Content uploaded by Rene A. Carmona
Author content
All content in this area was uploaded by Rene A. Carmona on Apr 10, 2015
Content may be subject to copyright.
A preview of the PDF is not available
Simulation of Implied Volatility Surfaces via Tangent Levy Models
Abstract and Figures
In this paper, we implement and test two types of market-based models for
European-type options, based on the tangent Levy models proposed recently by R.
Carmona and S. Nadtochiy. As a result, we obtain a method for generating Monte
Carlo samples of future paths of implied volatility surfaces. These paths and
the surfaces themselves are free of arbitrage, and are constructed in a way
that is consistent with the past and present values of implied volatility. We
use a real market data to estimate the parameters of these models and conduct
an empirical study, to compare the performance of market-based models with the
performance of classical stochastic volatility models. We choose the problem of
minimal-variance portfolio choice as a measure of model performance and compare
the two tangent Levy models to SABR model. Our study demonstrates that the
tangent Levy models do a much better job at finding a portfolio with smallest
variance, their predictions for the variance are more reliable, and the
portfolio weights are more stable. To the best of our knowledge, this is the
first example of empirical analysis that provides a convincing evidence of the
superior performance of the market-based models for European options using real
market data.
Figures - uploaded by Rene A. Carmona
Author content
All figure content in this area was uploaded by Rene A. Carmona
Content may be subject to copyright.
... Market models of implied volatility (Babbar 2001;Carmona, Ma, and Nadtochiy 2017;Cohen, Reisinger, and Wang 2023;Cont and da Fonseca 2002;Cont, Fonseca, and Durrleman 2002;Gatheral and Jacquier 2014;Martini and Mingone 2022;Schönbucher 1999;Schweizer and Wissel 2008) attempt to directly model the cross-section and dynamics of implied volatilities. One of the challenges in modeling implied volatility surfaces is to ensure that the absence of static arbitrage is satisfied. ...
... Indeed, the profile of the implied volatility surface cannot be arbitrary: static arbitrage constraints on the values of call and put options (Davis and Hobson 2007) put restrictions on the possible shape of the implied volatility surface. Analytical modeling has focused on obtaining parameterisations of implied volatility surfaces which guarantee that such arbitrage constraints are satisfied (Carmona, Ma, and Nadtochiy 2017;Cohen, Reisinger, and Wang 2023;Schweizer and Wissel 2008). Such models, however, are computationally challenging to implement, and even more challenging to calibrate to obtain realistic surface dynamics. ...
... These models are tractable and have been adopted for risk management applications, such as margin computations, but may lead to scenarios which are not compatible with arbitrage constraints. In parallel, analytical models have been developed with the goal of satisfying static (Gatheral and Jacquier 2014;Martini and Mingone 2022;Zhang, Li, and Zhang 2023), and dynamic arbitrage constraints (Carmona, Ma, and Nadtochiy 2017;Cohen, Reisinger, and Wang 2023;Schweizer and Wissel 2008;Wissel 2008). These models are computationally challenging to implement, simulate or estimate. ...
... Market models of implied volatility [3,5,7,9,10,15,23,22] attempt to directly model the cross-section and dynamics of implied volatilities. One of the challenges in modelling implied volatility surfaces is satisfying arbitrage conditions. ...
... Indeed, the profile of the implied volatility surface cannot be ar-bitrary: static arbitrage constraints on the values of call and put options [11] put restrictions on the possible shape of the implied volatility surface. Analytical modeling has focused on obtaining parameterisations of implied volatility surfaces which guarantee that such arbitrage constraints are satisfied [5,23,7]. Such models, however, are computationally challenging to implement, and even more challenging to calibrate to obtain realistic surface dynamics. ...
... These models are tractable and have been adopted for risk management applications -such as margin computations-but may lead to scenarios which are not compatible with arbitrage constraints. In parallel, analytical models have been developed with the goal of satisfying static [15] and dynamic arbitrage constraints [23,26,5,7]. These models are computationally challenging to implement, simulate or estimate. ...
We present a computationally tractable method for simulating arbitrage free implied volatility surfaces. We illustrate how our method may be combined with a factor model for the implied volatility surface to generate dynamic scenarios for arbitrage-free implied volatility surfaces. Our approach conciliates static arbitrage constraints with a realistic representation of statistical properties of implied volatility co-movements. We then introduce VolGAN, a nonparametric generative model for implied volatility surfaces.
... 2. Parameters (θ t0 , σ t0 , ρ t0 ) will move to another configuration (θ t1 , σ t1 , ρ t1 ), but, to enforce a smooth change between the first configuration and second, 3. Also the Lévy process will be adjusted and will compensate the changes in the parameters (θ t , σ t , ρ t ) to reproduce the same IVS. ...
... If this is the case, then we can model for indefinite time the evolution of an implied volatility surface without breaking any arbitrage constraints, neither static nor dynamic ones. To our knowledge, it is the first time that this achieved in an efficient way, one impressive other implementation has been presented in [3]. The algorithm used to accomplish that is outlined in Algorithm 1. ...
Consistent Recalibration models (CRC) have been introduced to capture in necessary generality the dynamic features of term structures of derivatives' prices. Several approaches have been suggested to tackle this problem, but all of them, including CRC models, suffered from numerical intractabilities mainly due to the presence of complicated drift terms or consistency conditions. We overcome this problem by machine learning techniques, which allow to store the crucial drift term's information in neural network type functions. This yields first time dynamic term structure models which can be efficiently simulated.
... This has been previously done in works on the so-called market models, see e.g. [34], [14], [16], [15], [13]. While these works focused on a fixed probabilistic setting, herein, we pursue the robust approach. ...
In this paper, we present a method for constructing a (static) portfolio of co-maturing European options whose price sign is determined by the skewness level of the associated implied volatility. This property holds regardless of the validity of a specific model – i.e. the method is robust. The strategy is given explicitly and depends only on one's beliefs about the future values of implied skewness, which is an observable market indicator. As such, our method allows to use the existing statistical tools to formulate the beliefs, providing a practical interpretation of the more abstract mathematical setting, in which the belies are understood as a family of probability measures. One of the applications of the results established herein is a method for trading one's views on the future changes in implied skew, largely independently of other market factors. Another application of our results provides a concrete improvement of the model-independent super-and sub-replication strategies for barrier options proposed in [10], which exploits the given beliefs on the implied skew. Our theoretical results are tested empirically, using the historical prices of S&P 500 options.
This paper introduces a new approach for generating sequences of implied volatility (IV) surfaces across multiple assets that is faithful to historical prices. We do so using a combination of functional data analysis and neural stochastic differential equations (SDEs) combined with a probability integral transform penalty to reduce model misspecification. We demonstrate that learning the joint dynamics of IV surfaces and prices produces market scenarios that are consistent with historical features and lie within the sub-manifold of surfaces that are free of static arbitrage.
We develop theory and applications of forward characteristic processes in
discrete time following a seminal paper of Jan Kallsen and Paul Kr\"uhner.
Particular emphasis is placed on the dynamics of volatility surfaces which can
be easily formulated and implemented from the chosen discrete point of view. In
mathematical terms we provide an algorithmic answer to the following question:
describe a rich, still tractable class of discrete time stochastic processes,
whose marginal distributions are given at initial time and which are free of
arbitrage. In terms of mathematical finance we can construct models with
pre-described (implied) volatility surface and quite general volatility surface
dynamics. In terms of the works of Rene Carmona and Sergey Nadtochiy, we
analyze the dynamics of tangent affine models. We believe that the discrete
approach due to its technical simplicity will be important in term structure
modelling.
In this paper we present an arbitrage pricing framework for valuing and hedging contingent equity index claims in the presence of a stochastic term and strike structure of volatility. Our approach to stochastic volatility is similar to the Heath-Jarrow-Morton (HJM) approach to stochastic interest rates. Starting from an initial set of index options prices and their associated local volatility surface, we show how to construct a family of continuous time stochastic processes which define the arbitrage-free evolution of this local volatility surface through time. The no-arbitrage conditions are similar to, but more involved than, the HJM conditions for arbitrage-free stochastic movements of the interest rate curve. They guarantee that even under a general stochastic volatility evolution the initial options prices, or their equivalent Black–Scholes implied volatilities, remain fair.
We introduce stochastic implied trees as discrete implementations of our family of continuous time models. The nodes of a stochastic implied tree remain fixed as time passes. During each discrete time step the index moves randomly from its initial node to some node at the next time level, while the local transition probabilities between the nodes also vary. The change in transition probabilities corresponds to a general (multifactor) stochastic variation of the local volatility surface. Starting from any node, the future movements of the index and the local volatilities must be restricted so that the transition probabilities to all future nodes are simultaneously martingales. This guarantees that initial options prices remain fair. On the tree, these martingale conditions are effected through appropriate choices of the drift parameters for the transition probabilities at every future node, in such a way that the subsequent evolution of the index and of the local volatility surface do not lead to riskless arbitrage opportunities among different option and forward contracts or their underlying index.
You can use stochastic implied trees to value complex index options, or other derivative securities with payoffs that depend on index volatility, even when the volatility surface is both skewed and stochastic. The resulting security prices are consistent with the current market prices of all standard index options and forwards, and with the absence of future arbitrage opportunities in the framework. The calculated options values are independent of investor preferences and the market price of index or volatility risk. Stochastic implied trees can also be used to calculate hedge ratios for any contingent index security in terms of its underlying index and all standard options defined on that index.
Brownian motion and normal distribution have been widely used in the Black-Scholes option-pricing framework to model the return of assets. However, two puzzles emerge from many empirical investigations: the leptokurtic feature that the return distribution of assets may have a higher peak and two (asymmetric) heavier tails than those of the normal distribution, and an empirical phenomenon called "volatility smile" in option markets. To incorporate both of them and to strike a balance between reality and tractability, this paper proposes, for the purpose of option pricing, a double exponential jump-diffusion model. In particular, the model is simple enough to produce analytical solutions for a variety of option-pricing problems, including call and put options, interest rate derivatives, and path-dependent options. Equilibrium analysis and a psychological interpretation of the model are also presented.
This paper presents a unifying theory for valuing contingent claims under a stochastic term strllcture of interest rates. The methodology, based on the equivalent martingale measure technique, takes as given an initial forward rate curve and a family of potential stochastic processe for its subsequent movements. A no arbitrage condition restricts this family of processes yielding valuation formulae for interest rate sensitive contingent claims which do not explicitly depend on the market prices of risk. Examples are provided to illustrate the key results. © 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
In the first chapter,which is a joint work with Mathieu Cambou and Philippe H.A. Charmoy, we study the distribution of the hedging errors of a European call option for the delta and variance-minimizing strategies. Considering the setting proposed by Heston (1993), we assess the error distribution by computing its moments under the real-world probability measure. It turns out that one is better off implementing either a delta hedging or a variance-minimizing strategy, depending on the strike and maturity of the option under consideration. In the second paper, which is a joint work with Damir Filipovic and Loriano Mancini, we develop a practicable continuous-time dynamic arbitrage-free model for the pricing of European contingent claims. Using the framework introduced by Carmona and Nadtochiy (2011, 2012), the stock price is modeled as a semi-martingale process and, at each time t , the marginal distribution of the European option prices is coded by an auxiliary process that starts at t and follows an exponential additive process. The jump intensity that characterizes these auxiliary processes is then set in motion by means of stochastic dynamics of Itô's type. The model is a modification of the one proposed by Carmona and Nadtochiy, as only finitely many jump sizes are assumed. This crucial assumption implies that the jump intensities are taken values in only a finitedimensional space. In this setup, explicit necessary and sufficient consistency conditions that guarantee the absence of arbitrage are provided. A practicable dynamic model verifying them is proposed and estimated, using options on the S&P 500. Finally, the hedging of variance swap contracts is considered. It is shown that under certain conditions, a variance-minimizing hedging portfolio gives lower hedging errors on average, compared to a model-free hedging strategy. In the third and last chapter, which is a joint work with Rémy Praz, we concentrate on the commodity markets and try to understand the impact of financiers on the hedging decisions. We look at the changes in the spot price, variance, production and hedging choices of both producers and financiers, when the mass of financiers in the economy increases. We develop an equilibrium model of commodity spot and futures markets in which commodity production, consumption, and speculation are endogenously determined. Financiers facilitate hedging by the commodity suppliers. The entry of new financiers thus increases the supply of the commodity and decreases the expected spot prices, to the benefits of the end-users. However, this entry may be detrimental to the producers, as they do not internalize the price reduction due to greater aggregate supply. In the presence of asymmetric information, speculation on the futures market serves as a learning device. The futures price and open interest reveal different pieces of private information regarding the supply and demand side of the spot market, respectively. When the accuracy of private information is low, the entry of new financiers makes both production and spot prices more volatile. The entry of new financiers typically increases the correlation between financial and commodity markets.
In this paper a stochastic volatility model is presented that directly prescribes the stochastic development of the implied Black-Scholes volatilities of a set of given standard options. Thus the model is able to capture the stochastic movements of a full term structure of implied volatilities. Conditions are derived that have to be satisfied to ensure absence of arbitrage in the model and its numerical implementation is discussed.
In some options markets (e.g. commodities), options are listed with only a
single maturity for each underlying. In others, (e.g. equities, currencies),
options are listed with multiple maturities. In this paper, we provide an
algorithm for calibrating a pure jump Markov martingale model to match the
market prices of European options of multiple strikes and maturities. This
algorithm only requires solutions of several one-dimensional root-search
problems, as well as application of elementary functions. We show how to
construct a time-homogeneous process which meets a single smile, and a
piecewise time-homogeneous process which can meet multiple smiles.
Designed for those individuals interested in the current state of development in the field of investment science, this book emphasizes the fundamental principles and how they can be mastered and transformed into solutions of important and interesting investment problems. The book examines what the essential ideas are behind investment science, how they are represented, and how they can be used in actual investment practice. The book also examines where the field might be headed in the future, and goes much further in terms of mathematical content, featuring varying levels of mathematical sophistication throughout. End-of-chapter exercises are also included to help individuals get a better grasp on investment science.