Local Volatility Research Articles

CALIBRATING LOCAL VOLATILITY MODELS WITH STOCHASTIC DRIFT AND DIFFUSION

We propose Monte Carlo calibration algorithms for three models: local volatility with stochastic interest rates, stochastic local volatility with deterministic interest rates and finally stochastic local volatility with stochastic interest rates. For each model, we include detailed derivations of the corresponding SDE systems and list the required input data and steps for calibration. We give conditions under which a local volatility can exist given European option prices, stochastic interest rate model parameters, and correlations. The models are posed in a foreign exchange setting. The drift term for the exchange rate is given as a difference of two stochastic short rates, domestic and foreign; each modeled by a Gaussian one-factor model with deterministic shift (G1[Formula: see text]) process. For stochastic volatility, we model the variance for the exchange rate by a Cox–Ingersoll–Ross (CIR) process. We include tests to show the convergence and the accuracy of the proposed algorithms.

Non-equilibrium skewness, market crises, and option pricing: Non-linear Langevin model of markets with supersymmetry

This paper presents a tractable model of non-linear dynamics of market returns using a Langevin approach. Due to non-linearity of an interaction potential, the model admits regimes of both small and large return fluctuations. Langevin dynamics are mapped onto an equivalent quantum mechanical (QM) system. Borrowing ideas from supersymmetric quantum mechanics (SUSY QM), a parameterized ground state wave function (WF) of this QM system is used as a direct input to the model, which also fixes a non-linear Langevin potential. Using a two-component Gaussian mixture as a ground state WF with an asymmetric double well potential produces a tractable low-parametric model with interpretable parameters, referred to as the NES (Non-Equilibrium Skew) model. Supersymmetry (SUSY) is then used to find time-dependent solutions of the model in an analytically tractable way. Additional approximations give rise to a final practical version of the NES model, where real-measure and risk-neutral return distributions are given by three component Gaussian mixtures. This produces a closed-form approximation for option pricing in the NES model by a mixture of three Black–Scholes prices, providing accurate calibration to option prices for either benign or distressed market environments, while using only a single volatility parameter. These results stand in stark contrast to the most of other option pricing models such as local, stochastic, or rough volatility models that need more complex specifications of noise to fit the market data.

Quantum pricing with a smile: implementation of local volatility model on quantum computer

Quantum algorithms for the pricing of financial derivatives have been discussed in recent papers. However, the pricing model discussed in those papers is too simple for practical purposes. It motivates us to consider how to implement more complex models used in financial institutions. In this paper, we consider the local volatility (LV) model, in which the volatility of the underlying asset price depends on the price and time. As in previous studies, we use the quantum amplitude estimation (QAE) as the main source of quantum speedup and discuss the state preparation step of the QAE, or equivalently, the implementation of the asset price evolution. We compare two types of state preparation: One is the amplitude encoding (AE) type, where the probability distribution of the derivative’s payoff is encoded to the probabilistic amplitude. The other is the pseudo-random number (PRN) type, where sequences of PRNs are used to simulate the asset price evolution as in classical Monte Carlo simulation. We present detailed circuit diagrams for implementing these preparation methods in fault-tolerant quantum computation and roughly estimate required resources such as the number of qubits and T-count.

A new representation of the risk-neutral distribution and its applications

This paper establishes a novel model-free representation of the risk-neutral density in terms of market-observed options prices by combining exact series representations of the Dirac Delta function and the Carr-Madan asset spanning formula. Compared to the widely used method for obtaining the risk-neutral densities via the Breeden–Litzenberger device, our method yields estimates of risk-neutral densities that are model-free, automatically smooth, and in closed-form. The closed-form feature of our new representation makes it ideal for many potential applications including a new model-free representation of the local volatility function in the Dupire's local volatility model. The validity of our method is demonstrated through simulation studies as well as an empirical application using S&P 500 index option data. Extension of the method to higher dimensions is also obtained by extending the spanning formula.

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>

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.

Local Volatility Model with Local Dividends

Local Volatility Model with Local Dividends

Academic Map and Hotspot Analysis of China's Legal Construction Research since the Eighteenth National Congress of the Communist Party of China: Visual Analysis Based on CiteSpace.

Since the 18th National Congress of the Communist Party of China, the research on the construction of the rule of law in China has received keen attention from scholars. In this study, CSSCI literature published from 2012 to 2022 included in CNKI, Beijing Wanfang, and Chongqing VIP Chinese databases were used as data templates. NoteExpress software was used to deduplicate and organize, and 4,603 literature related to the construction of the rule of law in China were obtained. The citespace5.8 software has carried out a visual analysis, and the results show that: (1) The research on the “rule of law China” as the overall concept in the academic circle first appeared in 2013, and the overall stability was stable, and the local volatility increased. (2) The relationship between researchers and research institutions in this field is relatively loose, and the high-yield author group has not yet been formed. (3) Through the visualization of keyword co-occurrence map and keyword clustering map and the analysis of high-frequency words and centrality, four academic maps and research hotspots are mainly proposed: core cognition of the construction of the rule of law in China, internal theory, external Practice orientation and construction path. (4) Future research on the construction of the rule of law in China strengthens the legal response to scientific and technological information, pays attention to the research on foreign-related fields of the construction of the rule of law in China, and insists on promoting the combination of “domestic rule of law” and “foreign rule of law.”

Pricing with À La Carte Joint Distributions: Multi-Asset Derivatives Via Copulas and the Collocating Local Volatility Model

Pricing with À La Carte Joint Distributions: Multi-Asset Derivatives Via Copulas and the Collocating Local Volatility Model

Vega KT for the Local Volatility Model: An AD Approach

Vega KT for the Local Volatility Model: An AD Approach

Effective Stochastic Local Volatility Models

Effective Stochastic Local Volatility Models

A Quantum Mechanics for interest rate derivatives markets

We present a model-free axiomatic formulation of Option Pricing Theory for interest rate derivatives. In this setting, completely analogous to axiomatic Quantum Mechanics, the role of the wave function is played by the discounted zero-coupon bond price. The theory is linked to term structure models through the Hamiltonian operator, and we show that its associated Schrödinger equation is consistent with the [25] model. We also find the quantum-mechanical equivalent of the standard risk-neutral option pricing formula.In order to take advantage of this quantum framework, we develop a time-dependent perturbation theory and we apply it to obtain an approximate closed-form expression for the price of a call option under the [26] model with a small local volatility perturbation.

Density estimates and short-time asymptotics for a hypoelliptic diffusion process

We study a system of n differential equations, each in dimension d. Only the first equation is forced by a Brownian motion and the dependence structure is such that, under a local weak Hörmander condition, the noise propagates to the whole system. We prove upper bounds for the transition density (heat kernel) and its derivatives of any order. Then we give precise short-time asymptotics of the density at a suitable central limit time scale. Both these results account for the different non-diffusive scales of propagation in the various components. Finally, we provide a valuation formula for short-maturity at-the-money Asian basket options under correlated local volatility dynamics.

Model risk and model choice in the case of barrier options and bonus certificates

In the pricing of exotic options, model risk arises when different models yield different prices, even though they are calibrated to the same observable prices of plain vanilla options. We analyze model risk in the case of barrier options and bonus certificates. This study uses an empirical data set of over 40,000 certificates to analyze the real market extent of model risk for traded barrier options. In particular, applying the local volatility model, the Heston model, and the Bates model, theoretical model risk amounts to about 8.5% (median) of the barrier option value. In contrast, the median empirical model risk, based on the range of market prices, is only 2.2%. We find evidence that the majority of issuers prefer stochastic volatility over local volatility models. Model risk is a factor priced into issuers’ margin policy—that is, they let retail customers pay for their model risk.

Application of Itô processes and Schwartz distributions to local volatility for Margrabe options

The main aim of this paper is a generalization and extension of the Dupire formula to the case of the Margrabe option. The aforementioned financial derivative is not a special case of a plain vanilla option, and therefore, our generalization is significant from both a theoretical and practical point of view. We use advanced stochastic methods, the theory of probability, and the theory of Schwartz distributions to provide a mathematically rigorous proof of the generalized Dupire formula in the space of distributions. We apply the obtained result in the classical Dupire local volatility setting. Furthermore, we discuss its application to interest rate swaptions.

Relationship Between Exchange Rate and Stock Market Volatilities in India

With increasing globalisation and integration of national stock exchanges, for the global investor, the portfolio risk increases not only from the local stock market volatility but also in the exchange rate risk. This paper examines the exchange rate volatility effect on volatility in stock market return from India’s perspective for the period January 2010 to December 2015, applying ARCH and GARCH estimation. The daily data of the BSE SENSEX returns, exchange rates of US dollar/rupee, British pound/rupee, Euros/rupee are used. It is estimated that the Euro/rupee exchange rate volatility has a significant positive effect on the BSE SENSEX return volatility, while the effect of the US dollar/rupee and British pound/rupee exchange rate the volatilities are insignificantly negative. The larger GARCH parameter over the ARCH term indicates that the own lagged values of the stock return cause more volatility in stock returns than the innovations. There exists a highly persistent effect of shocks to the BSE SENSEX return and the volatility effect wanes only slowly

A local volatility model for intraday volume prediction and its empirical study

Based on clarifying the theoretical points of three intraday trading volume forecasting models, the additive model, the product model and the decomposition-optimization model, and conducting the empirical tests using intraday volume data from China stock market, a new intraday trading volume prediction model, the so-called local volatility model was proposed. In terms of transaction volume forecasting and VWAP (volume weighted average price) strategy, the local volatility model is more stable than the historical average estimation. Compared to its fast calculation speed and compatibleness with dynamic forecasting, it has satisfactory prediction accuracy. Its robustness lies between the product model and the decomposition-optimization model, which is better than the production model. Moreover, the model excels at large-cap data and the data with high frequency and low transaction volume fluctuation.

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.