A note on essential smoothness in the Heston model

In [2] the authors study the limiting behaviour of the implied volatility in the Heston model as maturity tends to infinity. The main aim of this note is to give a rigorous account of the relationship between the concept of essential smoothness and the large deviation principle for the family of random variables (X t /t ± E λ /t) t≥1 , where the process X denotes the log-spot in Heston model ( 5) and E λ is an exponential random variable with parameter λ > 0 independent of X. This note fills a gap in the proof of Corollary 2.4 in [2] and hence completes the proof of the main result in [2], which describes the limiting behaviour of the implied volatility smile in the Heston model far from maturity.

The note is organized as follows. Section 2 describes the relevant concepts of the large deviation theory and discusses how the effective domain changes when a family of random variables is perturbed by an independent exponential random variable. Section 3 discusses the failure of essential smoothness when the Heston model is perturbed by an independent exponential, which is what causes the gap in the proof of Corollary 2.4 in [2]. Section 3 also proves Theorem 3, which fills the gap.

We briefly recall the basic facts of the large deviation theory in R (see monograph [1, Ch. 2] for more details). Let (Z t ) t≥1 be a family of random variables with Z t ∈ R. J is a rate function if it is lower semicontinuous and J(R) ⊂ [0, ∞] holds. The family (Z t ) t≥1 satisfies the large deviation principle (LDP) with the rate function J if for every Borel set B ⊂ R we have

where D Λ := {u ∈ R : Λ(u) < ∞} is the effective domain of Λ and D • Λ is its interior. The Fenchel-Legendre transform Λ * of the convex function Λ is defined by the formula

Under the assumption in (2), Λ * is lower semicontinuous with compact level sets {x : Λ * (x) ≤ α} (see [1,Lemma 2.3.9(a)]) and Λ * (R) ⊂ [0, ∞] and hence satisfies the definition of a good rate function.

We now state the Gärtner-Ellis theorem (see [1,Section 2.3] for its proof).

Theorem 1. Let the random variables (Z t ) t≥1 satisfy the assumption in (2). If Λ is essentially smooth and lower semicontinuous, then LDP holds for (Z t ) t≥1 with the good rate function Λ * .

Λ is a strict subset of R, which is the case in the setting of [2] (see also Section 3 below), essential smoothness, which plays a key role in the proof of Theorem 1, is not automatic.

The following question is of central importance in [2]: does the LDP persist if a family of random variables (Z t ) t≥1 is perturbed by an independent exponential random variable E 1 ? It is implicitly assumed in the proof of Corollary 2.4 in [2] (see the last line on page 17 and lines 4 and 14 on page 18) that if (Z t ) t≥1 satisfies the assumptions of Theorem 1, then so do the families (Y 1+ t ) t≥1 and (Y 1- t ) t≥1 , where

and the LDP is applied. In particular the authors in [2] assume that the limiting cumulant generating functions of (Y 1± t ) t≥1 are essentially smooth. However the following simple lemma holds. Lemma 2. Let (Z t ) t≥1 satisfy the assumption in (2) with a limiting cumulant generating function Λ. Let λ > 0 and E λ an exponential random variable independent of (Z t ) t≥1 with E[E λ ] = 1/λ and let

Then the families of random variables (Y λ± t ) t≥1 satisfy the assumption in (2) and the corresponding limiting cumulant generating functions are given by

Remarks. (a) Let (Z t ) t≥1 satisfy the assumption in (2) and assume further that Λ is differentiable in

The inequality u ≥ λ implies that, since Λ t (tu) > -∞, we have Λ λ+ t (tu) = ∞ for all t and hence

This proves the lemma for (Y λ+ t ) t≥1 . The case of (Y λ- t ) t≥1 is analogous.

The Heston model S = e X is a stochastic volatility model with the log-stock process X given by

where κ, θ, σ > 0, Y 0 = y 0 > 0, X 0 = x 0 ∈ R and W 1 , W 2 are standard Brownian motions with correlation ρ ∈ (-1, 1). The standing assumption ρσκ < 0, (6) is made in [2] (see equation (2.2) in Theorem 2.1 on page 5 of [2]). In particular the inequality in (6) implies that S is a strictly positive true martingale and allows the definition of the share measure P via the Radon-Nikodym derivative d P/dP = e Xt-x 0 .

The authors’ aim in [2] is to obtain the limiting implied volatility smile as maturity tends to infinity at the strike K = S 0 e xt for any x ∈ R in the Heston model. Their main formula is given in Corollary 3.1 of [2]. A key step in the proof of [2, Corollary 3.1] is given by [2,Corollary 2.4]. In the proof of [2, Corollary 2.4] (see last line on page 17 and lines 4 and 14 on page 18) it is implicitly assumed that the LDP for (X t /t t≥1 implies the LDP for the family (X t /t ± E 1 /t) t≥1 . However, as we have seen in Section 2 (see remarks following Lemma 2), Theorem 1 cannot be applied directly to the family (X t /t ± E 1 /t) t≥1 , even if (X t /t) t≥1 satisfies its assumptions. We start with a precise description of the problem and present the solution in Theorem 3.

…(Full text truncated)…

This content is AI-processed based on ArXiv data.