Local Stochastic Volatility Models Research Articles

Deep calibration with random grids

We propose a neural network-based approach to calibrating stochastic volatility models, which combines the pioneering grid approach by Horvath et al. [Deep learning volatility: A deep neural network perspective on pricing and calibration in (rough) volatility models. Quant. Finance, 2021, 21(1), 11–27]. with the pointwise two-stage calibration of Bayer and Stemper [Deep calibration of rough stochastic volatility models. Working Paper, arXiv:1810.03399, 2018] and Liu et al. [A neural network-based framework for financial model calibration. J. Math. Ind., 2019, 9(1), 1–28]. Our methodology inherits robustness from the former while not suffering from the need for interpolation/extrapolation techniques, a clear advantage ensured by the pointwise approach. The crucial point to the entire procedure is the generation of implied volatility surfaces on random grids, which one dispenses to the network in the training phase. We support the validity of our calibration technique with several empirical and Monte Carlo experiments for the rough Bergomi and Heston models under a simple but effective parametrization of the forward variance curve. The approach paves the way for valuable applications in financial engineering—for instance, pricing under local stochastic volatility models—and extensions to the fast-growing field of path-dependent volatility models.

Continuous-time Markov chain approximation for pricing Asian options under rough stochastic local volatility models

We propose a general framework for pricing both discretely and continuously monitored arithmetic average Asian options whose underlying asset price satisfies the rough stochastic local volatility model. We use a semimartingale approximation approach for the rough stochastic local volatility model to obtain its Markovian representation. For each type of Asian options, we use the double-layer continuous-time Markov chain to approximate the transformed underlying asset price, and derive the double transform of the Asian option price in terms of the unique bounded solution to a related functional equation. We invert the analytical double transforms of Asian option prices under the approximate continuous-time Markov chain numerically to obtain the approximations for Asian option prices. Monte Carlo simulation demonstrates the accuracy and efficiency of the proposed method for Asian option pricing.

Power utility maximization problem under rough stochastic local volatility models with continuous-time Markov chain approximation

This paper studies the portfolio optimization problem based on rough stochastic local volatility models. Due to the challenge that this model is neither a semimartingale nor a Markov process, we first introduce the semimartingale approximate optimization problem and present the convergence analysis. We use the classic stochastic control method and continuous-time Markov chain approximation approach to solve the approximate optimization problem respectively. Then the optimal investment strategy and the HJB equation satisfied by the value function are obtained respectively. Furthermore, for the classic stochastic control method, we demonstrate the existence of solutions for the associated partial differential equation via the nonlinear Feynman–Kac formula. For continuous-time Markov chain approximation method, we prove the convergence of the value function. Finally, we provide a numerical analysis to illustrate the above results by using the power utility function. The numerical analysis shows that continuous-time Markov chain approximation method is effective in solving the portfolio optimization problem under rough stochastic local volatility models.

A general approach for lookback option pricing under Markov models

We propose a computationally efficient method for pricing various types of lookback options under Markov models. We utilize the model-free representations of lookback option prices as integrals of first passage probabilities. We combine efficient numerical quadrature with continuous-time Markov chain approximation for the first passage problem to price lookbacks. Our method is applicable to a variety of models, including one-dimensional time-homogeneous and time-inhomogeneous Markov processes, regime-switching models and stochastic local volatility models. We demonstrate the efficiency of our method through various numerical examples.

SPX, VIX and scale-invariant LSV*

Local Stochastic Volatility (LSV) models have been used for pricing and hedging derivatives positions for over 20 years. An enormous body of literature covers analytical and numerical techniques for calibrating the m model to market data. However, the literature misses a potent approach commonly used in physics and works with absolute (dimensional) variables rather than with relative (nondimensional) ones. While model parameters defined in absolute terms are counter-intuitive for trading desks and tend to be heavily time-dependent, relative parameters are intuitive and stable, making it easy to steer the model adequately and consistently with its Profit and Loss (PnL) explanation power. We propose a specification that first explores historical data and uses physically well-defined relative quantities to design the model. We then develop an efficient hybrid method to price derivatives under this specification. We also show how our method can be used for robust scenario generation purposes�an important risk management task vital for buy-side firms.

A stochastic-local volatility model with [formula omitted] jumps for pricing derivatives

We introduce a new mixed model unifying local volatility, pure stochastic volatility, and Le´vy type of jumps in this paper. Our model framework allows the pure stochastic volatility and the jump intensity to be functions of fast (and slowly) varying stochastic processes and the local volatility to be given in a constant elasticity of variance type of parametric form. We use asymptotic analysis to derive a system of partial integro-differential equations for the prices of European derivatives and use Fourier analysis to obtain an explicit formula for the prices. Our result is an extension of a stochastic-local volatility model from no jumps to Le´vy jumps and an extension of a multiscale stochastic volatility model with Le´vy jumps from the zero elasticity of variance to the non-zero elasticity of variance. We find that our model outperforms two benchmark models in view of fitting performance when the time-to-maturities of European options are relatively short and outperforms one benchmark model in terms of pricing time cost.

A general framework for a joint calibration of VIX and VXX options

We analyze the VIX futures market with a focus on the exchange-traded notes written on such contracts, in particular we investigate the VXX notes tracking the short-end part of the futures term structure. Inspired by recent developments in commodity smile modelling, we present a multi-factor stochastic-local volatility model that is able to jointly calibrate plain-vanilla options both on VIX futures and VXX notes, thus going beyond the failure of purely stochastic or simply local-volatility models. We discuss numerical results on real market data by highlighting the impact of model parameters on implied volatilities.

Arbitrage-Free Neural-SDE Market Models

Modelling joint dynamics of liquid vanilla options is crucial for arbitrage-free pricing of illiquid derivatives and managing risks of option trade books. This paper develops a nonparametric model for the European options book respecting underlying financial constraints and while being practically implementable. We derive a state space for prices which are free from static (or model-independent) arbitrage and study the inference problem where a model is learnt from discrete time series data of stock and option prices. We use neural networks as function approximators for the drift and diffusion of the modelled SDE system, and impose constraints on the neural nets such that no-arbitrage conditions are preserved. In particular, we give methods to calibrate neural SDE models which are guaranteed to satisfy a set of linear inequalities. We validate our approach with numerical experiments using data generated from a Heston stochastic local volatility model.

A Heston local-stochastic volatility model for optimal investment–reinsurance strategy with a defaultable bond in an ambiguous environment

A Heston local-stochastic volatility model for optimal investment–reinsurance strategy with a defaultable bond in an ambiguous environment

A dynamic Heston local–stochastic volatility model and Legendre transform dual-asymptotic solution for optimal investment strategy problems with CARA utility

In this paper, we consider a Heston local–stochastic volatility (HLSV) model to study an optimal investment strategy problem, and analyze the optimal strategy when the volatility component of the model obeys a slow varying process and a fast varying process, respectively. For the optimal investment objective with a constant absolute risk aversion (CARA) utility function, the analytical solution under the HLSV model cannot be obtained due to the complicated nonlinearity of the partial differential equation. In this paper we employ a dual method, Legendre transformation, and an asymptotic expansion technique to derive an asymptotic solution. We also apply a Monte Carlo method to compute the optimal strategy, which can be compared with the asymptotic solution. Finally, numerical examples are provided to support our theoretical results.

Three Strikes and You�re Out

Suggests a new methodology for approximating barrier option prices in stochastic local volatility models. For touch options the method only requires the input of the level, slope and curvature around the barrier in addition to the stochastic volatility parameters. Three strikes are enough.

Joint calibration of S&P 500 and VIX options under local stochastic volatility models

AbstractIt is extremely challenging to design a model calibrating both SPX and VIX option prices. A long‐standing conjecture due to Julien Guyon is that it may not be possible to calibrate these two quantities with a continuous model. So far, most studied continuous time models are affine, so we investigate the conjecture among 14 well‐known non‐affine local stochastic volatility models in this article. First, we propose a unified efficient willow tree method for S&P500 and VIX option pricing under non‐affine models. Second, we compare the joint calibration performance on these 14 models on the S&P500 and VIX option prices data from 2006 to 2019. We find that the VIX option price data can provide extra information of the variance dynamics of the models, and the non‐affine structure and the volatility with linear, rather than square‐root, diffusion process provide a better fit for the both data sets than the affine counterparts. Among the 14 stochastic models, the SABR model provides the best in‐ and out‐of‐sample performance (in terms of mean square error) regardless of the state of the economy. Nevertheless, even for the best‐fitted SABR model, the relative error on the VIX option is still around 18%, still quite significant. Therefore, we found the non‐affine and local volatility structure improve the joint calibration but are still far from satisfactory.

Dynamics and synchronization of global equilibrium interest rates

ABSTRACT Empirical estimates of equilibrium real interest rates are so far mostly limited to advanced economies, since no statistical procedure suitable for a larger set of countries is available. This is surprising, as equilibrium rates have strong policy implications in emerging markets and developing economies as well. Current estimates of the global equilibrium rate rely on only a few countries but estimates for a more diverse set of countries can improve understanding of its dynamics and drivers. This paper uses a model and estimation strategy that decompose ex-ante real interest rates into permanent and transitory components even with short samples and high volatility. This is done with an unobserved component local level stochastic volatility model, which is used to estimate equilibrium rates for 50 countries with Bayesian methods. Equilibrium rates were lower in emerging markets and developing economies than in advanced economies in the 1980s, similar in the 1990s, and have been higher since 2000. In line with economic integration and rising global capital markets, synchronization has been rising over time and is higher among advanced economies. Equilibrium rates of countries with stronger trade linkages and similar demographic and economic trends are more synchronized.

Pricing autocallables under local-stochastic volatility

This paper investigates the pricing of single-asset autocallable barrier reverse convertibles in the Heston local-stochastic volatility (LSV) model. Despite their complexity, autocallable structured notes are the most traded equity-linked exotic derivatives. The autocallable payoff embeds an early-redemption feature generating strong path- and model-dependency. Consequently, the commonly-used local volatility (LV) model is overly simplified for pricing and risk management. Given its ability to match the implied volatility smile and reproduce its realistic dynamics, the LSV model is, in contrast, better suited for exotic derivatives such as autocallables. We use quasi-Monte Carlo methods to study the pricing given the Heston LSV model and compare it with the LV model. In particular, we establish the sensitivity of the valuation differences of autocallables between the two models with respect to payoff features, model parameters, underlying characteristics, and volatility regimes. We find that the improved spot-volatility dynamics captured by the Heston LSV model typically result in higher prices, demonstrating the dependence of autocallables on the forward-skew and vol-of-vol risk. Moreover, we show that the parameters of the stochastic component of LSV models enable controlling for the autocallables price while leaving the fit to European options unaffected.

Delta family approach for the stochastic control problems of utility maximization

In this paper, we propose a new approach for stochastic control problems arising from utility maximization. The main idea is to directly start from the dynamical programming equation and compute the conditional expectation using a novel representation of the conditional density function through the Dirac Delta function and the corresponding series representation. We obtain an explicit series representation of the value function, whose coefficients are expressed through integration of the value function at a later time point against a chosen basis function. Thus we are able to set up a recursive integration time-stepping scheme to compute the optimal value function given the known terminal condition, e.g. utility function. Due to tensor decomposition property of the Dirac Delta function in high dimensions, it is straightforward to extend our approach to solving high-dimensional stochastic control problems. The backward recursive nature of the method also allows for solving stochastic control and stopping problems, i.e. mixed control problems. We illustrate the method through solving some two-dimensional stochastic control (and stopping) problems, including the case under the classical and rough Heston stochastic volatility models, and stochastic local volatility models such as the stochastic alpha beta rho (SABR) model.

Stochastic local volatility models and the Wei-Norman factorization method

<p style='text-indent:20px;'>In this paper, we show that a time-dependent local stochastic volatility (SLV) model can be reduced to a system of autonomous PDEs that can be solved using the heat kernel, by means of the Wei-Norman factorization method and Lie algebraic techniques. Then, we compare the results of traditional Monte Carlo simulations with the explicit solutions obtained by said techniques. This approach is new in the literature and, in addition to reducing a non-autonomous problem into an autonomous one, allows for shorter time in numerical computations.</p>

Effective Stochastic Local Volatility Models

Effective Stochastic Local Volatility Models

Convergence analysis for continuous-time Markov chain approximation of stochastic local volatility models: Option pricing and Greeks

This paper establishes the precise second order convergence rates of the continuous-time Markov chain (CTMC) approximation method for pricing options under the general framework of stochastic local volatility (SLV) models. The stochastic local volatility models studied in this paper include Heston model, 4/2 model, α-Hypergeometric model, stochastic alpha beta rho (SABR) model, Heston-SABR model and quadratic SLV model. Using the stochastic flow theorem, the closed-form CTMC approximation formula for the Greeks are obtained and the second order convergence rates are proved. Numerical examples confirm the theoretical findings.

Pricing commodity index options

We present a stochastic local volatility model for derivative contracts on commodity futures. The aim of the model is to recover the prices of derivative claims both on future contracts and on indices on future strategies. Numerical examples for calibration and pricing are provided for the S&P GSCI Crude Oil excess-return index.

Effective stochastic local volatility models

If a high degree of accuracy and market consistency is required for option pricing, stochastic local volatility models are often the approach of choice. When calibrating these types of models, one of the major challenges lies in the proper fitting of the leverage function. This often requires an optimization procedure in terms of computationally intensive numerical methods, such as Monte Carlo simulation, or methods not well suited to local volatility formulations, such as Fourier transform pricing. In this article, we provide an alternative approach using an effective stochastic volatility technique, which provides an efficient semi-analytical approximation of the PDE for the density function of the underlying. This approach allows efficient direct calibration of the leverage function for a large class of stochastic local volatility models, which includes stochastic volatility models such as the SABR, ZABR or Heston model as the underlying base model. We provide calibration and computational schemes and illustrate our approach using numerical experiments.