Uncertain Volatility Model Research Articles

Model Uncertainty: A Reverse Approach

In robust finance, Knightian uncertainty is often captured by sets of probability measures on the future states of the world. If these measures are nondominated, this usually comes at the cost of losing tractability, and advanced functional-analytic tools are often not available anymore. This tends to be mitigated by ad hoc assumptions that guarantee a certain degree of tractability, for instance concerning the aggregation of consistent random variables. The present paper instead investigates from a reverse perspective what implications the validity of certain functional-analytic tools has. In this vein, we categorise the Kreps-Yan property, robust variants of the Brannath-Schachermayer Bipolar Theorem and the Grothendieck Lemma, and uncertain volatility models. By doing so, we also uncover connections to robust statistics.

Valuation of lookback option under uncertain volatility model

Lookback option is an exotic option whose payoff depends on the maximum or minimum price of underlying asset within the period of validity. This paper mainly studies the valuation of lookback option under an uncertain volatility model. The pricing formulas of lookback option are derived and some algorithms are given to calculate the option prices numerically. Finally, we apply the minimum cover estimation approach to compute the parameters of the uncertain volatility process and the corresponding numerical example is provided to illustrate the method.

$\alpha$-Hypergeometric Uncertain Volatility Models and their Connection to 2BSDEs

In this article we propose a $\alpha$-hypergeometric model with uncertain volatility (UV) where we derive a worst-case scenario for option pricing. The approach is based on the connexion between a certain class of nonlinear partial differential equations of HJB-type (G-HJB equations), that govern the nonlinear expectation of the UV model and that provide an alternative to the difficult model calibration problem of UV models, and second-order backward stochastic differential equations (2BSDEs). Using asymptotic analysis for the G-HJB equation and the equivalent 2BSDE representation, we derive a limit model that provides an accurate description of the worst-case price scenario in cases when the bounds of the UV model are slowly varying. The analytical results are tested by numerical simulations using a deep learning based approximation of the underlying 2BSDE.

Double barrier American put option pricing under uncertain volatility model

This paper aims to study the asymptotic behavior of double barrier American-style put option prices under an uncertain volatility model, which degenerates to a single point. We give an approximation of the double barrier American-style option prices with a small volatility interval, expressed by the Black–Scholes–Barenblatt equation. Then, we propose a novel representation for the early exercise boundary of American-style double barrier options in terms of the optimal stopping boundary of a single barrier contract.

PRICING AMERICAN OPTIONS WITH THE RUNGE–KUTTA–LEGENDRE FINITE DIFFERENCE SCHEME

This paper presents the Runge–Kutta–Legendre (RKL) finite difference scheme, allowing for an additional shift in its polynomial representation. A short presentation of the stability region, comparatively to the Runge–Kutta–Chebyshev scheme follows. We then explore the problem of pricing American options with the RKL scheme under the one factor Black–Scholes and the two factor Heston stochastic volatility models, as well as the pricing of butterfly spread and digital options under the uncertain volatility model, where a Hamilton–Jacobi–Bellman partial differential equation needs to be solved. We explore the order of convergence in these problems, as well as the option greeks stability, compared to the literature and popular schemes such as Crank–Nicolson, with Rannacher time-stepping.

A Hilbert transform approach for controlled jump-diffusions with financial applications

We propose a new computational method for a class of controlled jump-diffusions for financial applications. In the first step of our method, we apply piecewise constant policy approximation where we partition the time horizon into small time intervals and the control is constant on each interval. In the second step, we develop a Hilbert transform approach to solve a discrete time dynamic programming problem. We provide rigorous error bounds for the piecewise constant policy approximation for controlled jump-diffusions, generalizing previous results for diffusions. We also apply our method to solve two classical types of financial problems: option pricing under uncertain volatility and/or correlation models and optimal investment, including utility maximization and mean-variance portfolio selection. Through various numerical examples, we demonstrate the properties of our method and show that it is a computationally efficient choice for low-dimensional problems. Our method also compares favorably with some popular approaches.

Asian Option Pricing under an Uncertain Volatility Model

In this paper, we study the asymptotic behavior of Asian option prices in the worst-case scenario under an uncertain volatility model. We derive a procedure to approximate Asian option prices with a small volatility interval. By imposing additional conditions on the boundary condition and splitting the obtained Black–Scholes–Barenblatt equation into two Black–Scholes-like equations, we obtain an approximation method to solve a fully nonlinear PDE.

European option pricing under multifactor uncertain volatility model

This paper presents an uncertain stock model under the multifactor uncertain volatility framework. Based on the uncertainty theory, some closed-form and analytical formulas presented to value a European call and put option under the multifactor uncertain volatility model. Numerical tests are reported to highlight how the proposed model provides interesting results on pricing a European option. Finally, we summarize the theoretical results and some numerical experiments.

Quasi-closed-form solution and numerical method for currency option with uncertain volatility model

There exist some non-stochastic factors in the financial market, so the dynamics of the exchange rate highly depends on human uncertainty. This paper investigates the pricing problems of foreign currency options under the uncertain environment. First, we propose an currency model under the assumption that exchange rate, volatility, domestic interest rate and foreign interest rate are all driven by uncertain differential equations; especially, the exchange rate exhibits mean reversion. Since the analytical solutions of nested uncertain differential equations cannot always be obtained, we design a new numerical method, Runge–Kutta-99 hybrid method, for solving nested uncertain differential equations. The accuracy of the designed numerical method is investigated by comparison with the analytical solution. Subsequently, the quasi-closed-form solutions are derived for the prices of both European and American foreign currency options. Finally, in order to illustrate the rationality and the practicability of the proposed currency model, we design several numerical algorithms to calculate the option prices and analyze the price behaviors of foreign currency options across strike price and maturity.

Vulnerable options pricing under uncertain volatility model

In this paper, we consider the pricing problem of options with counterparty default risks. We study the asymptotic behavior of vulnerable option prices in the worst case scenario under an uncertain volatility model which contains both corporate assets and underlying assets. We propose a method to estimate the price of vulnerable options when the volatility of the underlying assets is within a small interval. By imposing additional conditions on the boundary condition and cutting the obtained Black–Scholes–Barenblatt equation into two Black–Scholes-like equations, we obtain an approximate method for solving the fully nonlinear partial differential equation satisfied by the price of vulnerable options under the uncertain volatility model.

The two-dimensional tree–grid method

In this paper, we introduce a novel, explicit, wide-stencil, two-dimensional (2D) tree–grid method for solving stochastic control problems (SCPs) with two space dimensions and one time dimension, or, equivalently, the corresponding Hamilton– Jacobi–Bellman equation. This new method can be seen as a generalization of the tree–grid method for SCPs with one space dimension that was recently developed by the authors. The method is unconditionally stable and no 2D interpolation is needed in the stencil construction. We prove the convergence of the method and exemplify it in our application to a two-factor uncertain volatility model.

Uncertain Volatility Models with Stochastic Bounds

In this paper, we study a class of uncertain volatility models with stochastic bounds. Like in the regular uncertain volatility models, we know only that volatility stays between two bounds, but in...

The multivariate mixture dynamics: Consistent no-arbitrage single-asset and index volatility smiles

ABSTRACTWe introduce a new arbitrage-free multivariate dynamic asset pricing model that allows us to reconcile single name and index/basket volatility smiles using a tractable and explicit dependence structure that goes beyond instantaneous correlation. Each asset volatility smile is modeled according to a density-mixture dynamical model while the same property holds for the multivariate process of all assets whose density is a mixture of multivariate basic densities. After introducing the model, we derive tractable index option smile formulas resulting from the model and related closed-form solutions for multivariate densities taking the form of multivariate mixtures. Using Markovian projection techniques, we relate our model to a multivariate uncertain volatility model and show a consistency result with geometric baskets with hints on possible uses in investigating triangular relationships between foreign exchange rates and the related smiles in practice. We also derive closed-form solutions for a number of terminal statistics of dependence and derive a precise relationship with a simpler, but less tractable, model based on a basic instantaneous correlation structure. Finally, closed-form solutions for volatility/asset correlations illuminating the relationship with the uncertain volatility model are introduced. The model tractability makes it particularly suited for calibration and risk management applications, where speed of calculations and tractability are essential. A few numerical examples on basket and spread options pricing conclude the article.

Valuation of European option under uncertain volatility model

Valuation of an option plays an important role in modern finance. As the financial market for derivatives continues to grow, the progress and the power of option pricing models at predicting the value of option premium are under investigations. In this paper, we assume that the volatility of the stock price follows an uncertain differential equation and propose an uncertain counterpart of the Heston model. This study also focuses on deriving a numerical method for pricing a European option under uncertain volatility model, and some numerical experiments are presented. Numerical experiments confirm that the developed methods are very efficient.

The pricing of average options with jump diffusion processes in the uncertain volatility model

The pricing equations of the average options with jump diffusion processes can be formulated as two-dimensional partial integro-differential equations (PIDEs). In the uncertain volatility model, for options with non-convex and non-concave payoffs, such as the butterfly spread, the PIDEs are nonlinear. We use the semi-Lagrangian method to reduce the two-dimensional nonlinear PIDE to a one-dimensional nonlinear PIDE along the trajectory of the average price, and use a Newton-type iteration to guarantee the convergence of the discrete solution to the viscosity solution. Monotonicity and stability as well as the convergence results are derived. Numerical tests of convergence for a variety of cases, including average butterfly spread and ordinary butterfly spread, are presented.

An unconditionally monotone numerical scheme for the two-factor uncertain volatility model

Under the assumption that two asset prices follow an uncertain volatility model, the maximal and 2 minimal solution values of an option contract are given by a two dimensional Hamilton-Jacobi-Bellman 3 (HJB) PDE. A fully implicit, unconditionally monotone nite dierence numerical scheme is developed 4 in this paper. Consequently, there are no time step restrictions due to stability considerations. The 5 discretized algebraic equations are solved using policy iteration. Our discretization method results in a 6 local objective function which is a discontinuous function of the control. Hence some care must be taken 7 when applying policy iteration. The main diculty in designing a discretization scheme is development 8 of a monotonicity preserving approximation of the cross derivative term in the PDE. We derive a hybrid 9 numerical scheme which combines use of a xed point stencil and a wide stencil based on a local coordinate 10 rotation. The algorithm uses the xed point stencil as much as possible to take advantage of its accuracy 11 and computational eciency. The analysis shows that our numerical scheme is l1 stable, consistent in 12 the viscosity sense, and monotone. Thus, our numerical scheme guarantees convergence to the viscosity 13 solution. 14

Stochastic Volatility Modeling: Chapter 2 - Local Volatility

This is Chapter 2 of Stochastic Volatility Modeling, published by CRC/Chapman & Hall.In this chapter the local volatility model is surveyed as a market model for the underlying together with its associated vanilla options.First, relationships of implied to local volatilities are derived, as well as approximations for skew and curvature. Exact and approximate techniques for taking dividends into account are presented.We then turn to the dynamics of the local volatility model. We introduce the Skew Tickiness Ratio (SSR) and derive approximate formulas for the SSR and volatilities of volatilities in the local volatility model.We also examine future skews.We then consider the delta and carry P&L of a hedged option position. We derive the expression of the market-model delta of the local volatility model and discuss the relationship between sticky-strike and market-model deltas. We characterize the gamma/theta break-even levels of a hedged position and show that the local volatility model is indeed a market model.We then derive the expression of the vega-hedge portfolio.Markov-functional models are considered next.Finally, we survey the Uncertain Volatility Model and its usage.A digest summarizes key points.

A Dual Algorithm for Stochastic Control Problems: Applications to Uncertain Volatility Models and CVA

We derive an algorithm in the spirit of Rogers [SIAM J. Control Optim., 46 (2007), pp. 1116--1132] and Davis and Burstein [Stochastics Stochastics Rep., 40 (1992), pp. 203--256] that leads to upper bounds for stochastic control problems. Our bounds complement lower biased estimates recently obtained in Guyon and Henry-Labordere [J. Comput. Finance, 14 (2011), pp. 37--71]. We evaluate our estimates in numerical examples motivated by mathematical finance.

A Dual Algorithm for Stochastic Control Problems: Applications to Uncertain Volatility Models and CVA

A Dual Algorithm for Stochastic Control Problems: Applications to Uncertain Volatility Models and CVA

Approximation for Option Prices under Uncertain Volatility

In this paper, we study the asymptotic behavior of the worst case scenario option prices as the volatility interval in an uncertain volatility model (UVM) degenerates to a single point and then provide an approximation procedure for the worst case scenario prices in a UVM with a small volatility interval. Numerical experiments show that this approximation procedure performs well even when the size of the volatility band is not so small.