Archil Gulisashvili

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

In this paper, we obtain asymptotic formulas with error estimates for the implied volatility associated with a European call pricing function. We show that these formulas imply Lees moment formulas for the implied volatility and the tail-wing formulas due to Benaim and Friz. In addition, we analyze Pareto-type tails of stock price distributions in uncorrelated Hull-White, Stein-Stein, and Heston models and find asymptotic formulas with error estimates for call pricing functions in these models.

On refined volatility smile expansion in the Heston model

It is known that Hestons stochastic volatility model exhibits moment explosion, and that the critical moment s + can be obtained by solving (numerically) a simple equation. This yields a leading-order expansion for the implied volatility at large strikes: σBS(k, T)2 T ∼ Ψ(s + − 1) × k (Roger Lees moment formula). Motivated by recent ‘tail-wing’ refinements of this moment formula, we first derive a novel tail expansion for the Heston density, sharpening previous work of Drăgulescu and Yakovenko [Quant. Finance, 2002, 2(6), 443–453], and then show the validity of a refined expansion of the type σBS(k, T)2 T = (β1 k 1/2 + β2 + ···)2, where all constants are explicitly known as functions of s +, the Heston model parameters, the spot vol and maturity T. In the case of the ‘zero-correlation’ Heston model, such an expansion was derived by Gulisashvili and Stein [Appl. Math. Optim., 2010, 61(3), 287–315]. Our methods and results may prove useful beyond the Heston model: the entire quantitative analysis is based on affine principles and at no point do we need knowledge of the (explicit, but cumbersome) closed-form expression of the Fourier transform of log S T (equivalently the Mellin transform of S T ). What matters is that these transforms satisfy ordinary differential equations of the Riccati type. Secondly, our analysis reveals a new parameter (the ‘critical slope’), defined in a model-free manner, which drives the second- and higher-order terms in tail and implied volatility expansions.

Asymptotic Behavior of Distribution Densities in Models with Stochastic Volatility. I

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 a geometric Brownian motion and the density of the stock price process in the Hull–White 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 price of an Asian option are obtained.

The Jain–Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory

We discuss stochastic calculus for large classes of Gaussian processes, based on rough path analysis. Our key condition is a covariance measure structure combined with a classical criterion due to Jain and Monrad [Ann. Probab. 11 (1983) 46–57]. This condition is verified in many examples, even in absence of explicit expressions for the covariance or Volterra kernels. Of special interest are random Fourier series, with covariance given as Fourier series itself, and we formulate conditions directly in terms of the Fourier coefficients. We also establish convergence and rates of convergence in rough path metrics of approximations to such random Fourier series. An application to SPDE is given. Our criterion also leads to an embedding result for Cameron–Martin paths and complementary Young regularity (CYR) of the Cameron–Martin space and Gaussian sample paths. CYR is known to imply Malliavin regularity and also Ito-like probabilistic estimates for stochastic integrals (resp., stochastic differential equations) despite their (rough) pathwise construction. At last, we give an application in the context of non-Markovian Hormander theory.

Tail behavior of sums and differences of log-normal random variables

We present sharp tail asymptotics for the density and the distribution function of linear combinations of correlated log-normal random variables, that is, exponentials of components of a correlated Gaussian vector. The asymptotic behavior turns out to depend on the correlation between the components, and the explicit solution is found by solving a tractable quadratic optimization problem. These results can be used either to approximate the probability of tail events directly, or to construct variance reduction procedures to estimate these probabilities by Monte Carlo methods. In particular, we propose an efficient importance sampling estimator for the left tail of the distribution function of the sum of log-normal variables. As a corollary of the tail asymptotics, we compute the asymptotics of the conditional law of a Gaussian random vector given a linear combination of exponentials of its components. In risk management applications, this finding can be used for the systematic construction of stress tests, which the financial institutions are required to conduct by the regulators. We also characterize the asymptotic behavior of the Value at Risk for log-normal portfolios in the case where the confidence level tends to one.

Implied Volatility in the Hull-White Model

We study the implied volatility K↦I(K) in the Hull–White model of option pricing, and obtain asymptotic formulas for this function as the strike price K tends to infinity or zero. We also prove that the function I is convex near zero and concave near infinity, and characterize the behavior of the first two derivatives of this function.

LEFT-WING ASYMPTOTICS OF THE IMPLIED VOLATILITY IN THE PRESENCE OF ATOMS

The paper considers the asymptotic behavior of the implied volatility in stochastic asset price models with atoms. In such models, the asset price distribution has a singular component at zero. Examples of models with atoms include the constant elasticity of variance (CEV) model, jump-to-default models, and stochastic models described by processes stopped at the first hitting time of zero. For models with atoms, the behavior of the implied volatility at large strikes is similar to that in models without atoms. On the other hand, the behavior of the implied volatility at small strikes is influenced significantly by the atom at zero. De Marco, Hillairet, and Jacquier found an asymptotic formula for the implied volatility at small strikes with two terms and also provided an incomplete description of the third term. In the present paper, we obtain a new asymptotic formula for the left wing of the implied volatility, which is qualitatively different from the De Marco–Hillairet–Jacquier formula. The new formula contains three explicit terms and an error estimate. In the paper, we show how to derive the De Marco–Hillairet–Jacquier formula from the new formula, and compare the performance of the two formulas in the case of the CEV model. The graphs included in the paper show that the new asymptotic formula provides a notably better approximation to the implied volatility at small strikes in the CEV model than the De Marco–Hillairet–Jacquier formula.

ASYMPTOTIC EQUIVALENCE IN LEE'S MOMENT FORMULAS FOR THE IMPLIED VOLATILITY, ASSET PRICE MODELS WITHOUT MOMENT EXPLOSIONS, AND PITERBARG'S CONJECTURE

In this paper, we study the asymptotic behavior of the implied volatility in stochastic asset price models. We provide necessary and sufficient conditions for the validity of asymptotic equivalence in Lees moment formulas, and obtain new asymptotic formulas for the implied volatility in asset price models without moment explosions. As an application, we prove a modified version of Piterbargs conjecture. The asymptotic formula suggested by Piterbarg may be considered as a substitute for Lees moment formula for the implied volatility at large strikes in the case of models without moment explosions. We also characterize the asymptotic behavior of the implied volatility in several special asset price models, e.g., the CEV model, the finite moment log-stable model of Carr and Wu, the Heston model perturbed by a compound Poisson process with double exponential law for jump sizes, and SV1 and SV2 models of Rogers and Veraart.

Mass at zero in the uncorrelated SABR model and implied volatility asymptotics

We study the probability mass at the origin in the SABR stochastic volatility model, and derive several tractable expressions for it, in particular when time becomes small or large. In the uncorrelated case, tedious saddlepoint expansions allow for (semi) closed-form asymptotic formulae. As an application – the original motivation for this paper – we derive small-strike expansions for the implied volatility when the maturity becomes short or large. These formulae, by definition arbitrage-free, allow us to quantify the impact of the mass at zero on currently used implied volatility expansions. In particular we discuss how much those expansions become erroneous.

On the probability of hitting the boundary for Brownian motions on the SABR plane

Starting from the hyperbolic Brownian motion as a time-changed Brownian motion, we explore a set of probabilistic models--related to the SABR model in mathematical finance--which can be obtained by geometry-preserving transformations, and show how to translate the properties of the hyperbolic Brownian motion (density, probability mass, drift) to each particular model. Our main result is an explicit expression for the probability of any of these models hitting the boundary of their domains, the proof of which relies on the properties of the aforementioned transformations as well as time-change methods.