For option pricing models and heavy‐tailed distributions, this study proposes a continuous‐time stochastic volatility model based on an arithmetic Brownian motion: a one‐parameter extension of the normal stochastic alpha‐beta‐rho (SABR) model. Using two generalized Bougerol's identities in the literature, the study shows that our model has a closed‐form Monte Carlo simulation scheme and that the transition probability for one special case follows Johnson's SU distribution—a popular heavy‐tailed distribution originally proposed without stochastic process. It is argued that the SU distribution serves as an analytically superior alternative to the normal SABR model because the two distributions are empirically similar.
This study examine the theoretical and empirical perspectives of the symmetric Hawkes model of the price tick structure. Combined with the maximum likelihood estimation, the model provides a proper method of volatility estimation specialized in ultra-high-frequency analysis. Empirical studies based on the model using the ultra-high-frequency data of stocks in the S&P 500 are performed. The performance of the volatility measure, intraday estimation, and the dynamics of the parameters are discussed. A new approach of diffusion analogy to the symmetric Hawkes model is proposed with the distributional properties very close to the Hawkes model. As a diffusion process, the model provides more analytical simplicity when computing the variance formula, incorporating skewness and examining the probabilistic property. An estimation of the diffusion model is performed using the simulated maximum likelihood method and shows similar patterns to the Hawkes model. integer. This paper only considers the simple counting measure, i.e., k i = 1 for all i. A point process N can be regarded as a stochastic process by letting N (t, ω) = N ((−∞, t], ω). Consider a filtered probability space (Ω, {F t }, P), −∞ < t ≤ T , where the σ-field F t is generated by N (t). The Hawkes process is an orderly stationary point process N constructed by modeling the conditional intensity, λ. The conditional intensity function is represented as an adapted process toFor an M -dimensional Hawkes process (N 1 , . . . , N M is normally a deterministic function and called kernel. The integration of the r.h.s. is the stochastic integration defined pathwise. To apply the stochastic integration theory in the later, the Hawkes and intensity processes are considered to be right continuous processes with left limits.
Self and mutually excited HawkesThis subsection briefly reviews the self and mutually excited Hawkes model. Consider a two dimensional Hawkes process (N 1 , N 2 ) with exponential decay kernels in the conditional intensities with constants µ i , α ij and β ij , for 0 < t:α 12 e −β12(t−u) dN 2 (u) = µ 1 + λ 11 (0)e −β11t + λ 12 (0)e −β12t + t 0 α 11 e −β11(t−u) dN 1 (u) + t 0
We examine the effect of the oil and gas firms' use of derivatives for hedging risks on the marginal value of cash holdings. Analyzing 155 U.S. oil and gas producers from 1998 to 2017, we find that the use of derivatives for hedging risks, especially oil and gas‐related risk, reduces the marginal value of corporate cash holdings. Furthermore, the effect of using derivatives is stronger for firms exposed to higher risk. Our findings imply that cash holdings and derivatives use act as substitutes in hedging risk in this industry.
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