Simple Smiles For The Mixing Setup Joint work with D. Sloth Elisa - - PowerPoint PPT Presentation

Simple Smiles For The Mixing Setup

Joint work with D. Sloth Elisa Nicolato

Department of Economics and Business, Aarhus University

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 1 / 26

Few Words of Introduction

Analytical approximations of implied volatility have been and continue to be proposed, even for solvable models, for the need of Transparency Robustness Speed In this work, we propose an approximation of the implied volatility which can be used for a large variety of well-established models and is Transparent, as it decomposes the /smile into meaningful quantities associated with higher-order option risks. Simple, fast and easy to implement. Quite accurate where it matters.

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 2 / 26

The Mixing Setup

The Mixing Setup

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 3 / 26

The Mixing Setup

The Mixing Setup

The mixing setup is a natural generalization of the Black-Scholes model St = S0 exp

• −1

2σ2t + Wσ2t

• .
• btained by randomizing the spot S0 and the total variance σ2t via their

stochastic counterparts. The risk-neutral dynamics of the asset price are given by St = S eff

t exp

• −1

2V eff

t

+ WV eff

t

• ,

V eff

0 = 0, S eff 0 = S0,

where S eff is a positive martingale, V eff is an increasing process and W is a Brownian motion, independent of (S eff, V eff). The price process S is a conditionally log-normal martingale.

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 4 / 26

The Mixing Setup

Embedded subclasses: SV models

The mixing setup contains stochastic volatility models of the following type dSt = St√vt

• 1 − ρ2dW + ρdB
• dvt

= µ(vt, t)dt + σ(vt, t)dBt, v0 > 0, where W ⊥ B are Brownian motions and ρ is the correlation parameter. The Heston model, the 3/2 model, and the quadratic class specification are examples of solvable specifications. The mixing representation follows by setting V eff

t

= (1 − ρ2) t

0 vsds,

S eff

t

= S0 exp

• − ρ2

2

t

0 vsds + ρ

t √vsdBs

• ,

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 5 / 26

The Mixing Setup

Purely jumping models

Purely jumping models are obtained by setting V eff as an increasing and purely jumping semimartingale. The price dynamics are then specified as St = S eff

t exp

• − 1

2V eff t

+ WV eff

t

• ,

S eff

t

= S0 exp (−Kt(c) + cV eff

t ) ,

where the real parameter c allows for correlation between S and V eff, and K(c) is the cumulant exponent process. Exponential L´ evy models are obtained by modeling V eff as a drift-less L´ evy subordinator. Models of this class include, for instance, the VG model, the NIG model, and the CGMY model.

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 6 / 26

The Two Series Expansions

The Two Series Expansions

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 7 / 26

The Two Series Expansions

The S, V -expansion

By conditional log-normality, the price C(S0, K, τ) of a call with strike K and expiry τ > 0 admits the mixing representation C(S0, K, τ) = E[(Sτ − K)+] = E[CBS(S eff

τ , V eff τ )],

where CBS(S, V ) denotes the BS call-price in terms of total variance V = στ. First, we apply a 2-dimensional Taylor series expansion around the point (ES eff

τ , EV eff τ ) = (S0, EV eff τ )

E[CBS(S eff

τ , V eff τ )] = CBS(S0, EV eff τ ) + 1

2!E[(S eff

τ − S0)2]∂2CBS

∂S2 + 1 2!E[(V eff

τ − EV eff τ )2]∂2CBS

∂V 2 + E[(S eff

τ − S0)(V eff τ − EV eff τ )]∂2CBS

∂S∂V + · · ·

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 8 / 26

The Two Series Expansions

The Σ-expansion

Next, recall that by definition C(S0, K, τ) ≡ CBS(S0, Σ). where Σ = τI 2 denotes the implied total variance. Then expand this expression in the second variable Σ around EV eff

τ

using a

• ne-dimensional Taylor series.

C(S0, K, τ) = CBS(S0, EV eff

τ ) +

• Σ − EV eff

τ

∂CBS ∂V + 1 2!

• Σ − EV eff

τ

2 ∂2CBS ∂V 2 + · · ·

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 9 / 26

The Two Series Expansions

From Call Prices to Implied Vols

Finally truncate the Σ-expansion to the first order and the S, V -expansion to the q-th order, to approximate I 2 as I 2 ≈ EV eff

τ

τ + 1 τ

q

• k=2

k

• l=0

Dslvk−l l!(k − l)! E[(S eff

τ − S0)l(V eff τ − EV eff τ )k−l],

where Dsmvn ≡ ∂CBS

∂V

−1 ∂ m+nCBS

∂Sm∂V n denote the Vega-normalized Black-Scholes

derivatives. Application demands that E

• (V eff

t )m (S eff t )n

are easy to compute. This is a simple task whenever Leff

t (u, w) = E

• euX eff

t

+ wV eff

t

• with

X eff = log S eff, is available in (semi) closed-form. In this case E

• V eff

t

m S eff

t

n = ∂m Leff

t (u, w)

∂w m

• u=n,w=0,

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 10 / 26

The Two Series Expansions

A Simple Quadratic Approximation

The 2nd-order expansion yields a simple approximation I 2

Q(x, τ) of the

implied variance which is quadratic in x = log K/S. Specifically I 2 ≈ I 2

Q(x, τ) = EV eff τ

τ + Var[S eff

τ ]

2 τ Dss + Cov[S eff

τ , V eff τ ]

τ Dsv + Var[V eff

τ ]

2 τ Dvv , where the normalized Gamma Dss, Vanna Dsv and Volga Dvv are Dss = 2 S2 , Dsv = x SV + 1 2S , Dvv = x2 2V 2 − 1 8 − 1 2V . We see that

• the Gamma risk Dss only contributes to the level of smile,
• the Vanna term Dsv determines the slope
• the Volga term Dvv introduces convexity.

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 11 / 26

The Two Series Expansions

A first look at ATM term-structure

Re-arranging the terms we obtain I 2

Q(x, τ) = I0(τ) + I1(τ) x + I2(τ) x2,

where I.(τ) describe the term structure of the approximate smile:

• ATM Variance:

I0(τ) = EV eff

τ

τ + Var[S eff

τ ]

S2

0τ

+ Cov[S eff

τ , V eff τ ]

2S0τ − Var[V eff

τ ]

4τEV eff

τ

• 1 + 1

4EV eff

τ

• ATM Skew:

I1(τ) = 1 τ Cov[S eff

τ , V eff τ ]

S0 EV eff

τ

• ATM Curvature:

I2(τ) = 1 τ Var[V eff

τ ]

4(EV eff

τ )2

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 12 / 26

Does it work?

Does it work?

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 13 / 26

Does it work?

Illustration for SV Models

Naturally, we consider the Heston (1993) model dvt = κ(θ − vt)dt + εv 1/2

t

dBt. We also consider the 3/2 model with the instantaneous variance dvt = vtκ(θ − vt)dt + εv 3/2

t

dBt. Both models are solvable, as the joint Laplace transform LXV

t

• f X = log S

and V = ·

0 vsds has closed-form.

Also the relevant moments are computable, since it holds that Leff

t (u, w) = LXV t

• u , (1 − ρ2)(w + 1

2u − 1 2u2)

• ,

between the ”standard” and the ”effective” transforms. However, Fourier inversion is numerically not trivial in the 3/2 model, due to complex evaluations of the confluent hypergeometric function.

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 14 / 26

Does it work?

ATM Implied Vols and Skews

2 4 6 8 10 12 14 16 18 19.3% 19.4% 19.5% 19.6% 19.7% 19.8% 19.9% 20% 20.1% maturity ATM vols ATM Vols. Heston Fourier Quadratic 2 4 6 8 10 12 14 16 18 −4% −3.5% −3% −2.5% −2% −1.5% −1% −0.5% 0% maturity log−scale skews ATM skews Heston Fourier Quadratic 2 4 6 8 10 12 14 16 16.5% 17% 17.5% 18% 18.5% 19% 19.5% 20% maturity log−scale ATM vols ATM Vols 3/2 Fourier Quadratic 2 4 6 8 10 12 14 16 −4% −3.5% −3% −2.5% −2% −1.5% −1% −0.5% 0% maturity log−scale skews ATM skews 3/2 Fourier Quadratic

ATM vols (left) and skews (right) for the Heston (top) and the 3/2 (bottom). The maturity ranges from τ = 0.05 up to τ = 18 years. Parameters are as in Forde et al. (2012).

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 15 / 26

Does it work?

SV Models: The Smile at Short Maturities

The ATM accuracy of I 2

Q(x, τ) at short maturities is not coincidental.

For models with time-independent coefficients dvt = a(vt)dt + b(vt)dBt. it holds that lim

τ→0 I 2(0, τ) = v0

and lim

τ→0

∂I 2 ∂x2

• x=0 = 1

2 ρb(v0) √v0 see e.g., Lewis (2000), Lee (2001) and Medvedev and Scaillet (2007). The quadratic approximation is consistent with these results lim

τ→0 I0(τ) = v0

and lim

τ→0 I1(τ) = 1

2 ρb(v0) √v0 It is therefore tempting to compare I 2

Q(x, τ) with well-established asymptotic

results.

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 16 / 26

Does it work?

Heston model at short maturities

0.9 0.95 1 1.05 1.1 1.15 18.5% 19% 19.5% 20% 20.5% 21% 21.5% 22% moneyness implied vols τ=0.05 Fourier Quadratic FJL 0.9 0.95 1 1.05 1.1 1.15 18.5% 19% 19.5% 20% 20.5% 21% 21.5% 22% moneyness implied vols τ=0.1 Fourier Quadratic FJL 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 18% 18.5% 19% 19.5% 20% 20.5% 21% 21.5% 22% 22.5% moneyness implied vols τ=0.25 Fourier Quadratic FJL 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 18% 18.5% 19% 19.5% 20% 20.5% 21% 21.5% 22% 22.5% moneyness implied vols τ=0.5 Fourier Quadratic FJL

I 2

Q vs Forde et al. (2012), for Heston model at short maturities.

Maturities: τ = 0.05 to τ = 0.5. Moneyness: ±15%.

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 17 / 26

Does it work?

3/2 Model at Short Maturities

0.9 0.95 1 1.05 1.1 1.15 18.5% 19% 19.5% 20% 20.5% 21% 21.5% 22% moneyness implied vols τ=0.07 Fourier Quadratic Med−Sca 0.9 0.95 1 1.05 1.1 1.15 18.5% 19% 19.5% 20% 20.5% 21% 21.5% 22% moneyness implied vols τ=0.1 Fourier Quadratic Med−Sca 0.9 0.95 1 1.05 1.1 1.15 18.5% 19% 19.5% 20% 20.5% 21% 21.5% 22% moneyness implied vols τ=0.25 Fourier Quadratic Med−Sca 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 18% 18.5% 19% 19.5% 20% 20.5% 21% 21.5% 22% 22.5% moneyness implied vols τ=0.5 Fourier Quadratic Med−Sca

I 2

Q vs Medvedev and Scaillet (2007), for general SV (plus jumps),

short-maturities/small strikes. Maturities: τ = 0.07 to τ = 0.5. Moneyness: ±15%.

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 18 / 26

Does it work?

Mid-Long Maturities

Top: Heston model. Bottom: 3/2 model Maturities τ = 1, 3, 5. Moneyness: from ±30% to ±40%.

0.8 0.9 1 1.1 1.2 1.3 18% 18.5% 19% 19.5% 20% 20.5% 21% 21.5% 22% 22.5% moneyness implied vols Heston, τ=1 Fourier Quadratic 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 18% 18.5% 19% 19.5% 20% 20.5% 21% 21.5% moneyness implied vols Heston, τ=3 Fourier Quadratic 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 18% 18.5% 19% 19.5% 20% 20.5% 21% moneyness implied vols Heston, τ=5 Fourier Quadratic 0.8 0.9 1 1.1 1.2 1.3 13% 14% 15% 16% 17% 18% 19% 20% 21% 22% 23% moneyness implied vols 3/2 Model, τ=1 Fourier Quadratic 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 12% 13% 14% 15% 16% 17% 18% 19% moneyness implied vols 3/2 Model, τ=3 Fourier Quadratic 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 12.5% 13% 13.5% 14% 14.5% 15% 15.5% 16% 16.5% 17% moneyness implied vols 3/2 Model, τ=5 Fourier Quadratic

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 19 / 26

Does it work?

SV models: The smile at long maturities

At large maturities, the asymptotic behavior is lim

τ→∞ I 2(x, τ) = 8λ(k0)

and ∂I 2 ∂x

• x=0 ≈ −8ik0 + 4

τ + O(1/τ 2) as τ → ∞ where λ is the first eigenvalue of a differential operator and k0 is a complex

• number. See Lewis (2000), Jaquier (2007) and Tehranchi (2009).

For I 2

Q, if ρ = 0, it holds that

lim

τ→∞ I0(τ) = ∞

lim

τ→∞ I1(τ) = 0

lim

τ→∞ I2(τ) = 0

So the accuracy of the quadratic approximation is bound to deteriorate as the maturity increases. Luckily, this happens at a very slow rate, and the mismatch becomes

• bservable only at extremely long expiries.

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 20 / 26

Does it work?

ATM Implied Vols and Skews - Revisited

10

−1

10 10

1

19.3% 19.4% 19.5% 19.6% 19.7% 19.8% 19.9% 20% 20.1% maturity log−scale ATM vols ATM Vols. Heston Fourier Quadratic FJL 10

−1

10 10

1

−4% −3.5% −3% −2.5% −2% −1.5% −1% −0.5% 0% maturity log−scale skews ATM skews Heston Fourier Quadratic Asymptotic 10

−1

10 10

1

16.5% 17% 17.5% 18% 18.5% 19% 19.5% 20% maturity log−scale ATM vols ATM Vols 3/2 Fourier Quadratic Asymptotic 10

−1

10 10

1

−4% −3.5% −3% −2.5% −2% −1.5% −1% −0.5% 0% maturity log−scale skews ATM skews 3/2 Fourier Quadratic Asymptotic

ATM Vols (left) and Skews (right), for Heston (top) and 3/2 (bottom), but with added ATM asymptotic behaviors. Maturity from τ = 0.05 to τ = 18 years.

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 21 / 26

Exponential L´ evy models

Exponential L´ evy models

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 22 / 26

Exponential L´ evy models

Exponential L´ evy Models

True short-maturity behavior lim

τ→0 I 2(0, τ) = 0

and lim

τ→0 I 2(x, τ) = +∞,

for x = 0 True long-maturity behavior lim

τ→∞ I 2(x, τ) = A

and I 2(x, τ) ≈ A + B τ + C τ x for large τ In case of exponential L´ evy models, I 2

Q(x, τ) takes the form

I 2

Q(x, τ) = A(τ) + B

τ + C τ x + D τ 2 x2, with B < 0, limτ→0 A(τ) = A and limτ→∞ A(τ) = ∞ (unless c = 0). In spite of this ”anti-asymptotic” behavior, I 2

Q(x, τ) is nevertheless quite

useful.

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 23 / 26

Exponential L´ evy models

VG Model smiles

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 23% 24% 25% 26% 27% 28% 29% 30% 31% 32% 33% τ=0.25 moneyness implied vols Fourier Quadratic Jackel 0.7 0.8 0.9 1 1.1 1.2 1.3 24% 25% 26% 27% 28% 29% 30% 31% moneyness implied vols τ=0.5 Fourier Quadratic Jackel 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 24.5% 25% 25.5% 26% 26.5% 27% 27.5% 28% 28.5% 29% 29.5% moneyness K implied vols τ=1 Fourier Quadratic Jackel 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 25.4% 25.5% 25.6% 25.7% 25.8% 25.9% 26% 26.1% 26.2% 26.3% 26.4% moneyness K implied vols τ=5 Fourier Quadratic Jackel

I 2

Q(x, τ) vs J¨

ackel (2009) singular approximation. Maturity: τ = 0.25 up to τ = 5. Moneyness: ±30% up to ±50%. J¨ ackel (2009) parameters.

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 24 / 26

Exponential L´ evy models

VG Model Term Structure

10

−1

10 10

1

20% 21% 22% 23% 24% 25% 26% maturity ATM volatility ATM Vols VG Fourier Quadratic Jackel 10

−1

10 10

1

−0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 maturity ATM volatility ATM Skews VG Fourier Quadratic Jackel

While both approximations are singular around the expiry-date, they both capture the overall behavior of the surface for a large relevant region of the smile.

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 25 / 26

Exponential L´ evy models

Concluding Remarks

The approximation is simple, easy to implement, and quite accurate where it matters (i.e. liquid moneyness). The approximation decomposes the volatility smile into meaningful quantities associated with higher-order option risks. The approximation is largely generic in the sense that it may be used for a large variety of option pricing models. Finally, in the paper we explore two domains of application of the approximation.

1

We suggest to use the approximation as a control variate in Fourier

• ption pricing.

2

We propose a ’speedy’, approximation-based approach for model calibration to at-the-money volatilities and skews.

Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 26 / 26