It is impossible to discriminate between the commonly used stochastic volatility models of Heston, log-normal, and 3-over-2 on the basis of exponentially weighted averages of daily returns-even though it appears so at first sight. However, with a 5-min sampling frequency, the models can be differentiated and empirical evidence overwhelmingly favours a fast mean-reverting log-normal model.

We consider stochastic volatility models under parameter uncertainty and investigate how model derived prices of European options are affected. We let the pricing parameters evolve dynamically in time within a specified region, and formalise the problem as a control problem where the control acts on the parameters to maximise/minimise the option value. Through a dual representation with backward stochastic differential equations, we obtain explicit equations for Heston's model and investigate several numerical solutions thereof. In an empirical study, we apply our results to market data from the S&P 500 index where the model is estimated to historical asset prices. We find that the conservative model-prices cover 98% of the considered market-prices for a set of European call options.

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Derivative hedging under transaction costs has attracted considerable attention over the past three decades. Yet comparatively little effort has been made towards integrating this problem in the context of trading through a limit order book. In this paper, we propose a simple model for a wealth-optimizing option seller, who hedges his position using a combination of limit and market orders, while facing certain constraints as to how far he can deviate from a targeted (Bachelierian) delta strategy. By translating the control problem into a three-dimensional Hamilton–Jacobi–Bellman quasi-variational inequality (HJB QVI) and solving numerically, we are able to deduce optimal limit order quotes alongside the regions surrounding the targeted delta surface in which the option seller must place limit orders vis-à-vis the more aggressive market orders. Our scheme is shown to be monotone, stable, and consistent and thence, modulo a comparison principle, convergent in the viscosity sense.

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The local volatility model is a widely used for pricing and hedging financial derivatives. While its main appeal is its capability of reproducing any given surface of observed option prices-it provides a perfect fit-the essential component is a latent function which can be uniquely determined only in the limit of infinite data. To (re)construct this function, numerous calibration methods have been suggested involving steps of interpolation and extrapolation, most often of parametric form and with point-estimate representations. We look at the calibration problem in a probabilistic framework with a nonparametric approach based on a Gaussian process prior. This immediately gives a way of encoding prior beliefs about the local volatility function and a hypothesis model which is highly flexible yet not prone to overfitting. Besides providing a method for calibrating a (range of) point-estimate(s), we draw posterior inference from the distribution over local volatility. This leads to a better understanding of uncertainty associated with the calibration in particular, and with the model in general. Further, we infer dynamical properties of local volatility by augmenting the hypothesis space with a time dimension. Ideally, this provides predictive distributions not only locally, but also for entire surfaces forward in time. We apply our approach to S&P 500 market data.

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