Local Volatility Surface Research Articles

Option Pricing and Local Volatility Surface by Physics-Informed Neural Network

Option Pricing and Local Volatility Surface by Physics-Informed Neural Network

Formula omitted]-constrained implied transition densities

This paper presents a practical method of constructing an implied price surface and, more importantly, risk-neutral transition density functions from market option price data, which is the crucial building block of the local volatility surface. We preprocess the market data using the thin plate splines (TPS) in terms of the implied variances, which are the implied volatilities squares times the time to maturities. After prepossessing the market data, we build an implied price surface: cubic spline option price curves that satisfy no-arbitrage conditions. Furthermore, we extract the risk-neutral transition density functions with no extra cost. The extracted risk-neutral transition density functions accurately recover the market option prices.

Local volatility under rough volatility

AbstractSeveral asymptotic results for the implied volatility generated by a rough volatility model have been obtained in recent years (notably in the small‐maturity regime), providing a better understanding of the shapes of the volatility surface induced by rough volatility models, supporting their calibration power to SP500 option data. Rough volatility models also generate a local volatility surface, via the so‐called Markovian projection of the stochastic volatility. We complement the existing results on implied volatility by studying the asymptotic behavior of the local volatility surface generated by a class of rough stochastic volatility models, encompassing the rough Bergomi model. Notably, we observe that the celebrated “1/2 skew rule” linking the short‐term at‐the‐money skew of the implied volatility to the short‐term at‐the‐money skew of the local volatility, a consequence of the celebrated “harmonic mean formula” of [Berestycki et al. (2002). Quantitative Finance, 2, 61–69], is replaced by a new rule: the ratio of the at‐the‐money implied and local volatility skews tends to the constant (as opposed to the constant 1/2), where H is the regularity index of the underlying instantaneous volatility process.

The realized local volatility surface

The realized local volatility surface

VOLATILITY SMILE INTERPOLATION WITH RADIAL BASIS FUNCTIONS

The Radial Basis Functions (RBF) interpolation is a popular approximation technique used to smooth scattered data in various dimensions. This study uses RBF interpolation to interpolate the volatility skew of the S&P500 index options. The interpolated skews are used to construct the risk-neutral densities of the index and its local volatility surface. The RBF interpolation is contrasted throughout the study with the cubic spline interpolation. An analysis of the densities and the local volatility shows that RBF are an effective and practical tool for interpolating the implied volatility surface.

Reconstruction of local volatility surface from American options

Abstract In this paper, we discuss the reconstruction of a local volatility surface from American option prices. First of all, the American option prices are calculated by an accurate and fast finite difference scheme. Then the local volatility is obtained by minimizing the distance between theoretical prices and market option prices, which yields an optimization problem. The Bicubic spline regularization technique is used to overcome the ill-posedness of the reconstruction problem. We solve the nonlinear optimization problem by using a gradient-based optimization algorithm. Finally, we test our model with numerical examples and real market American put option data. The results show the good performance of our method.

Robust and accurate construction of the local volatility surface using the Black–Scholes equation

• We present an algorithm to reconstruct a local volatility (LV) function. • We reconstruct the local volatility surface using a nonlinear fitting function. • The results demonstrate the robustness and accuracy of the proposed algorithm. In this study, we develop a numerical method for the robust and accurate construction of a local volatility (LV) surface using the generalized Black–Scholes (BS) equation from the given option price data. The BS equation is a partial differential equation and has been used to model financial option pricing. Constant volatility was used in the classical BS model. However, it is well known that the constant volatility BS model is practically unsuitable because real financial market data demonstrate non-constant volatility behavior. The LV function is dependent on the asset prices and time. One of the difficulties in reconstructing an unknown LV surface is uniqueness. We extend a previous study of reconstructing time-dependent volatility, which is unique, to time- and space-dependent volatility surfaces. We propose an algorithm comprising four steps: the first step is estimating constant implied volatility; the second step is finding the influential region using the probability density function of a log-normal distribution; the third step is calculating the time-dependent volatility function; and the final step is reconstructing the LV surface. We use a finite difference method to numerically solve the BS model and a nonlinear fitting function to compute the LV surface. We perform computational experiments using synthetic and real market data. The numerical results demonstrate the robust and accurate construction of an unknown LV surface using the proposed method.

Short Communication: Beyond Surrogate Modeling: Learning the Local Volatility via Shape Constraints

We explore the abilities of two machine learning approaches for no-arbitrage interpolation of European vanilla option prices, which jointly yield the corresponding local volatility surface: a finit...

Deep Local Volatility

Deep learning for option pricing has emerged as a novel methodology for fast computations with applications in calibration and computation of Greeks. However, many of these approaches do not enforce any no-arbitrage conditions, and the subsequent local volatility surface is never considered. In this article, we develop a deep learning approach for interpolation of European vanilla option prices which jointly yields the full surface of local volatilities. We demonstrate the modification of the loss function or the feed forward network architecture to enforce (hard constraints approach) or favor (soft constraints approach) the no-arbitrage conditions and we specify the experimental design parameters that are needed for adequate performance. A novel component is the use of the Dupire formula to enforce bounds on the local volatility associated with option prices, during the network fitting. Our methodology is benchmarked numerically on real datasets of DAX vanilla options.

An Expanded Local Variance Gamma Model

The paper proposes an expanded version of the Local Variance Gamma model of Carr and Nadtochiy by adding drift to the governing underlying process. Still in this new model it is possible to derive an ordinary differential equation for the option price which plays a role of Dupire’s equation for the standard local volatility model. It is shown how calibration of multiple smiles (the whole local volatility surface) can be done in such a case. Further, assuming the local variance to be a piecewise linear function of strike and piecewise constant function of time this ODE is solved in closed form in terms of Confluent hypergeometric functions. Calibration of the model to market smiles does not require solving any optimization problem and, in contrast, can be done term-by-term by solving a system of non-linear algebraic equations for each maturity. This is much faster as compared with calibration which requires solving the Dupire PDE.

A splitting strategy for the calibration of jump-diffusion models

We present a detailed analysis and implementation of a splitting strategy to identify simultaneously the local volatility surface and the jump-size distribution from quoted European prices. The underlying model consists of a jump-diffusion driven asset with time- and price-dependent volatility. Our approach uses a forward Dupire-type partial integro-differential equation for the option prices to produce a parameter-to-solution map. The ill-posed inverse problem for this map is then solved by means of a Tikhonov-type convex regularisation. The proofs of convergence and stability of the algorithm are provided together with numerical examples that illustrate the robustness of the method both for synthetic and real data.

An Innovation for the Local Volatility Surface

An Innovation for the Local Volatility Surface

Efficient trinomial trees for local‐volatility models in pricing double‐barrier options

AbstractA local‐volatility (LV) model captures the volatility smile while retaining the preference freedom of the Black–Scholes model. Past attempts to construct a smile‐consistent tree for the LV surface do not guarantee validity. This paper presents an efficient and valid smile‐consistent tree for the LV model. The only assumption is that the LV surface be upper‐ and lower‐bounded. With this tree, double‐barrier options can be priced with fast convergence even in the presence of volatility smile. This is confirmed numerically. An implied tree is also presented. It recovers the LV surface reasonably well.

Calibration of Dupire local volatility model using genetic algorithm of optimization

The problem of calibration of local volatility model of Dupire has been formalized. It uses genetic algorithm as alternative to regularization approach with further application of gradient descent algorithm. Components that solve Dupire’s partial differential equation that represents dynamics of underlying asset’s price within Dupire model have been built. This price depends in particular on values of volatility parameters. Local volatility is parametrized in two dimensions (by Dupire model): time to maturity of the option and strike price (execution price). On maturity axis linear interpolation is used while on strike axis we use B-Splines. Genetic operators of mutation and selection are then applied to parameters of B-Splines. Resulting parameters allow us to obtain the values of local volatility both in knot points and intermediate points using interpolation techniques. Then we solve Dupire equation and calculate model values of option prices. To calculate cost function we simulate market values of option prices using classic Black-Scholes model. An experimental research to compare simulated market volatility and volatility obtained by means of calibration of Dupire model has been conducted. The goal is to estimate the precision of the approach and its usability in practice. To estimate the precision of obtained results we use a measure based on average deviation of modeled local volatility from values used to simulate market prices of the options. The research has shown that the approach to calibration using genetic algorithm of optimization requires some additional manipulations to achieve convergence. In particular it requires non-uniform discretization of the space of model parameters as well as usage of de Boor interpolation. Value 0.07 turns out to be the most efficient mutation parameter. Using this parameter leads to quicker convergence. It has been proved that the algorithm allows precise calibration of local volatility surface from option prices.

The calibration of stochastic local-volatility models: An inverse problem perspective

We tackle the calibration of the Stochastic Local-Volatility (SLV) model. This is the class of financial models that combines the local volatility and stochastic volatility features and has been subject of the attention by many researchers and practitioners recently. The corresponding inverse problem consists in finding certain (functional) coefficients in a class of parabolic partial differential equations from observed values of the solutions. More precisely, given a calibrated local volatility surface and a choice of stochastic volatility parameters, we calibrate the corresponding leverage function. Our approach makes use of regularization techniques from inverse-problem theory, respecting the integrity of the data and thus avoiding data interpolation. The result is a stable and efficient algorithm which is resilient to instabilities in the regions of low probability density of the spot price and of the instantaneous variance. We substantiate our claims with numerical experiments using synthetic and real data.

A connection between uniqueness of minimizers in Tikhonov-type regularization and Morozov-like discrepancy principles

We state sufficient conditions for the uniqueness of minimizers of Tikhonov-type functionals. We further explore a connection between such results and the well-posedness of Morozov-like discrepancy principle. Moreover, we find appropriate conditions to apply such results to the local volatility surface calibration problem.

From Model to Market Risks: The Implicit Function Theorem (IFT) Demystified

One persisting conundrum in the theory and practice of quantitative risk management models is the relationship of model risks (the risk sensitivities of a transaction or set of transactions to the parameters of the model, for example, in a Dupire (1992) model, the local volatility surface) and market risks (the sensitivities to the market variables, for example, the implied volatility surface). Model parameters are typically calibrated to market variables, sometimes analytically (see Dupire’s formula expressing a local volatility as a function of the implied volatilities) but mostly numerically, where the model parameters are iteratively set to minimize the (generally, squared) error to market instruments. In machine learning lingo, the model learns its parameters by calibration to market instruments (underlying assets and European options) and applies them to off-market instruments (exotics). The value of a transaction is an explicit (although, most of the time, numerical) function of the model parameters, so the model risks are easily computed by finite difference or AAD (automatic adjoint differentiation, see for instance Savine’s textbook, Wiley, 2018). It is however, in general, not advisable, for speed, memory footprint or accuracy, to proceed in the same manner for market risks. To differentiate through a numerical calibration is an inefficient process that may lead to unstable or wrong results. The solution, presented by a few authors (Giles, Capriotti, Naumann, Huge-Flyger-Savine), is the application of a multi-dimensional version of the implicit function theorem (IFT) to deduce market risks from model risks. This document explains, and hopefully demystifies the application of the IFT in this contexts, and outlines the steps of an efficient and accurate algorithm for the determination of the market risks, in particular in the context of AAD.

Modeling and implementation of local volatility surfaces in Bayesian framework

In this study, we focus on the reconstruction of volatility surfaces via a Bayesian framework. Apart from classical methods, such as, parametric and non-parametric models, we study the Bayesian analysis of the (stochastically) parametrized volatility structure in Dupire local volatility model. We systematically develop and implement novel mathematical tools for handling the classical methods of constructing local volatility surfaces. The most critical limitation of the classical methods is obtaining negative local variances due to the ill-posedness of the numerator and/or denominator in Dupire local variance equation. While several numerical techniques, such as Tikhonov regularization and spline interpolations have been suggested to tackle this problem, we follow a more direct and robust approach. With the Bayesian analysis, choosing a suitable prior on the positive plane eliminates the undesired negative local variances.

Local Volatility, Conditioned Diffusions, and Varadhan's Formula

We study classes of stochastic volatility models and derive asymptotic formulae for the associated local volatility surface. This gives new insight into the geometry of a reasonable local volatilit...

The calibration of volatility for option pricing models with jump diffusion processes

ABSTRACTThis paper is devoted to calibrate smooth local volatility surface under jump-diffusion processes. This calibration problem is posed as an inverse problem: given a finite set of observed European option prices, find a local volatility function such that the theoretical option prices matches the observed ones optimally with respect to a prescribed performance criterion. Firstly, we obtain an Euler-Lagrange equation for the calibration problem using Tikhonov regularization method. Then we solve the Euler–Lagrange equation using an iterative algorithm and obtain the volatility. Finally, numerical experiments show the effectiveness of the proposed method.