Hedging with Stochastic and Local Volatility

We derive the local volatility hedge ratios that are consistent with a stochastic instantaneous volatility and show that this ‘stochastic local volatility’ model is equivalent to the market model for implied volatilities. We also show that a common feature of all Markovian single factor stochastic volatility models, (log)normal mixture option pricing models and ‘sticky delta’ models is that they predict incorrect dynamics for implied volatility. As a result they over-hedge the Black-Scholes model in the presence of a market skew and this explains the poor delta hedging performance of these models reported in the literature. Whilst the traditional ‘sticky tree’ local volatility models do not possess this unfortunate property, they cannot be used for pricing without exogenous and ad hoc smoothing of results. However the stochastic local volatility framework allows one to extend a good pricing model into a good hedging model. The theoretical results are supported by an empirical analysis of the hedging performance of seven models, each with different volatility characteristics, on the SP500 index skew.Local volatility, stochastic volatility, implied volatility, hedging, dynamic delta hedging, volatility dymamics

Optimal Hedging and Scale Inavriance: A Taxonomy of Option Pricing Models

The assumption that the probability distribution of returns is independent of the current level of the asset price is an intuitive property for option pricing models on financial assets. This ‘scale invariance’ feature is common to the Black-Scholes (1973) model, most stochastic volatility models and most jump-diffusion models. In this paper we extend the scale-invariant property to other models, including some local volatility, Lévy and mixture models, and derive a set of equivalence properties that are useful for classifying their hedging performance. Bates (2005) shows that, if calibrated exactly to the implied volatility smile, scale-invariant models have the same ‘model-free’ partial price sensitivities for vanilla options. We show that these model-free price hedge ratios are not optimal hedge ratios for many scale-invariant models. We derive optimal hedge ratios for stochastic and local volatility models that have not always been used in the literature. An empirical comparison of well-known models applied to SP 500 index options shows that optimal hedges are similar in all the smile-consistent models considered and they perform better than the Black-Scholes model on average. The partial price sensitivities of scale-invariant models provide the poorest hedges.

Estimating the Leverage Parameter of Continuous-time Stochastic Volatility Models Using High Frequency S&P 500 and VIX

This paper proposes a new method for estimating continuous-time stochastic volatility (SV) models for the S&P 500 stock index process using intraday high-frequency observations of both the S&P 500 index and the Chicago Board of Exchange (CBOE) implied (or expected) volatility index (VIX). Intraday high-frequency observations data have become readily available for an increasing number of financial assets and their derivatives in recent years, but it is well known that attempts to directly apply popular continuous-time models to short intraday time intervals, and estimate the parameters using such data, can lead to nonsensical estimates due to severe intraday seasonality. A primary purpose of the paper is to provide a framework for using intraday high frequency data of both the index estimate, in particular, for improving the estimation accuracy of the leverage parameter, , that is, the correlation between the two Brownian motions driving the diffusive components of the price process and its spot variance process, respectively. As a special case, we focus on Heston’s (1993) square-root SV model, and propose the realized leverage estimator for , noting that, under this model without measurement errors, the “realized leverage,” or the realized covariation of the price and VIX processes divided by the product of the realized volatilities of the two processes, is in-fill consistent for  . Finite sample simulation results show that the proposed estimator delivers more accurate estimates of the leverage parameter than do existing methods.Continuous time, high frequency data, stochastic volatility, S&P 500, implied volatility, VIX.

Optimal Investment Strategies under Stochastic Volatility - Estimation and Applications

This paper studies the impact of stochastic volatility (SV) on optimal investment decisions. We consider three different SV models: an extended Stein/Stein model, the Heston Model and an extended Heston Model with a constant elasticity variance (CEV) process and derive the the long-term optimal investment strategies under each of these processes. Since volatility is not a directly observable quantity, extended Kalman filter techniques are adopted to deal with this partial information problem. Optimal investment strategies based on the CEV volatility model are obtained by adopting the Backward Markov Chain approximation method since analytical solutions are no longer available. We find in the empirical investigation that the Heston model is favored as a more parsimonious model compared with the other two models. All three investment strategies based on the three SV models contain a positive intertemporal hedging term in addition to the static mean-variance portfolio. However, in their details the three investment strategies differ from each other. We also ?nd that the investment strategies are sensitive to the CEV parameter.asset allocation; stochastic volatility; partial information problem; extended Kalman ?lter; the Heston model; CEV process

"Pricing Swaptions under the Libor Market Model of Interest Rates with Local-Stochastic Volatility Models"

This paper presents a new approximation formula for pricing swaptions and caps/floors under the LIBOR market model of interest rates (LMM) with the local and affine-type stochastic volatility. In particular, two approximation methods are applied in pricing, one of which is so called "drift-freezing" that fixes parts of the underlying stochastic processes at their initial values. Another approximation is based on an asymptotic expansion approach. An advantage of our method is that those approximations can be applied in a unified manner to a general class of local-stochastic volatility models of interest rates. To demonstrate effectiveness of our method, the paper takes CEVHeston LMM and Quadratic-Heston LMM as examples; it confirms sufficient flexibility of the models for calibration in a caplet market and enough accuracies of the approximation method for numerical evaluation of swaption values under the models.

Bid-ask spread modelling, a perturbation approach

Our objective is to study liquidity risk, in particular the so-called ``bid-ask spread'', as a by-product of market uncertainties. ``Bid-ask spread'', and more generally ``limit order books'' describe the existence of different sell and buy prices, which we explain by using different risk aversions of market participants. The risky asset follows a diffusion process governed by a Brownian motion which is uncertain. We use the error theory with Dirichlet forms to formalize the notion of uncertainty on the Brownian motion. This uncertainty generates noises on the trajectories of the underlying asset and we use these noises to expound the presence of bid-ask spreads. In addition, we prove that these noises also have direct impacts on the mid-price of the risky asset. We further enrich our studies with the resolution of an optimal liquidation problem under these liquidity uncertainties and market impacts. To complete our analysis, some numerical results will be provided