Pricing and hedging derivative securities in markets with uncertain volatilities
TLDR
In this paper, the authors present a model for pricing and hedging derivative securities and option portfolios in an environment where the volatility is not known precisely, but is assumed instead to lie between two extreme values σmin and σmax.Abstract:
We present a model for pricing and hedging derivative securities and option portfolios in an environment where the volatility is not known precisely, but is assumed instead to lie between two extreme values σminand σmax. These bounds could be inferred from extreme values of the implied volatilities of liquid options, or from high-low peaks in historical stock- or option-implied volatilities. They can be viewed as defining a confidence interval for future volatility values. We show that the extremal non-arbitrageable prices for the derivative asset which arise as the volatility paths vary in such a band can be described by a non-linear PDE, which we call the Black-Scholes-Barenblatt equation. In this equation, the ‘pricing’ volatility is selected dynamically from the two extreme values, σmin, σmax, according to the convexity of the value-function. A simple algorithm for solving the equation by finite-differencing or a trinomial tree is presented. We show that this model captures the importance of diversifi...read more
Citations
More filters
Journal ArticleDOI
Solving high-dimensional partial differential equations using deep learning
TL;DR: A deep learning-based approach that can handle general high-dimensional parabolic PDEs using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function.
Book ChapterDOI
Towards a Theory of Volatility Trading
Peter Carr,Dilip B. Madan +1 more
Abstract: White, and especially Robert Jarrow for useful discussions. They are not responsible for any errors.
Book ChapterDOI
G -Expectation, G -Brownian Motion and Related Stochastic Calculus of Itô Type
TL;DR: In this article, the authors introduce a nonlinear expectation generated by a heat equation with infinitesimal generator G. The G-standard normal distribution is introduced and the canonical process is a G-Brownian motion.
Posted Content
Nonlinear Expectations and Stochastic Calculus under Uncertainty
TL;DR: In this paper, a new approach of sublinear expectation is introduced to deal with the problem of probability and distribution model uncertainty, and a new type of normal distributions and the related central limit theorem under sublinear expectations are presented.
Journal ArticleDOI
Function Spaces and Capacity Related to a Sublinear Expectation: Application to G -Brownian Motion Paths
TL;DR: In this article, the authors give some basic and important properties of typical Banach spaces of functions of G-Brownian motion paths induced by a sublinear expectation, which can be applied in continuous time dynamic and coherent risk measures in finance, in particular for path-dependence risky positions under situations of volatility model uncertainty.
References
More filters
Journal ArticleDOI
The Pricing of Options and Corporate Liabilities
Fischer Black,Myron S. Scholes +1 more
TL;DR: In this paper, a theoretical valuation formula for options is derived, based on the assumption that options are correctly priced in the market and it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks.
Journal ArticleDOI
The Pricing of Options on Assets with Stochastic Volatilities
John Hull,Alan White +1 more
TL;DR: In this article, the option price is determined in series form for the case in which the stochastic volatility is independent of the stock price, and the solution of this differential equation is independent if (a) the volatility is a traded asset or (b) volatility is uncorrelated with aggregate consumption, if either of these conditions holds, the risk-neutral valuation arguments of Cox and Ross [4] can be used in a straightfoward way.
Journal ArticleDOI
Martingales and arbitrage in multiperiod securities markets
Book
Dynamic Asset Pricing Theory
TL;DR: The "Dynamic Asset Pricing Theory" (DAT) as discussed by the authors is a textbook for doctoral students and researchers on the theory of asset pricing and portfolio selection in multi-period settings under uncertainty.
Journal ArticleDOI
Implied Binomial Trees
TL;DR: In this article, a new method for inferring risk-neutral probabilities (or state-contingent prices) from the simultaneously observed prices of European options is developed. But this method requires the assumption that the underlying asset has a limited risk-free lognormal distribution.