Stochastic Volatility Jump Diffusion Model Research Articles

A Simulation and Empirical Study of the Maximum Likelihood Estimator for Stochastic Volatility Jump-Diffusion Models

Abstract We investigate the behaviour of the maximum likelihood estimator (MLE) for stochastic volatility jump-diffusion models commonly used in financial risk management. A simulation study shows the practical conditions under which the MLE behaves according to theory. In an extensive empirical study based on nine indices and more than 6000 individual stocks, we nonetheless find that the MLE is unable to replicate key higher moments. We then introduce a moment-targeted MLE – robust to model misspecification – and revisit both simulation and empirical studies. We find it performs better than the MLE, improving the management of financial risk.

Option Pricing with Fractional Stochastic Volatilities and Jumps

Empirical studies suggest that asset price fluctuations exhibit “long memory”, “volatility smile”, “volatility clustering” and asset prices present “jump”. To fit the above empirical characteristics of the market, this paper proposes a fractional stochastic volatility jump-diffusion model by combining two fractional stochastic volatilities with mixed-exponential jumps. The characteristic function of the log-return is expressed in terms of the solution of two-dimensional fractional Riccati equations of which closed-form solution does not exist. To obtain the explicit characteristic function, we approximate the pricing model by a semimartingale and convert fractional Riccati equations into a classic PDE. By the multi-dimensional Feynman-Kac theorem and the affine structure of the approximate model, we obtain the solution of the PDE with which the explicit characteristic function and its cumulants are derived. Based on the derived characteristic function and Fourier cosine series expansion, we obtain approximate European options prices. By differential evolution algorithm, we calibrate our approximate model and its two nested models to S&P 500 index options and obtain optimal parameter estimates of these models. Numerical results demonstrate the pricing method is fast and accurate. Empirical results demonstrate our approximate model fits the market best among the three models.

Robust consumption and portfolio choice with derivatives trading

This paper analyzes the optimal consumption and portfolio choice problem for an ambiguity-averse investor with recursive preferences. The investor has access to both the stock and derivatives markets. The stock price process follows a stochastic volatility jump-diffusion model and the investor can have different levels of ambiguity about diffusion and jump risks, respectively. We obtain an exact analytical solution for investors with unit elasticity of intertemporal substitution of consumption and an approximate solution otherwise. We find that the optimal consumption policy is more sensitive to ambiguity aversion with respect to diffusion risks than the jump risk. The optimal exposures to diffusion and jump risks are significantly affected by the corresponding ambiguity aversions in the complete market; however, the optimal stock investment is relatively insensitive to jump misspecification in the incomplete market. We also show that taking into consideration ambiguity aversion to diffusion risks and participating in the derivatives market are essential to reduce the potential welfare loss, while the impact of jump misspecification is marginal.

Investigating the effects of illiquidity on credit risks via new liquidity augmented stochastic volatility jump diffusion model

Liquidity is extremely important not only within the context of financial markets but also in every scale of economic transactions. In this study, within the realm of financial markets, we configure liquidity as an independent stochastic process moderating the fluidity of all transactions and hence dynamically changing asset values. This study’s asset value process ignoring liquidity is modelled with a stochastic volatility jump-diffusion () model and that model is augmented with the incorporation of a liquidity process. The new model is called liquidity augmented stochastic volatility jump-diffusion () model. The simulation results suggest that model outperforms the models without liquidity. The application of model on the estimation of probabilities of default and credit risk spreads, using actual financial data of the selected companies listed in Dow Jones 30, reveals that neglecting the liquidity dimension in asset valuation might lead to inefficient assessments of risks. The models without an illiquidity process underestimate the probabilities of default and credit spread risks in comparison to the liquidity augmented model, model, and we believe this might have accounted for the ignored risks that caused the 2007–2008 financial crisis in a great degree.

Investigating the Effects of Illiquidity on Credit Risks via New Liquidity Augmented Stochastic Volatility Jump Diffusion Model

Investigating the Effects of Illiquidity on Credit Risks via New Liquidity Augmented Stochastic Volatility Jump Diffusion Model

Option Pricing under Two-Factor Stochastic Volatility Jump-Diffusion Model

Empirical evidence shows that single-factor stochastic volatility models are not flexible enough to account for the stochastic behavior of the skew, and certain financial assets may exhibit jumps in returns and volatility. This paper introduces a two-factor stochastic volatility jump-diffusion model in which two variance processes with jumps drive the underlying stock price and then considers the valuation on European style option. We derive a semianalytical formula for European vanilla option and develop a fast and accurate numerical algorithm for the computation of the option prices using the fast Fourier transform (FFT) technique. We compare the volatility smile and probability density of the proposed model with those of alternative models, including the normal jump diffusion model and single-factor stochastic volatility model with jumps, respectively. Finally, we provide some sensitivity analysis of the model parameters to the options and several calibration tests using option market data. Numerical examples show that the proposed model has more flexibility to capture the implied volatility term structure and is suitable for empirical work in practice.

An Intelligent Learning and Ensembling Framework for Predicting Option Prices

ABSTRACT Estimating option prices and implied volatilities are critical for option risk management and trading. Common strategies in previous studies have relied on parametric models, including the stochastic volatility model (SV), jump-diffusion model (JD), and Black-Scholes model (BS). However, these models are built on several strict and idealistic assumptions, including lognormality and sample-path continuity. In addition, previous studies on option pricing mainly relied on its own market-level indicators without considering the effect of other concurrent options. To address these challenges, we propose an intelligent learning and ensembling framework based on convolutional neural network (CNN). Specifically, the customized nonparametric learning approach is first utilized to estimate option prices. Second, several traditional parametric models are also applied to estimate these prices. The estimated prices are combined by a CNN to obtain the final estimations. Our experiments based on Chinese SSE 50 ETF options demonstrate that the proposed intelligent framework outperforms the traditional SV model, JD model, and BS model with at least 41.52% performance enhancement in terms of RMSE.

LONG MEMORY VERSION OF STOCHASTIC VOLATILITY JUMP-DIFFUSION MODEL WITH STOCHASTIC INTENSITY

It is widely accepted that certain financial data exhibit long range dependence. We consider a fractional stochastic volatility jump diffusion model in which the stock price follows a double exponential jump diffusion process with volatility described by a long memory stochastic process and intensity rate expressed by an ordinary Cox, Ingersoll, and Ross (CIR) process. By calibrating the model with real data, we examine the performance of the model and also, we illustrate the role of long range dependence property by comparing our presented model with the Heston model.

European option pricing under stochastic volatility jump-diffusion models with transaction cost

In this paper, we consider an underlying general stochastic volatility jump-diffusion model. Option pricing under this general model with transaction costs will lead to handling with nonlinear partial integro-differential equations (here after PIDE). In this case, option replication in a discrete-time framework with transaction costs and the non-uniqueness option pricing in such incomplete market will be studied. Observing and introducing a traded proxy for the volatility in the modern market, we acquire a nonlinear PIDE in the advent of transaction costs. Under appropriate regularity conditions, the existence of the strong solution to this pricing problem has been proved. The corresponding self-financing and positions readjustment for pricing a portfolio will also be discussed in the end.

Numerical aspects of integration in semi-closed option pricing formulas for stochastic volatility jump diffusion models

ABSTRACTIn mathematical finance, a process of calibrating stochastic volatility (SV) option pricing models to real market data involves a numerical calculation of integrals that depend on several model parameters. This optimization task consists of large number of integral evaluations with high precision and low computational time requirements. However, for some model parameters, many numerical quadrature algorithms fail to meet these requirements. We can observe an enormous increase in function evaluations, serious precision problems and a significant increase of computational time. In this paper, we numerically analyse these problems and show that they are especially caused by inaccurately evaluated integrands. We propose a fast regime switching algorithm that tells if it is sufficient to evaluate the integrand in standard double arithmetic or if a higher precision arithmetic has to be used. We compare and recommend numerical quadratures for typical SV models and different parameter values, especially for problematic cases.

A Theory of Scenario Generation

We show how to reduce financial asset prices to low-dimensional scenario trees. Applied to time series, the scenarios and their probabilities become time-varying factors. From S&P 500 options, two or three time-varying scenarios suffice to forecast returns, implied variance or skewness on par, or better, than extant multivariate stochastic volatility jump-diffusion models, while reducing the computational effort to fractions of a second.

Use of Adapted Particle Filters in SVJD Models

Particle Filter algorithms for filtering latent states (volatility and jumps) of Stochastic-Volatility Jump-Diffusion (SVJD) models are being explained. Three versions of the SIR particle filter with adapted proposal distributions to the jump occurrences, jump sizes, and both are derived and their performance is compared in a simulation study to the un-adapted particle filter. The filter adapted to both the jump occurrences and jump sizes achieves the best performance, followed in their respective order by the filter adapted only to the jump occurrences and the filter adapted only to the jump sizes. All adapted particle filters outperformed the unadapted particle filter.

Option Pricing in Stochastic Volatility Models Driven by Fractional Jump-Diffusion Processes

In this paper, we propose a fractional stochastic volatility jump-diffusion model which extends the Bates(1996) model. Where we model the volatility as a fractional process. Extensive empirical studies show that the distributions of the logarithmic returns of financial asset usually exhibit properties of self-similarity and long-range dependence and since the fractional Brownian motion has these two important properties, it has the ability to capture the behavior of underlying asset price. Further incorporating jumps into the stochastic volatility framework gives further freedom to financial mathematicians to fit both the short and long end of the implied volatility surface. We propose a stochastic model which contains both fractional and jump process. Then we price options using Monte Carlo simulations along with a variance reduction technique(antithetic variates). We use market data from the S&P 500 index and we compare our results with the Heston and Bates model using error measures. The results show our model greatly outperforms previous models in terms of estimation accuracy.

Robust Portfolio Choice and Consumption with Derivative Trading Under Stochastic Volatility and Jumps

This paper analyzes the optimal consumption and portfolio choice problem for an ambiguity averse investor who has access to the stock and derivatives markets with recursive preferences. The stock process follows a stochastic volatility jump-diffusion model and the investor can have different levels of ambiguity about diffusion risks and the jump risk, respectively. We obtain an analytical solution which is exact when the investor has unit elasticity of intertemporal substitution of consumption, and approximate otherwise. We find that optimal exposures to diffusion risks and to the jump risk are significantly affected by the ambiguity aversion about the corresponding risk factors in the complete market. However, the optimal stock investment is insensitive to the ambiguity aversion about the jump risk in the incomplete market. We also find that considering ambiguity aversion with respect to diffusion risks and participating in the derivatives markets are essential to reduce the potential welfare loss, while the impact of ignoring the jump ambiguity is negligible.

Forecasting Foreign Exchange Rate Movements with k-Nearest-Neighbour, Ridge Regression and Feed-Forward Neural Networks

Three different classes of data mining methods (k-Nearest Neighbour, Ridge Regression and Multilayer Perceptron Feed-Forward Neural Networks) are applied for the purpose of quantitative trading on 10 simulated time series, as well as real world time series of 10 currency exchange rates ranging from 1.11.1999 to 12.6.2015. Each method is tested in multiple variants. The k-NN algorithm is applied alternatively with the Euclidian, Manhattan, Mahalanobis and Maximum distance function. The Ridge Regression is applied as Linear and Quadratic, and the Feed-Forward Neural Network is applied with either 1, 2 or 3 hidden layers. In addition to that Principal Component Analysis (PCA) is eventually applied for the dimensionality reduction of the predictor set and the meta-parameters of the methods are optimized on the validation sample. In the simulation study a Stochastic-Volatility Jump-Diffusion model, extended alternatively with 10 different non-linear conditional mean patterns, is used, to simulate the asset price behaviour to which the tested methods are applied. The results show that no single method was able to profit on all of the non-linear patterns in the simulated time series, but instead different methods worked well for different patterns. Alternatively, past price movements and past returns were used as predictors. In the case when the past price movements were used, quadratic ridge regression achieved the most robust results, followed by some of the k-NN methods. In the case when past returns were used, k-NN based methods were the most consistently profitable, followed by the linear ridge regression and quadratic ridge regression. Neural networks, while being able to profit on some of the time series, did not achieve profit on most of the others. No evidence was further found of the PCA method to improve the results of the tested methods in a systematic way. In the second part of the study, the models were applied to empirical foreign exchange rate time series. Overall the profitability of the methods was rather low, with most of them ending with a loss on most of the currencies. The most profitable currency was EURUSD, followed by EURJPY, GBPJPY and EURGBP. The most successful methods were the linear ridge regression and the Manhattan distance based k-NN method which both ended with profits for most of the time series (unlike the other methods). Finally, a forward selection procedure using the linear ridge regression was applied to extend the original predictor set with some technical indicators. The selection procedure achieved limited success in improving the out-sample results for the linear ridge regression model but not the other models.

Contingent Convertible Bonds with Floating Coupon Payments: Fixing the Equilibrium Problem

Contingent convertible bonds (CoCos) are increasingly popular financial instruments used by banks to satisfy capital requirements. CoCos with market-based conversion triggers in particular receive much attention in the literature. The pricing of CoCos with such a market trigger is problematic as the market value of equity itself depends on the firm’s capital structure. This results in a not-unique arbitrage-free price for the CoCos. We propose a new type of CoCos with a market based trigger and floating coupons. The coupons increase near the trigger value to compensate CoCo holders for the possibility of bankruptcy before conversion. This leads to a unique no-arbitrage price before conversion. The properties of the innovative CoCo contract are studied for different dynamic models of a bank’s assets, such as the (Black-Scholes) Merton and stochastic volatility jump diffusion model. In particular, we illustrate how the CoCo coupons vary as functions of jump intensities and volatilities.

Efficient pricing and hedging under the double Heston stochastic volatility jump-diffusion model

The performances of jump-risk mitigation under a double Heston stochastic volatility jump-diffusion (dbHJ) model are examined. The quadratic exponential (QE) scheme is used to simulate the -measure price paths supposed to follow the dbHJ model, whereas a Fourier-COS-expansion-based scheme (i.e. the COS formula, see [F. Fang and C.W. Oosterlee, A novel pricing method for European based on Fourier-cosine series expansions, SIAM J. Sci. Comput. 31 (2008), pp. 826–848] is employed to price options and to calculate Greeks. Numerical results from extensive dynamic hedging experiments suggest that, when facing a -measure market with stiff volatility skews and non-trivial jumps, the dbHJ model is better than the plain (double) Heston and Black–Scholes models, and slightly outperforms the Heston stochastic volatility jump-diffusion (HJ) model and the Merton model in mitigating the jump risk of an option. This conclusion holds independent of the model specification of -measure market.

Time-variations in commodity price jumps

In this paper, we study jumps in commodity prices. Unlike assumed in existing models of commodity price dynamics, a simple analysis of the data reveals that the probability of tail events is not constant but depends on the time of the year, i.e. exhibits seasonality. We propose a stochastic volatility jump–diffusion model to capture this seasonal variation. Applying the Markov Chain Monte Carlo (MCMC) methodology, we estimate our model using 20 years of futures data from four different commodity markets. We find strong statistical evidence to suggest that our model with seasonal jump intensity outperforms models featuring a constant jump intensity. To demonstrate the practical relevance of our findings, we show that our model typically improves Value-at-Risk (VaR) forecasts.

Time-Variations in Commodity Price Jumps

In this paper, we study jumps in commodity prices. Unlike assumed in existing models of commodity price dynamics, a simple analysis of the data reveals that the probability of tail events is not constant but depends on the time of the year, i.e. exhibits seasonality. We propose a stochastic volatility jump-diffusion model to capture this seasonal variation. Applying the Markov Chain Monte Carlo (MCMC) methodology, we estimate our model using 20 years of futures data from four different commodity markets. We find strong statistical evidence to suggest that our model with seasonal jump intensity outperforms models featuring a constant jump intensity. To demonstrate the practical relevance of our findings, we show that our model typically improves Value-at-Risk (VaR) forecasts.

ABC of SV: Limited information likelihood inference in stochastic volatility jump-diffusion models

We develop novel methods for estimation and filtering of continuous-time models with stochastic volatility and jumps using so-called Approximate Bayesian Computation which build likelihoods based on limited information. The proposed estimators and filters are computationally attractive relative to standard likelihood-based versions since they rely on low-dimensional auxiliary statistics and so avoid computation of high-dimensional integrals. Despite their computational simplicity, we find that estimators and filters perform well in practice and lead to precise estimates of model parameters and latent variables. We show how the methods can incorporate intra-daily information to improve on the estimation and filtering. In particular, the availability of realized volatility measures help us in learning about parameters and latent states. The method is employed in the estimation of a flexible stochastic volatility model for the dynamics of the S&P 500 equity index. We find evidence of the presence of a dynamic jump rate and in favor of a structural break in parameters at the time of the recent financial crisis. We find evidence that possible measurement error in log price is small and has little effect on parameter estimates. Smoothing shows that, recently, volatility and the jump rate have returned to the low levels of 2004–2006.