In this short note, we prove by an appropriate change of variables that the
SVI implied volatility parameterization presented in Gatheral's book and the
large-time asymptotic of the Heston implied volatility agree algebraically,
thus confirming a conjecture from Gatheral as well as providing a simpler
expression for the asymptotic implied volatility in the Heston model. We show
how this result can help in interpreting SVI parameters.Comment: 5 page

Gatheral, Jim

Jacquier, Antoine

Quantitative Finance

English

arXiv

Gatheral, J

Jacquier, A

Spiral - Imperial College Digital Repository

arXiv:1002.3633v1  [q-fin.PR]  18 Feb 2010CONVERGENCE OF HESTON TO SVIJIM GATHERAL AND ANTOINE JACQUIERAbstract. In this short note, we prove by an appropriate change of variables that the SVI implied volatilityparameterization presented in [2] and the large-time asymptotic of the Heston implied volatility derived in [1]agree algebraically, thus confirming a conjecture from [2] as well as providing a simpler expression for theasymptotic implied volatility in the Heston model. We show how this result can help in interpreting SVIparameters.1. IntroductionThe stochastic volatility inspired or SVI parameterization of the implied volatility surface was originallydevised at Merrill Lynch in 1999. This parameterization has two key properties that have led to its subsequentpopularity with practitioners:• For a fixed time to expiry T , the implied Black-Scholes variance σ2BS(k, T ) is linear in the log-strike kas |k| → ∞ consistent with Roger Lee’s moment formula [4].• It is relatively easy to fit listed option prices whilst ensuring no calendar spread arbitrage.1The result we prove in this note shows that SVI is an exact solution for the implied variance in the Hestonmodel in the limit T → ∞ thus providing a direct interpretation of the SVI parameters in terms of the parametersof 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 anSVI fit to an empirically observed volatility smile.2. NotationsFrom [2], recall that the SVI parameterization for the implied variance reads(1) σ2SV I (x) =ω12(1 + ω2ρx+√(ω2x+ ρ)2+ 1− ρ2), for all x ∈ R,where x represents the time-scaled log-moneyness, and consider the Heston model where the stock price process(St)t≥0 satisfies the following stochastic differential equation:dSt =√vtStdWt, S0 ∈ R∗+dvt = κ (θ − vt) dt+ σ√vtdZt, v0 ∈ R∗+d〈W,Z〉t = ρ dt,with ρ ∈ [−1, 1], κ, θ, σ and v0 are strictly positive real numbers satisfying 2κθ ≥ σ2 (this is the Feller conditionensuring that the process (vt)t≥0 never reaches zero almost surely). We further make the following assumptionas in [1], under which the Heston asymptotic implied volatility is derived.Assumption 1. κ− ρσ > 0.Note that this assumption is usually assumed in the literature, either explicitly or implicitly when assuminga negative correlation ρ < 0 between the spot and the volatility as observed in equity markets. When thiscondition is not satisfied, the stock price process is still a true martingale, but moments greater than one willcease to exist after a certain amount of time, as pointed out in [6], which refers to this special case as theThe authors would like to thank Aleksandar Mijatović for useful discussions.1It is seemingly impossible to eliminate the possibility of butterfly arbitrage but this is rarely a problem in practice.12 JIM GATHERAL AND ANTOINE JACQUIERlarge correlation regime. Let us now consider the following choice of SVI parameters in terms of the Hestonparameters,(2) ω1 :=4κθσ2 (1− ρ2)(√(2κ− ρσ)2 + σ2 (1− ρ2)− (2κ− ρσ)), and ω2 :=σκθ.Now we know from [1] that the implied variance in the Heston model in the large time limit T → ∞ takesthe following form:(3) σ2∞ (x) = 2(2V ∗(x)− x+ 2(1x∈(−θ/2,θ̄/2) − 1x∈R\(−θ/2,θ̄/2))√V ∗(x)2 − xV ∗(x)), for all x ∈ R,where θ̄ := κθ/ (κ− ρσ), and the function V ∗ : R → R+ is defined by(4) V ∗(x) := p∗ (x) x− V (p∗ (x)) , for all x ∈ R,whereV (p) :=κθσ2(κ− ρσp− d(p)), for all p ∈ (p−, p+) ,d(p) :=√(κ− ρσp)2 + σ2p (1− p2), for all p ∈ (p−, p+) ,p∗ (x) :=σ − 2κρ+ (κθρ+ xσ) η(x2σ2 + 2xκθρσ + κ2θ2)−1/22σρ̄2, for all x ∈ R,η :=√4κ2 + σ2 − 4κρσ, p± :=(−2κρ+ σ ±√σ2 + 4κ2 − 4κρσ)/(2σρ̄2), and ρ̄ :=√1− ρ2.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 S0 exp (xT ), where T ≥ 0 represents thematurity of the option.3. The saddle-point conditionIn this section, we give a non-rigorous motivation for the conjecture in [2] that the T → ∞ limit of the Hestonvolatility smile should be SVI.Consider equation (5.7) on page 60 of [3] which relates the implied volatility σBS(k, T ) at log-strike k andexpiration T to the characteristic function φT (·) of the log-stock price. We rewrite this equation in the form(5)∫ ∞−∞duu2 + 14e−iu kφT (u− i/2) =∫ ∞−∞duu2 + 14e−iu k e−12 (u2+ 14 )σ2BS(k,T )T .In the limit T → ∞, the Heston characteristic function has the formφT (u− i/2) ∼ e−ψ(u)T .Then, as pointed out on page 186 of [5], we may apply the saddle-point method to both sides in equation (5)to obtain(6) e−i k ũe−ψ(ũ)Tũ2 + 14√2 πψ′′(ũ)T∼ 4 exp{−v T8− k22 v T}√2 πv T,where v is short-form notation for σ2BS(k, T )T and ũ is such thatψ′(ũ) = −i kT,so that ũ (which is in general a function of k) is a saddle-point, which in the Heston model at least, may becomputed explicitly as in Lemma 5.3 of [1].Defining k := xT and equating the arguments of the exponentials in equation (6), the dependence on Tcancels and we obtain(7)v(x)8+x22 v(x)= ψ(ũ(x)) + i x ũ(x),where we have reinstated explicit dependence on x for emphasis.CONVERGENCE OF HESTON TO SVI 3With the help of e.g. Mathematica, one can verify that in the T → ∞ limit of the Heston model and withthe choice (2) of SVI parameters, expression (1) exactly solves the saddle-point condition (7):σ2SV I(x)8+x22 σ2SV I(x)= ψ(ũ(x)) + i x ũ(x).We are thus led to conjecture that σ2SV I (x) = σ2∞ (x) so that the T → ∞ limit of implied variance in the Hestonmodel is SVI.4. Main result and proofWe now state and prove the main result of this note,Proposition 1. Under Assumption 1 and the choice of SVI parameters (2), σ2SV I (x) = σ2∞ (x) for all x ∈ R.Proof. Let us now introduce the following notations: ∆(x) :=√σ2x2 + 2κθρσx+ κ2θ2, where η and ρ̄ aredefined in Section 2. Under the change of variables (2), the SVI implied variance takes the form(8) σ2SV I (x) =2σ2ρ̄2(η − (2κ− ρσ))(κθ + ρσx+∆(x)), for all x ∈ R.We now move on to simplify the expression for σ2∞ as written in (3). We first start by the expression for V∗(x)appearing in (3). We haveV ∗(x) =A (x)∆ (x) +B(x)η2σ2ρ̄2∆(x),withA(x) := xσ2 − 2xκρσ − 2κ2θ + κθρσ, and B(x) := 2xσκθρ+ x2σ2 + κ2θ2ρ2 + κ2θ2ρ̄2.Note that B(x) = ∆2 (x), so that V ∗ (x) = (A(x) + ∆(x)η) /(2σ2ρ̄2). We further have(9) 2V ∗ (x)− x = A (x) + ∆ (x) η − xσ2ρ̄2σ2ρ̄2=∆(x) η − (2κ− ρσ) (κθ + xρσ)σ2ρ̄2,where we use the factorisation A (x)− xσ2ρ̄2 = − (2κ− ρσ) (κθ + xρσ).Now, back to (3), where we denote Φ (x) := V ∗(x)2 − xV ∗(x). We haveΦ (x) =(∆(x)η2σ2ρ̄2)2+ α (x)∆ (x) + β (x) ,whereα (x) := −η (2κ− ρσ) (κθ + xρσ)2σ4ρ̄4, and β (x) :=14σ4ρ̄4{(2κ− ρσ)2 (κθ + xρσ)2 − x2σ4ρ̄4}.We now use the following factorisations:(10) ∆2 (x) = (κθ + xρσ)2+ x2σ2ρ̄2, and η2 = (2κ− ρσ)2 + σ2ρ̄2,so that we can write β (x) =(4σ4ρ̄4)−1((2κ− ρσ)2 ∆2 (x)− x2σ2ρ̄2η2)and henceΦ (x) =14σ4ρ̄4{[(2κ− ρσ)2 + σ2ρ̄2]∆2 (x) + a (x)∆ (x) +(η2 − σ2ρ̄2) (∆2 (x)− x2σ2ρ̄2)− x2σ4ρ̄4}=14σ4ρ̄4{(2κ− ρσ)2 ∆2 (x) + a (x)∆ (x) + η2 (κθ + xρσ)2}=14σ4ρ̄4{η (κθ + xρσ) − (2κ− ρσ)∆ (x)}2,(11)where, for convenience, we denote a (x) := 4σ4ρ̄4α (x). To complete the proof, we need to take the square rootof Φ (x), i.e. we need to study the sign of the expression under the square in (11). Using again (10), we haveη (κθ + xρσ) − (2κ− ρσ)∆ (x)= (κθ + xρσ)√(2κ− ρσ)2 + σ2ρ̄2 − (2κ− ρσ)√(κθ + xρσ)2+ x2σ2ρ̄2=√γ (x) + σ2ρ̄2 (κθ + xρσ)2 −√γ (x) + x2σ2ρ̄2 (2κ− ρσ)2,4 JIM GATHERAL AND ANTOINE JACQUIERwhere γ (x) := (2κ− ρσ)2 (κθ + xρσ)2. Now, because γ (x) ≥ 0 for all x ∈ R, then the sign of this wholeexpression is simply given by the sign of the difference ψ (x) := σ2ρ̄2 (κθ + xρσ)2 − x2σ2ρ̄2 (2κ− ρσ)2. Notefurther that we actually have ψ (x) = κσ2ρ̄2 (2x+ θ) (2xρσ + κθ − 2κx), that this polynomial has exactly tworeal roots −θ/2 and θ̄/2, and that its second-order coefficient reads −4κσ2ρ̄2 (κ− ρσ) < 0 under Assumption 1.So, plugging (9) and (11) into (3), we exactly obtain (8) and the proposition follows. 5. Interpretation of SVI parametersIn this section, we use our result in Proposition 1 to help interpret the SVI parameters. From [2], the standardSVI parameterization in terms of the log-strike k reads(12) σ2SV I(k) = a+ b{ρ̃ (k −m) +√(k −m)2 + σ̃2}.Equating (12) with (1) and with the parameter choice (2):ω1 :=4κθσ2 (1− ρ2)(√(2κ− ρσ)2 + σ2 (1− ρ2)− (2κ− ρσ)), and ω2 :=σκθ,we find the following correspondence between SVI parameters and Heston parameters;a =ω12(1 − ρ2),b =ω1 ω22T,ρ̃ = ρ,m = −ρ Tω2,σ̃ =√1− ρ2 Tω2.(13)For concreteness, imagine that we are given an SVI fit to the implied volatility smile generated from the Hestonmodel with T very large so that we have the SVI parameters a, b, ρ̃, m and σ̃. Our first observation is thatthe SVI parameter ρ̃ is exactly the correlation ρ between changes in instantaneous variance v and changes inthe underlying S in the Heston process. That is, we can read off correlation directly from the orientation of thevolatility smile. In particular, the smile is symmetric when ρ = 0.The parameter b gives the angle between the asymptotes of the implied variance smile. We see from (12)and (13) that the angle between the asymptotes of the total variance smile σ2SV I(k)T is constant for large Tbut that the overall level increases with T .From equation (1), ω1 is the at-the-money implied variance σ2SV I(0, T ). From (2), in the limit σ ≪ κ , wehaveω1 = θ{1 +ρ σκ+O((σκ)2)},so that the at-the-money volatility is given directly by θ when the volatility of volatility is small. In the limitσ ≫ κ , we haveω1 =4 κ θσ (1 − ρ){1− 2 κσ+O((κσ)2)},showing that at-the-money volatility decreases as the volatility-of-volatility increases and as the volatility be-comes less correlated with the underlying.Finally, the minimum of the variance smile is attained at x = −2 ρ/ω2, providing a simple interpretation ofthe parameter ω2. In particular, if ρ = 0, the minimum is exactly the at-the-money point. The minimum shiftsto the upside x > 0 if ρ < 0 and to the downside x < 0 if ρ > 0.References[1] M. Forde, A. Jacquier, A. Mijatovic (2009), Asymptotic formulae for implied volatility in the Heston model. Submitted.[2] J. Gatheral (2004), A parsimonious arbitrage-free implied volatility parameterization with application to the valua-tion of volatility derivatives. Presentation at Global Derivatives & Risk Management, Madrid, May 2004, available atwww.math.nyu.edu/fellows fin math/gatheral/madrid2004.pdf.[3] J. Gatheral (2006), The Volatility Surface: A Practitioner’s Guide, John Wiley & Sons, Hoboken, NJ.[4] Roger W. Lee (2004), The Moment Formula for Implied Volatility at Extreme Strikes. Mathematical Finance, 14, pp 469–480.CONVERGENCE OF HESTON TO SVI 5[5] Alan L. Lewis (2000), Option Valuation under Stochastic Volatility with Mathematica Code. Finance Press, Newport Beach,CA.[6] Zeliade Systems (2010), Heston 2009. White paper Zeliade. Available at www.zeliade.com/whitepapers/zwp-0004.pdf.Bank of America Merrill Lynch and Baruch College, CUNYDepartment of Mathematics, Imperial College London and Zeliade Systems, ParisE-mail address: ajacquie@imperial.ac.uk.

Convergence of Heston to SVI

Abstract

Similar works

Full text

Available Versions

Spiral - Imperial College Digital Repository