Local Volatility Surface Research Articles

Estimation and prediction under local volatility jump–diffusion model

Volatility is an important factor in operating a company and managing risk. In the portfolio optimization and risk hedging using the option, the value of the option is evaluated using the volatility model. Various attempts have been made to predict option value. Recent studies have shown that stochastic volatility models and jump–diffusion models reflect stock price movements accurately. However, these models have practical limitations. Combining them with the local volatility model, which is widely used among practitioners, may lead to better performance. In this study, we propose a more effective and efficient method of estimating option prices by combining the local volatility model with the jump–diffusion model and apply it using both artificial and actual market data to evaluate its performance. The calibration process for estimating the jump parameters and local volatility surfaces is divided into three stages. We apply the local volatility model, stochastic volatility model, and local volatility jump–diffusion model estimated by the proposed method to KOSPI 200 index option pricing. The proposed method displays good estimation and prediction performance.

CONVEX REGULARIZATION OF LOCAL VOLATILITY ESTIMATION

We apply convex regularization techniques to the problem of calibrating Dupire’s local volatility surface model taking into account the practical requirement of discrete grids and noisy data. Such requirements are the consequence of bid and ask spreads, quantization of the quoted prices and lack of liquidity of option prices for strikes far away from the at-the-money level. We obtain convergence rates and results comparable to those obtained in the idealized continuous setting. Our results allow us to take into account separately the uncertainties due to the price noise and those due to discretization errors, thus, allowing estimating better discretization levels both in the domain and in the image of the parameter to solution operator by a Morozov’s discrepancy principle. We illustrate the results with simulated as well as real market data. We also validate the results by comparing the implied volatility prices of market data with the computed prices of the calibrated model.

Data driven recovery of local volatility surfaces

This paper examines issues of data completion and location uncertainty, popular in many practical PDE-based inverse problems, in the context of option calibration via recovery of local volatility surfaces. While real data is usually more accessible for this application than for many others, the data is often given only at a restricted set of locations. We show that attempts to 'complete missing data' by approximation or interpolation, proposed and applied in the literature, may produce results that are inferior to treating the data as scarce. Furthermore, model uncertainties may arise which translate to uncertainty in data locations, and we show how a model-based adjustment of the asset price may prove advantageous in such situations. We further compare a carefully calibrated Tikhonov-type regularization approach against a similarly adapted EnKF method, in an attempt to fine-tune the data assimilation process. The EnKF method offers reassurance as a different method for assessing the solution in a problem where information about the true solution is difficult to come by. However, additional advantage in the latter approach turns out to be limited in our context.

Local Volatility with Multi-Factor Markovian Short Rate

The aim of this article is to provide an efficient procedure of calibration of the local volatility surface in presence of a d-dimensional stochastic rate process. The classical approach is to evaluate the correction to be applied on the local volatility obtained in a deterministic rates framework by solving the (d 1)-dimensional forward Kolomogorov PDE and integrate the Green function against a chosen function at each step time of the solver, which is a lot of effort for such small correction.

Bi-cubic B-spline fitting-based local volatility model with mean reversion process

This paper studies the traditional local volatility model and proposes: A novel local volatility model with mean-reversion process. The larger is the gap between local volatility and its mean level, the higher will be the rate at which local volatility will revert to the mean. Then, a B-spline method with proper knot control is applied to interpolate the local volatility matrix. The bi-cubic B-spline is used to recover the local volatility surface from this local volatility matrix. Finally, empirical tests show that the proposed mean-reversion local volatility model offers better prediction performance than the traditional local volatility model.

Smooth Calibration in Local Volatility with Jumps: The Stock Jumps, Local Volatility Doesn't

The aim of this article is to provide tools to calibrate a smooth local volatility surface in the presence of jumps. First we provide techniques to approximate the value of European options in a local volatility model with jumps, then we propose a quick and robust fixed point algorithm combined with this method to build smooth local volatility surfaces.

A GARCH Parameterization of the Volatility Surface

For more than 40 years, option traders have used the Black–Scholes (BS) formula to extract implied volatilities (IVs) from market option prices, allowing a different IV for each option. The result is the IV surface in strike price and maturity space. In BS-family models, IV is the market’s risk-neutral expectation of average volatility from the present through option expiration. In the 1990s, Dupire, Derman, and Kani developed “local volatility” models that produce the observed option prices from an implied set of time- and state-dependent future volatilities. However, the specific s(St, t) values for future volatilities that arise in the model do not come from a well-developed dynamic volatility process. Making use of several nice mathematical properties, Mazzoni shows, for each option, how to go from a GARCH model for variance along a stock-price path constrained to end up just at the money, through the corresponding local volatility surface, which is used to compute implied volatility, to get the option value. The system is calibrated against the full set of option prices observed in the market. Models that try to parameterize the entire volatility surface are rare, but in an empirical investigation with DAX index options, the GARCH-based procedure performs considerably better than the SABR and SIV model alternatives.

Implied and Local Volatility Surfaces for South African Index and Foreign Exchange Options

Certain exotic options cannot be valued using closed-form solutions or even by numerical methods assuming constant volatility. Many exotics are priced in a local volatility framework. Pricing under local volatility has become a field of extensive research in finance, and various models are proposed in order to overcome the shortcomings of the Black-Scholes model that assumes a constant volatility. The Johannesburg Stock Exchange (JSE) lists exotic options on its Can-Do platform. Most exotic options listed on the JSE’s derivative exchanges are valued by local volatility models. These models needs a local volatility surface. Dupire derived a mapping from implied volatilities to local volatilities. The JSE uses this mapping in generating the relevant local volatility surfaces and further uses Monte Carlo and Finite Difference methods when pricing exotic options. In this document we discuss various practical issues that influence the successful construction of implied and local volatility surfaces such that pricing engines can be implemented successfully. We focus on arbitrage-free conditions and the choice of calibrating functionals. We illustrate our methodologies by studying the implied and local volatility surfaces of South African equity index and foreign exchange options.

On the Uniqueness of Tikhonov Regularized Local Volatility Surfaces

We state sufficient conditions for the uniqueness of minimizers of the Tikhonov functional corresponding to the local volatility surface calibration problem. We use two different approaches. In the first one, using smoothness properties of the forward operator, we prove that, if the regularization parameter is sufficiently large, then, uniqueness hold. Secondly, we find sufficient conditions for the local uniqueness of Tikhonov minimizers by analyzing the geometrical properties of the image through the forward operator of line segments. We also study some theoretical consequences, such as, convergence-rate results with respect to the noise level, as well as, the well-posedness of an a posteriori rule for choosing the regularization parameter. In addition, we provide some numerical examples.

A penalty-based method from reconstructing smooth local volatility surface from American options

This paper is devoted to develop a robust penalty-based method of reconstructingsmooth local volatility surface from the observed American optionprices. This reconstruction problem is posed as an inverse problem:given a finite set of observed American option prices, find a localvolatility function such that the theoretical option prices matchesthe observed ones optimally with respect to a prescribed performancecriterion. The theoretical American option prices are governed bya set of partial differential complementarity problems (PDCP). Wepropose a penalty-based numerical method for the solution of the PDCP.Typically, the reconstruction problem is ill-posed and a bicubic splineregularization technique is thus proposed to overcome this difficulty.We apply a gradient-based optimization algorithm to solve this nonlinearoptimization problem, where the Jacobian of the cost function is computedvia finite difference approximation. Two numerical experiments: asynthetic American put option example and a real market American putoption example, are performed to show the robustness and effectivenessof the proposed method to reconstructing the unknown volatility surface.

Local Volatility Modeling of JSE Exotic Can-Do Options

Can-Do Options are derivative products listed on the JSE’s derivative exchanges | mostly equity derivative products listed on Safex and currency derivative products listed on YieldX. These products give investors the advantages of listed derivatives with the exibility of \over the counter (OTC) contracts. Investors can negotiate the terms for all option contracts, choosing the type of option, underlying asset and the expiry date. Many exotic options and even exotic option structures are listed. Exotic options cannot be valued using closed-form solutions or even by numerical methods assuming constant volatility. Most exotic options on Safex and Yield-X are valued by local volatility models. Pricing under local volatility has become a eld of extensive research in nance and various models are proposed in order to overcome the shortcomings of the Black-Scholes model that assumes the volatility to be constant. In this document we discuss various topics that inuence the successful construction of implied and local volatility surfaces in practice. We focus on arbitrage-free conditions, choice of calibrating functionals and selection of numerical algorithms to price options. We illustrate our methodologies by studying the local volatility surfaces of South African index and foreign exchange options. Numerical experiments are conducted using Excel and MATLAB.

Online local volatility calibration by convex regularization

We address the inverse problem of local volatility surface calibration from market given option prices. We integrate the ever-increasing flow of option price information into the well-accepted local volatility model of Dupire. This leads to considering both the local volatility surfaces and their corresponding prices as indexed by the observed underlying stock price as time goes by in appropriate function spaces. The resulting parameter to data map is defined in appropriate Bochner-Sobolev spaces. Under this framework, we prove key regularity properties. This enable us to build a calibration technique that combines online methods with convex Tikhonov regularization tools. Such procedure is used to solve the inverse problem of local volatility identification. As a result, we prove convergence rates with respect to noise and a corresponding discrepancy-based choice for the regularization parameter. We conclude by illustrating the theoretical results by means of numerical tests.

Decoding the American Vanilla Prices

We calculate the American local volatility. New insights on the American vanilla, call and put options in this paper are presented. In particular, the sensitivities of these products to the local volatility surface are illustrated. The Automatic Algorithmic Differentiation (AAD) pseudo code is presented for the American options in line with the paper [CLP] and [AP]. We propose a generalisation formula for the American vanilla option in the presence of smile. A gradient descent algorithm is presented and applied to the calibration task. A deep analysis of the difference between market practitionners and the true solution is presented and exploited to generate good initial guess for the calibration algorithm.

Non-parametric calibration of the local volatility surface for European options using a second-order Tikhonov regularization

We calibrate the local volatility surface for European options across all strikes and maturities of the same underlying. There is no interpolation or extrapolation of either the option prices or the volatility surface. We do not make any assumption regarding the shape of the volatility surface except to assume that it is smooth. Due to the smoothness assumption, we apply a second-order Tikhonov regularization. We choose the Tikhonov regularization parameter as one of the singular values of the Jacobian matrix of the Dupire model. Finally we perform extensive numerical tests to assess and verify the aforementioned techniques for both volatility models with known analytical solutions of European option prices and real market option data.

Non Parametric Calibration of the Local Volatility Surface for European Options Using a Second Order Tikhonov Regularization

We calibrate the local volatility surface for European options across all strikes and maturities of the same underlying. There is no interpolation or extrapolation of either the option prices or the volatility surface. We do not make any assumption regarding the shape of the volatility surface except to assume that it is smooth. Due to the smoothness assumption, we apply a second order Tikhonov regularization. We choose the Tikhonov regularization parameter as one of the singular values of the Jacobian matrix of the Dupire model. Finally we perform extensive numerical tests to assess and verify the aforementioned techniques for both volatility models with known analytical solutions of European option prices and real market option data.

The tangential cone condition for the iterative calibration of local volatility surfaces

The tangential cone condition for the iterative calibration of local volatility surfaces

Convex Regularization of Local Volatility Estimation in a Discrete Setting

We apply convex regularization techniques to the problem of calibrating Dupire's local volatility surface model taking into account the practical requirement of discrete grids and noisy data. Such requirements are the consequence of bid and ask spreads, quantization of the quoted prices and lack of liquidity of option prices for strikes far way from the at-the-money level. We obtain convergence rates and results comparable to those obtained in the idealized continuous setting. Our results allow us to take into account separately the uncertainties due to the price noise and those due to discretization errors. Thus allowing estimating better discretization levels both in the domain and in the image of the parameter to solution operator by a Morozov's discrepancy principle.We illustrate the results with simulated as well as real market data. We also validate the results by comparing the implied volatility prices of market data with the computed prices of the calibrated model.

Stable local volatility function calibration using spline kernel

We propose an optimization formulation using the l 1 norm to ensure accuracy and stability in calibrating a local volatility function for option pricing. Using a regularization parameter, the proposed objective function balances calibration accuracy with model complexity. Motivated by the support vector machine learning, the unknown local volatility function is represented by a spline kernel function and the model complexity is controlled by minimizing the 1-norm of the kernel coefficient vector. In the context of support vector regression for function estimation based on a finite set of observations, this corresponds to minimizing the number of support vectors for predictability. We illustrate the ability of the proposed approach to reconstruct the local volatility function in a synthetic market. In addition, based on S&P 500 market index option data, we demonstrate that the calibrated local volatility surface is simple and resembles the observed implied volatility surface in shape. Stability is illustrated by calibrating local volatility functions using market option data from different dates.

No-arbitrage implied volatility functions: Empirical evidence from KOSPI 200 index options

Implied and local volatility are very important variables to market practitioners because such variables can be exploited in numerous option models for the pricing and hedging of diverse exotic options. In the present study, we propose a method to implement no-arbitrage constraints in estimating the implied and local volatility surfaces extracted from data on option prices. With the aid of multiple local bandwidths, we increase the functional flexibility of estimators and provide a simple method by which to construct no-arbitrage volatility surfaces, such that the ready-in-computation advantage of the derivatives in local quadratic smoothing is preserved. To show the effectiveness of the arbitrage-free models, we perform a comprehensive empirical study on the performance of the competing models using the KOSPI 200 index options from January 2001 through December 2010. Using experiments, we examine the performance of the models based on three measures: in-sample pricing, out-of-sample pricing, and hedging errors. We find that implied and local volatility modeling under arbitrage-free conditions show better performance in terms of estimation, pricing, and hedging near the out-of-the-money with short maturities. From the range depicted in the findings, we often observe clear differences between the models with and without no-arbitrage conditions imposed. ► Implementation of no-arbitrage implied volatility surfaces is proposed. ► Comprehensive empirical study using the KOSPI200 index options is given. ► No-Arbitrage local volatility models show better performance for OTM and ITM.

The Most Likely Stream

The most likely stream is an extension of the most likely path. It solves at a small algorithmic cost the known defect of the most likely path (i.e. underestimation of the convexity). It offers a fast and robust method to transform a local volatility surface into an implied volatility one. This paper presents new theoretical results accompanied with graphs for better intuition on the problem.