the small-maturity smile for exponential lévy models · 2015-08-20 · close-to-expiration option...

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

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"

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 );

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

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.

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;

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

x/2s(u)du

+ 2Z − 1

−∞

x−ys(u)s(y)dudy

−2s(x)

Z12 x<|y|<1

ys(y)dy + 2s(x)

Z 12 x

− 12 x

x−y(s(u)− s(x))s(y)dudy .

P(Xt ≥ x) = d1(x)t +d2(x)

2t2 + · · ·+ dn(x)

n!tn + O(tn+1).

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

x/2s(u)du

+ 2Z − 1

−∞

x−ys(u)s(y)dudy

−2s(x)

Z12 x<|y|<1

ys(y)dy + 2s(x)

Z 12 x

− 12 x

P(Xt ≥ x) = d1(x)t +d2(x)

2t2 + · · ·+ dn(x)

n!tn + O(tn+1).

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

x/2s(u)du

+ 2Z − 1

−∞

x−ys(u)s(y)dudy

−2s(x)

Z12 x<|y|<1

ys(y)dy + 2s(x)

Z 12 x

− 12 x

P(Xt ≥ x) = d1(x)t +d2(x)

2t2 + · · ·+ dn(x)

n!tn + O(tn+1).

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

x/2s(u)du

+ 2Z − 1

−∞

x−ys(u)s(y)dudy

−2s(x)

Z12 x<|y|<1

ys(y)dy + 2s(x)

Z 12 x

− 12 x

P(Xt ≥ x) = d1(x)t +d2(x)

2t2 + · · ·+ dn(x)

n!tn + O(tn+1).

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

x/2s(u)du

+ 2Z − 1

−∞

x−ys(u)s(y)dudy

−2s(x)

Z12 x<|y|<1

ys(y)dy + 2s(x)

Z 12 x

− 12 x

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);

∣∣St = S]

= EQ[(

S0eXT − K)+

∣∣∣St = S]

= SEQ[(

eXτ − eκ)+

]= ΠLévy

0 (T − t ,K ,S, r),

∣∣St = S]

= EQ[(

S0eXT − K)+

∣∣∣St = S]

= SEQ[(

eXτ − eκ)+

]= ΠLévy

0 (T − t ,K ,S, r),

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∗).

1 In conclusion,

∫|x|≤1

x (ex − 1) s(x)dx + σ2.

∫ ∞−∞

2S[d∗2 (κ)−eκd2(κ)

]+O(τ3),

1 In conclusion,

∫|x|≤1

x (ex − 1) s(x)dx + σ2.

∫ ∞−∞

2S[d∗2 (κ)−eκd2(κ)

]+O(τ3),

1 In conclusion,

∫|x|≤1

x (ex − 1) s(x)dx + σ2.

∫ ∞−∞

2S[d∗2 (κ)−eκd2(κ)

]+O(τ3),

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) =

τ log( 1τ )

[1 + V1(τ, κ) + o(

1log 1

)] (τ → 0),

where, denoting a0(κ) :=∫∞−∞(ex − eκ)+s(x)dx ,

V1(τ, κ) =1

log( 1τ )

[4√πa0(κ)e−κ/2

)]3/2].

[τ log(τ−1)]12 σ̂τ (κ) ∼ |κ|/

√2; (κ > 0, τ → 0);

σ̂2τ (κ) =

τ log( 1τ )

[1 + V1(τ, κ) + o(

1log 1

)] (τ → 0),

V1(τ, κ) =1

log( 1τ )

[4√πa0(κ)e−κ/2

)]3/2].

[τ log(τ−1)]12 σ̂τ (κ) ∼ |κ|/

√2; (κ > 0, τ → 0);

σ̂2τ (κ) =

τ log( 1τ )

[1 + V1(τ, κ) + o(

1log 1

)] (τ → 0),

V1(τ, κ) =1

log( 1τ )

[4√πa0(κ)e−κ/2

)]3/2].

Numerical examples Variance Gamma Model

0 5 10 15 20 25 30

Time−to−maturity (in Days)

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.

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

Time−to−Maturity (in Days)

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.

Numerical examples CGMY model

0 5 10 15 20 25 30

CGMY ModelApproximation of Implied Volatility with k=0.2

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.

Conclusions

0 5 10 15 20 25 30

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

Approximation of Implied Volatility for the CGMYTerm Structure of Relative Error

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.

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) =

0Y (u)du, Y ⊥ Z ;

4 We then have

ΠLévyt = τEY0S

∫ ∞−∞

0 S[d∗2 (κ)−eκd2(κ)

]+O(τ3).

Conclusions

T (t) =

0Y (u)du, Y ⊥ Z ;

4 We then have

ΠLévyt = τEY0S

∫ ∞−∞

0 S[d∗2 (κ)−eκd2(κ)

]+O(τ3).

Conclusions

T (t) =

0Y (u)du, Y ⊥ Z ;

4 We then have

ΠLévyt = τEY0S

∫ ∞−∞

0 S[d∗2 (κ)−eκd2(κ)

]+O(τ3).

Conclusions

T (t) =

0Y (u)du, Y ⊥ Z ;

4 We then have

ΠLévyt = τEY0S

∫ ∞−∞

0 S[d∗2 (κ)−eκd2(κ)

]+O(τ3).

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.

the small-maturity smile for exponential lévy models · 2015-08-20 · close-to-expiration option...

Documents