Local Stochastic Volatility Research Articles

Approximate pricing of European and Barrier claims in a local-stochastic volatility setting

We derive asymptotic expansions for the prices of a variety of European and barrier-style claims in a general local-stochastic volatility setting. Our method combines Taylor series expansions of the diffusion coefficients with an expansion in the correlation parameter between the underlying asset and volatility process. Rigorous accuracy results are provided for European-style claims. For barrier-style claims, we include several numerical examples to illustrate the accuracy and versatility of our approximations.

The Hybrid Stochastic-Local Volatility Model with Applications in Pricing FX Options

This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.

Equity Modeling with Stochastic Dividends

In this paper, we show how to incorporate exactly discrete stochastic dividends in (hybrid) local stochastic volatility models. Our algorithm is illustrated by various numerical examples. We also specify a dividend model which is easy simulable and consistent with the market dividend smile.

An Empirical Method for Calibrating LSV Mixing Weight

In this discussion paper, we propose a method for calibration mixing weight of Local Stochastic Volatility model (LSV Hereafter). We also propose a method of computing backbone of given LSV in the context of PDE.

A finite volume – alternating direction implicit approach for the calibration of stochastic local volatility models

ABSTRACTCalibration of stochastic local volatility (SLV) models to their underlying local volatility model is often performed by numerically solving a two-dimensional nonlinear forward Kolmogorov equation. We propose a novel finite volume discretization in the numerical solution of general 1D and 2D forward Kolmogorov equations. The finite volume method does not require a transformation of the partial differential equation (PDE). This constitutes a main advantage in the calibration of SLV models as the pertinent PDE coefficients are often non-smooth. Moreover, the finite volume discretization has the crucial property that the total numerical mass is conserved. Applying the finite volume discretization in the calibration of SLV models yields a nonlinear system of ordinary differential equations (ODEs). Numerical time stepping is performed by the Hundsdorfer–Verwer alternating direction implicit scheme to increase the computational efficiency. The nonlinearity in the system of ODEs is handled by introducing an inner iteration. Ample numerical experiments are presented that illustrate the effectiveness of the calibration procedure.

A novel Monte Carlo approach to hybrid local volatility models

We present in a Monte Carlo simulation framework, a novel approach for the evaluation of hybrid local volatility [Risk, 1994, 7, 18–20], [Int. J. Theor. Appl. Finance, 1998, 1, 61–110] models. In particular, we consider the stochastic local volatility model—see e.g. Lipton et al. [Quant. Finance, 2014, 14, 1899–1922], Piterbarg [Risk, 2007, April, 84–89], Tataru and Fisher [Quantitative Development Group, Bloomberg Version 1, 2010], Lipton [Risk, 2002, 15, 61–66]—and the local volatility model incorporating stochastic interest rates—see e.g. Atlan [ArXiV preprint math/0604316, 2006], Piterbarg [Risk, 2006, 19, 66–71], Deelstra and Rayée [Appl. Math. Finance, 2012, 1–23], Ren et al. [Risk, 2007, 20, 138–143]. For both model classes a particular (conditional) expectation needs to be evaluated which cannot be extracted from the market and is expensive to compute. We establish accurate and ‘cheap to evaluate’ approximations for the expectations by means of the stochastic collocation method [SIAM J. Numer. Anal., 2007, 45, 1005–1034], [SIAM J. Sci. Comput., 2005, 27, 1118–1139], [Math. Models Methods Appl. Sci., 2012, 22, 1–33], [SIAM J. Numer. Anal., 2008, 46, 2309–2345], [J. Biomech. Eng., 2011, 133, 031001], which was recently applied in the financial context [Available at SSRN 2529691, 2014], [J. Comput. Finance, 2016, 20, 1–19], combined with standard regression techniques. Monte Carlo pricing experiments confirm that our method is highly accurate and fast.

A weak approximation with Malliavin weights for local stochastic volatility model

This paper introduces a new efficient and practical weak approximation for option price under local stochastic volatility model as marginal expectation of stochastic differential equation, using iterative asymptotic expansion with Malliavin weights. The explicit Malliavin weights for SABR model are shown. Numerical experiments confirm the validity of our discretization with a few time steps.

An adjoint method for the exact calibration of stochastic local volatility models

This paper deals with the exact calibration of semidiscretized stochastic local volatility (SLV) models to their underlying semidiscretized local volatility (LV) models. Under an SLV model, it is common to approximate the fair value of European-style options by semidiscretizing the backward Kolmogorov equation using finite differences. In the present paper we introduce an adjoint semidiscretization of the corresponding forward Kolmogorov equation. This adjoint semidiscretization is used to obtain an expression for the leverage function in the pertinent SLV model such that the approximated fair values defined by the LV and SLV models are identical for non-path-dependent European-style options. In order to employ this expression, a large non-linear system of ODEs needs to be solved. The actual numerical calibration is performed by combining ADI time stepping with an inner iteration to handle the non-linearity. Ample numerical experiments are presented that illustrate the effectiveness of the calibration procedure.

The Swap Market Model with Local Stochastic Volatility

The aim of this paper is to present the multi-factor swap market model with non-parametric local volatility functions and stochastic volatility scaling factors. We provide a Dupire-like formula with which calibration can be carried out with the particle algorithm in an efficient manner. We also discuss how the calibration method can be made applicable in the context of Libor Market Model. We show high accuracy of our calibration algorithm by numerical experiments.

Small-Time Asymptotics under Local-Stochastic Volatility with a Jump-to-Default: Curvature and the Heat Kernel Expansion

We compute a sharp small-time estimate for implied volatility under a general uncorrelated local-stochastic volatility model. For this we use the Bellaiche \cite{Bel81} heat kernel expansion combined with Laplace's method to integrate over the volatility variable on a compact set, and (after a gauge transformation) we use the Davies \cite{Dav88} upper bound for the heat kernel on a manifold with bounded Ricci curvature to deal with the tail integrals. If the correlation $\rho < 0$, our approach still works if the drift of the volatility takes a specific functional form and there is no local volatility component, and our results include the SABR model for $\beta=1, \rho \le 0$. \bl{For uncorrelated stochastic volatility models, our results also include a SABR-type model with $\beta=1$ and an affine mean-reverting drift, and the exponential Ornstein-Uhlenbeck model.} We later augment the model with a single jump-to-default with intensity $\lm$, which produces qualitatively different behaviour for the short-maturity smile; in particular, for $\rho=0$, log-moneyness $x > 0$, the implied volatility increases by $\lm f(x) t +o(t) $ for some function $f(x)$ which blows up as $x \searrow 0$. Finally, we compare our result with the general asymptotic expansion in Lorig, Pagliarani \& Pascucci \cite{LPP15}, and we verify our results numerically for the SABR model using Monte Carlo simulation and the exact closed-form solution given in Antonov \& Spector \cite{AS12} for the case $\rho=0$.

Calibrating a market model with stochastic volatility to commodity and interest rate risk

Based on the multi-currency LIBOR Market Model, this paper constructs a hybrid commodity interest rate market model with a stochastic local volatility function allowing the model to simultaneously fit the implied volatility surfaces of commodity and interest rate options. Since liquid market prices are only available for options on commodity futures, rather than forwards, a convexity correction formula for the model is derived to account for the difference between forward and futures prices. A procedure for efficiently calibrating the model to interest rate and commodity volatility smiles is constructed. Finally, the model is fitted to an exogenously given correlation structure between forward interest rates and commodity prices (cross-correlation). When calibrating to options on forwards (rather than futures), the fitting of cross-correlation preserves the (separate) calibration in the two markets (interest rate and commodity options), while in the case of futures a (rapidly converging) iterative fitting procedure is presented. The fitting of cross-correlation is reduced to finding an optimal rotation of volatility vectors, which is shown to be an appropriately modified version of the ‘orthonormal Procrustes’ problem in linear algebra. The calibration approach is demonstrated in an application to market data for oil futures.

A Weak Approximation with Malliavin Weights for Local Stochastic Volatility Model

This paper introduces a new efficient and practical weak approximation for option price under local stochastic volatility model as marginal expectation of stochastic differential equation, using iterative asymptotic expansion with Malliavin weights. The explicit Malliavin weights for SABR model are shown. Numerical experiments confirm the validity of our discretization with a few time steps.

Calibration of local correlation models to basket smiles

Allowing correlation to be local, i.e., state-dependent, in multi-asset models allows better hedging by incorporating correlation moves in the Delta. When options on a basket - be it a stock index, a cross-foreign exchange rate or an interest rate spread - are liquidly traded, one may want to calibrate a local correlation to these option prices. Only two particular solutions have been suggested so far in the literature. Both impose a particular dependency of the correlation matrix on the asset values that one has no reason to undergo. They may also fail to be admissible, i.e., positive semi-definite. We explain how, by combining the particle method presented in smile calibration problem solved by Guyon and Henry-Labordere (2011) with a simple affine transform, we can build all the calibrated local correlation models. The two existing models appear as special cases (if admissible). For the first time, one can now choose a calibrated local correlation in order to fit a view on the correlation skew, or reproduce historical correlation, or match some exotic option prices, thus improving the pricing, hedging and risk-management of multi-asset derivatives. This technique is generalized to calibrate models that combine stochastic interest rates, stochastic dividend yield, local stochastic volatility and local correlation. Numerical results show the wide variety of calibrated local correlations and give insight into a difficult (still unsolved) problem: finding lower bounds/upper bounds on general multi-asset option prices given the whole surfaces of implied volatilities of a basket and its constituents.

Local and Stochastic Volatility Under Stochastic Interest Rates Using Mixture Models and the Multidimensional Fractional FFT

Stochastic volatility, local volatility and stochastic interest rates are three of the most important extensions to the standard Black-Scholes framework. Although much work has been done on models incorporating one or two of these extensions, very little has been done on the combination of all three. We show how to efficiently calibrate and simulate such a model by utilizing a mixture diffusion based approach, which takes advantage of the multidimensional fractional FFT.

A general control variate method for multi-dimensional SDEs: An application to multi-asset options under local stochastic volatility with jumps models in finance

This paper presents a new control variate method for general multi-dimensional stochastic differential equations (SDEs) including jumps in order to reduce the variance of Monte Carlo method. Our control variate method is based on an asymptotic expansion technique, and does not require an explicit characteristic function of SDEs. This is an extension of previous researches using asymptotic expansions to obtain the control variates for such general models. Moreover, in our control variate method, the regression estimators can be chosen for each number of jump times with a stratified sampling, and improve the efficiency of the variance reduction. This paper also provides the asymptotic bias and variance of our method in terms of its terminal time and a small noise parameter used in an asymptotic expansion method.For an application of our method, we evaluate multi-asset options whose payoffs are expressed as linear functions of the underlying asset price processes in general local stochastic volatility with jumps models, and show a calculation scheme of control variates for Greeks.In numerical experiments, we apply the new control variate method to pricing basket, spread, and average options and Delta of basket options under a ZABR-type local stochastic volatility model with jumps, and confirm that our method works very well.

Short-time at-the-money skew and rough fractional volatility

The Black–Scholes implied volatility skew at the money of SPX options is known to obey a power law with respect to the time to maturity. We construct a model of the underlying asset price process which is dynamically consistent to the power law. The volatility process of the model is driven by a fractional Brownian motion with Hurst parameter less than half. The fractional Brownian motion is correlated with a Brownian motion which drives the asset price process. We derive an asymptotic expansion of the implied volatility as the time to maturity tends to zero. For this purpose, we introduce a new approach to validate such an expansion, which enables us to treat more general models than in the literature. The local-stochastic volatility model is treated as well under an essentially minimal regularity condition in order to show such a standard model cannot be dynamically consistent to the power law.

LSV models with stochastic interest rates and correlated jumps

ABSTRACTPricing and hedging exotic options using local stochastic volatility models drew a serious attention within the last decade, and nowadays became almost a standard approach to this problem. In this paper we show how this framework could be extended by adding to the model stochastic interest rates and correlated jumps in all three components. We also propose a new fully implicit modification of the popular Hundsdorfer and Verwer and Modified Craig–Sneyd finite-difference schemes which provides second-order approximation in space and time, is unconditionally stable and preserves positivity of the solution, while still has a linear complexity in the number of grid nodes.

Calibrating a Market Model to Commodity and Interest Rate Risk

Based on the multi-currency LIBOR Market Model (LMM) this paper constructs a hybrid commodity interest rate market model with a time-dependent stochastic local volatility function allowing the model to simultaneously fit the implied volatility surfaces of commodity and interest rate options. Since liquid market prices are only available for options on commodity futures, rather than forwards, a convexity correction formula for the model is derived to account for the difference between forward and futures prices. A procedure for efficiently calibrating the model to interest rate and commodity volatility smiles is constructed. Finally, the model is fitted to an exogenously given correlation structure between forward interest rates and commodity prices (cross-correlation). When calibrating to options on forwards (rather than futures), the fitting of cross-correlation preserves the (separate) calibration in the two markets (interest rate and commodity options), while in the case of futures a (rapidly converging) iterative fitting procedure is presented. The fitting of cross-correlation is reduced to finding an optimal rotation of volatility vectors, which is shown to be an appropriately modified version of the "orthonormal Procrustes" problem in linear algebra. The calibration approach is demonstrated in an application to market data for oil futures.

From Bond Yield to Macroeconomic Instability: The Effect of Negative Interest Rates

We present a hybrid Heston model with a local stochastic volatility to describe government bond yield dynamics. The model is analytically tractable and, therefore, can be efficiently estimated using the maximum likelihood approach. Twofold is the model contribution. First, it captures changes in the yield volatility and predict future yield values of Germany, French, Italy and Spain. The result is an early-warning indicator which anticipates phases of instability characterizing the time series investigated. Then, the model describes convergence/divergence phenomena among European government bond yields and explores the countries'reactions to a common monetary policy described through the EONIA interbank rate.

B-spline techniques for volatility modeling

This paper is devoted to the application of B-splines to volatility modeling, specifically the calibration of the leverage function in stochastic local volatility (SLV) models and the parameterization of an arbitrage-free implied volatility surface calibrated to sparse option data. We use an extension of classical B-splines obtained by including basis functions with infinite support. We first discuss the application of shape-constrained B-splines to the estimation of conditional expectations, not merely from a scatter plot but also from the given marginal distributions. One application is the Monte Carlo calibration of SLV models by Markov projection. We then present a new technique for the calibration of an implied volatility surface to sparse option data. We use a B-spline parameterization of the Radon–Nikodym derivative of the underlying’s risk-neutral probability density with respect to a roughly calibrated base model. We show that this method provides smooth arbitrage-free implied volatility surfaces. Finally, we sketch a Galerkin method with B-spline finite elements to the solution of the partial differential equation satisfied by the Radon–Nikodym derivative.