Stochastic Volatility Research Articles

Are exchange rates absorbers of global oil shocks? A generalized structural analysis

This paper studies the impact of global oil market shocks on the level, volatility, and correlation dynamics of real effective exchange rates over time. We find that the USD and the EURO act as shock absorbers, as shocks to the global oil market explain substantial proportions of the volatility and correlation dynamics of each exchange rate – compared to the small or even negligible contributions of exogenous interest rate and effective exchange rate shocks. Further we find an interesting “puzzle” that the Euro Area unexpectedly behaves as if it were an oil exporting country - the EURO appreciates in response to a flow demand shock that increases oil price. Our findings are garnered from a generalized time-varying SVAR model with stochastic volatility that allows for correlated disturbances between observation and transition equations and among transition equations themselves. This approach of enriching dynamics between the first and second moments of endogenous variables is particularly suitable for capturing the transmission of structural level oil shocks to the volatility of exchange rates and their correlation with the global oil market.

Investigating Stock Market Volatility and other Volatility Sources using Stochastic Volatility Models

Investigating Stock Market Volatility and other Volatility Sources using Stochastic Volatility Models

Discrepancy between regulations and practice in initial margin calculation

Counterparty risk remains an issue in over-the-counter derivative transactions following the 2008 financial crisis. While the margin for a derivative transaction can only be transferred until just before the counterparty’s default, the exposure of the derivative transaction can vary stochastically during the margin period of risk, that is, the period from the counterparty’s default to the actual closing-out of the transaction. Thus, the anticipated positive exposure may not be recognized, resulting in counterparty risk. Considering it is difficult to calculate the initial margin (IM) according to the regulations, IM has been calculated in practice using a simplified method proposed by the International Swaps and Derivatives Association (ISDA), which is called the ISDA Standard Initial Margin Model (“ISDA SIMM”). In this study, we derive an approximate formula for some counterparty risk indicators for a stochastic volatility model and illustrate numerical analyses for a call option in the SABR model as an example to examine the effect of the discrepancy between regulations and practices in margin calculation. Our results imply that the IM calculated in practice may be insufficient for counterparty risk management, particularly when the market is volatile.

Market turbulence and investor decision-making in currency option market

This research delves into the effects of recent crises on investor behavior within the currency options market, particularly focusing on the relationship with underlying exchange rates. Analyzing daily EUR/USD currency call pair data from May 4, 2011, to June 19, 2023, we employ a genetic algorithm to calculate stochastic volatility in line with the Garman and Kohlhagen model. Through the application of the STAR model, we identified shifts in investor behavior across various crisis periods. These shifts are linked to inherent asymmetries in the time series data, illustrating the diverse strategies of different investor types, such as fundamentalists and chartists. Our findings reveal how each investor group tailors its approach to these market asymmetries, showcasing distinct strategies and responses to fluctuating market conditions and crises. This study contributes to the financial literature by offering a more nuanced understanding of how crises influence investor behavior and the dynamics of currency markets. Ultimately, it sheds light on the complexities of investor behavior during economic challenges.

Closed‐Form Formulae for Variance and Volatility Swaps Under Stochastic Volatility With Stochastic Liquidity Risks

ABSTRACTWe construct a stochastic volatility model considering stochastic liquidity risks when valuing variance and volatility swaps with discrete sampling. We base our model on Heston stochastic volatility, which is adopted for the modeling of stock prices when the market is perfectly liquid. Stock dynamics are further revised by discounting their prices through the employment of mean reverting market liquidity. We convert the stock dynamics under the physical measure into the one under a risk‐neutral measure via measure transform, with which the analytical valuation of variance and volatility swaps is realized. By taking the limit of sampling frequency, we further consider how both swaps with continuous sampling can be priced. Numerical implementation is finally carried out, with which the capability of the constructed model in capturing the influence of the two common types of financial risks can be clear.

Evolution of option pricing models: From Black-Scholes to Heston and beyond

This essay explores the evolution of option pricing models, tracing their development from the foundational Black-Scholes model to more advanced frameworks such as the Heston model and beyond. Beginning with an introduction to option pricing theory, the essay discusses the origins of the Black-Scholes model and its assumptions, as well as the challenges and limitations it faces. It then examines the extension of the Black-Scholes model, so-called the Black-Scholes-Merton model. It incorporates dividends, and lays the groundwork for further research into options pricing and financial derivatives. Then various stochastic volatility models emerge, and the essay chooses the Heston model as a typical example for analysis, highlighting its advantages and applications in option pricing. Furthermore, the essay compares the Heston model with other option pricing models, including the SABR model and Bates model. At the end of the essay, recent advances and future directions in option pricing are introduced and discussed. Through this comprehensive exploration, readers can gain a deeper understanding of the evolution of option pricing models and their significance in modern finance.

The [formula omitted] stochastic volatility model

This article introduces a new class of stochastic volatility models called logGARCH Stochastic Volatility models (logGARCH-SV). We establish the strict stationarity and second-order stationarity properties of this model class. Additionally, we provide conditions for the existence of higher-order moments. To estimate the parameters of the proposed model, we utilize a sequential Monte Carlo method. Finally, we assess the performance of the suggested estimation method through a simulation study.

Option pricing in a sentiment-biased stochastic volatility model

Option pricing in a sentiment-biased stochastic volatility model

The time-varying effects of geopolitical risk on mutual fund risk taking.

Based on a time-varying parameter vector autoregression model with stochastic volatility (TVP-VAR-SV), this paper investigates the dynamic effects of geopolitical risk on mutual fund risk taking in China across three-time horizons and at three selected time points. Overall, the impulse responses are time-varying and we find a negative effect of geopolitical risk on mutual fund risk taking until 2015, with the short-term effect being the most pronounced, suggesting that when professional investors such as mutual fund managers are faced with the stock valuation uncertainty due to a geopolitical shock, they choose to reduce market risk exposures. After 2015, the short-term effect begins to diminish and gradually turns positive, which could be explained by the fact that with the increasing abundance and diversification of investment instruments, fund managers have more effective investment tools and more sophisticated trading strategies to hedge against geopolitical risk, rather than reducing market risk exposure. Further, we explore the heterogeneous effects of eight types of geopolitical risk and three types of mutual fund. The results indicate that the effect of geopolitical actions is stronger than that of geopolitical threats, while the effect of narrow geopolitical risk is stronger than that of broad geopolitical risk. Moreover, we find that the response of the risk taking of growth funds to the geopolitical risk is weaker than that of balanced and income funds.

Partisan conflict, trade policy uncertainty, and the energy market

This paper investigates the time-varying effects of partisan conflict (PC) and trade policy uncertainty (TPU) on the energy market from January 2000 to July 2021 by employing a time-varying parameter vector autoregressive model with stochastic volatility. The results show that the pass-through effects of the PC and TPU shocks on energy return and volatility varied over time and are different across energy sub-sectors. PC and TPU had a greater impact on energy market return in the short-term, whereas the impact on energy market volatility was spread across relatively long periods. Generally, the impact of TPU on energy return and volatility was greater than that of PC at different lag periods and different points. Further, under different energy frameworks, the trend of the time-varying relationship between PC and TPU was roughly the same, indicating the robustness of our results.

Application of Stochastic Processes in Financial Market Models

This paper explores the significant role of stochastic processes in financial modeling, tracing the evolution from basic Brownian motion to sophisticated stochastic differential equations used in modern financial markets. Beginning with the historical development of Brownian motion, identified by Robert Brown and later mathematically modeled by Louis Bachelier for stock price fluctuations, the paper outlines its foundational influence on the Efficient Market Hypothesis and the random walk theory. The extension of these concepts in the Black-Scholes model for option pricing highlights the practical applications of these theories in predicting financial outcomes. The discussion progresses to geometric Brownian motion (GBM) and its crucial role in stock price modeling, emphasizing its use in Monte Carlo simulations for option pricing. The limitations of the Black-Scholes model and GBM in dealing with real market conditions such as stochastic volatility and jump-diffusion processes are addressed, showcasing the evolution of more complex models like the Heston model and GARCH. Interest rate models like the Vasicek and Cox-Ingersoll-Ross models are evaluated for their real-world applicability, particularly in scenarios of low and negative interest rates. This comprehensive review not only underscores the theoretical advancements in financial modeling but also its practical implications in contemporary financial markets.

Empirical analysis of crude oil dynamics using affine vs. non-affine jump-diffusion models

This paper investigates the dynamics of the United States oil (USO) exchange traded fund (ETF). Daily USO returns are modelled using stochastic volatility (SV) frameworks derived from three different model classes: SV models with contemporaneous jumps in returns and volatility (SVCJ); SV model with jumps in returns only (SVJ); and a pure SV model class without jumps. Six affine and non-affine models are considered within each model class that depend on specification of the drift and the diffusion terms in the variance process, resulting in a total of 18 models that are estimated using particle Markov Chain Monte Carlo (PMCMC) approach. Model evaluation is conducted using the Deviance Information Criterion (DIC), Bayes factors, probability plots, and deviation measures to assess the discrepancy between the estimated volatility and key benchmarks, the crude oil ETF volatility index (OVX) and the realised volatility (RV). Our analysis indicates that models incorporating jumps, particularly the SVCJ-PLY-0.5 and SVCJ-PLY-1.0, more accurately capture USO dynamics than standard SV models. The SVCJ-PLY-0.5 model ranks highest based on DIC statistics and Bayes factors, and both models excel in aligning their estimated volatility with the OVX and RV benchmarks. Overall, the statistical criteria employed in our comparison favour models with jumps over the standard SV model class, suggesting that models incorporating jumps in both return and variance processes (SVCJ) are superior to those with jumps solely in the return process (SVJ). The affine models SVJ-LIN-0.5 and SVCJ-LIN-0.5 with linear variance drift and square root diffusion that are particularly interesting for theoretical finance applications are highly ranked among considered frameworks, outperforming several non-affine alternatives. Our analysis of the regression model for volatility forecasting reveals a significant predictive accuracy in the evaluated models, demonstrating their effectiveness in anticipating future volatility trends.

On the implied volatility skew outside the at-the-money point

The small-maturity implied volatility of an asset pricing model is fully determined by the asymptotics of traded option prices, and thus model-free expressions are available. We show how by sharpening one such expression it is possible to derive a novel general formula for the leading order of the in-the-money and out-of-the money (ITM/OTM) implied volatility skew. We apply this formula to find expressions of the small maturity limiting skew of the Heston stochastic volatility model, of exponential Lévy models and their time changes, as well as that of some recently proposed pricing models with independent log returns.

Pricing vulnerable options under cross-asset markov-modulated jump-diffusion dynamics

In this study, the dynamics of the underlying asset price and the counterparty's asset price are governed by a cross-asset Markov-modulated jump-diffusion model that captures the time-inhomogeneity and systematic cojumps. Additionally, the forward interest rate processes are driven by a Markov-modulated Heath–Jarrow–Morton model to depict stochastic volatilities. Under an incomplete-market setting, we apply the Markov-modulated Esscher transform technique to determine a risk-neutral martingale measure. After determining the Markov-modulated Esscher parameters, we obtain an integral expression on the prices of vulnerable European-style Black–Scholes options. The numerical illustrations indicate that the findings are consistent with Klein (1996) and contribute to the extant literature on cojumping impacts on vulnerable option prices.

Efficient likelihood estimation of Heston model for novel climate-related financial contracts valuation

We propose novel Bitcoin-denominated derivatives contracts on carbon bonds. We consider a futures contract on carbon bonds where its price is expressed in terms of bitcoins. Then, we put forward options on a futures contract of the former type. Governments can use such contracts to hedge climate change and influence the prices of carbon bonds and cryptocurrencies. We show how these derivatives transfer volatility to the bitcoin market without a negative effect in the carbon bonds market. Since the aforementioned options are not yet traded in the market, we price them by assuming that the underlying is driven by the well-known Heston model, where the model parameters are estimated by a novel method based on Shannon wavelets. Heston model belongs to the class of stochastic volatility (SV) models. The discrete observations from the SV model can be seen as a state-space model, that is, a stochastic model in discrete-time which contains two sets of equations, the state equation and the observation equation. While the first describes the transition of a latent process in time, the second shows how an observer measures the latent process at each time period. We infer the properties of the latent variable by means of a filtering algorithm, and we estimate the parameters of the model via maximum likelihood. The evaluation of the likelihood function is a time-consuming task that involves updating and prediction steps of the state variable, leading to the computation of complicated integrals. We calculate these integrals by means of an integration method based on Shannon wavelets, and compare the root mean square error (RMSE) of the estimation with state-of-the-art methods. The results show that the RSME is dramatically reduced in a short CPU time with the use of wavelets.

On Pricing Options Under Two Stochastic Volatility Processes

On Pricing Options Under Two Stochastic Volatility Processes

Pricing of geometric basket Options with Default risk under Stochastic Volatility and Jump diffusion Model

Pricing of geometric basket Options with Default risk under Stochastic Volatility and Jump diffusion Model

A novel term-structure-based Heston model for implied volatility surface

It remains a challenge for existing financial models to accurately capture the shapes of implied volatility (IV) of options with all maturities simultaneously. Inspired by Chen et al.’s empirical finding, ‘the shorter the option expiry, the higher the IV is’, we propose a novel and simple approach to solve this challenge, by introducing a term-structure-based correction (i.e. an exponential increase function of the options expiry) to the volatility of volatility (vol–vol) term of the classical Heston stochastic volatility model. We derive an approximate formula for the IV under the corrected model with the perturbation method and further apply the formula to predict the IVs of options written on the Shanghai Stock Exchange 50 ETF. Numerical experiments and empirical results show that the introduction of a term-structure-based correction function surely overcomes the deficiency of the classical Heston model in capturing the short-term IVs, thus improving notably its performance of IV forecasting.

A note on the determinants of non‐fungible tokens returns

AbstractWe aim to identify the determinants of non‐fungible tokens (NFTs) returns. The 10 most popular NFTs based on their price, trading volume, and market capitalisation are examined. Twenty‐three potential drivers of the returns of each NFT are considered. We employ a Bayesian LASSO model which takes into account stochastic volatility and leverage effect. The results indicate that NFTs returns are primarily driven by volatility and ethereum returns. We find a weak connection between NFTs returns and conventional assets, such as stock, oil, and gold markets.

The non-linear impact of monetary policy on shifts in economic policy uncertainty: evidence from the United States of America

A stochastic volatility estimation of VIX index’s latent volatility is used for the United States of America, as a proxy for the adjustments in the levels of investors’ uncertainty related to current and future economic policies. The impact of monetary policy stance on such measure is examined in the framework of the distributed lag non-linear models (DLNM). We place this analysis in the literature stream emphasizing the various sources of heterogeneity concerning investors’ expectations. The main finding is that the monetary policy does impact non-linearly the adjustments in investors’ predictions. While a tighter monetary policy does generally contribute to an increase in VIX’s latent volatility, the shape of such effect varies across different GLM and GAM specifications of DLNM. This outcome remains robust, even if: (1) we control for the global price of Brent crude and consumers’ confidence; (2) we use, instead of the stochastic framework, a Markov-switching GARCH-based estimator; or (3) we replace the monetary policy instrument with monetary policy uncertainty. We argue that accounting for its nonlinear effects on financial markets is of critical importance for the design of a monetary policy pursuing global financial stability.