We consider uncorrelated Stein-Stein, Heston, and Hull-White models and their perturbations by compound Poisson processes with jump amplitudes distributed according to a double exponential law. Similar perturbations of the Black-Scholes model were studied by S. Kou. For perturbed stochastic volatility models, we obtain two-sided estimates for the stock price distribution density and compare the tail behavior of this density before and after perturbation. It is shown that if the value of the parameter, characterizing the right tail of the double exponential law, is small, then the stock price density in the perturbed model decays slower than the density in the original model. On the other hand, if the value of this parameter is large, then there are no significant changes in the behavior of the stock price distribution density.
Deep Dive into Two-sided estimates for stock price distribution densities in jump-diffusion models.
We consider uncorrelated Stein-Stein, Heston, and Hull-White models and their perturbations by compound Poisson processes with jump amplitudes distributed according to a double exponential law. Similar perturbations of the Black-Scholes model were studied by S. Kou. For perturbed stochastic volatility models, we obtain two-sided estimates for the stock price distribution density and compare the tail behavior of this density before and after perturbation. It is shown that if the value of the parameter, characterizing the right tail of the double exponential law, is small, then the stock price density in the perturbed model decays slower than the density in the original model. On the other hand, if the value of this parameter is large, then there are no significant changes in the behavior of the stock price distribution density.
It is assumed in the celebrated Black-Scholes model that the volatility of a stock is constant. However, empirical studies do not support this assumption. In more recent models, the volatility of a stock is represented by a stochastic process. Well-known examples of stochastic volatility models are the Hull-White, the Stein-Stein, and the Heston model. The volatility processes in these models are a geometric Brownian motion, the absolute value of an Ornstein-Uhlenbeck process, and a Cox-Ingersoll-Ross process, respectively. For more information on stochastic volatility models, see [5] and [6].
A stock price model with stochastic volatility is called uncorrelated if standard Brownian motions driving the stock price equation and the volatility equation are independent. In [7], [9], and [10], sharp asymptotic formulas were found for the distribution density of the stock price in uncorrelated Hull-White, Stein-Stein, and Heston models. Various applications of these formulas were given in [8] and [11]. The results obtained in [9] and [10] will be used in the present paper.
It is known that the stock price distribution density in an uncorrelated stochastic volatility model possesses a certain structural symmetry (see formula (14) below). This implies a similar symmetry in the Black-Scholes implied volatility, which does not explain the volatility skew observed in practice. To improve the performance of an uncorrelated model, one can either assume that the stock price process and the volatility process are correlated, or add a jump component to the stock price equation or to the volatility equation. The stock price distribution in the resulting model fits the empirical stock price distribution better than in the uncorrelated case. However, passing to a correlated model or adding a jump component may sometimes lead to similar effects or may have different consequences (see e.g. [1] and [2]). Examples of stock price models with jumps can be found in [3], [15], and [16]. We refer the reader to [4] for more information about stock price models with jumps. An interesting discussion of the effect of adding jumps to the Heston model in contained in [14].
An important jump-diffusion model was introduced and studied by Kou (see [15] and [16]). This model can be described as a perturbation of the Black-Scholes model by a compound Poisson process with double-exponential law for the jump amplitudes. In the present paper, we consider similar perturbations of stochastic volatility models. Our main goal is to determine whether significant changes may occur in the tail behavior of the stock price distribution after such a perturbation. We show that the answer depends on the relations between the parameters defining the original model and the characteristics of the jump process. For instance, no significant changes occur in the behavior of the distribution density of the stock price in a perturbed Heston or Stein-Stein model if the value of the parameter characterizing the right tail of the double exponential law is large. On the other hand, if this value is small, then the distribution density of the stock price in the perturbed model decreases slower than in the original model. For the Hull-White model, there are no significant changes in the tail behavior of the stock price density, since this density decays extremely slowly.
We will next briefly overview the structure of the present paper. In Section 2, we describe classical stochastic volatility models and their perturbations by a compound Poisson process. In Section 3 we formulate the main results of the paper and discuss what follows from them. Finally, in Section 4, we prove the theorems formulated in Section 3.
In the present paper, we consider perturbations of uncorrelated Stein-Stein, Heston, and Hull-White models by compound Poisson processes. Our goal is to determine whether the behavior of the stock price distribution density in the original models changes after such a perturbation.
The stock price process X and the volatility process Y in the Stein-Stein model satisfy the following system of stochastic differential equations:
This model was introduced and studied in [18]. The process Y , solving the second equation in (1), is called an Ornstein-Uhlenbeck process. We assume that µ ∈ R, q ≥ 0, m ≥ 0, and σ > 0. The Heston model was developed in [12]. In this model, the processes X and Y satisfy
where µ ∈ R, q > 0, m ≥ 0, and c > 0. The volatility equation in ( 2) is uniquely solvable in the strong sense, and the solution Y is a non-negative stochastic process. This process is called a Cox-Ingersoll-Ross process.
The stock price process X and the volatility process Y in the Hull-White model are determined from the following system of stochastic differential equations:
In (3), µ ∈ R, ν ∈ R, and ξ > 0. The Hull-White model was introduced in [13]. The volatility process in this model is a geometric Brownian motion.
It will be assumed throughout the paper that s
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