Convergence of Heston to SVI

• For a fixed time to expiry T , the implied Black-Scholes variance σ 2 BS (k, T ) is linear in the log-strike k as |k| → ∞ consistent with Roger Lee’s moment formula [4].

• It is relatively easy to fit listed option prices whilst ensuring no calendar spread arbitrage. 1 The result we prove in this note shows that SVI is an exact solution for the implied variance in the Heston model in the limit T → ∞ thus providing a direct interpretation of the SVI parameters in terms of the parameters of the Heston model.

In Section 2, we present our notation. In Section 3, we motivate the conjecture which we prove in Section 4. We conclude in Section 5 by showing how our result can help us interpret SVI parameters resulting from an SVI fit to an empirically observed volatility smile.

From [2], recall that the SVI parameterization for the implied variance reads (1)

where x represents the time-scaled log-moneyness, and consider the Heston model where the stock price process (S t ) t≥0 satisfies the following stochastic differential equation:

, κ, θ, σ and v 0 are strictly positive real numbers satisfying 2κθ ≥ σ 2 (this is the Feller condition ensuring that the process (v t ) t≥0 never reaches zero almost surely). We further make the following assumption as in [1], under which the Heston asymptotic implied volatility is derived.

Note that this assumption is usually assumed in the literature, either explicitly or implicitly when assuming a negative correlation ρ < 0 between the spot and the volatility as observed in equity markets. When this condition is not satisfied, the stock price process is still a true martingale, but moments greater than one will cease to exist after a certain amount of time, as pointed out in [6], which refers to this special case as the large correlation regime. Let us now consider the following choice of SVI parameters in terms of the Heston parameters,

(2)

and ω 2 := σ κθ .

Now we know from [1] that the implied variance in the Heston model in the large time limit T → ∞ takes the following form:

where θ := κθ/ (κ -ρσ), and the function V * : R → R + is defined by

where

, for all x ∈ R,

Note that in this asymptotic Heston form for the implied volatility, x corresponds to a time-scaled log-moneyness, i.e. the implied volatility corresponds to call/put options with strike S 0 exp (xT ), where T ≥ 0 represents the maturity of the option.

In this section, we give a non-rigorous motivation for the conjecture in [2] that the T → ∞ limit of the Heston volatility smile should be SVI.

Consider equation (5.7) on page 60 of [3] which relates the implied volatility σ BS (k, T ) at log-strike k and expiration T to the characteristic function φ T (•) of the log-stock price. We rewrite this equation in the form

In the limit T → ∞, the Heston characteristic function has the form

Then, as pointed out on page 186 of [5], we may apply the saddle-point method to both sides in equation ( 5) to obtain

where v is short-form notation for σ 2 BS (k, T ) T and ũ is such that

so that ũ (which is in general a function of k) is a saddle-point, which in the Heston model at least, may be computed explicitly as in Lemma 5.3 of [1]. Defining k := x T and equating the arguments of the exponentials in equation ( 6), the dependence on T cancels and we obtain

where we have reinstated explicit dependence on x for emphasis.

With the help of e.g. Mathematica, one can verify that in the T → ∞ limit of the Heston model and with the choice (2) of SVI parameters, expression (1) exactly solves the saddle-point condition (7):

We are thus led to conjecture that σ 2 SV I (x) = σ 2 ∞ (x) so that the T → ∞ limit of implied variance in the Heston model is SVI.

We now state and prove the main result of this note, Proposition 1. Under Assumption 1 and the choice of SVI parameters (2), σ 2 SV I (x) = σ 2 ∞ (x) for all x ∈ R.

Proof. Let us now introduce the following notations: ∆(x) := σ 2 x 2 + 2κθρσx + κ 2 θ 2 , where η and ρ are defined in Section 2. Under the change of variables (2), the SVI implied variance takes the form

We now move on to simplify the expression for σ 2 ∞ as written in (3). We first start by the expression for V * (x) appearing in (3). We have

where we use the factorisation

Now, back to (3), where we denote Φ (x) := V * (x) 2 -xV * (x). We have

where

We now use the following factorisations:

where, for convenience, we denote a (x) := 4σ 4 ρ4 α (x). To complete the proof, we need to take the square root of Φ (x), i.e. we need to study the sign of the expression under the square in (11). Using again (10), we have

where γ (x) := (2κ -ρσ) 2 (κθ + xρσ) 2 . Now, because γ (x) ≥ 0 for all x ∈ R, then the sign of this whole expression is simply given by the si

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