Asymptotic Behavior of the Stock Price Distribution Density and Implied Volatility in Stochastic Volatility Models

Abstract

We study the asymptotic behavior of distribution densities arising in stock price models with stochastic volatility. The main objects of our interest in the present paper are the density of time averages of the squared volatility process and the density of the stock price process in the Stein-Stein and the Heston model. We find explicit formulas for leading terms in asymptotic expansions of these densities and give error estimates. As an application of our results, sharp asymptotic formulas for the implied volatility in the Stein-Stein and the Heston model are obtained.Keywords Stein-Stein model · Heston model · Mixing distribution density · Stock price · Bessel processes · Ornstein-Uhlenbeck processes · CIR processes · Asymptotic formulas · Implied volatility where W t and Z t are standard Brownian motions. We will assume that µ ∈ R, a ≥ 0, b < 0, and c > 0. The initial conditions for X t and Y t are denoted by x 0 and y 0 , respectively. The volatility equation in (2) can be 0 A. Gulisashvili rewritten in the mean-reverting form. This giveswhere r = −b and m = − a b . The volatility equation in (2) is uniquely solvable in the strong sense, and the solution Y t is a positive stochastic process. This process is called a Cox-Ingersoll-Ross process (a CIRprocess). This process was studied in [6]. Interesting results concerning the Heston model were obtained in [7].It will be assumed throughout the present paper that the models described by (1) and (2) are uncorrelated. This means that the Brownian motions W t and Z t driving the stock price equation and the volatility equation in (1) and (2) are independent. In the analysis of the probability distribution of the stock price X t , the mean-square averages of the volatility process over finite time intervals play an important role. For the Stein-Stein model, we set α t = 1 t t 0

Asymptotic Formulas with Error Estimates for Call Pricing Functions and the Implied Volatility at Extreme Strikes

“…Then G ∈ P F ∞ ∩ P F 0 , and hence the implied volatility I C associated with C and the implied volatility I G associated with G exist for all T > 0 and K > 0. Replacing σ by I C (K) in (21) and taking into account the equality (22), we see that…”

“…For special stochastic volatility models, sharp asymptotic formulas for the implied volatility were established in the papers of E. M. Stein and the author (see [20,22,19,21]). In sections 4 and 5, we compare formulas (1) and (2) with known asymptotic formulas for the implied volatility.…”

“…The idea to express the asymptotic properties of the implied volatility in terms of the behavior of the distribution density of the stock price has also been exploited in [22] and [21] in the case of special stock price models with stochastic volatility.…”

“…The models we are going to consider are the uncorrelated Stein-Stein, Heston, and Hull-White models. We will first formulate several results obtained in [20,19,21]. Recall that the stock price process X t and the volatility process Y t in the Hull-White model satisfy the following system of stochastic differential equations:…”

Large-Maturity Regimes of the Heston Forward Smile

“…Roger Lee [55] was the first to study extreme strike asymptotics, and further works on this have been carried out by Benaim and Friz [6,7] and in [39,40,41,31,23,19]. Large-maturity asymptotics have only been studied in [67,27,46,45,29] using large deviations and saddlepoint methods.…”