Days and Drawbacks Stochastic-Local-Volatility Versions of Vanna-Volga Mixture Local Volatility Model Design and Consistency Issues Summary Review Vanna Volga: Early Days 1 Pricing in Vanna Volga (VV) model is popular in FX as it still allows to compute prices nearly as fast as in the BS model, using analytical formulas.
    [Show full text]
  • Model Validation of Xva Using ORE ORE User Meeting
    Model Validation of xVA using ORE ORE User Meeting Patrick Büchel, Commerzbank AG; Jörg Kienitz, Quaternion Risk Management | 23/11/2018 Overview 1 Hybrid multi-asset IR/FX model 2 Prototype of a Cheyette Model 3 Appendix ORE for Validation 2/35 Disclaimer Patrick Büchel is a Managing Director at Commerzbank in Frankfurt. The views expressed in this presentation are the personal views of the speaker and do not necessarily reflect those of Commerzbank. Important legal information and additional information are set out in the Legal Notices contained in this presentation. Chatham House Rules apply The Open Source Risk Engine’s objective is to provide a free/open source platform for risk analytics and xVA. It is based on QuantLib and grew from work developed by market professionals and academics. The aim is to build on QuantLib’s simulation models, instruments, and engines, and work by professionals and academics. It brings large-scale open-source risk models to the public domain to facilitate research and understanding through simple interfaces for trade/market data and system configurations, and simple launchers, see opensourcerisk.org. ORE for Validation 3/35 1 Hybrid multi-asset IR/FX model Overview 1 Hybrid multi-asset IR/FX model Validation of Risk Factor Models and Calibration Exposure Benchmarking using ORE Regression Test based on Hull-White Model Collateralization 2 Prototype of a Cheyette Model 3 Appendix Markovian Representation European swaption pricing in TS ORE for Validation 4/35 1 Hybrid multi-asset IR/FX model Validation Tasks using ORE Exposure simulation and xVA calculation Method based on Least Squares Monte-Carlo (LSM) simulation IR and FX models and products were considered ORE was applied for: Modeling assumptions of the risk factors (RF) Calibration methodology of RF models Analysis / Benchmarking of LSM - in particularly the regression error Benchmarking exposure profiles for collateralized / un-collateralized netting sets The validation scope: IR/FX RF models and products.
    [Show full text]
  • Non-Affine Stochastic Volatility Jump Diffusion Models
    Non-affine stochastic volatility jump diffusion models Griselda Deelstra∗ Ahmed Ezziney Jacques Janssenz Universit´eLibre de Bruxelles Campus Plaine CP 210 B - 1050 Brussels Belgium Abstract This paper proposes an alternative option pricing model in which the stock prices follow a diffusion process with non-affine stochastic volatility and random jumps. Approximative European option pricing formulae are derived by transforming a non-linear PDE in an approximate linear PDE which is explicitly solved by using Fourier transformations. We check that these approximative prices are close to the Monte Carlo estimates and compare them with the prices in an affine stochastic volatility jump diffusion model. Model parameters are estimated by using the method of simulated moments. We evaluate the impact of the different submodels on option prices and on implied volatility. Keywords: jump-diffusion, non-affine stochastic volatility, method of simulated moments, implied volatility curves. ∗D´ept.de Math, ISRO and ECARES. [email protected] yD´ept.de Math and ISRO. [email protected] zC.E.S.I.A.F. [email protected] 1 1 Introduction In the last decade, there has been evidence that stochastic volatility processes with jumps in the returns are important to model index return and volatility, but some recent studies on option pricing prove that most models still are incapable to capture some empirical characteristics observed in data. In particular the mean squared errors by the parameter estimation remain high and the derived implied volatility graphs are not satisfactory. Some authors use the square root process to represent the dynamics of the instantaneous variance (e.g.
    [Show full text]