Uncertain Volatility Models — Theory and Application

Buff, Robert

doi:10.1007/978-3-642-56323-2

Cited by 17 publications

(9 citation statements)

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“…A European call option with transaction cost and uncertain volatility is considered in Dokuchaev and Savkin (1998). Barrier options under uncertain volatility were studied in Avellaneda and Buff (1999) and Buff (2002), as well as American options and a portfolio of uncertain volatility options. American options were also studied in Smith (2002).…”

Section: Uncertain Volatility Modelmentioning

confidence: 99%

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

2016

IMA J Numer Anal

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

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Section: Uncertain Volatility Modelmentioning

confidence: 99%

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

2016

IMA J Numer Anal

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“…Notice that various studies of mathematical finance (e.g., pricing and hedging [2,25,29]) have remarkably focused on markets with uncertain volatility. In [5], uncertain volatility models are introduced to evaluate a scenario where the volatility coefficient of the pricing model cannot be determined exactly. Therefore, a practical motivation here is, in many decision problems, a large number of coupled decision markers share a common noise but with uncertain volatility on it.…”

Section: Volatility Uncertainty With Common Noisementioning

confidence: 99%

Social Optima in Mean Field Linear-Quadratic-Gaussian Control with Volatility Uncertainty

Huang

Wang

Yong

2019

Preprint

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This paper examines mean field linear-quadratic-Gaussian (LQG) social optimum control with volatility-uncertain common noise. The diffusion terms in the dynamics of agents contain an unknown volatility process driven by a common noise. We apply a robust optimization approach in which all agents view volatility uncertainty as an adversarial player. Based on the principle of person-by-person optimality and a two-step-duality technique for stochastic variational analysis, we construct an auxiliary optimal control problem for a representative agent. Through solving this problem combined with a consistent mean field approximation, we design a set of decentralized strategies, which are further shown to be asymptotically social optimal by perturbation analysis.

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“…The governing equations for V 0 and V 1 are derived in the literature by considering a best and worst case scenario for the value of a portfolio as the volatility and interest rates are allowed to vary freely within their assigned ranges [3], [4], [12]. However, they are already implied by the Black Scholes equation (2.1) and the standard maximum principle for parabolic equations.…”

Section: Since Limmentioning

confidence: 99%

The Black Scholes Barenblatt Equation for Options With Uncertain Volatility and Its Application to Static Hedging

Meyer

2006

Int. J. Theor. Appl. Finan.

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The Black Scholes Barenblatt (BSB) equation for the envelope of option prices with uncertain volatility and interest rate is derived from the Black Scholes equation with the maximum principle for diffusion equations and shown to be equivalent to a readily solvable standard Black Scholes equation with a nonlinear source term. Analogous arguments yield the envelope for the delta of option prices. We then interpret the concept of static hedging for narrowing the envelope in terms of partial differential equations and define the optimal static hedge and computable approximations to it. We apply the BSB equation to find numerically some optimally hedged portfolios of representative European and American options.

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Uncertain Volatility Models — Theory and Application

Cited by 17 publications

References 0 publications

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

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

Social Optima in Mean Field Linear-Quadratic-Gaussian Control with Volatility Uncertainty

The Black Scholes Barenblatt Equation for Options With Uncertain Volatility and Its Application to Static Hedging

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