the small-maturity smile for exponential lévy modelsfigueroa/talks/slidessiam-seas.pdf · for...
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The small-maturity smile for exponential Lévy
models
José Enrique Figueroa-López1
1Department of Statistics
Purdue University
SIAM-SEAS
Minisymposium in Mathematical Finance
UNC Charlotte
March 27, 2011
Joint work with Martin Forde, Dublin City University
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Motivation Empirical features of implied volatility
From the book “Trading Options at Expiration: Strategies and Models
for Winning the Endgame" by Jeff Augen:
“The final hours of each expiration cycle are characterized by
unusual market forces and price distortions caused by the
breakdown of traditional pricing calculations"
![Page 3: The small-maturity smile for exponential Lévy modelsfigueroa/Talks/SlidesSIAM-SEAS.pdf · for Winning the Endgame" by Jeff Augen: “The final hours of each expiration cycle are](https://reader036.vdocument.in/reader036/viewer/2022071417/6114cecc1ca34c7e5a33d9d6/html5/thumbnails/3.jpg)
Motivation Empirical features of implied volatility
Empirical features of implied volatilities
1 Implied volatility σ̂(K ,T ) as a function of K is
• U-shaped with minimum around S0 (“smirk") ; or
• Decreasing (skew);
2 Market charges a premium for OTM puts (K < S0) above their BS price
computed with ATM implied vol σ̂(S0,T );
3 Smile flattens out as T →∞;
4 “Significant" skew for short-term options (t → T );
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Motivation Close-to-expiration smile for continuous models
Known results for continuous models
1 [Gatheral et al. (2010)]. In a local volatility model:
dSt = St {rdt + σ(St )dWt} ⇐⇒ logSt+h
St
D≈ N
([r − σ2(St )
2]h, σ2(St )h
),
the implied volatility satisfies:
σ̂t (T ,K ) = σ0(K ) + σ1 × (T − t) + O((T − t)2),
where σ0(K ) =(
1ln(S/K )
∫ KS
1uσ(u)du
)−1.
2 In particular, as t → T , implied volatility smile behaves like σ0(K );
3 Similar behavior for other popular continuous model including Heston
model, SABR model, etc.
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Close-to-expiration option prices for exponential Lévy models (ELM) Description of the problem
The problem
1 Set-up: Suppose that St = S0eXt follows an exponential Lévy model
under the (risk-neutral) pricing measure Q;
2 Goal 1: Analyze the behavior of the option prices close-to-expiration:
ΠLévyt (T ,K ,S, r) := e−r(T−t)EQ((ST − K )+|St = S),
3 Goal 2: Obtain close-to-expiration asymptotics for the implied volatility
σ̂ := σ̂t (T ,K ):
ΠBSt (σ̂,T ,K ,S, r) = ΠLévy
t (T ,K ,S, r).
4 Applications:
• Calibration of the model parameters to market option prices near expiration;
• Quick and stable pricing of options near expiration;
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Close-to-expiration option prices for exponential Lévy models (ELM) A fundamental preliminary result
Small-time asymptotic behavior of tail distributions
1 Let (b, σ2, s) be the triple of the Lévy process X with a smooth jump
intensity function s (a.k.a Lévy density);
2 Theorem: [F-L & Houdré, 2009, Rüschendorf & Woerner, 2002]
For any n ≥ 0 and x > 0,
P(Xt ≥ x) = d1(x)t +d2(x)
2t2 + · · ·+ dn(x)
n!tn + O(tn+1).
3 The coefficients:• d1(x) = limt→0
1t P(Xt ≥ x) =
R∞x s(u)du;
• d2(x) = limt→02t
˘ 1t P(Xt ≥ x)− d1(x)
¯= −σ2s′(x) + 2bs(x)
−„Z ∞
xs(u)du
«2
+
Z x
x/2s(u)du
!2
+ 2Z − 1
2 x
−∞
Z x
x−ys(u)s(y)dudy
−2s(x)
Z12 x<|y|<1
ys(y)dy + 2s(x)
Z 12 x
− 12 x
Z x
x−y(s(u)− s(x))s(y)dudy .
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Close-to-expiration option prices for exponential Lévy models (ELM) A fundamental preliminary result
Small-time asymptotic behavior of tail distributions
1 Let (b, σ2, s) be the triple of the Lévy process X with a smooth jump
intensity function s (a.k.a Lévy density);
2 Theorem: [F-L & Houdré, 2009, Rüschendorf & Woerner, 2002]
For any n ≥ 0 and x > 0,
P(Xt ≥ x) = d1(x)t +d2(x)
2t2 + · · ·+ dn(x)
n!tn + O(tn+1).
3 The coefficients:• d1(x) = limt→0
1t P(Xt ≥ x) =
R∞x s(u)du;
• d2(x) = limt→02t
˘ 1t P(Xt ≥ x)− d1(x)
¯= −σ2s′(x) + 2bs(x)
−„Z ∞
xs(u)du
«2
+
Z x
x/2s(u)du
!2
+ 2Z − 1
2 x
−∞
Z x
x−ys(u)s(y)dudy
−2s(x)
Z12 x<|y|<1
ys(y)dy + 2s(x)
Z 12 x
− 12 x
Z x
x−y(s(u)− s(x))s(y)dudy .
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Close-to-expiration option prices for exponential Lévy models (ELM) A fundamental preliminary result
Small-time asymptotic behavior of tail distributions
1 Let (b, σ2, s) be the triple of the Lévy process X with a smooth jump
intensity function s (a.k.a Lévy density);
2 Theorem: [F-L & Houdré, 2009, Rüschendorf & Woerner, 2002]
For any n ≥ 0 and x > 0,
P(Xt ≥ x) = d1(x)t +d2(x)
2t2 + · · ·+ dn(x)
n!tn + O(tn+1).
3 The coefficients:• d1(x) = limt→0
1t P(Xt ≥ x) =
R∞x s(u)du;
• d2(x) = limt→02t
˘ 1t P(Xt ≥ x)− d1(x)
¯= −σ2s′(x) + 2bs(x)
−„Z ∞
xs(u)du
«2
+
Z x
x/2s(u)du
!2
+ 2Z − 1
2 x
−∞
Z x
x−ys(u)s(y)dudy
−2s(x)
Z12 x<|y|<1
ys(y)dy + 2s(x)
Z 12 x
− 12 x
Z x
x−y(s(u)− s(x))s(y)dudy .
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Close-to-expiration option prices for exponential Lévy models (ELM) A fundamental preliminary result
Small-time asymptotic behavior of tail distributions
1 Let (b, σ2, s) be the triple of the Lévy process X with a smooth jump
intensity function s (a.k.a Lévy density);
2 Theorem: [F-L & Houdré, 2009, Rüschendorf & Woerner, 2002]
For any n ≥ 0 and x > 0,
P(Xt ≥ x) = d1(x)t +d2(x)
2t2 + · · ·+ dn(x)
n!tn + O(tn+1).
3 The coefficients:• d1(x) = limt→0
1t P(Xt ≥ x) =
R∞x s(u)du;
• d2(x) = limt→02t
˘ 1t P(Xt ≥ x)− d1(x)
¯= −σ2s′(x) + 2bs(x)
−„Z ∞
xs(u)du
«2
+
Z x
x/2s(u)du
!2
+ 2Z − 1
2 x
−∞
Z x
x−ys(u)s(y)dudy
−2s(x)
Z12 x<|y|<1
ys(y)dy + 2s(x)
Z 12 x
− 12 x
Z x
x−y(s(u)− s(x))s(y)dudy .
![Page 10: The small-maturity smile for exponential Lévy modelsfigueroa/Talks/SlidesSIAM-SEAS.pdf · for Winning the Endgame" by Jeff Augen: “The final hours of each expiration cycle are](https://reader036.vdocument.in/reader036/viewer/2022071417/6114cecc1ca34c7e5a33d9d6/html5/thumbnails/10.jpg)
Close-to-expiration option prices for exponential Lévy models (ELM) A fundamental preliminary result
Small-time asymptotic behavior of tail distributions
1 Let (b, σ2, s) be the triple of the Lévy process X with a smooth jump
intensity function s (a.k.a Lévy density);
2 Theorem: [F-L & Houdré, 2009, Rüschendorf & Woerner, 2002]
For any n ≥ 0 and x > 0,
P(Xt ≥ x) = d1(x)t +d2(x)
2t2 + · · ·+ dn(x)
n!tn + O(tn+1).
3 The coefficients:• d1(x) = limt→0
1t P(Xt ≥ x) =
R∞x s(u)du;
• d2(x) = limt→02t
˘ 1t P(Xt ≥ x)− d1(x)
¯= −σ2s′(x) + 2bs(x)
−„Z ∞
xs(u)du
«2
+
Z x
x/2s(u)du
!2
+ 2Z − 1
2 x
−∞
Z x
x−ys(u)s(y)dudy
−2s(x)
Z12 x<|y|<1
ys(y)dy + 2s(x)
Z 12 x
− 12 x
Z x
x−y(s(u)− s(x))s(y)dudy .
![Page 11: The small-maturity smile for exponential Lévy modelsfigueroa/Talks/SlidesSIAM-SEAS.pdf · for Winning the Endgame" by Jeff Augen: “The final hours of each expiration cycle are](https://reader036.vdocument.in/reader036/viewer/2022071417/6114cecc1ca34c7e5a33d9d6/html5/thumbnails/11.jpg)
Close-to-expiration option prices for exponential Lévy models (ELM) Our main result
Prices of out-the-money Call Options
1 WLOG, we assume that r = 0;
2 Equivalent formulation:
ΠLévyt (T ,K ,S, r) := EQ [ (ST − K )+
∣∣St = S]
= EQ[(
S0eXT − K)+
∣∣∣St = S]
= SEQ[(
eXτ − eκ)+
]= ΠLévy
0 (T − t ,K ,S, r),
where τ = T − t (time-to-maturity) and κ = log(K/S) (log-moneyness).
3 Next, as it is done with the B-S formula,
EQ(eXτ − eκ)+ = EQ(eXτ − eκ)1Xτ≥κ
= EQ(eXτ 1Xτ≥κ)− eκQ(Xτ ≥ κ) = Q∗(Xτ ≥ κ)− eκQ(Xτ ≥ κ),
where Q∗(A) := EQ{1AeXt} if A ∈ Ft (Esscher or Share measure);
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Close-to-expiration option prices for exponential Lévy models (ELM) Our main result
Prices of out-the-money Call Options
1 WLOG, we assume that r = 0;
2 Equivalent formulation:
ΠLévyt (T ,K ,S, r) := EQ [ (ST − K )+
∣∣St = S]
= EQ[(
S0eXT − K)+
∣∣∣St = S]
= SEQ[(
eXτ − eκ)+
]= ΠLévy
0 (T − t ,K ,S, r),
where τ = T − t (time-to-maturity) and κ = log(K/S) (log-moneyness).
3 Next, as it is done with the B-S formula,
EQ(eXτ − eκ)+ = EQ(eXτ − eκ)1Xτ≥κ
= EQ(eXτ 1Xτ≥κ)− eκQ(Xτ ≥ κ) = Q∗(Xτ ≥ κ)− eκQ(Xτ ≥ κ),
where Q∗(A) := EQ{1AeXt} if A ∈ Ft (Esscher or Share measure);
![Page 13: The small-maturity smile for exponential Lévy modelsfigueroa/Talks/SlidesSIAM-SEAS.pdf · for Winning the Endgame" by Jeff Augen: “The final hours of each expiration cycle are](https://reader036.vdocument.in/reader036/viewer/2022071417/6114cecc1ca34c7e5a33d9d6/html5/thumbnails/13.jpg)
Close-to-expiration option prices for exponential Lévy models (ELM) Our main result
Prices of out-the-money Call Options
1 WLOG, we assume that r = 0;
2 Equivalent formulation:
ΠLévyt (T ,K ,S, r) := EQ [ (ST − K )+
∣∣St = S]
= EQ[(
S0eXT − K)+
∣∣∣St = S]
= SEQ[(
eXτ − eκ)+
]= ΠLévy
0 (T − t ,K ,S, r),
where τ = T − t (time-to-maturity) and κ = log(K/S) (log-moneyness).
3 Next, as it is done with the B-S formula,
EQ(eXτ − eκ)+ = EQ(eXτ − eκ)1Xτ≥κ
= EQ(eXτ 1Xτ≥κ)− eκQ(Xτ ≥ κ) = Q∗(Xτ ≥ κ)− eκQ(Xτ ≥ κ),
where Q∗(A) := EQ{1AeXt} if A ∈ Ft (Esscher or Share measure);
![Page 14: The small-maturity smile for exponential Lévy modelsfigueroa/Talks/SlidesSIAM-SEAS.pdf · for Winning the Endgame" by Jeff Augen: “The final hours of each expiration cycle are](https://reader036.vdocument.in/reader036/viewer/2022071417/6114cecc1ca34c7e5a33d9d6/html5/thumbnails/14.jpg)
Close-to-expiration option prices for exponential Lévy models (ELM) Our main result
Prices of out-the-money Call Options
1 In conclusion,
ΠLévyt (K ,T ,S) = SQ∗(Xτ ≥ κ)− SeκQ(Xτ ≥ κ)
2 Under Q∗, {Xt} is again a Lévy process with triple (b∗, σ2, s∗):
s∗(x) = exs(x) and b∗ = b +
∫|x|≤1
x (ex − 1) s(x)dx + σ2.
3 Hence, when κ > 0 (OTM), we can apply the asymptotic result for the tail
distributions of Lévy processes.
4 Theorem: [F-L & Forde (2010)]. For κ := log K/S > 0 (OTM options),
ΠLévyt (K ,T ,S) = τS
∫ ∞−∞
(ex−eκ)+s(x)dx +τ2
2S[d∗2 (κ)−eκd2(κ)
]+O(τ3),
where τ = T − t > 0, d2(κ) = d2(κ; b, σ2, s), and d∗2 (κ) = d2(κ; b∗, σ2, s∗).
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Close-to-expiration option prices for exponential Lévy models (ELM) Our main result
Prices of out-the-money Call Options
1 In conclusion,
ΠLévyt (K ,T ,S) = SQ∗(Xτ ≥ κ)− SeκQ(Xτ ≥ κ)
2 Under Q∗, {Xt} is again a Lévy process with triple (b∗, σ2, s∗):
s∗(x) = exs(x) and b∗ = b +
∫|x|≤1
x (ex − 1) s(x)dx + σ2.
3 Hence, when κ > 0 (OTM), we can apply the asymptotic result for the tail
distributions of Lévy processes.
4 Theorem: [F-L & Forde (2010)]. For κ := log K/S > 0 (OTM options),
ΠLévyt (K ,T ,S) = τS
∫ ∞−∞
(ex−eκ)+s(x)dx +τ2
2S[d∗2 (κ)−eκd2(κ)
]+O(τ3),
where τ = T − t > 0, d2(κ) = d2(κ; b, σ2, s), and d∗2 (κ) = d2(κ; b∗, σ2, s∗).
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Close-to-expiration option prices for exponential Lévy models (ELM) Our main result
Prices of out-the-money Call Options
1 In conclusion,
ΠLévyt (K ,T ,S) = SQ∗(Xτ ≥ κ)− SeκQ(Xτ ≥ κ)
2 Under Q∗, {Xt} is again a Lévy process with triple (b∗, σ2, s∗):
s∗(x) = exs(x) and b∗ = b +
∫|x|≤1
x (ex − 1) s(x)dx + σ2.
3 Hence, when κ > 0 (OTM), we can apply the asymptotic result for the tail
distributions of Lévy processes.
4 Theorem: [F-L & Forde (2010)]. For κ := log K/S > 0 (OTM options),
ΠLévyt (K ,T ,S) = τS
∫ ∞−∞
(ex−eκ)+s(x)dx +τ2
2S[d∗2 (κ)−eκd2(κ)
]+O(τ3),
where τ = T − t > 0, d2(κ) = d2(κ; b, σ2, s), and d∗2 (κ) = d2(κ; b∗, σ2, s∗).
![Page 17: The small-maturity smile for exponential Lévy modelsfigueroa/Talks/SlidesSIAM-SEAS.pdf · for Winning the Endgame" by Jeff Augen: “The final hours of each expiration cycle are](https://reader036.vdocument.in/reader036/viewer/2022071417/6114cecc1ca34c7e5a33d9d6/html5/thumbnails/17.jpg)
Close-to-expiration option prices for exponential Lévy models (ELM) Our main result
Prices of out-the-money Call Options
1 In conclusion,
ΠLévyt (K ,T ,S) = SQ∗(Xτ ≥ κ)− SeκQ(Xτ ≥ κ)
2 Under Q∗, {Xt} is again a Lévy process with triple (b∗, σ2, s∗):
s∗(x) = exs(x) and b∗ = b +
∫|x|≤1
x (ex − 1) s(x)dx + σ2.
3 Hence, when κ > 0 (OTM), we can apply the asymptotic result for the tail
distributions of Lévy processes.
4 Theorem: [F-L & Forde (2010)]. For κ := log K/S > 0 (OTM options),
ΠLévyt (K ,T ,S) = τS
∫ ∞−∞
(ex−eκ)+s(x)dx +τ2
2S[d∗2 (κ)−eκd2(κ)
]+O(τ3),
where τ = T − t > 0, d2(κ) = d2(κ; b, σ2, s), and d∗2 (κ) = d2(κ; b∗, σ2, s∗).
![Page 18: The small-maturity smile for exponential Lévy modelsfigueroa/Talks/SlidesSIAM-SEAS.pdf · for Winning the Endgame" by Jeff Augen: “The final hours of each expiration cycle are](https://reader036.vdocument.in/reader036/viewer/2022071417/6114cecc1ca34c7e5a33d9d6/html5/thumbnails/18.jpg)
Close-to-expiration implied volatility smile for ELM
Small-time asymptotics for Implied Volatilities
1 σ̂τ (k) be the implied volatility at log-moneyness κ and time-to-maturity τ
under the exponential Lévy models;
2 First-order approximation for σ̂t (κ): [Tankov (2009) & F-L & Forde (2010)]
[τ log(τ−1)]12 σ̂τ (κ) ∼ |κ|/
√2; (κ > 0, τ → 0);
Hence, (rescaled) implies volatility is V-shaped independent of s;
3 Correction term or Second-order approximation:
σ̂2τ (κ) =
12κ
2
τ log( 1τ )
[1 + V1(τ, κ) + o(
1log 1
τ
)] (τ → 0),
where, denoting a0(κ) :=∫∞−∞(ex − eκ)+s(x)dx ,
V1(τ, κ) =1
log( 1τ )
log
[4√πa0(κ)e−κ/2
κ
[log(
1τ
)]3/2].
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Close-to-expiration implied volatility smile for ELM
Small-time asymptotics for Implied Volatilities
1 σ̂τ (k) be the implied volatility at log-moneyness κ and time-to-maturity τ
under the exponential Lévy models;
2 First-order approximation for σ̂t (κ): [Tankov (2009) & F-L & Forde (2010)]
[τ log(τ−1)]12 σ̂τ (κ) ∼ |κ|/
√2; (κ > 0, τ → 0);
Hence, (rescaled) implies volatility is V-shaped independent of s;
3 Correction term or Second-order approximation:
σ̂2τ (κ) =
12κ
2
τ log( 1τ )
[1 + V1(τ, κ) + o(
1log 1
τ
)] (τ → 0),
where, denoting a0(κ) :=∫∞−∞(ex − eκ)+s(x)dx ,
V1(τ, κ) =1
log( 1τ )
log
[4√πa0(κ)e−κ/2
κ
[log(
1τ
)]3/2].
![Page 20: The small-maturity smile for exponential Lévy modelsfigueroa/Talks/SlidesSIAM-SEAS.pdf · for Winning the Endgame" by Jeff Augen: “The final hours of each expiration cycle are](https://reader036.vdocument.in/reader036/viewer/2022071417/6114cecc1ca34c7e5a33d9d6/html5/thumbnails/20.jpg)
Close-to-expiration implied volatility smile for ELM
Small-time asymptotics for Implied Volatilities
1 σ̂τ (k) be the implied volatility at log-moneyness κ and time-to-maturity τ
under the exponential Lévy models;
2 First-order approximation for σ̂t (κ): [Tankov (2009) & F-L & Forde (2010)]
[τ log(τ−1)]12 σ̂τ (κ) ∼ |κ|/
√2; (κ > 0, τ → 0);
Hence, (rescaled) implies volatility is V-shaped independent of s;
3 Correction term or Second-order approximation:
σ̂2τ (κ) =
12κ
2
τ log( 1τ )
[1 + V1(τ, κ) + o(
1log 1
τ
)] (τ → 0),
where, denoting a0(κ) :=∫∞−∞(ex − eκ)+s(x)dx ,
V1(τ, κ) =1
log( 1τ )
log
[4√πa0(κ)e−κ/2
κ
[log(
1τ
)]3/2].
![Page 21: The small-maturity smile for exponential Lévy modelsfigueroa/Talks/SlidesSIAM-SEAS.pdf · for Winning the Endgame" by Jeff Augen: “The final hours of each expiration cycle are](https://reader036.vdocument.in/reader036/viewer/2022071417/6114cecc1ca34c7e5a33d9d6/html5/thumbnails/21.jpg)
Numerical examples Variance Gamma Model
0 5 10 15 20 25 30
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time−to−maturity (in Days)
Impl
ied
Vola
tility
Variance Gamma ModelApproximation of Implied Volatility with k=0.2
"True" implied volatility1st order approx.2nd order approx.
Figure: Term structure of implied volatility approximations for the Variance Gamma
model (i.e. s(x) = αx e−x/β+1x>0 + α
|x|e−|x|/β−1x<0 and σ = 0) with κ = 0.2.
![Page 22: The small-maturity smile for exponential Lévy modelsfigueroa/Talks/SlidesSIAM-SEAS.pdf · for Winning the Endgame" by Jeff Augen: “The final hours of each expiration cycle are](https://reader036.vdocument.in/reader036/viewer/2022071417/6114cecc1ca34c7e5a33d9d6/html5/thumbnails/22.jpg)
Numerical examples CGMY model
0 5 10 15 20 25 30
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
Time−to−Maturity (in Days)
Rela
tive
Erro
r ( ! a−
! ) /
!
Approximation of Implied Volatility for VGTerm Structure of Relative Error
1st, k=0.32nd, k=0.31st, k=0.22nd, k=0.21st, k=0.11st, k=0.1
Figure: Relative errors σ̂τ (κ)−στ (κ)στ (κ)
of the implied volatility approximations for the
Variance Gamma Model.
![Page 23: The small-maturity smile for exponential Lévy modelsfigueroa/Talks/SlidesSIAM-SEAS.pdf · for Winning the Endgame" by Jeff Augen: “The final hours of each expiration cycle are](https://reader036.vdocument.in/reader036/viewer/2022071417/6114cecc1ca34c7e5a33d9d6/html5/thumbnails/23.jpg)
Numerical examples CGMY model
0 5 10 15 20 25 30
0.3
0.4
0.5
0.6
0.7
0.8
CGMY ModelApproximation of Implied Volatility with k=0.2
Time−to−maturity (in Days)
Impl
ied
Vola
tility
"True" implied volatility1st order approx.2nd order approx.
Figure: Term structure of implied volatility approximations for the CGMY model (i.e.
s(x) = CxY+1 e−x/M1x>0 + C
|x|1+Y e−|x|/G1x<0 and σ = 0) with κ = 0.2.
![Page 24: The small-maturity smile for exponential Lévy modelsfigueroa/Talks/SlidesSIAM-SEAS.pdf · for Winning the Endgame" by Jeff Augen: “The final hours of each expiration cycle are](https://reader036.vdocument.in/reader036/viewer/2022071417/6114cecc1ca34c7e5a33d9d6/html5/thumbnails/24.jpg)
Conclusions
0 5 10 15 20 25 30
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
Approximation of Implied Volatility for the CGMYTerm Structure of Relative Error
Time−to−maturity (in Days)
Rela
tive
Erro
r ( ! a−
! ) /
!
1st, k=0.32nd, k=0.31st, k=0.22nd, k=0.21st, k=0.12nd, k=0.1
Figure: Relative errors σ̂τ (κ)−στ (κ)στ (κ)
of the implied volatility approximations for the
CGMY model.
![Page 25: The small-maturity smile for exponential Lévy modelsfigueroa/Talks/SlidesSIAM-SEAS.pdf · for Winning the Endgame" by Jeff Augen: “The final hours of each expiration cycle are](https://reader036.vdocument.in/reader036/viewer/2022071417/6114cecc1ca34c7e5a33d9d6/html5/thumbnails/25.jpg)
Conclusions
Conclusions and extensions
1 The numerical results show that the second order significantly improves
the first order approximation for mid range values of κ (say, κ ≤ .2);
2 For κ > 0.2, it seems that τ has to be extremely small for the second
approximations to work well;
3 Similar results hold for time-changed Lévy models of the form:
St = S0eXt ; Xt = ZT (t),
T (t) =
∫ t
0Y (u)du, Y ⊥ Z ;
4 We then have
ΠLévyt = τEY0S
∫ ∞−∞
(ex−eκ)+s(x)dx +τ2
2EY 2
0 S[d∗2 (κ)−eκd2(κ)
]+O(τ3).
![Page 26: The small-maturity smile for exponential Lévy modelsfigueroa/Talks/SlidesSIAM-SEAS.pdf · for Winning the Endgame" by Jeff Augen: “The final hours of each expiration cycle are](https://reader036.vdocument.in/reader036/viewer/2022071417/6114cecc1ca34c7e5a33d9d6/html5/thumbnails/26.jpg)
Conclusions
Conclusions and extensions
1 The numerical results show that the second order significantly improves
the first order approximation for mid range values of κ (say, κ ≤ .2);
2 For κ > 0.2, it seems that τ has to be extremely small for the second
approximations to work well;
3 Similar results hold for time-changed Lévy models of the form:
St = S0eXt ; Xt = ZT (t),
T (t) =
∫ t
0Y (u)du, Y ⊥ Z ;
4 We then have
ΠLévyt = τEY0S
∫ ∞−∞
(ex−eκ)+s(x)dx +τ2
2EY 2
0 S[d∗2 (κ)−eκd2(κ)
]+O(τ3).
![Page 27: The small-maturity smile for exponential Lévy modelsfigueroa/Talks/SlidesSIAM-SEAS.pdf · for Winning the Endgame" by Jeff Augen: “The final hours of each expiration cycle are](https://reader036.vdocument.in/reader036/viewer/2022071417/6114cecc1ca34c7e5a33d9d6/html5/thumbnails/27.jpg)
Conclusions
Conclusions and extensions
1 The numerical results show that the second order significantly improves
the first order approximation for mid range values of κ (say, κ ≤ .2);
2 For κ > 0.2, it seems that τ has to be extremely small for the second
approximations to work well;
3 Similar results hold for time-changed Lévy models of the form:
St = S0eXt ; Xt = ZT (t),
T (t) =
∫ t
0Y (u)du, Y ⊥ Z ;
4 We then have
ΠLévyt = τEY0S
∫ ∞−∞
(ex−eκ)+s(x)dx +τ2
2EY 2
0 S[d∗2 (κ)−eκd2(κ)
]+O(τ3).
![Page 28: The small-maturity smile for exponential Lévy modelsfigueroa/Talks/SlidesSIAM-SEAS.pdf · for Winning the Endgame" by Jeff Augen: “The final hours of each expiration cycle are](https://reader036.vdocument.in/reader036/viewer/2022071417/6114cecc1ca34c7e5a33d9d6/html5/thumbnails/28.jpg)
Conclusions
Conclusions and extensions
1 The numerical results show that the second order significantly improves
the first order approximation for mid range values of κ (say, κ ≤ .2);
2 For κ > 0.2, it seems that τ has to be extremely small for the second
approximations to work well;
3 Similar results hold for time-changed Lévy models of the form:
St = S0eXt ; Xt = ZT (t),
T (t) =
∫ t
0Y (u)du, Y ⊥ Z ;
4 We then have
ΠLévyt = τEY0S
∫ ∞−∞
(ex−eκ)+s(x)dx +τ2
2EY 2
0 S[d∗2 (κ)−eκd2(κ)
]+O(τ3).
![Page 29: The small-maturity smile for exponential Lévy modelsfigueroa/Talks/SlidesSIAM-SEAS.pdf · for Winning the Endgame" by Jeff Augen: “The final hours of each expiration cycle are](https://reader036.vdocument.in/reader036/viewer/2022071417/6114cecc1ca34c7e5a33d9d6/html5/thumbnails/29.jpg)
Appendix Bibliography
For Further Reading I
Figueroa-Lopez & Houdré.
Small-time expansions for the transition distributions of Lévy processes.
Stochastic Processes and Their Applications, 119:3862-3889, 2009.
Figueroa-López and Forde.
The small-maturity smile for exponential Lévy models
Preprint, 2010.
Figueroa-López, Gong, and Houdré.
Small-time expansions of the distributions, densities, and option prices of
stochastic volatility models with Lévy jump
Preprint, 2010.