Calibration and hedging under jump diffusion

A jump diffusion model coupled with a local volatility function has been suggested by Andersen and Andreasen (2000). By generating a set of option prices assuming a jump diffusion with known parameters, we investigate two crucial challenges intrinsic to this type of model: calibration of parameters and hedging of jump risk. Even though the estimation problem is ill-posed, our results suggest that the model can be calibrated with sufficient accuracy. Two different strategies are explored for hedging jump risk: a semi-static approach and a dynamic technique. Simulation experiments indicate that each of these methods can sharply reduce risk exposure. JEL Classification G12 · G13     

Insiders' hedging in a jump diffusion model

In this paper, we formulate the optimal hedging problem when the underlying stock price has jumps, especially for insiders who have more information than the general public. The jumps in the underlying price process depend on another diffusion process, which models a sequence of firm-specific information. This diffusion process is observed only by insiders. Nevertheless, the market is incomplete to insiders as well as to the general public. We use the local risk minimization method to find an optimal hedging strategy for insiders. We also numerically compare the value of the insider's hedging portfolio with the value of an honest trader's hedging portfolio for a simulated sample path of a stock price.     

Lookback options and diffusion hitting times: A spectral expansion approach

Lookback options have payoffs dependent on the maximum and/or minimum of the underlying price attained during the options lifetime. Based on the relationship between diffusion maximum and minimum and hitting times and the spectral decomposition of diffusion hitting times, this paper gives an analytical characterization of lookback option prices in terms of spectral expansions. In particular, analytical solutions for lookback options under the constant elasticity of variance (CEV) diffusion are obtained.Received: 1 October 2003, Mathematics Subject Classification: 60J35, 60J60, 60G70JEL Classification: G13The author thanks Phelim Boyle for bringing the problem of pricing lookback options under the CEV process to his attention and for useful discussions and Viatcheslav Gorovoi for computational assistance. This research was supported by the U.S. National Science Foundation under grants DMI-0200429 and DMS-0223354.     

The Markov-switching jump diffusion LIBOR market model

In this paper, we introduce an extension to the LIBOR Market Model (LMM) that is suitable to incorporate both sudden market shocks as well as changes in the overall economic climate into the interest rate dynamics. This is achieved by substituting the simple diffusion process of the original LMM by a regime-switching jump diffusion. We demonstrate that the new Markov-switching jump diffusion (MSJD) LMM can be embedded into a generalized regime-switching Heath–Jarrow–Morton model and prove that the considered market is arbitrage-free. We derive pricing formulas for caps, floors and swaptions using Fourier pricing techniques and show how the model can be calibrated to real market data.     

Smart expansion and fast calibration for jump diffusions

Using Malliavin calculus techniques, we derive an analytical formula for the price of European options, for any model including local volatility and Poisson jump processes. We show that the accuracy of the formula depends on the smoothness of the payoff function. Our approach relies on an asymptotic expansion related to small diffusion and small jump frequency/size. Our formula has excellent accuracy (the error on implied Black–Scholes volatilities for call options is smaller than 2 bp for various strikes and maturities). Additionally, model calibration becomes very rapid.      

A slightly depressing jump model: intraday volatility pattern simulation

Hawkes processes have been finding more applications in diverse areas of science, engineering and quantitative finance. In multi-frequency finance various phenomena have been observed, such as shocks, crashes, volatility clustering, turbulent flows and contagion. Hawkes processes have been proposed to model those challenging phenomena appearing across asset prices in various exchanges. The original Hawkes process is an intensity-based model for series of events with path dependence and self-exciting or mutual-exciting mechanisms. This paper introduces a slightly depressing process to model the reverse phenomenon of self-exciting mechanisms. Such a process models the decline in the intensity of jumps observed in market regimes. The proposed birth-immigration-death process captures the decline in jump intensity observed at the start of a daily trading regime while the classical immigration-birth process models an increase in jump intensity towards the close of daily trading. Each of these processes can be expressed as a special case of a simple bivariate Hawkes process.     

On the first hitting time density for a reducible diffusion process

In this paper, we study the classical problem of the first hitting time density to a moving boundary for a diffusion process, which satisfies the Cherkasov condition, and hence, can be reduced to a standard Wiener process. We give two complementary (forward and backward) formulations of this problem and provide semi-analytical solutions for both. By using the method of heat potentials, we show how to reduce these problems to linear Volterra integral equations of the second kind. For small values of t, we solve these equations analytically by using Abel equation approximation; for larger t we solve them numerically. We illustrate our method with representative examples, including Ornstein–Uhlenbeck processes with both constant and time-dependent coefficients. We provide a comparison with other known methods for finding the hitting density of interest, and argue that our method has considerable advantages and provides additional valuable insights. We also show applications of the problem and our method in various areas of financial mathematics.     

Macroeconomic fundamentals,jump dynamics and expected volatility

In this paper, we develop a new volatility model capturing the effects of macroeconomic variables and jump dynamics on the stock volatility. The proposed GARCH-Jump-MIDAS model is applied to the S&P 500 index. Our in-sample results indicate that macroeconomic activities have important impacts on aggregate market volatility. Out-of-sample evidence suggests that our model with macroeconomic variables significantly outperform a wide range of competitors including the original GARCH(1,1), GARCH-MIDAS and GJR-A-MIDAS models. The volatility timing results also show that the information from jumps and macroeconomic activity is helpful for improving the portfolio performance.     

A practical approach to semideviation and its time scaling in a jump-diffusion process

One of the most popular risk-adjusted fund return measures in the asset management industry is the Sortino ratio. It is an alternative to the Sharpe ratio that differentiates harmful volatility from general volatility by taking into account the standard deviation of negative asset returns, a quantity called semideviation. Indeed, the semideviation is generally preferred to the standard deviation when the distribution of the returns is skewed. A common method to annualize it is to use the square-root-of-time rule, where an estimated quantile of a return distribution is scaled to a lower frequency by the square root of the time horizon. However, this relation does not generally hold for this risk measure and often gives a terrible estimation of it. The aim of this paper is to provide a practical approach to semideviation by explaining how it should be computed. We propose and justify the use of a new model, which delivers a more accurate estimation of the downside risk. It is a generalization of the Ball-Torous approximation of a jump-diffusion process, which can be applied when the volatility is constant or stochastic. In the latter case, we use Markov Chain Monte Carlo (MCMC) methods to fit our stochastic volatility model. We also derive an exact formula for the semideviation when the volatility is kept constant, explaining how it should be scaled when considering a lower frequency. For the tests, we apply our methodology to a highly skewed set of returns based on the Barclays US High Yield Index, where we compare different time scalings for the semideviation. Our work shows that the square-root-of-time rule provides a poor approximation of the semideviation, and that the simplification brought by Ball and Torous should be replaced by our new methodology, as it gives much better results.     

Jump-diffusion volatility models for variance swaps: An empirical performance analysis

This paper studies a class of tractable jump-diffusion models, including stochastic volatility models with various specifications of jump intensity for stock returns and variance processes. We employ the Markov chain Monte Carlo (MCMC) method to implement model estimation, and investigate the performance of all models in capturing the term structure of variance swap rates and fitting the dynamics of stock returns. It is evident that the stochastic volatility models, equipped with self-exciting jumps in the spot variance and linearly-dependent jumps in the central-tendency variance, can produce consistent model estimates, aptly explain the stylized facts in variance swaps, and boost pricing performance. Moreover, our empirical results show that large self-exciting jumps in the spot variance, as an independent risk source, facilitate term structure modeling for variance swaps, whilst the central-tendency variance may jump with small sizes, but signaling substantial regime changes in the long run. Both types of jumps occur infrequently, and are more related to market turmoils over the period from 2008 to 2021.     

The euro and pound volatility dynamics: An investigation from conditional jump process

This paper investigates the variations of the volatility of euro and pound after the introduction of euro. A GARJI model is employed to analyze the impact of the news arrivals on the exchange rate volatility. The results are robust to the data-splitting schemes and indicate: (1) the conditional variance of euro is larger than that of pound. (2) The stability of euro exchange rates has made progress in recent years, which is accomplished by the decreases in the jump innovations. This paper supports the arguments on the determinants of exchange rate stability claimed by Mundell (1998) and Mussa (2000).     

ESO compensation: The roles of default risk, employee sentiment, and insider information

This paper derives a pricing model for employee stock options (ESO) that includes default risk and considers employee sentiment. Using ESO data from 1992 to 2004, the study finds that the average executive's subjective value is about 55% of the Black-Scholes value. Only employees who over-estimate firm returns (or insiders who know that the firm is under-valued) by about 10% per annum will prefer ESOs over cash compensation. Our model also shows that work incentives offered by ESOs may be far lower than those implied by Black-Scholes but that ESOs may induce less risk-taking behavior, contrary to typical moral hazard arguments. Findings may impact relevant accounting regulations as well as compensation decisions.     

Modelling jumps in electricity prices: theory and empirical evidence

Objective of this paper is to enhance the understanding of modelling jumps and to analyse the model risk based on the jump component in electricity markets. We provide a common modelling framework that allows to incorporate the main jump patterns observed in electricity spot prices and compare the effectiveness of different jump specifications. To this end, we calibrate the models to daily European Energy Exchange (EEX) market data through Markov Chain Monte Carlo based methods. To assess the quality of the estimated jump processes, we analyse their trajectorial and statistical properties. Moreover, even when the models are calibrated to a cross-section of derivative prices substantial model risk remains.      

Pricing derivatives with modeling CO2 emission allowance using a regime-switching jump diffusion model: with regime-switching risk premium

Carbon markets trade the spot European Union Allowance (EUA), with one EUA providing the right to emit one tone of carbon dioxide (CO₂). We examine the spot EUA returns in BlueNext that exhibit jumps and a volatility clustering feature. We propose a regime-switching jump diffusion model (RSJM) with a hidden Markov chain to capture not only a volatility clustering feature, but also the dynamics of the spot EUA returns that are influenced by change in the CO₂ emission economic conditions. In addition, the switching jump intensities of the RSJM are shown to be affected by change in the carbon-market macroeconomic environment. We further derive the theoretical futures-option prices with a constant convenience yield under the RSJM via the generalized Esscher transform where regime-switching risk is priced with a risk premium. The empirical study shows that the derived futures-option pricing model under the RSJM with regime-switching risk is a more complete model than a jump diffusion model for pricing CO₂ options.     

Stochastic volatility: A tale of co-jumps,non-normality,GMM and high frequency data

In this article we introduce a linear–quadratic volatility model with co-jumps and show how to calibrate this model to a rich dataset. We apply GMM and more specifically match the moments of realized power and multi-power variations, which are obtained from high-frequency stock market data. Our model incorporates two salient features: the setting of simultaneous jumps in both return process and volatility process and the superposition structure of a continuous linear–quadratic volatility process and a Lévy-driven Ornstein–Uhlenbeck process. We compare the quality of fit for several models, and show that our model outperforms the conventional jump diffusion or Bates model. Besides that, we find evidence that the jump sizes are not normally distributed and that our model performs best when the distribution of jump-sizes is only specified through certain (co-) moment conditions. Monte Carlo experiments are employed to confirm this.     

A closed-form solution to American options under general diffusion processes

This paper investigates American option pricing under general diffusion processes. Specifically, the underlying asset price is assumed to follow a diffusion process in which both the dividend yield and volatility are functions of time and the underlying asset price. Using the generalized homotopy analysis method, the determination of the early exercise boundary is separated from the valuation procedure of American options. Then, an exact and explicit solution for American options on a dividend-paying stock is derived as a Maclaurin series. In addition, the corresponding optimal early exercise boundary and the Greeks are obtained in closed-form solutions. A nonlinear sequence transformation, the Padé technique, is used to effectively accelerate the convergence of the partial sums of the infinite series. As the homotopy constructed in this paper is based on a generalized deformation with a shape parameter and kernel function, the error of the homotopic approximation could be reduced further for a fixed order. Numerical examples demonstrate the validity, effectiveness, and flexibility of the proposed approach.     

A revisited and stable Fourier transform method for affine jump diffusion models

In the last decade, fast Fourier transform methods (i.e. FFT) have become the standard tool for pricing and hedging with affine jump diffusion models (i.e. AJD), despite the FFT theoretical framework is still in development and it is known that the early solutions have serious problems in terms of stability and accuracy. This fact depends from the relevant computational gain that the FFT approach offers with respect to the standard Fourier transform methods that make use of a canonical inverse Levy formula. In this work we revisit a classic FT method and find that changing the quadrature algorithm and using alternative, less flawed, representation for the pricing formulas can improve the computational performance up to levels that are only three time slower than FFT can achieve. This allows to have at the same time a reasonable computational speed and the well known stability and accuracy of canonical FT methods.     

Interpreting financial market crashes as earthquakes: A new Early Warning System for medium term crashes

We propose a modeling framework which allows for creating probability predictions on a future market crash in the medium term, like sometime in the next five days. Our framework draws upon noticeable similarities between stock returns around a financial market crash and seismic activity around earthquakes. Our model is incorporated in an Early Warning System for future crash days. Testing our EWS on S&P 500 data during the recent financial crisis, we find positive Hanssen–Kuiper Skill Scores. Furthermore our modeling framework is capable of exploiting information in the returns series not captured by well known and commonly used volatility models. EWS based on our models outperform EWS based on the volatility models forecasting extreme price movements, while forecasting is much less time-consuming.     

The Feller diffusion,filter rules and abnormal stock returns

We determine the conditional expected logarithmic (i.e. continuously compounded) return on a stock whose price evolves in terms of the Feller diffusion and then use it to demonstrate how one must know the exact probability density that describes a stock’s return before one can determine the correct way to calculate the abnormal returns that accrue on the stock. We show in particular that misspecification of the stochastic process which generates a stock’s price will lead to systematic biases in the abnormal returns calculated on the stock. We examine the implications this has for the proper conduct of empirical work and for the evaluation of stock and portfolio performance.