Calibration Of Volatility Research Articles

Investment Skill and Consistent Long-Term Alpha

How should an active investor calibrate their investment decisions, when the goal is to reach an outperformance target over the long term, without succumbing to large drawdowns in the interim? Closed-form expressions for the probability of consistent long-term alpha are derived and show that maximizing this probability requires a thoughtful calibration of skill, volatility, decision frequency, and costs. The article derives the minimum skill level for which the probability of consistent long-term alpha trends to its maximum for long investment horizons. This minimum skill level is not typically large, at only marginally above a fair coin toss. Unfortunately, the maximum probability of consistent long-term alpha is also not very large, even when skill is high and the investment horizon is long. The reason is that, although the probability of reaching the long-term goal rises with the time horizon for sufficiently high skill, so does the probability of experiencing the intertemporal loss that can force a stop-out in the interim.

Calibration of time-dependent volatility for European options under the fractional Vasicek model

<abstract><p>In this paper, we calibrate the time-dependent volatility function for European options under the fractional Vasicek interest rate model. A fully implicit finite difference method is applied to solve the partial differential equation of option pricing numerically. To find the volatility function, we minimize a cost function that is the sum of the squared errors between the theoretical prices and market prices with Tikhonov $ L_2 $ regularization and $ L_{1/2} $ regularization respectively. Finally numerical experiments with simulated and real market data verify the efficiency of the proposed methods.</p></abstract>

An alternative tree method for calibration of the local volatility

<p style='text-indent:20px;'>In this paper, we combine the traditional binomial tree and trinomial tree to construct a new alternative tree pricing model, where the local volatility is a deterministic function of time. We then prove the convergence rates of the alternative tree method. The proposed model can price a wide range of derivatives efficiently and accurately. In addition, we research the optimization approach for the calibration of local volatility. The calibration problem can be transformed into a nonlinear unconstrained optimization problem by exterior penalty method. For the optimization problem, we use the quasi-Newton algorithm. Finally, we test our model by numerical examples and options data on the S & P 500 index. Numerical results confirm the excellent performance of the alternative tree pricing model.</p>

Forecasting volatility with a stacked model based on a hybridized Artificial Neural Network

An appropriate calibration and forecasting of volatility and market risk are some of the main challenges faced by companies that have to manage the uncertainty inherent to their investments or funding operations such as banks, pension funds or insurance companies. This has become even more evident after the 2007–2008 Financial Crisis, when the forecasting models assessing the market risk and volatility failed. Since then, a significant number of theoretical developments and methodologies have appeared to improve the accuracy of the volatility forecasts and market risk assessments. Following this line of thinking, this paper introduces a model based on using a set of Machine Learning techniques, such as Gradient Descent Boosting, Random Forest, Support Vector Machine and Artificial Neural Network, where those algorithms are stacked to predict S&P500 volatility. The results suggest that our construction outperforms other habitual models on the ability to forecast the level of volatility, leading to a more accurate assessment of the market risk.

On full calibration of hybrid local volatility and regime‐switching models

Calibrating local regime‐switching models is a challenging problem, especially when the volatility functions are assumed to depend on both of the underlying price and time. In this paper, the inverse problem of determining local volatility functions is firstly established and then solved through the Tikhonov regularization to obtain the optimal solution, which is achieved iteratively through a newly designed numerical algorithm. While our numerical tests with artificial data show that our algorithm can provide quite accurate and stable results, its performance with the involvement of real market data have been further demonstrated using options written on the S&P 500 index.

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.

The Alternating Direction Explicit (ADE) Method for One-Factor Problems

In this article we apply the ADE method to a number of partial differential equations in option pricing using one-factor models (Black–Scholes, local volatility, uncertain volatility). We first give an introduction to ADE. We discuss the stability, accuracy, and performance of ADE for a generic one-factor partial differential equation. Of particular importance is how we transform a problem on an unbounded domain to one on a bounded domain, thus avoiding complex mathematical techniques to find the optimal truncated boundary and the determination of the corresponding numerical boundary conditions. The second part of the article examines a number of applications. We show that oscillation-free Greeks make ADE a suitable candidate for uncertain volatility models. We also apply ADE to the problem of calibration of local volatility using the Kolmogorov forward equation and show that the performance and the accuracy obtained make ADE the method of choice instead of the popular Crank–Nicolson method. Copyright © 2011 Wilmott Magazine Ltd.

Numerical Procedure for Calibration of Volatility with American Options

In finance, the price of an American option is obtained from the price of the underlying asset by solving a parabolic variational inequality. The calibration of volatility from the prices of a family of American options yields an inverse problem involving the solution of the previously mentioned parabolic variational inequality. In this paper, the discretization of the variational inequality by finite elements is studied in detail. Then, a calibration procedure, where the volatility belongs to a finite‐dimensional space (finite element or bicubic splines) is described. A least square method, with suitable regularization terms is used. Necessary optimality conditions involving adjoint states are given and the differentiability of the cost function is studied. A parallel algorithm is proposed and numerical experiments, on both academic and realistic cases, are presented.

An Inverse Problem for a Parabolic Variational Inequality Arising in Volatility Calibration with American Options

In finance, the price of an American option is obtained from the price of the underlying asset by solving a parabolic variational inequality. The free boundary associated with this variational inequality can be interpreted as the price for which the option should be exercised. The calibration of volatility from the observations of the prices of an American option yields an inverse problem for the previously mentioned parabolic variational inequality. After studying the variational inequality and the exercise price, we give results concerning the sensitivity of the option price and of the exercise price with respect to the variations of the volatility. The inverse problem is addressed by a least square method, with suitable regularization terms. We give necessary optimality conditions involving an adjoint state for a simplified inverse problem and we study the differentiability of the cost function. Optimality conditions are also given for the genuine calibration problem.

A variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm

In this work, we develop a reduced-basis approach for the efficient computation of parametrized expected values, for a large number of parameter values, using the control variate method to reduce the variance. Two algorithms are proposed to compute online, through a cheap reduced-basis approximation, the control variates for the computation of a large number of expectations of a functional of a parametrized Ito stochastic process (solution to a parametrized stochastic differential equation). For each algorithm, a reduced basis of control variates is pre-computed offline, following a so-called greedy procedure, which minimizes the variance among a trial sample of the output parametrized expectations. Numerical results in situations relevant to practical applications (calibration of volatility in option pricing, and parameter-driven evolution of a vector field following a Langevin equation from kinetic theory) illustrate the efficiency of the method.