Smile Expansions

D. Sloth∗

This version: April 5, 2013

Abstract We derive a closed-form expansion of implied volatility and at-the-money skew. The expan- sion is simple, easy to interpret and implement, and accurate where it matters. We focus on the Variance Gamma model and its extensions to allow for stochastic volatility. However, the expansion may be used for any option pricing model with time-changed underlying dynamics. Finally, we explore two domains of application of the expansion: one as a control variate in Fourier option pricing, and another one as a fast, rst-order approach for calibration to at-the-money volatilities and skews.

Keywords: Analytical approximation, implied volatility, Lévy processes, stochastic volatil- ity, control variate, calibration.

1 Introduction

Take it hard to the hoop! This basketball expression beautifully captures what analytical approx- imations are all about. They are rough - sure, this is what makes them approximations - but they often constitute the most lucid and direct approach. In quantitative nance, good approxi- mations help model handling by speeding up calibration to market-observed quantities and, even more importantly, they enhance our understanding of the model's analytical features. Numerical methods for option pricing are extremely powerful in terms of accuracy. Nevertheless, they are hermetically closed. They fail to provide a clear link between the structural properties of the model and its eects on the characteristics of the generated implied volatility surface. For these reasons, analytical approximations, often with the term expansion attached, have been and continue to be proposed in the literature. In the context of diusion-based stochastic volatility models, Lewis (2000) derives an asymp- totic expansion of implied volatility assuming small volatility of volatility. Corresponding results were obtained by Lee (2001) assuming slow mean-reversion of volatility, while Fouque, Papani- colaou, and Sircar (2000) study the case of fast mean-reverting volatility. Hagan et al. (2002) derived a short-expiry expansion in the SABR model via singular pertubations. Medvedev and Scaillet (2007) derive an asymptotic expansion in a jump-diusion stochastic volatility model. Similar work can be found in Berestycki et al. (2004), Osajima (2007) and Forde (2008), whereas Benhamou, Gobet, and Miri (2009) and Larsson (2012) suggest pertubative expansions using . More recently, Kristensen and Mele (2011) and Drimus (2011) have derived analytical approximations of option prices based on diusions. Nicolato and Sloth (2012) study similar expansions under more general dynamics allowing for jumps and stochastic volatility. An- other strain of literature focuses on approximating the risk-neutral density, see e.g. Abadir and Rockinger (2003), Aït-Sahalia (2002), Egorov, Li, and Xu (2003) and Yu (2007). In addition, sad- dlepoint approximations applied to option pricing are studied in Rogers and Zane (1999), Xiong, Wong, and Salopek (2005) and Aït-Sahalia and Yu (2006). Sartorelli (2010) and Poulsen and Ribeiro (2012) consider Edgeworth expansions.

∗Department of Economics and Business, Aarhus University, Denmark. Email: [email protected]

1 In this paper we derive a closed-form expansion of implied volatility and at-the-money skew. The expansion builds on and extends the recent work of Nicolato and Sloth (2012) which considers approximations of option prices under time-changed dynamics. We illustrate the use of the implied volatility expansion for the Variance Gamma model (Madan, Carr, and Chang (1998)) and its extensions to allow for stochastic volatility. Nonetheless, the expansion is largely generic in the sense that it can be used for any option pricing model based on time-changed . This includes popular jump models such as the Normal Inverse Gaussian model (Barndor-Nielsen (1998)) and the CGMY model (Carr et al. (2002)) as well as their stochastic volatility counterparts proposed in Carr et al. (2003). In addition, the expansion is very simple, easy to implement, and fast. A key insight oered by the expansion is that implied volatility can be understood as the expected volatility adjusted for risks of movements in underlying quantities. The risk adjustment terms are governed by higher-order option risks, such as convexity in volatility and dependence of the delta on the level of volatility. Yet, the magnitudes of the correction terms are purely determined by the characteristics of the model. This way of interpreting the volatility smile is consistent with industry practice. Market participants cope with smile eects by hedging not only classical risks, like Delta risk, but also higher-order risks such as Volga and Vanna risk. The decomposition of the volatility smile elucidates how the analytical features of the un- derlying model are translated into implied volatilities. This is not to be underestimated since understanding how a model responds to moves of its parameters or underlying quantities is crucial for its introduction and use on the trading oor. Moreover, the decomposition allows for sys- tematic comparison of complex option pricing models by quantifying how option risks aect the volatility smile across dierent models. For example, in a simplied model assuming zero correla- tion between the stock price and its volatility, risks associated with movements in the stock price do not aect implied volatility, whereas these risks will have a nonzero eect in models allowing for correlation. We explore two domains of application of the expansion. First, we suggest an expansion-based control variate for option pricing based on Fourier transform methods to improve convergence properties of the embedded numerical integration. Andersen and Andreasen (2002) suggest using the Black-Scholes price as a control. This, unfortunately, constitutes a Catch-22 as to which volatility one should use in the Black-Scholes pricing formula. In practice, the choice has usually been arbitrary. We argue that the expansion of implied volatility gives a consistent choice of the input volatility. Our numerical results conrm that this approach results in a signicant speed- up by reducing the number of function evaluations by more than half. Secondly, we propose a fast, rst-order model calibration approach. In particular, we suggest using the expansion to jointly calibrate to at-the-money volatility and skew. Besides a gain in computational speed, the advantage of this approach is that it requires fewer data points compared to the usual procedure of calibrating to the whole surface of option prices. The paper is organized as follows. In Section 2 we outline the model setup. In Section 3 we develop the main theoretical results underlying the expansion. In Section 4 we assess the accuracy and stability of the approximation under various model assumptions. In Section 5 we study two dierent domains of application. Finally, in Section 6 we make some concluding remarks. Appendices gather further numerical results and technical details.

2 Model Setup

We consider a general market model specied via a bivariate process (X,V ) that lives on a ltered probability space ∗ , is continuous in probability, and takes (Ω, F, (Ft)06t6T , Q) values in . The positive component denotes the total variance in the model. is assumed R×R+ V V to be an increasing process with V0 = 0 and dierential characteristics (b, 0,F (du)) where the drift is a positive predictable process, while is a transition kernel from to with b F (Ω × R+, P) (R+, B) V R F ({0}) = 0 and (1 ∧ u)F.(du) < +∞. By Kallsen and Shiryaev (2002), the Laplace cumulant . R+

2 process K(ζ) of V is then given by Z t Z t Z ζ u  Kt(ζ) = ζbsds + e − 1 Fs(du)ds (1) 0 0 R+ and it is well dened for all where . Furthermore, we assume that ζ ∈ C− C− = {z ∈ C :

X −K (1) + Xt St = S0 e t , (6) where the cumulant process X 1 2 (7) Kt (ζ) = Kt(ω ζ + 2 ζ ) is well dened for those ζ ∈ C with <ζ ∈ [0, −2ω]. For notational convenience later on, we shall use the re-parameterization c = ω+1/2 and refer to c as the skewness parameter. We shall also impose the simplifying constraint that the skewness parameter satises c 6 0. This is not restrictive in practice but consistent with the negative skew of volatility surfaces observed in equities markets. Under the independence between W and V , the process S in (6) is a true martingale and therefore describes the dynamics of the asset price under the risk-neutral measure Q. In addition, a key feature of this model framework is that the price process S can be represented as the product of two positive martingales 1 − Vt + WV St = S0 ξt e 2 t , (8) and the spot-correcting martingale ξ is given by

−Kt(c) + cVt ξt = e . (9) The model setup outlined above embeds a variety of popular option pricing models. First, the simplest case is obtained if the total variance process V is specied as a Lévy subordinator, i.e., an increasing process with stationary and increasing increments1. For such a process the dierential triplet, termed the Lévy triplet, is deterministic and time-independent with

bt = β and Ft(du) = ν(du) for all t ≥ 0, where β ≥ 0 and Lévy measure ν(du) such that R (1 ∧ u)ν(du) < ∞. The cumulant process is R+ also deterministic and takes the form Kt(ζ) = t · κ(ζ), where κ is the Lévy exponent Z ζu (10) κ(ζ) = ζβ + (e − 1)ν(du), ζ ∈ C−. R+ 1For a comprehensive account on Lévy processes and their use in nancial modeling, we refer the reader to Cont and Tankov (2004).

3 It follows that the associated X process is a Lévy process with triplet (βX , γX , νX ) computed according to equations (3)-(5) Z Z βX = ω β + ν(du) h(x)µ(dx; ω u, u) (11) R+ γX = β, (12) Z X ν (B) = µ(B; ω u, u)ν(du),B ∈ B(R). (13) R+ Thus, in this case the asset-pricing model in (6) belongs to the class of exponential Lévy models. Popular models that can be obtained in our setup include the Hyperbolic models suggested by Eberlein and Keller (1995), the Normal Inverse Gaussian model proposed in Barndor-Nielsen (1998), the Variance Gamma model introduced by Madan and Seneta (1990), the Meixner model studied in Schoutens (2002), and CGMY model considered in Carr et al. (2002) (see also Bo- yarchenko and Levendorskii (2002)). While the exponential Lévy models are able to capture the implied volatility smile for a single maturity, empirics document that they have a hard time accommodating the term structure of volatility. Carr et al. (2003) propose the construction of a set of stochastic volatility models with jumps by time-changing a Lévy process. Here, the driving source of randomness X is obtained by subordinating a general Lévy process L to an absolutely continuous random clock Y

Z t with (14) Xt = LYt , Yt = ys ds, 0 where the instantaneous activity rate y is given by some positive process independent of L. Y may be seen as reecting some kind of intrinsic market time that represents business activity, and it is the randomness in business activity that generates stochastic volatility. Intuitively, a more active business day, captured by a higher rate of activity, generates higher volatility in the market. In a similar manner we can obtain models belonging to the Lévy stochastic volatility class within our model framework by specifying the total variance process V as a subordinator A evaluated at an independent and absolutely continuous stochastic time Y

Z t with (15) Vt = AYt Yt = ys ds. 0 In this case the dierential triplet (b, 0,F ) and the cumulant process K of V are given by

bt = β yt,Ft(du) = ν(du) yt and Kt(ζ) = κ(ζ) Yt where (β, 0, ν) denotes the Lévy triplet of A, while κ is its Levy exponent given as in (10). The triplet (bX , cX ,F X ) for the corresponding process X in (2) is as follows

X X X X X X bt = β yt, ct = γ yt,Ft (du) = ν (du) yt, where (βX , γX , νX ) is the Lévy triplet given in (11)-(13). Therefore, the driving process X admits the Lévy stochastic volatility process representation (14) with L being the Lévy process char- acterized by (βX , γX , νX ). The combination of jumps in returns and stochastic volatility allows calibration of the volatility surface across both strikes and maturities without the need to introduce explicit time dependence in the parameters (Cont and Tankov (2004)).

2.1 The Variance Gamma Model The Variance Gamma (VG) option pricing model studied in Madan, Carr, and Chang (1998), Madan and Seneta (1990) and Madan and Milne (1991) is embedded in our model setup. The Variance displays paths of nite variation and is obtained by specifying the total

4 variance V process as a Γ-subordinator, i.e., by setting the unit-time distribution V1 as a Gamma law Γ(a, b) with density function

ba f (x) = xa−1e−bx, x > 0, a, b > 0 . (16) Γ(a) The Laplace transform of V is ζVt t κ(ζ) (17) LV (ζ) ≡ E[e ] = e , where the corresponding Lévy exponent is given by

  Z ∞ −bu 1 ζu ae (18) κ(ζ) = log −1 a = (e − 1) du, (1 − ζb ) 0 u implying that the drift , while the Lévy measure ae−bu . γ = 0 ν(du) = u du The associated background-driving random process X becomes a VG process characterized by the parameters a, b > 0 and c 6 0. However, the VG process is commonly parameterized in terms of (σ, ν, θ) from the equivalent time-change representation

Xt = θVt + σWVt where the unit-time V1 is specied as a Gamma law with mean value 1 and variance ν, see e.g. Schoutens (2003). We obtain the following useful relationship between the two parameterizations

ra 1 a  1 σ = , ν = , θ = c − , (19) b a b 2 where σ denotes the standard deviation, ν is a parameter, and θ is a skewness parameter.

2.2 Variance Gamma with Stochastic Volatility To obtain stochastic volatility Variance Gamma (VGSV) models, we specify the total variance process V as a time-changed Gamma subordinator

Z t with (20) Vt = ΓYt Yt = ys ds, 0 where Γ is a Gamma subordinator with density given in (16).

VG-CIR The most common choice in the literature is to let y follow the square-root diusion √ dyt = κ(η − yt)dt + λ ytdWt, (21) where the parameters κ, η, and λ are positive constants and W is a standard Brownian motion. Often referred to as the CIR process due to Cox, Ingersoll, and Ross (1985), this process is positive and mean-reverting, and the Laplace transform of the integrated CIR process Y given y0 is known in explicit form as

 2  exp κ ηt   λ2 2y0z zYt (22) LY (z) = E[e ] = 2κt exp $t , z ∈ C−, $t κ $t 2  λ κ + $ coth 2 cosh 2 + $ sinh 2 √ where $ := κ2 − 2λ2z.

5 VG-ΓOU Another alternative process for y is an Ornstein-Uhlenbeck (OU) driven by a Lévy subordinator (Barndor-Nielsen and Shephard (2001)). Here, we consider the Gamma-OU process. The rate of time change is then the solution of the stochastic dierential equation

dyt = −λytdt + dZλt, (23) where λ > 0 and Z is a . Then y is a strictly positive, with a marginal law that follows a Γ(ς, γ)-distribution and the Laplace transform of the integrated Gamma-OU process Y given y0 is  −λt     zy0(1 − e ) λς γ L (z) = [ezYt ] = exp + γ log − zt , (24) Y E λ z − λγ γ − z(1 − e−λt)/λ for all . z ∈ C− 3 Implied Volatility Expansions

From the martingale-product representation of the asset price process in (8), we make the impor- tant observation that the asset prices St are log-normally distributed conditional on each realization of the total variance process V . As a result, the price of a European call or put option admits a mixing solution (Hull and White (1987) and Romano and Touzi (1997)). It can be represented as a risk-neutral expectation of the Black-Scholes price evaluated in the spot-corrected price of the underlying asset S0 ξt and the total variance Vt. For a European call option with strike K and expiry τ > 0, we obtain the following mixing representation + (25) C(S, K, τ) = E[(Sτ − K) ] = E[Cb(S0ξτ ,Vτ )], where the function Cb(·, ·) is dened as  q  BS 1 (26) Cb(S, V ) ≡ C S, τ V and CBS(S, σ) denotes the usual Black-Scholes call price formula as a function of spot price and volatility. In complete analogy to Renault and Touzi (1996), the mixing solution (25) allows us to see that the generated implied volatility smile is symmetric around at-the-money when the skewness parameter c = 0. On the other hand, letting c < 0 enables the model to capture the empirically-observed negative volatility skew. We make two expansions of the true price of a European call option. First, following Nicolato and Sloth (2012), we consider a two-dimensional Taylor series expansion of the mixing represen- tation of the call price around the point . This yields (S, EVτ ) Denition 1 (The hξ, V i-expansion).

1 ∂2Cb 1 ∂2Cb C(S, K, τ) = C(S, V ) + S2 [(ξ − 1)2] + [(V − V )2] b E τ E τ 2 E τ E τ 2 2! ∂S (S,EVτ ) 2! ∂V (S,EVτ ) ∂2C b (27) + SE[(ξτ − 1)(Vτ − EVτ )] + ··· ∂S∂V (S,EVτ ) Xu and Taylor (1994) and Lewis (2000) consider similar mixing solution expansions in the special case of diusion models with uncorrelated volatility and stock processes. Drimus (2011) considers expansions for the Heston stochastic volatility model. Next, we note that by denition the call price C(S, K, τ) is equal to the Black-Scholes price for the same option characteristics evaluated at the implied volatility I = I(S, K, τ). If we denote 2 the implied total variance by Στ = τI , we have that

C(S, V ) ≡ Cb(S, Στ ). We then expand this expression in the second variable around the expected total variance EVτ using a one-dimensional Taylor series. This step yields the second expansion

6 Denition 2 (The hΣi-expansion).

2 ∂Cb 1 2 ∂ Cb C(S, V ) = C(S, V ) + (Σ − V ) + (Σ − V ) + ··· (28) b E τ τ E τ τ E τ 2 ∂V (S,EVτ ) 2! ∂V (S,EVτ ) Equating the two expansions gives us

2 ∂Cb 1 2 ∂ Cb (Σ − V ) + (Σ − V ) + ··· τ E τ ∂V 2! τ E τ ∂V 2 1 ∂2Cb 1 ∂2Cb = S2 [(ξ − 1)2] + [(V − V )2] (29) 2! E τ ∂S2 2!E τ E τ ∂V 2 ∂2Cb + S [(ξ − 1)(V − V )] + ··· , E τ τ E τ ∂S∂V where the common argument of the Black-Scholes derivatives has been omitted. By (S, EVτ ) dividing both sides by the derivative and solving for , we obtain an analytical ∂C/∂Vb (Στ − EVτ ) though formal expansion of the implied variance I2. Truncating the hΣi-expansion to the pth order and the hξ, V i-expansion to the qth order, we refer to the truncated implied variance expansion as of order O(p, q). As noted by Carr (2000), Taylor expansions of Black-Scholes prices in the underlying spot price and volatility have a nite radius of convergence, while in the hξ, V i-expansion and hΣi- expansion Black-Scholes prices are expanded over the entire domain. Thus, there is no guarantee that higher-order expansions will result in more accurate approximations of the implied variance or volatility. Finally, we note that the relative Black-Scholes derivatives entering the expansion,

!−1 m n ∂Cb  ∂   ∂   m n (30) Ds v ≡ Cb , n, m > 0, ∂V (S,EV ) ∂S ∂V (S,EV ) can be computed analytically and fast using standard mathematical software.2 In the sequel, we will refer to these Vega-normalized derivatives as normalized . They can be viewed as measures of higher-order option risks per unit of Vega risk.

3.1 Expansion Moments The expansion is only meaningful under the assumption that all the moments entering the expres- sion are nite. Moreover, the practical applicability of the expansion relies on the possibility of computing expectations of the type m n , preferably in closed form. From well-known results E[Vτ ξτ ] of , we know that these issues are related to the regularity and tractability of the joint Laplace transform of the total variance Vτ and its cumulant Kτ (c)

uKτ (c) + vVτ (31) LK,V (u, v) = E[e ], dened in the domain n o 2 uKτ (c) + vVτ CK,V = (u, v) ∈ C : E[ |e | ] < +∞ .

Assume that for a given both the points and lie in the interior domain ˚ n > 0 (0, 0) (−n, nc) CK,V of the joint Laplace transform , then m k for all , and it is given LK,V E[Vτ ξτ ] < +∞ m > 0 0 6 k 6 n by ∂m m k (32) E[Vτ ξτ ] = LK,V (u, v) . ∂vm u=−k,v=kc

2In the case n > 0, m = 0, Lewis (2000) derives a recursive scheme for computing the relative Black-Scholes derivatives.

7 Let us look at our particular model cases. In the Variance Gamma model the cumulant process of the Gamma subordinator is deterministic and the joint Laplace transform LK,V reduces to

uτκ(c) LK,V (u, v) = e LV (v) (33)

t κ(v) with LV (v) = e . We observe that the Lévy exponent κ(ζ) in (18) is nite for < ζ < b and existence of the moments m n follows for all . E[Vτ ξτ ] m, n > 0 Next, we consider the case of Variance Gamma stochastic volatility models. Due to the inde- pendence between the Gamma subordinator and the stochastic clock Y in (20), the joint Laplace transform can be expressed in terms of LY as follows

LK,V (u, v) = LY (uκ(c) + κ(v)) . (34) The moments m n for all whenever the point lies in E[Vτ ξτ ] < +∞ m, n > 0 κ(nc) − nκ(c) > 0 the interior domain C˚Y of the Laplace transform of Y . For the VG-CIR and VG-ΓOU models, moments of Y can be obtained in closed-form through direct dierentiation of LY . Alternatively, when Y is an integrated CIR process, the recursive scheme outlined in Dufresne (2001) oers a fast way to compute the moments.

3.2 A Simple Quadratic Approximation

A particularly simple approximation of implied volatility is obtained by truncating the hΣi- expansion (27) to the rst order and the hξ, V i-expansion (28) to the second order. Equating the truncated expansions yields the following simple implied variance approximation of order O(1, 2)

V S2 ar[ξ ] ar[V ] S ov[ξ ,V ] I2(S, K, τ) ≈ E τ + V τ D + V τ D + C τ τ D , (35) τ 2 τ ss 2 τ vv τ sv where Dss denotes the normalized Gamma, Dvv denotes the normalized Volga, and Dsv denotes the normalized Vanna. The Gamma of an option measures the rate of change in the Delta with respect to changes in the underlying price and is computed as the second derivative of the option value with respect to the underlying price. Volga is the second derivative of the option value with respect to the volatility, while Vanna is the second derivative of the option value, once to the underlying spot price and once to volatility. The moments entering the approximation can be computed explicitly under VG or VGSV dynamics and are reported in Table 1.

Table 1: Moments of the quadratic approximation

Model Moments VG VGSV

2 2 E[Vτ ] σ τ σ EYτ 2 4 2 2 4 2 Var[Vτ ] νσ τ σ EYτ + νσ EYτ − σ (EYτ ) τ  (1−ψ)2  ν  1 (1−ψ)2  Var[ξτ ] 1−2ψ − 1 L ν log 1−2ψ − 1  ψ  2  ψ  2 Cov[ξτ ,Vτ ] 1−ψ σ τ 1−ψ σ EYτ Remark: The following notation has been established in the table. First, we dene 1 2. denotes the Laplace transform of the underlying ψ := νθ + 2 νσ L(u) stochastic clock Y evaluated in u ∈ C−. L is given by (22) for the VG-CIR model and by (24) for the VG-ΓOU model.

Approximation (35) gives a direct relationship between the Black-Scholes implied volatility and higher-order option risks, such as the convexity in volatility and the dependence of the option's Delta on the level of volatility. Figure 1 plots the Black-Scholes Greeks and the normalized Greeks against moneyness. While normalized Gamma risk only aects the level of volatility in the approximation, normalized Vanna and Volga risks determine the shape of the approximate

8 volatility smile. In particular, the approximation's Volga term determines upward or downward movements in the wings of the volatility smile with respect to the at-the-money level, while the Vanna term determines the slope of the smile, i.e., a twist of the wings with respect to the at-the- money level. This is consistent with common market understanding; to cope with smile eects, traders not only hedge their exposures to classical Greeks but also to higher-order sensitivities like, in fact, the Vanna and Volga risks. Often one is not just interested in the volatility smile itself but also in its at-the-money skew, which directly relates the smile's asymmetric shape to the correlation between the asset price and its volatility. Dierentiating (35) with respect to log-moneyness, we obtain the following explicit approximation for the at-the-money implied variance skew

2 ∂I Var[Vτ ] ˙ S Cov[ξτ ,Vτ ] ˙ ≈ Dvv + Dsv. ∂x x=0 2 τ τ where ˙ ∂Dvv and ˙ ∂Dsv . One may think of ˙ and ˙ as the at-the-money Dvv = ∂x |x=0 Dsv = ∂x |x=0 Dvv Dsv sensitivities of the option's Volga and Vanna to small changes in moneyness. Analogously, the curvature of the skew is approximately given by

2 2 ∂ I Var[Vτ ] ¨ ≈ Dvv. ∂x2 x=0 2 τ

2 where ¨ ∂ Dvv captures the convexity in the option's Volga risk with respect to money- Dvv = ∂x2 |x=0 ness.

2.6 10 8

2.4 6 5 Volga 2.2 4

2 0 Normalized Volga 2 1.8 −5 0 1.6 −2 1.4 −10 Gamma −4 1.2 Vanna Normalized Gamma −15 Normalized Vanna 1 −6

0.8 −20 −8 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness Moneyness Moneyness

Figure 1: Black-Scholes Greeks versus the corresponding Vega-normalized Greeks for τ = 1/2, S0 = 1, r = q = 0 and a volatility of 25%.

Next, having established how the approximation associates implied volatility of an option to the option sensitivities with respect to movements of underlying quantities, we rewrite the implied variance approximation in terms of log-moneyness K . After computing the normalized x ≡ log S Greeks D• in (35) explicitly and making some rearrangements, we obtain the following analytic approximation

2 2 I (x, τ) ≈ I0(τ) + I1(τ) x + I2(τ) x , (36) where   EVτ Var[Vτ ] 1 2 Var[ξτ ] Cov[ξτ ,Vτ ] I0(τ) = − 2 EVτ + (EVτ ) + + , τ 4τ(EVτ ) 4 τ 2τ Cov[ξτ ,Vτ ] I1(τ) = , τ(EVτ ) Var[Vτ ] I2(τ) = 2 . 4τ(EVτ ) The rst observation we make is that, for a given expiry τ, the implied variance is approx- imately a quadratic polynomial in log-moneyness x with minimum implied variance in x˜ =

9 . Secondly, the functionals have a clear interpretation. − 2 EVτ Cov[ξτ ,Vτ ] / Var[Vτ ] I•(τ) I0(τ) approximates the at-the-money implied variance as a function of time-to-expiry, while the at-the- money skew is approximately given by I1(τ) as a function of τ. Finally, the implied variance curvature is approximately ∂2I2 ≈ 2 I (τ). (37) ∂x2 2 One observes that the at-the-money skew and its curvature diminishes as time-to-expiry increases. This is consistent with empirical facts. Alternative quadratic approximations of implied volatility have been proposed in Backus et al. (1997) and Zhang and Xiang (2008). By establishing relationships between implied volatility and the return distribution, their approximations can be used to identify the conditional, risk-neutral distribution of returns. However, in contrast to (3.2), such approaches cannot determine the process governing the underlying dynamics. Finally, we look at the special case of zero correlation between the price process of the under- lying asset and its volatility process. In this particular case, the quadratic approximation reduces to

V ar[V ] I2(S, K, τ) ≈ E τ + V τ D . (38) τ 2 τ vv Clearly, the implied volatility smile only depends on Volga risk, i.e., the convexity in volatility, and the at-the-money skew ∂I2 . This is consistent with the results of Renault and Touzi ∂x |x=0 = 0 (1996) who show that the smile is symmetric around at-the-money under zero correlation, that is I(−x, τ) = I(x, τ) for I increasing in x > 0. Moreover, rewriting the expression in terms of x gives   2 EVτ 1 1 Var[V ] 2 1 2 (39) I (x, τ) ≈ + x − EVτ − (EVτ ) , τ 4 τ (EV )2 4 which is equivalent to the quadratic approximation considered in Ball and Roma (1994), Lewis (2000) and Lee (2005).

4 Numerics

In this section we study the numerical accuracy of the expansion. We consider three simple approx- imations obtained as special cases of the general formal expansion. Specically, the approximations are obtained by truncating the hΣi-expansion (27) to rst order, while the hξ, V i-expansion (28) is truncated to the second, third and fourth orders. Next, equating the truncated expansions leaves us with three explicit implied volatility approximations of orders O(1, 2), O(1, 3) and O(1, 4). We recall that O(1, 2)-order is the quadratic approximation studied in the previous section. We investigate the numerical performance under Variance Gamma dynamics. The approxi- mated implied volatilities are compared to the ones based on the Fourier transform approach of Lewis (2001) and Lipton (2002). Similarly, the accuracy of the implied volatility skew approxima- tions is studied.

4.1 VG Dynamics

Figure 2 plots the implied volatilities and their approximations for time-to-expiry τ = 5, 1, 1/2, 1/4. In many practical applications the kurtosis parameter ν is set to be a small number (Jäckel (2009)). We use Variance Gamma parameters σ = 25%, ν = 1/10, θ = −1/4. Let us observe from Figure 2 that the approximations seem fairly accurate over any reasonable strike range. For instance, for 6 months to expiry, the mean absolute error over the strike range [0.7, 1.6] - which corresponds to 30% out to 60% in the money calls - is only 13 volatility basis points. However, Figure 2 also makes it clear that the strike range of validity of the approximations

10 27% 30%

Fourier 29% Fourier O(1,2) O(1,2) 26.5% O(1,3) O(1,3) 28% O(1,4) O(1,4)

26% 27% Implied volatility Implied volatility 26%

25.5% 25%

25% 24% 0.5 0.75 1 1.25 1.5 1.75 2 0.5 0.75 1 1.25 1.5 1.75 2 Moneyness Moneyness (a) τ = 5 years (b) τ = 1 year 31% 33%

32% Fourier 30% Fourier O(1,2) 31% O(1,2) O(1,3) O(1,3) 29% 30% O(1,4) O(1,4) 29% 28% 28% 27% 27% Implied volatility Implied volatility

26% 26%

25% 25% 24%

24% 23% 0.6 0.8 1 1.2 1.4 1.6 1.8 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Moneyness Moneyness (c) τ = 1/2 year (d) τ = 1/4 year

Figure 2: Implied volatilities as a function of moneyness K/S0 for the Variance Gamma model. shrinks as time-to-expiry decreases. In addition, one makes the observation that while higher- orders appear better to capture the wings of the smile, the quadratic approximation is the most accurate close to at-the-money. Figure 3 illustrates the quality of the analytical approximations for the at-the-money skew through time. Clearly, the higher-order approximations become erratic for short-to-expiry options. However, the quadratic approximation seems much more stable for short expiries, and overall it gives a fairly accurate approximation of the at-the-money implied volatility skew. Finally, in Figure 4 we study the accuracy of the quadratic approximation through time under various parameter assumptions for the Variance Gamma model. We make the general observation that the at-the-money accuracy of the approximation diminishes for short maturities. A more careful analysis of the approximation's stability with respect to the Variance Gamma parameters reveals that it is particularly sensitive to the kurtosis parameter ν. For an increase in ν, the approximation deteriorates faster when τ → 0 and the quality of the approximation breaks down for τ close to or less than ν. Unless for very short times to expiry, a low value of ν results in a very peaked gamma density. The gamma density loses its uni-modal shape for τ . ν. 4.2 VGSV Dynamics Next, we extend the analysis by allowing for stochastic volatility in the Variance Gamma model. As a starting point, we consider a Variance Gamma process with parameters σ = 25%, ν = 1/20, and θ = −1/5 governed by a stochastic time-change Y as described in Section 2.2. In the rst

11 −3 −10

Fourier O(1,2) O(1,3)

−2 O(1,4) −10

−1

Implied volatility skew −10

0 −10 −1 0 1 10 10 10 Time to expiry

Figure 3: At-the-money implied volatility skew as a function of time-to-expiry τ for the Variance Gamma model. case, Y is chosen as an integrated CIR process with parameters κ = 5/4, η = 1, and λ = 3/4, while, in the second case, Y is an integrated Gamma Ornstein-Uhlenbeck process with parameters , , and . The parameters are chosen so that . λ = 3/4 ς = 5/2 γ = 5/2 E[Y1|y0] = 1 In Figure 5 we plot the approximations against the true implied volatilities for selected ma- turities. Again, we get fairly good approximations for reasonable strike ranges and, similarly to the pure Variance Gamma case, the strike range of validity of the approximations shrinks as time- to-expiry decreases. Moreover, for the given parameter values, it is evident that the quadratic approximation seems to perform best. Especially, the O(1, 3)-approximation appears to have problems capturing the smile around at-the-money for shorter maturities. Let us recall here that the expansion is, in fact, just formal and convergence is not guaranteed a priori. This is, however, not just a shortcoming of this particular expansion but a common trait of similar expansions. One may conjecture that approximations of orders O(2, 2), O(3, 3) and O(4, 4) would be more accurate because terms would balance out. Yet, the potentially higher accuracy would come at the cost of more numerical legwork and, more critically, the neat explicitness inherent in the O(1, ·)-approximations would be somewhat lost. Figure 6 illustrates the quality of the analytical approximations for the at-the-money skew. Clearly, the quadratic approximation seems much more stable for short expiries than the higher- order approximations and gives an overall fairly accurate approximation of the at-the-money im- plied volatility skew.

5 Applications

We discuss two potential applications of the proposed expansion of implied volatility. First, the expansion may be used as a control variate in Fourier pricing of options. Second, it may be used as a fast, rst-order model calibration approach. In particular, we look at joint calibration to the at-the-money volatility and skew. We will exemplify the applications using the Variance Gamma model. However, any option pricing model embedded in the model framework of Section 2 admits, in principle, the following applications.

5.1 Control Variate Fourier transform methods have been introduced to option pricing by several authors, see e.g. Carr and Madan (1999) or Lewis (2001). Schmelzle (2010) provides a comprehensive overview

12 −3 −3 x 10 x 10 26% 6 22% 6

25% 5 21% 5

24% 4 29% 4

23% 3 19% 3 Absolute error Approximation Absolute error

Implied volatility Approximation Implied volatility 18% 2 22% Fourier 2 Fourier Error Error 21% 1 17% 1

20% 0 16% 0 −1 0 1 −1 0 1 10 10 10 10 10 10 Time to expiry Time to expiry (a) σ = 25% (b) σ = 20% −3 −3 x 10 x 10 27% 9 26% 4

26% 8

25% 7 25% 3

24% 6

23% 5 24% 2 22% 4 Absolute error Absolute error Implied volatility Implied volatility 21% Approximation 3 Approximation Fourier 23% Fourier 1 20% Error 2 Error 19% 1

18% 0 22% 0 −1 0 1 −1 0 1 10 10 10 10 10 10 Time to expiry Time to expiry (c) ν = 0.15 (d) ν = 0.05 −3 −3 x 10 x 10 26% 6 26% 6

25% 5 25% 5

24% 4 24% 4

23% 3 23% 3 Absolute error Absolute error Implied volatility Implied volatility 22% 2 22% 2 Approximation Approximation Fourier Fourier Error 21% 1 21% Error 1

20% 0 20% 0 −1 0 1 −1 0 1 10 10 10 10 10 10 Time to expiry Time to expiry (e) θ = −0.20 (f) θ = −0.15

Figure 4: At-the-money implied volatilities as a function of time τ for the Variance Gamma model under various parameter assumptions. The base-case parameters are σ = 25%, ν = 1/10, θ = −1/4. of Fourier transform methods with applications to option pricing. The approach suggested in Lewis (2001) and independently derived in Lipton (2002) stipulates that given the (normalized)

13 29% 29%

28% Fourier 28% Fourier O(1,2) O(1,2) O(1,3) O(1,3) 27% O(1,4) 27% O(1,4)

26% 26% Implied volatility Implied volatility

25% 25%

24% 24% 0.5 0.75 1 1.25 1.5 1.75 2 0.5 0.75 1 1.25 1.5 1.75 2 Moneyness Moneyness (a) VG-CIR model and τ = 1 year (b) VG-ΓOU model and τ = 1 year

30% 30%

Fourier Fourier 29% 29% O(1,2) O(1,2) O(1,3) O(1,3) 28% 28% O(1,4) O(1,4)

27% 27%

Implied volatility 26% Implied volatility 26%

25% 25%

24% 24% 0.6 0.8 1 1.2 1.4 1.6 1.8 0.6 0.8 1 1.2 1.4 1.6 1.8 Moneyness Moneyness (c) VG-CIR model and τ = 1/2 year (d) VG-ΓOU model and τ = 1/2 year

Figure 5: Implied volatilities as a function of moneyness K/S0 for the VG-CIR model (Left) and the VG-ΓOU model (Right).

characteristic function iu log Sτ the price of a call option is φτ (u) ≡ Ee Z iα+∞ K −izx dz (40) C(S, K, τ) = S − e φτ (−z) 2 , 2π iα−∞ z − iz for α = =z ∈ (0, 1), assuming zero interest rates and dividends for simplicity. In practice, one often chooses the integration contour α = 1/2, and, after some rearrangements, the formula becomes √ SK Z ∞ du C(S, K, τ) = S − < e−iuxφ u − i  . (41) τ 2 2 1 π 0 u + 4 To improve convergence, Andersen and Andreasen (2002) suggest incorporating a control variate when computing the inverse Fourier integral in (41). This idea has been further elaborated in Sepp (2003), Itkin (2005), and Andersen and Piterbarg (2011). The common approach is to use the Black-Scholes price as a control variate. Specically, via Fourier transform, one computes the dierence between the option price and the Black-Scholes price for the same option characteristics and a suitable volatility. The price of a call option would then be given by

C(S, K, τ) = CBS(S, K, τ; σ) (42) √ Z ∞ SK 1 h −iux  i  − 1 (u2+ 1 )τσ2 i − < e φ u − − e 2 4 du. 2 1 τ 2 π 0 u + 4

14 −3 −3 −10 −10

−2 −2 −10 −10

−1 −1

Implied volatility skew −10 Implied volatility skew −10 Fourier Fourier O(1,2) O(1,2) O(1,3) O(1,3) O(1,4) O(1,4)

0 0 −10 −10 −1 0 1 −1 0 1 10 10 10 10 10 10 Time to expiry Time to expiry

Figure 6: At-the-money implied volatility skew as a function of time-to-expiry τ for the VG-CIR model (Left) and the VG-ΓOU model (Right).

Eectively, this reduces the number of function evaluations required in numerically integrating the inverse Fourier transform with a certain accuracy or error tolerance. However, this strategy also poses a tautological question as to which volatility σ to use in the Black-Scholes pricing formula. Thus, the choice of volatility has generally been arbitrary; a popular choice being the at-the-money spot volatility. Instead, we suggest using the quadratic approximation to compute an approximate value for the option's implied volatility. In other words, we set σ = IQA(x, τ) in (42) with IQA denoting the implied volatility obtained by (36), and we refer to this approach as the QA control. As documented in the previous section, the quadratic approximation is reasonably accurate for such applications. The QA control will further reduce the numerical integration error compared to the common case of arbitrarily chosen volatility, the BS control. The QA approach does not really add more complexity to the problem as the quadratic approximation is just an elementary function of the option's moneyness and expiry. In addition, by appropriate vectorization, the whole implied volatility surface can be approximated in just one sweep providing us with a surface of suitable control variates. Finally, the QA control variate may also be used in model calibration based on Fourier inversion. No extra parameters need to be calibrated as the parameters of the quadratic approximation are exactly the same as those dening the model's characteristic function. In the numerical experiments, we compare three cases: (i) Fourier pricing without a control variate, (ii) Fourier pricing with the BS control, and (iii) Fourier pricing with the QA control. Table 2 reveals the number of function evaluations required for numerical integration of the inverse Fourier transform under the Variance Gamma model with parameters σ = 25%, ν = 1/10, θ = −1/4 and τ = 1 year. In the BS control case the Black-Scholes volatility is put equal to the σ parameter of the VG model. We use two adaptive quadratures for the numerical integration, a recursive Simpson's rule and a Gauss-Lobatto quadrature rule, and, for both cases, we choose an absolute error tolerance of 1.0e-9. Compared to the case without a control variate, the QA control variate reduces the average number of functional evaluations by 64% for Simpson's rule and 59% for the Gauss-Lobatto quadrature. The number of function evaluations are 28% (Simpson's) and 33% (Gauss-Lobatto) less, on average, compared to the case with the BS control. Tables 5-9 in the Appendix provide results of further tests of the numerical performance of the QA control variate for various option expiries and parameters of the VG model. The overall picture is that the QA control reduces the number of functional evaluations by > 60% (Simpson's) and > 50% (Gauss-Lobatto), on average, compared to not using a control variate at all, while the average reductions are > 20% (Simpson's) and > 30% (Gauss-Lobatto) compared to using the BS control.

15 Table 2: Number of function evaluations required for nu- merical integration of the inverse Fourier transform

Control Variate Strike No Control BS Control QA Control 600 397 225 189 [618] [408] [318] 700 397 209 153 [618] [408] [288] 800 397 197 141 [588] [378] [228] 900 401 189 109 [558] [318] [168] 1000 401 189 125 [588] [348] [198] 1100 405 189 141 [618] [348] [228] 1200 393 185 133 [618] [378] [258] 1300 405 197 141 [618] [378] [258] 1400 413 221 161 [618] [378] [288] Mean Reduction 50% 64% [39%] [59%] Remark: The experiment is based on a VG model with parameters σ = 0.25, ν = 0.10, θ = −0.25. Moreover, S0 = 1000, τ = 1 year, and r = q = 0. In the BS Control case, we use the Black-Scholes price with a volatility of σ as control variate, while we use the quadratic approximation as control variate in the QA Control case. We consider two adaptive quadrature rules for numerical integration: the recursive Simpson's rule (without brackets) and the Gauss-Lobatto quadrature (within brackets). In all cases, we use an absolute error tolerance of 1.0e-9.

5.2 Calibration We propose a fast, rst-order calibration approach. Specically, we propose to jointly calibrate the model to the at-the-money volatility and skew using the corresponding approximations presented in this paper. We conduct a series of calibration experiments to document the use of the approach. We consider three data cases. In Case I, we generate at-the-money volatilities and values of the skew from a VG model with parameters σ = 0.25, ν = 0.10, and θ = −0.25. In Case II, we change the parameters to σ = 0.25, ν = 0.10, and θ = −0.15. Finally, in Case III, we use the parameters σ = 0.20, ν = 0.15, and θ = −0.25. The data was generated using the Fourier transform approach discussed in the previous section. First, note that we have the following implicit relationship between the implied volatility surface and the characteristic function, Z ∞ h −iux  i  − 1 (u2+ 1 )τI2 i du < e φ u − − e 2 4 = 0, (43) τ 2 2 1 0 u + 4 allowing us to numerically compute the true implied volatilities. Then, following Gatheral (2006), we compute the at-the-money implied volatility skew as

∞ ∂I τI2 1 Z u 8 2  2  i  (44) = −e πτ 1 = φτ u − 2 du. x=0 2 ∂x 0 u + 4 The generated experimental data is shown in Table 4 in the Appendix.

16 Next, we use our approximation to calibrate the Variance Gamma model to the generated at-the-money volatilities and at-the-money skew values jointly. We use the Levenberg-Marquardt algorithm to minimize the sum of squared errors. Moreover, we only concentrate on the quadratic approximation. From Section 4.1 we saw that while higher-order approximations oered more accuracy in the wings of the volatility smile, the quadratic approximation was more accurate around at-the-money which is also where we have most faith in the option quotes. Finally, the approximations deteriorate for very short maturities, hence we exclude the rst observation (τ = 0.8) from the experiments. This leaves us with 2 × 13 observation points to calibrate to in each experimental case.

Table 3: Calibrated model parameters

Case I Case II Case III True Calibrated True Calibrated True Calibrated σ 0.25 0.2505 0.25 0.2498 0.20 0.2017 ν 0.1 0.1091 0.1 0.1028 0.15 0.1554 θ -0.25 -0.2458 -0.15 -0.1618 -0.25 -0.2431

Table 3 gives the calibrated as well as the true parameters in the three cases. We see that the suggested expansion-based calibration approach gives parameter values that are reasonably close to the true parameters from which the data was generated. Besides a gain in computational speed, the advantage of the calibration approach is that fewer data points are required in the calibration compared to calibrating to an option price surface. Consistency with the observed at-the-money volatilities controls the level of the model implied volatility surface, while consistency with the at-the-money skews secures that the model volatility surface has the correct smile/skew shape.

−3 −3 −3 x 10 x 10 x 10

1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4 Calibration error Calibration error Calibration error 0.2 0.2 0.2

0 0 0 3 3 3

2 1.2 2 1.2 2 1.2 1.1 1.1 1.1 1 1 1 1 1 1 0.9 0.9 0.9 0.8 0.8 0.8 0 0.7 0 0.7 0 0.7 Time to expiry Moneyness Time to expiry Moneyness Time to expiry Moneyness

Figure 7: Calibration error surfaces measured in 10 volatility basis points. (Left) Case I, (Middle) Case II, and (Right) Case III.

To further support the calibrated parameters, we look at the absolute errors between the true implied volatilities and implied volatilities based on the calibrated parameters for call options with strikes from 30% out of the money to 30% in the money. The calibration errors are illustrated in Figure 7. In the three cases the mean absolute calibration errors are less than 6 volatility basis points. Even for liquid equity options, the bid-ask spread is often not much less than 100 volatility basis points. In other words, the dierences between the true and the calibrated parameters are not economically signicant.

6 Conclusion

In this paper we derive a closed-form expansion of the implied volatility surface. The expansion is simple and easy to interpret and implement. In addition, it decomposes the volatility surface into meaningful quantities that directly relate its smile/skew shape to higher-order option risks and analytical features of the underlying model. We apply the expansion to the popular Variance

17 Gamma model and its extensions which allow for stochastic volatility. However, the expansion is largely generic in the sense that it may be used for any option pricing model with time-changed underlying dynamics. Finally, we explore two domains of potential application. First, we suggest using the expansion as a control variate in Fourier option pricing. Our numerical experiments show that this eectively reduces the number of function evaluations required for computing the inverse Fourier transform. Secondly, we propose a speedy, expansion-based approach for model calibration to at-the-money volatilities and skews.

18 References

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21 A Appendix

Table 4: Generated data for calibration experiments

Case I Case II Case III Expiry (yrs) ATM Vol ATM Skew ATM Vol ATM Skew ATM Vol ATM Skew 0.08 22.54% -0.7058 21.84% -0.4659 18.17% -1.1652 0.17 24.12% -0.2870 23.48% -0.1721 19.81% -0.5069 0.25 24.63% -0.1850 24.01% -0.1084 20.36% -0.3317 0.50 25.19% -0.0871 24.59% -0.0501 21.00% -0.1575 0.75 25.39% -0.0569 24.79% -0.0325 21.22% -0.1029 1.00 25.49% -0.0422 24.89% -0.0241 21.33% -0.0764 1.25 25.55% -0.0336 24.95% -0.0191 21.40% -0.0607 1.50 25.58% -0.0279 24.99% -0.0159 21.44% -0.0504 1.75 25.61% -0.0238 25.02% -0.0136 21.48% -0.0431 2.00 25.63% -0.0208 25.05% -0.0118 21.50% -0.0376 2.25 25.65% -0.0185 25.06% -0.0105 21.52% -0.0334 2.50 25.66% -0.0166 25.08% -0.0094 21.54% -0.0300 2.75 25.68% -0.0151 25.09% -0.0086 21.55% -0.0272 3.00 25.68% -0.0138 25.10% -0.0078 21.56% -0.0249

22 Table 5: Number of function evaluations required for numer- ical integration of the inverse Fourier transform for τ = 1/2 year and VG parameters σ = 0.25, ν = 0.10, θ = −0.25.

Control Variate Strike No Control BS Control QA Control 600 449 257 249 [648] [438] [378] 700 437 237 181 [648] [408] [288] 800 421 201 161 [618] [348] [288] 900 421 201 121 [588] [318] [198] 1000 429 201 121 [588] [318] [198] 1100 429 205 145 [618] [318] [198] 1200 421 197 161 [618] [318] [258] 1300 433 205 161 [618] [318] [258] 1400 441 229 169 [618] [378] [258] Mean Reduction 50% 62% [43%] [58%]

Table 6: Number of function evaluations required for numer- ical integration of the inverse Fourier transform for τ = 1 year and VG parameters σ = 0.25, ν = 0.10, θ = −0.15.

Control Variate Strike No Control BS Control QA Control 600 393 221 169 [618] [408] [318] 700 389 201 133 [618] [408] [258] 800 393 193 129 [588] [378] [228] 900 398 185 113 [588] [348] [198] 1000 401 189 121 [588] [348] [198] 1100 401 189 133 [618] [348] [228] 1200 393 195 133 [618] [378] [228] 1300 409 201 133 [618] [378] [258] 1400 417 225 141 [618] [378] [288] Mean Reduction 50% 66% [38%] [60%]

23 Table 7: Number of function evaluations required for numer- ical integration of the inverse Fourier transform for τ = 1/2 year and VG parameters σ = 0.25, ν = 0.10, θ = −0.15.

Control Variate Strike No Control BS Control QA Control 600 449 253 221 [648] [438] [378] 700 449 249 185 [618] [378] [258] 800 425 205 153 [588] [318] [228] 900 425 205 121 [588] [318] [198] 1000 429 201 125 [588] [318] [198] 1100 417 193 133 [618] [318] [198] 1200 425 201 153 [618] [318] [228] 1300 433 205 157 [618] [318] [228] 1400 437 225 165 [648] [408] [318] Mean Reduction 50% 64% [43%] [60%]

Table 8: Number of function evaluations required for numer- ical integration of the inverse Fourier transform for τ = 1 year and VG parameters σ = 0.20, ν = 0.15, θ = −0.15.

Control Variate Strike No Control BS Control QA Control 600 421 245 221 [618] [438] [378] 700 417 213 173 [618] [408] [348] 800 405 201 153 [588] [348] [258] 900 413 197 129 [588] [318] [198] 1000 409 189 117 [588] [288] [198] 1100 413 193 145 [618] [318] [228] 1200 405 189 145 [618] [348] [228] 1300 413 197 149 [618] [348] [258] 1400 421 205 177 [618] [378] [348] Mean Reduction 51% 62% [42%] [55%]

24 Table 9: Number of function evaluations required for numer- ical integration of the inverse Fourier transform for τ = 1/2 year and VG parameters σ = 0.20, ν = 0.15, θ = −0.15.

Control Variate Strike No Control BS Control QA Control 600 505 309 305 [738] [528] [468] 700 477 265 225 [708] [438] [378] 800 453 237 197 [708] [438] [378] 900 433 189 141 [648] [348] [258] 1000 441 193 137 [618] [288] [228] 1100 441 189 153 [678] [348] [258] 1200 437 209 173 [678] [378] [288] 1300 453 221 181 [648] [348] [318] 1400 449 229 201 [678] [378] [348] Mean Reduction 50% 58% [43%] [52%]

25