Local Volatility Models Research Articles

Pricing Variable Annuity Contract with GMAB Guarantee Under a Regime Switching Local Volatility Model

Pricing Variable Annuity Contract with GMAB Guarantee Under a Regime Switching Local Volatility Model

A general valuation framework for rough stochastic local volatility models and applications

Rough volatility models are a new class of stochastic volatility models that have been shown to provide a consistently good fit to implied volatility smiles of SPX options. They are continuous-time stochastic volatility models, whose volatility process is driven by a fractional Brownian motion with the corresponding Hurst parameter less than a half. Albeit the empirical success, the valuation of derivative securities under rough volatility models is challenging. The reason is that it is neither a semi-martingale nor a Markov process. This paper proposes a novel valuation framework for rough stochastic local volatility (RSLV) models. In particular, we introduce the perturbed stochastic local volatility (PSLV) model as the semi-martingale approximation for the RSLV model and establish its existence, uniqueness, Markovian representation and convergence. Then we propose a fast continuous-time Markov chain (CTMC) approximation algorithm to the PSLV model and establish its convergence. Numerical experiments demonstrate the convergence of our approximation method to the true prices, and also the remarkable accuracy and efficiency of the method in pricing European, barrier and American options. Comparing with existing literature, a significant reduction in the CPU time to arrive at the same level of accuracy is observed.

Joint calibration of local volatility models with stochastic interest rates using semimartingale optimal transport

We develop and implement a non-parametric method for joint exact calibration of a local volatility model and a correlated stochastic short rate model using semimartingale optimal transport. The method relies on the duality results established in Joseph et al. (Calibration of local volatility models with stochastic interest rates using optimal transport, 2023). and jointly calibrates the whole equity-rate dynamics. It uses an iterative approach which starts with a parametric model and tries to stay close to it, until a perfect calibration is obtained. We demonstrate the performance of our approach on market data using European SPX options and European cap interest rate options. Finally, we compare the joint calibration approach with the sequential calibration, in which the short rate model is calibrated first and frozen.

Extreme ATM skew in a local volatility model with discontinuity: joint density approach

This paper concerns a local volatility model in which the volatility takes two possible values, and the specific value depends on whether the underlying price is above or below a given threshold. The model is known, and a number of results have been obtained for it. In particular, option pricing formulas and a power-law behaviour of the implied volatility skew have been established in the case when the threshold is taken at the money. In this paper, we derive an alternative representation of option pricing formulas. In addition, we obtain an approximation of option prices by the corresponding Black–Scholes prices. Using this approximation streamlines obtaining the aforementioned behaviour of the skew. Our approach is based on the natural relationship of the model with skew Brownian motion and consists of the systematic use of the joint distribution of this stochastic process and some of its functionals.

Convergence of the Two Point Flux Approximation method and the fitted Two Point Flux Approximation method for options pricing with local volatility function

In this paper, we deal with numerical approximations for solving the Black–Scholes Partial Differential Equation (PDE) for European and American options pricing with local volatility. This PDE is well-known to be degenerated. Local volatility model is a model where the volatility depends locally of both stock price and time. In contrast to constant volatility or time-dependent volatility models for which analytical representations of the exact solution is known for European Call options, there is no analytical solution for local volatility. The space discretization is performed using the classical finite volume method with Two-Point Flux Approximation (TPFA) and a novel scheme called Fitted Two-Point Flux Approximation (FTPFA). The Fitted Two-Point Flux Approximation (FTPFA) combines the fitted finite volume method and the standard TPFA method. More precisely the fitted finite volume method is used when the stock price approaches zero with the goal to handle the degeneracy of the PDE while the TPFA method is used on the rest of space domain. This combination yields our fitted TPFA scheme. The Euler method is used for the time discretization. We provide the rigorous convergence proofs of the two fully discretized schemes. Numerical experiments to support theoretical results are provided.

A local volatility correction to mean-reverting stochastic volatility model for pricing derivatives

Generally, in the real market, empirical findings suggest that either local volatility (LV) or stochastic volatility (SV) models have a limit to capture the full dynamics and geometry of the implied volatilities of the given equity options. In this study, to overcome the disadvantage of such LV and SV models, we propose a special type of hybrid stochastic-local volatility (SLV∗) model in which the volatility is given by the squared logarithmic function of the underlying asset price added to a function of a fast mean-reverting process. By making use of asymptotic analysis and Mellin transform, we derive analytic pricing formulas for European derivatives with both smooth and non-smooth payoffs under the SLV∗ model. We run numerical experiments to verify the accuracy of the pricing formulas using a Monte-Carlo simulation method and to display that the proposed new model fits the geometry of the market implied volatility more closely than other models such as the Heston model, the stochastic elasticity of variance (SEV) model, the hybrid stochastic and CEV type local volatility (SVCEV) model and the multiscale stochastic volatility (MSV) model, especially for short time-to-maturity options.

A reproducing kernel Hilbert space approach to singular local stochastic volatility McKean–Vlasov models

Motivated by the challenges related to the calibration of financial models, we consider the problem of numerically solving a singular McKean–Vlasov equation dXt=σ(t,Xt)XtvtE[vt|Xt]dWt,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ d X_{t}= \\sigma (t,X_{t}) X_{t} \\frac{\\sqrt{v}_{t}}{\\sqrt{\\mathbb{E}[v_{t}|X_{t}]}}dW_{t}, $$\\end{document} where W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$W$\\end{document} is a Brownian motion and v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$v$\\end{document} is an adapted diffusion process. This equation can be considered as a singular local stochastic volatility model. While such models are quite popular among practitioners, its well-posedness has unfortunately not yet been fully understood and in general is possibly not guaranteed at all. We develop a novel regularisation approach based on the reproducing kernel Hilbert space (RKHS) technique and show that the regularised model is well posed. Furthermore, we prove propagation of chaos. We demonstrate numerically that a thus regularised model is able to perfectly replicate option prices coming from typical local volatility models. Our results are also applicable to more general McKean–Vlasov equations.

Neural network empowered liquidity pricing in a two-price economy under conic finance settings

In the article at hand neural networks are used to model liquidity in financial markets, under conic finance settings, in two different contexts. That is, on the one hand this paper illustrates how the use of neural networks within a two-price economy allows to obtain accurate pricing and Greeks of financial derivatives, enhancing computational performances compared to classical approaches such as (conic) Monte Carlo. The methodology proposed for this purpose is agnostic of the underlying valuation model, and it easily adapts to all models suitable for pricing in conic financial markets. On the other hand, this article also investigates the possibility of valuing contingent claims under conic assumptions, using local stochastic volatility models, where the local volatility is approximated by means of a (combination of) neural network(s). Moreover, we also show how it is possible to generate hybrid families of distortion functions to better fit the implied liquidity of the market, as well as we introduce a conic version of the SABR model, based on the Wang transform, that still allows for analytical bid and ask pricing formulae.

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.

Consistent asset modelling with random coefficients and switches between regimes

We explore a stochastic model that enables capturing external influences in two specific ways. The model allows for the expression of uncertainty in the parametrisation of the stochastic dynamics and incorporates patterns to account for different behaviours across various times or regimes. To establish our framework, we initially construct a model with random parameters, where the switching between regimes can be dictated either by random variables or deterministically. Such a model is highly interpretable. We further ensure mathematical consistency by demonstrating that the framework can be elegantly expressed through local volatility models taking the form of standard jump diffusions. Additionally, we consider a Markov-modulated approach for the switching between regimes characterised by random parameters. For all considered models, we derive characteristic functions, providing a versatile tool with wide-ranging applications. In a numerical experiment, we apply the framework to the financial problem of option pricing. The impact of parameter uncertainty is analysed in a two-regime model, where the asset process switches between periods of high and low volatility imbued with high and low uncertainty, respectively.

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.

Calibration of Local Volatility Models under the Implied Volatility Criterion

Calibration of Local Volatility Models under the Implied Volatility Criterion

The valuation of real options for risky barrier to entry with hybrid stochastic and local volatility and stochastic investment costs

Real options are a type of investment choice that supports decision-makers in making better strategic management decisions while simultaneously reducing uncertainty in investments. This study presents a new model to help investors flexibly handle uncertain investment environments. First, we adopt a hybrid stochastic and local volatility model to efficiently describe the uncertain effects of the external environment on the project value in decision-making cases. Second, we set the investment cost (or sunk cost) as a geometric Brownian motion (GBM) to illustrate the opportunity costs that originating from discarding alternatives to invest in complex decision-making circumstances. We derive partial differential equations (PDEs) for the value of real options, and then use asymptotic analysis to obtain analytical solutions. Additionally, we analyze the price accuracy of the approximate formulas compared with that of the solutions obtained from a Monte Carlo simulation. Finally, we investigate the numerical effects of various parameters related to stochastic volatility on real options to observe their economic implications.

SHORT-MATURITY ASYMPTOTICS FOR OPTION PRICES WITH INTEREST RATE EFFECTS

We derive the short-maturity asymptotics for option prices in the local volatility model in a new short-maturity limit [Formula: see text] at fixed [Formula: see text], where r is the interest rate and q is the dividend yield. In the case of practical relevance [Formula: see text] being small, however, our result holds for any fixed [Formula: see text]. The result is a generalization of the Berestycki–Busca–Florent formula ( Berestycki et al. , 2002 ) [Asymptotics and calibration of local volatility models, Quantitative Finance 2, 61–69] for the short-maturity asymptotics of the implied volatility which includes interest rates and dividend yield effects of [Formula: see text] to all orders in n. We obtain the analytical results for the ATM volatility and skew in this asymptotic limit. Explicit results are derived for the CEV model. The asymptotic result is tested numerically against the exact evaluation in the square-root model [Formula: see text], which demonstrates that the new asymptotic result is in very good agreement with the exact evaluation in a wide range of model parameters relevant for practical applications.

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.

Monotonicity of implied volatility for perpetual put options

AbstractWe define and study properties of implied volatility for American perpetual put options. In particular, we show that if the market prices are derived from a local volatility model with a monotone volatility function, then the corresponding implied volatility is also monotone as a function of the strike price.

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.

Implementation of variance reduction techniques applied to the pricing of investment certificates

Certificates are structured financial instruments that aim to provide investors with investment solutions tailored to their needs. Certificates can be modeled using a bond component and a derivative component, typically an options strategy. The pricing of certificates is typically performed using the Monte Carlo numerical methodology. Such method allows for projections of the underlying using series of random numbers. The results obtained display an error (standard deviation) that depends on the number of simulations used and on the specific characteristics of the structured product. This work has the objective of minimizing the experimental error, and, consequently, of accelerating the speed of convergence using statistical techniques known in the literature as variance reduction methods. The most popular stochastic dynamics have been analyzed, like the classical Black and Scholes model, the Local Volatility model and the Heston model. Three certificates are analyzed in the paper and they are characterized by different payoffs. The variance reduction techniques, implemented in different programming languages (Python, Matlab and R), are: Latin Hypercube, Stratified Sampling, Antithetic Variables, Importance Sampling, Moment Matching and Control Variates

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.