Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices

This paper examines the empirical performance of jump diffusion models of stock price dynamics from joint options and stock markets data. The paper introduces a model with discontinuous correlated jumps in stock prices and stock price volatility, and with state-dependent arrival intensity. We discuss how to perform likelihood-based inference based upon joint options/returns data and present estimates of risk premiums for jump and volatility risks. The paper finds that while complex jump specifications add little explanatory power in fitting options data, these models fare better in fitting options and returns data simultaneously.     

Invisible Parameters in Option Prices

This paper characterizes contingent claim formulas that are independent of parameters governing the probability distribution of asset returns. While these parameters may affect stock, bond, and option values, they are “invisible” because they do not appear in the option formulas. For example, the Black-Scholes ( 1973 ) formula is independent of the mean of the stock return. This paper presents a new formula based on the log-negative-binomial distribution. In analogy with Cox, Ross, and Rubinstein's ( 1979 ) log-binomial formula, the log-negative-binomial option price does not depend on the jump probability. This paper also presents a new formula based on the log-gamma distribution. In this formula, the option price does not depend on the scale of the stock return, but does depend on the mean of the stock return. This paper extends the log-gamma formula to continuous time by defining a gamma process. The gamma process is a jump process with independent increments that generalizes the Wiener process. Unlike the Poisson process, the gamma process can instantaneously jump to a continuum of values. Hence, it is fundamentally “unhedgeable.” If the gamma process jumps upward, then stock returns are positively skewed, and if the gamma process jumps downward, then stock returns are negatively skewed. The gamma process has one more parameter than a Wiener process; this parameter controls the jump intensity and skewness of the process. The skewness of the log-gamma process generates strike biases in options. In contrast to the results of diffusion models, these biases increase for short maturity options. Thus, the log-gamma model produces a parsimonious option-pricing formula that is consistent with empirical biases in the Black-Scholes formula.     

The Statistical and Economic Role of Jumps in Continuous-Time Interest Rate Models

This paper analyzes the role of jumps in continuous‐time short rate models. I first develop a test to detect jump‐induced misspecification and, using Treasury bill rates, find evidence for the presence of jumps. Second, I specify and estimate a nonparametric jump‐diffusion model. Results indicate that jumps play an important statistical role. Estimates of jump times and sizes indicate that unexpected news about the macroeconomy generates the jumps. Finally, I investigate the pricing implications of jumps. Jumps generally have a minor impact on yields, but they are important for pricing interest rate options.     

Jump Diffusion Option Valuation in Discrete Time

We develop a simple, discrete time model to value options when the underlying process follows a jump diffusion process. Multivariate jumps are superimposed on the binomial model of Cox, Ross, and Rubinstein (1979) to obtain a model with a limiting jump diffusion process. This model incorporates the early exercise feature of American options as well as arbitrary jump distributions. It yields an efficient computational procedure that can be implemented in practice. As an application of the model, we illustrate some characteristics of the early exercise boundary of American options with certain types of jump distributions.     

A self-exciting switching jump diffusion: properties,calibration and hitting time

A way to model the clustering of jumps in asset prices consists in combining a diffusion process with a jump Hawkes process in the dynamics of the asset prices. This article proposes a new alternative model based on regime switching processes, referred to as a self-exciting switching jump diffusion (SESJD) model. In this model, jumps in the asset prices are synchronized with changes of states of a hidden Markov chain. The matrix of transition probabilities of this chain is designed in order to approximate the dynamics of a Hawkes process. This model presents several advantages compared to other jump clustering models. Firstly, the SESJD model is easy to fit to time series since estimation can be performed with an enhanced Hamilton filter. Secondly, the model explains various forms of option volatility smiles. Thirdly, several properties about the hitting times of the SESJD model can be inferred by using a fluid embedding technique, which leads to closed form expressions for some financial derivatives, like perpetual binary options.     

Pricing American interest rate options under the jump-extended constant-elasticity-of-variance short rate models

This paper demonstrates how to value American interest rate options under the jump-extended constant-elasticity-of-variance (CEV) models. We consider both exponential jumps (see Duffie et al., 2000) and lognormal jumps (see Johannes, 2004) in the short rate process. We show how to superimpose recombining multinomial jump trees on the diffusion trees, creating mixed jump-diffusion trees for the CEV models of short rate extended with exponential and lognormal jumps. Our simulations for the special case of jump-extended Cox, Ingersoll, and Ross (CIR) square root model show a significant computational advantage over the Longstaff and Schwartz’s (2001) least-squares regression method (LSM) for pricing American options on zero-coupon bonds.     

Exchange options under clustered jump dynamics

Exchange options are one of the most popular exotic options, and have important implications for many common financial arrangements and for implied beta as a measure of systematic risk. In this study, we extend the existing literature on exchange options to allow for clustered jump contagion dynamics in each single asset, as well as across assets, using the Hawkes jump-diffusion model. We derive the analytical pricing formulae, the Greeks, and the optimal hedging strategy via Fourier transforms. Using an illustrative numerical analysis, we present the relationship between the exchange option price and clustered jump intensities and jump sizes in the underlying assets. We discuss the managerial insights on financial arrangements with exchange option characteristics. Furthermore, we discuss the implications of incorporating clustered jumps into the estimation of implied beta with exchange options, in which the applications can be insightful and useful in finance practice.     

EXACT FORMULAS FOR PRICING BONDS AND OPTIONS WHEN INTEREST RATE DIFFUSIONS CONTAIN JUMPS

I develop Heath‐Jarrow‐Morton extensions of the Vasicek and Jamshidian pure‐diffusion models, extend these models to incorporate Poisson‐Gaussian interest rate jumps, and obtain closed‐form models for valuing default‐free, zero‐coupon bonds and European call and put options on default‐free, zero‐coupon bonds in a market where interest rates can experience discontinuous information shocks. The jump‐diffusion pricing models value the instrument as the probability‐weighted average of the pure‐diffusion model prices, each conditional on a specific number of jumps occurring during the life of the instrument. I extend the models to coupon‐bearing instruments by applying Jamshidian's serial‐decomposition technique.     

Parisian options with jumps: a maturity–excursion randomization approach

This paper introduces an analytically tractable method for the pricing of European and American Parisian options in a flexible jump–diffusion model. Our contribution is threefold. First, using a double Laplace–Carson transform with respect to the option maturity and the Parisian (excursion) time, we obtain closed-form solutions for different types of Parisian contracts. Our approach allows us also to analytically disentangle contributions of the jump and diffusion components for Parisian options in the excursion region. Second, we provide numerical examples and quantify the impact of jumps on the option price and the Greeks. Finally, we study the non-monotonic effects of volatility and jump intensity close to the excursion barrier, which are important for shareholders’ investment policy decisions in a levered firm.     

News Arrival, Jump Dynamics, and Volatility Components for Individual Stock Returns

This paper models components of the return distribution, which are assumed to be directed by a latent news process. The conditional variance of returns is a combination of jumps and smoothly changing components. A heterogeneous Poisson process with a time‐varying conditional intensity parameter governs the likelihood of jumps. Unlike typical jump models with stochastic volatility, previous realizations of both jump and normal innovations can feed back asymmetrically into expected volatility. This model improves forecasts of volatility, particularly after large changes in stock returns. We provide empirical evidence of the impact and feedback effects of jump versus normal return innovations, leverage effects, and the time‐series dynamics of jump clustering.     

Contingent Claims on Foreign Assets Following Jump-Diffusion Processes

In this paper we price contingent claims on several foreign assetsthat follow jump-diffusion processes. Discontinuities (jumps) arise dueto the assets' movement in the respective countries, or the exchangerates, or both. We assume the existence of multiple classes (sources)of jumps. Each jump can affect one or more state-variables and is definedby its intensity of arrival and by the joint probability distributionof its magnitude. The existence of jumps gives rise to significant deviationsfrom the joint lognormality assumptions of the multivariate geometricBrownian motion, and affords more flexibility in capturing the empiricallyobserved asymmetry and fat tails in asset returns. Analytic solutionsare provided for the European option on the best of several assets withoutor with exchange rate (quanto-type) protection. A Markov-chainnumerical method that can also handle American claims is given and itsaccuracy is demonstrated. Neglecting the effect of jumps causes seriousmisspricing and leads to erroneous decision-making when purchasing orexercising such options.     

Detecting and modelling the jump risk of CO2 emission allowances and their impact on the valuation of option on futures contracts

Modelling CO₂ emission allowance prices is important for pricing CO₂ emission allowance linked assets in the emissions trading scheme (ETS). Some statistical properties of CO₂ emission allowance prices have been discovered in the literature ignoring price jumps. By employing real data from the ETS, this research first detects the jump risk using a jump test and then verifies jump effects in modelling CO₂ emission allowance prices by comparing the in-sample and out-of-sample model performance. We suggest a model which can capture the statistical properties of autocorrelation, volatility clustering and jump effects is more appropriate for modelling CO₂ emission allowance prices. We establish a general framework for pricing CO₂ emission allowance options on futures contracts with these properties and find that the jump risk significantly affects the value of the CO₂ emission allowance option on futures contracts. More importantly, we demonstrate that the dynamic jump ARMA–GARCH model can provide more accurate valuations of the CO₂ emission allowance options on futures than other models in terms of pricing error.     

A NO-ARBITRAGE MARTINGALE ANALYSIS FOR JUMP-DIFFUSION VALUATION

This study presents a jump-diffusion valuation framework using the no-arbitrage martingale approach. Equilibrium conditions needed to support a jump-diffusion pricing standard process are derived. The results are a generalized jump-diffusion security market line and its corresponding equilibrium valuation relation that prices both jump and diffusion risk. To value options, a fundamental formula is derived that includes existing jump-diffusion option valuation formulas as special cases. 1 find Merton's (1976a) assumption of diversifiable jump risk to be consistent with no-arbitrage only when the aggregate consumption flow does not jump. Simulation shows that Merton's formula undervalues/overvalues options on hedging/cyclical assets. When the jump arrival frequency is larger, the mispricing is larger/smaller for in-the-money/out-of-the-money options.     

A jump-diffusion approach to modelling vulnerable option pricing

Following the framework of Klein [1996. Journal of Banking and Finance 20, 1211–1229], this paper presents an improved method of pricing vulnerable options under jump diffusion assumptions about the underlying stock prices and firm values which are appropriate in many business situations. In contrast to Klein [1996. Journal of Banking and Finance 20, 1211–1229] model, jumps can be used to model sudden changes in stock prices and firm values. Further, with the jump risk, a firm can default instantaneously because of an unexpected drop in its value. Therefore, our model is able to provide sufficient conceptual insights about the economic mechanism of vulnerable option pricing. The numerical results show that a jump occurrence in firm values can increase the likelihood of default and reduce the vulnerable option prices.     

The impact of co-jumps in the oil sector

We study the dynamics of the oil sector using a new multivariate stochastic volatility model with a structure of common factors subjected to jumps in mean and conditional variance. This model contributes to the literature allowing the estimation of spillover effects between assets in a multivariate framework through joint jumps (co-jumps), identifying the permanent and transitory effects through a structure defined by Bernoulli processes. The jump structure introduced in the article can be interpreted as a regime-switching model with an endogenous number of states, avoiding the difficulties associated with models with a fixed number of regimes. We apply the model to oil prices and stock prices of integrated oil companies. The jump structure allows dating the relevant events in the oil sector in the period 2000–2019. The period analyzed encompasses important events in the oil market such as the price escalation in 2008 and the falling prices in 2014. We also apply the model to estimate risk management measures and portfolio allocation and perform a comparison with other multivariate models of conditional volatility, showing the good properties of the model in these applications.     

Modelling conditional heteroscedasticity and jumps in Australian short-term interest rates

The present paper explores a class of jump–diffusion models for the Australian short‐term interest rate. The proposed general model incorporates linear mean‐reverting drift, time‐varying volatility in the form of LEVELS (sensitivity of the volatility to the levels of the short‐rates) and generalized autoregressive conditional heteroscedasticity (GARCH), as well as jumps, to match the salient features of the short‐rate dynamics. Maximum likelihood estimation reveals that pure diffusion models that ignore the jump factor are mis‐specified in the sense that they imply a spuriously high speed of mean‐reversion in the level of short‐rate changes as well as a spuriously high degree of persistence in volatility. Once the jump factor is incorporated, the jump models that can also capture the GARCH‐induced volatility produce reasonable estimates of the speed of mean reversion. The introduction of the jump factor also yields reasonable estimates of the GARCH parameters. Overall, the LEVELS–GARCH–JUMP model fits the data best.     

Dynamic jump intensities and risk premiums: Evidence from S&P500 returns and options

We build a new class of discrete-time models that are relatively easy to estimate using returns and/or options. The distribution of returns is driven by two factors: dynamic volatility and dynamic jump intensity. Each factor has its own risk premium. The models significantly outperform standard models without jumps when estimated on S&P500 returns. We find very strong support for time-varying jump intensities. Compared to the risk premium on dynamic volatility, the risk premium on the dynamic jump intensity has a much larger impact on option prices. We confirm these findings using joint estimation on returns and large option samples.     

Exact solutions for bond and option prices with systematic jump risk

A variety of realistic economic considerations make jump-diffusion models of interest rate dynamics an appealing modeling choice to price interest-rate contingent claims. However, exact closed-form solutions for bond prices when interest rates follow a mixed jump-diffusion process have proved very hard to derive. This paper puts forward two new models of interest-rate dynamics that combine infrequent, discrete changes in the interest-rate level, modeled as a jump process, with short-lived, mean reverting shocks, modeled as a diffusion process. The two models differ in the way jumps affect the central tendency of interest rates; in one case shocks are temporary, in the other shocks are permanent. We derive exact closed-form solutions for the price of a discount bond and computationally tractable schemes to price bond options.     

The role of time-varying jump risk premia in pricing stock index options

This paper examines out-of-sample option pricing performances for the affine jump diffusion (AJD) models by using the S&P 500 stock index and its associated option contracts. In particular, we investigate the role of time-varying jump risk premia in the AJD specifications. Our empirical analysis shows strong evidence in favor of time-varying jump risk premia in pricing cross-sectional options. We also find that, during a period of low volatility, the role of jump risk premia becomes less pronounced, making the differences across pricing performances of the AJD models not as substantial as during a period of high volatility. This finding can possibly explain poor pricing perfomances of the sophisticated AJD models in some previous studies whose sample periods can be characterized by low volatility.     

The economic value of volatility timing with realized jumps

This paper comprehensively investigates the role of realized jumps detected from high frequency data in predicting future volatility from both statistical and economic perspectives. Using seven major jump tests, we show that separating jumps from diffusion improves volatility forecasting both in-sample and out-of-sample. Moreover, we show that these statistical improvements can be translated into economic value. We find that a risk-averse investor can significantly improve her portfolio performance by incorporating realized jumps into a volatility timing based portfolio strategy. Our results hold true across the majority of jump tests, and are robust to controlling for microstructure effects and transaction costs.