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Outline Introduction FC-GARCH Conditional Esscher transform Parametric cases Simulation Results Conclusions References Option Pricing under a Nonlinear and Nonnormal GARCH Renato Costa 1 ´ Alvaro Veiga 1 Tak Kuen Siu 2 1 Department of Electrical Engineering, Pontifical Catholic University of Rio de Janeiro 2 Department of Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney January 27, 2012 Renato Alencar Adelino da Costa, ´ Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

FCGARCH auckland 27012012

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"Option Pricing under a Nonlinear and Nonnormal GARCH" in the Australia and New Zealand Probability Workshop, University of Auckland, Auckland, New Zealand, January 22-27, 2012.

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Page 1: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Option Pricing under a Nonlinear and Nonnormal

GARCH

Renato Costa 1 Alvaro Veiga 1 Tak Kuen Siu2

1Department of Electrical Engineering,Pontifical Catholic University of Rio de Janeiro

2Department of Actuarial Studies,Faculty of Business and Economics, Macquarie University, Sydney

January 27, 2012

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 2: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Option Pricing under a Nonlinear and Nonnormal

GARCH

Renato Costa 1 Alvaro Veiga 1 Tak Kuen Siu2

1Department of Electrical Engineering,Pontifical Catholic University of Rio de Janeiro

2Department of Actuarial Studies,Faculty of Business and Economics, Macquarie University, Sydney

January 27, 2012

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 3: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

IntroductionRisk-Neutral Valuation

FC-GARCH

Conditional Esscher transform

Parametric casesNormal InnovationsShifted-Gamma Innovations

Simulation Results

Conclusions

References

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 4: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Risk-Neutral Valuation

Call Prices and Put Prices

C (ST ,T ) =

0 , se ST ≤ KST − K , se ST > K

= max (ST − K , 0)Notation

= (ST − K )+

P(ST ,T ) =

0 , se ST ≥ KK − ST , se ST < K

= max(K − ST , 0)Notation

= (K − ST )+

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 5: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Risk-Neutral Valuation

In the bible, Genesis Chapter 29, it is possible to find the firstoption trading and the first default on derivatives. When Labangave a Call option to Jacob under which the underlying asset wasRachel, his youngest daughter. The exercise price was working forhim for 7 years. At the expiration time, he decided to exercise theoption but Laban didn’t allow him to marry her youngest daughter.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 6: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Risk-Neutral Valuation

Black & Scholes

Ct = StN(d1) + Ke−r(T−t)N(d2)

where

d1 =ln (St

K) + (r + σ2

2 )(T − t)

σ√T − t

and

d2 =ln (St

K) + (r − σ2

2 )(T − t)

σ√T − t

ord2 = d1 − σ

√T − t

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 7: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Risk-Neutral Valuation

References:

Duan, Jin-Chuan. 1995. “The GARCH Option PricingModel,” Mathematical Finance 5: 13-32.

Siu, Tak Kuen, Howell Tong, and Hailiang Yang. 2004.“OnPricing Derivatives Under GARCH Models: A DynamicGerber-Shiu Approach,” North American Actuarial Journal.8(3): 17-31.

Medeiros, Marcelo C. and Alvaro Veiga. 2009. “ModelingMultiple Regimes in Financial Volatility with a FlexibleCoefficient GARCH Model”,Econometric Theory.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 8: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Risk-Neutral Valuation

Main ideas and contributions:

Extension from GARCH to FCGARCH

Shifted-Gamma case with negative sign

Simulation results, greeks and curiosities

Conclusions

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 9: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Risk-Neutral Valuation

Risk-Neutral Valuation

In general people are risk-averse, then we can expect:

π0er < EP [π1]

which intuitively means that nobody would like to risk keeping arisk asset in order to receive the same amount of money as aperson that keeps a risk-free asset.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 10: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Risk-Neutral Valuation

Under the risk-neutral measure we can bring the expectation to thepresent value using the risk-free interest rate, i.e. for example:

πo = e−rEQ [π1]

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 11: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

FC-GARCH

ǫt |t ∈ T \0 are i.i.d. r.v.’s with common distributionD(0, 1) .

S := Stt∈T .

Yt := ln(St/St−1),

Then we assume that the return process Y := Yt |t ∈ T followsa Flexible Coefficient Generalized Autoregressive ConditionalHeteroscedastic model with m = H + 1 limiting regimes,henceforth, FC-GARCH (m, 1, 1):

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 12: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Yt = µt + ξt ; ξt = h1/2t ǫt ,

ht = α0 + β0ht−1 + λ0ξ2t−1 (1)

+H∑

i=1

[αi + βiht−1 + λiξ2t−1]f (st ; γi , ci ) . (2)

where for each i = 1, 2, . . . ,H, the logistic function

f (st , γi , ci ) :=1

1 + e−γi (st−ci );

Here, we consider a simple case that st = Yt−1.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 13: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

We assume that under P,

Yt = µt + ξt .

where ξt is an i.i.d innovation process having distribution D(0, ht)and µt and ht are Ft−1-measurable. Here we are going to assumethat ξt has a moment generation function and is infinitely divisibleso that we can use the Conditional Esscher Transform Method.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 14: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Advantages of the FC-GARCH of Medeiros and Veiga (2009):

The model describe the sign and size asymmetries in financialvolatility.

The model nests some of the previous specifications found inthe literature.(GARCH, LST-GARCH, DTGARCH/DTARCH,ANSTGARCH, etc..)

One of the few Non-linear models to account for more thantwo regimes.

Stationarity restriction on the parameters is relatively weak. Itis shown that the model may have explosive regimes but canstill be strictly stationary and ergodic.

The model can generate series with high kurtosis and lowfirst-order autocorrelation of the squared observations andexhibit the so-called Taylor effect, even with Gaussian errors.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 15: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Conditional Esscher transform

We now review the conditional Esscher transform.

Let θt |t ∈ T \0 be an F -predictable, real-valued, process on(Ω,F ,P).

Denote, for each t ∈ T \0, the moment generating function ofYt given Ft−1 under P evaluated at z ∈ ℜ by MY (t, z); that is,

MY (t, z) := E [ezYt |Ft−1] .

Here E is expectation under P.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 16: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Assume that, for each t ∈ T \0 and z ∈ ℜ, MY (t, z) < ∞.Consider an F -adapted process Λt |t ∈ T on (Ω,F ,P) withΛ0 = 1, P-a.s., defined by:

Λt :=t∏

k=1

eθkYk

MY (k , θk), t ∈ T \0 .

Then, it is easy to check that Λtt∈τ is an (F ,P)-martingale, i.e.,E [Λt |Ft−1] = Λt−1.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 17: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Now we define a new probability measure Pθ equivalent to P onFT by setting

dPθ

dP

FT

:= ΛT . (1)

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 18: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Theorem(Baye’s rule) Let µ and ν be two probability measures on ameasurable space (Ω,G) such that

dν(ω) = f (ω)dµ(ω)

for some f ∈ L1(µ). Let X be a ν-integrable random variable on(Ω,G)Let H be a σ-algebra, H ⊂ G. Then,

Eν [X |H] =Eµ[fX |H]

Eµ[f |H]a.s. (2)

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 19: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Let MθY (t, z) be the moment generating function of the return Yt

given Ft−1 under the new measure Pθ. Write E θ[·] for expectationunder Pθ. Then, by the Bayes’ rule, it is easy to check that

MθY (t, z) =

MY (t, θt + z)

MY (t, θt). (3)

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 20: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Indeed, by the Bayes’ rule,

MθY (t, z) := E θ[ezYt |Ft−1]

=E [ezYtΛt |Ft−1]

E [Λt |Ft−1]

= E

[

Λt

Λt−1ezYt |Ft−1

]

=E [e(z+θt)Yt |Ft−1]

MY (t, θt)

=MY (t, θt + z)

MY (t, θt). (4)

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 21: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Now we write St := e−rtSt , which is the discounted asset price attime t, for each t ∈ T . Then in our case, the martingale conditionis:

Su = E θ[St |Fu] , for all u, t ∈ T with u ≤ t . (5)

Here E θ is expectation under Pθ.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 22: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Consider an European-style option with payoff V (ST ) at maturityT . Then, a conditional price of the option at time t given Ft isdetermined as:

Vt = e−r(T−t)E θ[V (ST )|Ft ] (6)

(7)

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 23: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Proposition : The martingale condition is satisfied if and only ifthere exists an F -predictable process θt |t ∈ T \0 such that

r = lnMY (t, θt + 1)− lnMY (t, θt) . (8)

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 24: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Normal InnovationsShifted-Gamma Innovations

Parametric cases

Now we are going to deal with two parametric cases.We are going to apply the method of Siu et al.(2004) to theNormal and Shifted-Gamma innovations.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 25: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Normal InnovationsShifted-Gamma Innovations

Normal Innovations

Firstly, under P,

Yt = µt + ξt , ξt ∼ N(0, ht)

ht = α0 + β0ht−1 + λ0ξ2t−1 +

H∑

i=1

(αi + βiht−1 + λiξ2t−1)f (st ; γi , ci ) ,

where

f (st , γi , ci ) :=1

1 + e−γi (st−ci ).

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 26: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Normal InnovationsShifted-Gamma Innovations

Then, under P, Yt |Ft−1 ∼ N(µt , ht). Consequently,

ln(MθYt |Ft−1

(1, θt)) = r = ln

(

MYt |Ft−1(1 + θt)

MYt |Ft−1(θt)

)

= ln

eµt(1+θt)+(1+θt )

2ht2

eµtθt+θ2t ht2

= µt + htθt +ht2

.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 27: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Normal InnovationsShifted-Gamma Innovations

Then, the martingale condition implies that

θt =r − µt − ht

2

ht.

It is not difficult to see that

MθYt |Ft−1

(z , θt) =MYt |Ft−1

(z + θt)

MYt |Ft−1(θt)

= ez(r−ht2 )+

z2ht2 .

This is the moment generating function of a normal distributionwith mean r − ht

2 and variance ht .

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

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OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Normal InnovationsShifted-Gamma Innovations

Let ǫθt := ξt − r + ht2 + µt , for each t ∈ T \0. Then under Pθ,

ǫθt |Ft−1 ∼ N(0, ht). Further, if we take µt := r + λ√ht − 1

2ht as inDuan (1995),

ǫθt = ξt + λ√

ht .

Consequently, under Pθ, the conditional variance dynamics aregiven by:

ht = α0 + β0ht−1 + λ0(ǫθt−1 − λ

ht−1)2 (1)

+H∑

i=1

[

αi + βiht−1 + λi (ǫθt−1 − λ

ht−1)2]

f (st ; γi , ci ) .(2)

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 29: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Normal InnovationsShifted-Gamma Innovations

TheoremLet the model under P be:

Yt = r + λ√

ht −1

2ht + ξt

ξt |Ft−1 = N(0, ht )

ht = α0 + β0ht−1 + λ0ξ2t−1 +

H∑

i=1

(αi + βi ht−1 + λiξ2t−1)f (st ; γi , ci ).

Then, under the risk neutral measure the model is

Yt = r −1

2ht + ǫt

ǫt |Ft−1 ∼ Nθ

P(0, ht )

ht = α0 + β0ht−1 + λ0(ǫt−1 − λ√

ht−1)2

+H∑

i=1

[

αi + βi ht−1 + λi (ǫt−1 − λ√

ht−1)2]

f (st ; γi , ci )

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 30: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Normal InnovationsShifted-Gamma Innovations

Shifted-Gamma Innovations

The skewness of the Gamma distribution is strictly positivewhilst financial time series can present both signs.

In practice, check if there is any skewness to be modeled.Otherwise, the Normal model should suffice.

Then, one should check for the sign of the skewness in orderto chose the correct formulation of the shifted-gammainnovations.

The positive case is similar and for the GARCH case it isalready documented in Siu et al.(2004).

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 31: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Normal InnovationsShifted-Gamma Innovations

Suppose that for each t ∈ T \0, Xt ∼ Ga(a, b), where Ga(a, b)represents a Gamma distribution with shape parameter a and scaleparameter b. We now suppose that the innovation at time t isgiven by:

ξt := −√

ht

(

Xt − a/b√

a/b2

)

. (3)

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 32: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Normal InnovationsShifted-Gamma Innovations

Then, under P,

Yt = r + λ√

ht −1

2ht + ξt

ht = α0 + β0ht−1 + λ0ξ2t−1 +

H∑

i=1

(αi + βiht−1 + λiξ2t−1)f (st ; γi , ci ) ,

where

f (st , γi , ci ) :=1

1 + e−γi (st−ci ).

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 33: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Normal InnovationsShifted-Gamma Innovations

The return process Y can be expressed as:

Yt = r + λ√

ht −1

2ht +

aht − b

htaXt .

Note that b√

htaXt ∼ Ga(a,

aht), and that if W ∼ Ga(a, b) is a

gamma random variable, the moment generation function of −W

is M−W (t) =(

bb+θ

)a

. Then the moment generation function of

Yt |F is given by

MYt |Ft−1(t, θt) =

aht

aht

+ θt

a

e(r+λ√ht− 1

2ht+

√aht)θt (4)

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 34: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Normal InnovationsShifted-Gamma Innovations

Again, using the following formula,

MθY (t, z , θt) =

MY (t, θt + z)

MY (t, θt), (5)

it is not difficult to show that

MθYt |Ft−1

(t, z , θt) =

aht

+ θt√

aht

+ θt + z

a

e(r+λ√ht− 1

2ht+

√aht)z ,(6)

Consequently, the martingale condition implies that

θqt =1

eλ√

ht−12 ht+

√aht

a − 1

−√

a

ht(7)

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 35: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Normal InnovationsShifted-Gamma Innovations

Now if we take bt :=√

aht

and bθt := 1

eλ√

ht−12 ht+

√aht

a −1

, then

bθt = θt + bt .

Under Pθ,

Yt ∼ r + λ√

ht −1

2ht +

aht + X θt .

where X θt ∼ − 1

bθtGa(a, 1), and

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

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OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Normal InnovationsShifted-Gamma Innovations

ht = α0 + β0ht−1 + λ0(Xθt−1 +

aht−1)2

+H∑

i=1

[αi + βiht−1 + λi (Xθt−1 +

aht−1)2]f (st ; γi , ci ) .

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 37: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Normal InnovationsShifted-Gamma Innovations

TheoremLet the model under P be

Yt = r + λ√

ht −1

2ht + ξt ,

ξt |Ft−1 ∼ −SGa(0, ht ),

ht = α0 + β0ht−1 + λ0ξ2t−1 +

H∑

i=1

(αi + βi ht−1 + λiξ2t−1)f (st ; γi , ci ),

Then, under the risk neutral measure, the model is

Yt ∼ r + λ√

ht −1

2ht +

aht + Xθ

t

t ∼ −1

bθt

Ga(a, 1)

ht = α0 + β0ht−1 + λ0(Xθ

t−1 +√

aht−1)2

+H∑

i=1

[αi + βi ht−1 + λi (Xθ

t−1 +√

aht−1)2]f (st ; γi , ci ) .

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 38: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Simulation results

FC-GARCH Parameters

α [9.77× 10−16, 5.14× 10−7, 1.81× 10−5]β [1.21,−0.32,−0.25]λ [0.06,−0.01,−0.04]γ [2.52, 2.85]c [−0.72, 1.56]

Risk Premium 0.0349a (Gamma case) 100

Table: Parameters for the FC-GARCH including the values of α, β and λin the three different regimes.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

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OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

For the estimation of the GARCH parameters we use an iteratedtwo-stage method. Initially, we suppose ht a constant equal to thesample variance. Then, we estimate the risk premium by weightedleast squares(WLS). Next, we fit a GARCH(1,1) model to theresiduals of the WLS by performing a Quasi Maximum Likelihood.We iterate these two steps until convergence is attained.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 40: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

GARCH Parameters in Normal Case

α 7.4079× 10−7

β 0.9375λ 0.0445

Risk Premium 0.0288

Table: Estimated GARCH Parameters in the Normal Case

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 41: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

We estimated the parameters using the two stage procedure asdescribed before and then for finding a we used the method ofmoments as in Siu et al(2004) to obtain the expression:

a =

[

2∑T

t=1 h3/2t

∑Tt=1 ξ

3t

]2

(8)

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 42: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

GARCH Parameters in Shifted-Gamma Case

α 5.4959× 10−7

β 0.9461λ 0.0397

Risk Premium 0.0363a 79.4022

Table: Estimated GARCH Parameters in the Gamma Case

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 43: FCGARCH auckland 27012012

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Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Call Prices with IV=1.0 for Artificial Normal FCGARCH Series

K/S0 BS FC-Normal GARCH-Normal FC-Gamma GARCH-Gamma

0.80 20.0002 20.0009 19.9978 20.0149 20.01030.85 15.0063 15.0110 15.0079 15.0292 15.02380.90 10.0957 10.0939 10.1022 10.1216 10.12270.95 5.6537 5.5895 5.6292 5.6255 5.65651.00 2.4175 2.2812 2.3714 2.2757 2.37621.05 0.7397 0.6704 0.7163 0.6449 0.68451.10 0.1575 0.1515 0.1667 0.1362 0.14401.15 0.0234 0.0298 0.0310 0.0223 0.02941.20 0.0025 0.0049 0.0045 0.0032 0.0050

Table: Call Prices with IV=1.0 for Normal FCGARCH Series and T=90.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 44: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Figure: Graphs of Call Prices ratios with IV=1.0 for NormalFCGARCH Series

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 45: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Call Prices with IV=1.0 for Artificial Shifted-Gamma FCGARCH Series

K/S0 BS FC-Gamma GARCH-Gamma FC-Normal GARCH-Normal

0.80 20.0001 20.0149 19.8948 20.0009 20.00790.85 15.0045 15.0292 14.9062 15.0110 15.01760.90 10.0803 10.1216 9.9955 10.0939 10.10310.95 5.6017 5.6255 5.5110 5.5895 5.61361.00 2.3402 2.2757 2.2413 2.2812 2.32801.05 0.6831 0.6449 0.6171 0.6704 0.69071.10 0.1349 0.1362 0.1209 0.1515 0.15371.15 0.0180 0.0223 0.0227 0.0298 0.03141.20 0.0017 0.0032 0.0041 0.0049 0.0049

Table: Call Prices with IV=1.0 for Shifted-Gamma FCGARCH Series andT=90.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 46: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Figure: Graphs of Call Prices ratios with IV=1.0 for Shifted-GammaFCGARCH Series

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 47: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Put Prices with IV=1.0 for Artificial Normal FCGARCH Series

K/S0 BS FC-Normal GARCH-Normal FC-Gamma GARCH-Gamma

0.80 0.0002 0.0001 0.0008 0.0024 0.00100.85 0.0063 0.0083 0.0097 0.0169 0.01070.90 0.0957 0.0893 0.0940 0.1095 0.10410.95 0.6537 0.5753 0.6230 0.6088 0.65231.00 2.4175 2.2835 2.3685 2.3003 2.36661.05 5.7397 5.6568 5.7088 5.5981 5.67281.10 10.1575 10.1466 10.1593 10.0596 10.11181.15 15.0234 15.0281 15.0358 14.9438 14.97011.20 20.0025 20.0059 20.0135 19.9292 19.9468

Table: Put Prices with IV=1.0 for Artificial Normal FCGARCH Seriesand T=90.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 48: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Figure: Graphs of Put Prices ratios with IV=1.0 for Normal FCGARCH Series

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 49: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Put Prices with IV=1.0 for Artificial Shifted-Gamma FCGARCH Series

K/S0 BS FC-Gamma GARCH-Gamma FC-Normal GARCH-Normal

0.80 0.0001 0.0024 0.0009 0.0001 0.00020.85 0.0045 0.0169 0.0099 0.0083 0.00850.90 0.0803 0.1095 0.0946 0.0893 0.09480.95 0.6017 0.6088 0.5760 0.5753 0.61491.00 2.3402 2.3003 2.2122 2.2835 2.33981.05 5.6831 5.5981 5.4999 5.6568 5.69521.10 10.1349 10.0596 9.9490 10.1466 10.15051.15 15.0180 14.9438 14.8426 15.0281 15.02131.20 20.0017 19.9292 19.8236 20.0059 19.9995

Table: Put Prices with IV=1.0 for Artificial Shifted-Gamma FCGARCHSeries and T=90.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 50: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Figure: Graphs of Put Prices ratios with IV=1.0 for Shifted-GammaFCGARCH Series

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 51: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Put option price ratios have their largest values deep in the money.

The more pronounced effect is in the FC-Gamma scheme. The values arelarger than in any other scheme.

On the other hand, the Call ratios their largest values deep out themoney , although its effect is not as large as in the put case.

In the call options, the Normal models overprice the other models whilstin the put options, the Shifted-Gamma models do.

This behavior may be explained by the negative asymmetry we introducedchanging the sign of the innovation, in the Shifted-Gamma case.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 52: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Curiosities

Table: Average rate of exercising

Model/rate Risk Neutral Measure Physical Measure

FCGARCH Normal 0.4881 0.6197FCGARCH Gamma 0.5032 0.6198Gamma GARCH 0.4940 0.6338Normal Garch 0.4885 0.5939

GARCH-Gamma(Normal data) 0.4805 0.5994GARCH-Normal(Gamma data) 0.4870 0.6238

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 53: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Average rate of exercising (Gamma GARCH)

K/S0 Risk Neutral measure Physical measure

0.80 0.9991 1.00000.85 0.9932 0.99780.90 0.9491 0.97840.95 0.7922 0.88851.00 0.4908 0.63361.05 0.2004 0.29911.10 0.0552 0.08621.15 0.0095 0.01621.20 0.0015 0.0023

Table: Average rate of exercising for Artificial Shifted-Gamma GARCHSeries and T=90.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 54: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Numerical Greeks

We evaluate numerically the Greeks under the GARCH-typemodels considered in this paper. It is known that when we hedgeusing only the Delta, a lot of money is spent on transaction costs.So, we need to look on other sensibility measures before decidinghow to hedge. Three Greeks are considered here, namely, theDelta, the Gamma and the Vega.

The Delta tells us how the option values change when theunderlying asset prices vary, while the Gamma shows how Deltavaries. The Vega conveys information about changes in optionvalues when the volatility changes.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 55: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

80 84 88 92 96 100 104 108 112 116 1200

0.1

0.2

0.3

0.4

0.5

0.6

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Stock Price

Del

ta

BSFCGARCHGARCH

Figure: The Delta with Normal innovations

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 56: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

80 84 88 92 96 100 104 108 112 116 1200

0.1

0.2

0.3

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0.5

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Stock Price

Del

ta

BSFCGARCHGARCH

Figure: The Delta with Gamma innovations

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 57: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

80 84 88 92 96 100 104 108 112 116 1200

0.1

0.2

0.3

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0.5

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Del

ta

BSFCGARCHGARCH

Figure: Delta withNormal innovations

80 84 88 92 96 100 104 108 112 116 1200

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Del

ta

BSFCGARCHGARCH

Figure: Delta withGamma innovations

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 58: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

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80 85 90 95 100 105 110 115 120−0.01

0

0.01

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Stock Price

Gam

ma

BSFCGARCHGARCH

Figure: The Gamma with Normal innovations

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 59: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

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80 85 90 95 100 105 110 115 120−0.01

0

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Gam

ma

BSFCGARCHGARCH

Figure: The Gamma with Gamma innovations

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 60: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

80 85 90 95 100 105 110 115 120−0.01

0

0.01

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Gam

ma

BSFCGARCHGARCH

Figure: Gamma withNormal innovations

80 85 90 95 100 105 110 115 120−0.01

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Gam

ma

BSFCGARCHGARCH

Figure: Gamma withGamma innovations

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 61: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

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Simulation ResultsConclusionsReferences

0.0672 0.1172 0.1672 0.2172 0.2672 0.0672 0.1172 0.1672 0.21726

8

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Volatility

Veg

a

BSFCGARCHGARCH

Figure: The Vega with Normal innovations

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 62: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

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Simulation ResultsConclusionsReferences

0.0684 0.0934 0.1184 0.1434 0.1684 0.1934 0.2184−1000

−800

−600

−400

−200

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200

Volatility

Veg

a

BSFCGARCHGARCH

Figure: The Vega with Gamma innovations

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 63: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

0.0672 0.1172 0.1672 0.2172 0.2672 0.0672 0.1172 0.1672 0.21726

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Volatility

Veg

a

BSFCGARCHGARCH

Figure: Vega withNormal innovations

0.0684 0.0934 0.1184 0.1434 0.1684 0.1934 0.2184−1000

−800

−600

−400

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0

200

Volatility

Veg

a

BSFCGARCHGARCH

Figure: Vega withGamma innovations

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 64: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

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Simulation ResultsConclusionsReferences

From these Figures we can see that the Deltas from the three optionvaluation models are close to each other in both the normal and shiftedGamma cases.

In the case of normally distributed innovations, the Gammas and Vegasarising from the GARCH model and those obtained from the FC-GARCHmodel appear to be similar. However, when the innovations follow ashifted-Gamma distribution, the Gammas and Vegas arising from theGARCH model and those obtained from the FC-GARCH model differquite significantly.

This reveals that the flexible coefficient effect may have a significantimpact on hedging volatility risk when the return data exhibit conditionalnon-normality.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 65: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Conclusions

In this paper we adopted the method of Siu et al(2004) to find the riskneutral version of the FC-GARCH with two different innovations, thenormal and the shifted-gamma cases. In the Gamma case we introducedthe possibility of a negative noise so that skewness can taken intoaccount properly.

We also performed simulations and showed tables comparing the BlackScholes price and the GARCH price to our simulation results of theFC-GARCH.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 66: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Conclusions II

Put option price ratios have their largest values deep in the money.

The more pronounced effect is in the FC-Gamma scheme. The values arelarger than in any other scheme.

On the other hand, the Call ratios their largest values deep out themoney , although its effect is not as large as in the put case.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 67: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Conclusions III

In the call options, the Normal models overprice the other models whilstin the put options, the Shifted-Gamma models do. This behavior may beexplained by the negative asymmetry we introduced changing the sign ofthe innovation, in the Shifted-Gamma case.

We perform a sensibility analysis to understand how the change of someparameters affect the results. The GARCH parameters for the regimezero and the first regime were the most sensible to disturbances while theGARCH parameters of the third regime, the logistic paramter γ, and therisk premium λ have little or no effect on option prices.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 68: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Summary

Extension of Pricing results from GARCH to FCGARCH

Shifted-Gamma case with negative sign

Simulation results, greeks and curiosities

Although using this nonlinear model has an insignificant gainin prediction, in option pricing the effect is significant.

The flexible coefficient effect may have a significant impact onhedging volatility risk when the return data exhibit conditionalnon-normality.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

Page 69: FCGARCH auckland 27012012

OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

References:

Duan, Jin-Chuan. 1995. “The GARCH Option PricingModel,” Mathematical Finance 5: 13-32.

Siu, Tak Kuen, Howell Tong, and Hailiang Yang. 2004.“OnPricing Derivatives Under GARCH Models: A DynamicGerber-Shiu Approach,” North American Actuarial Journal.8(3): 17-31.

Medeiros, Marcelo C. and Alvaro Veiga. 2009. “ModelingMultiple Regimes in Financial Volatility with a FlexibleCoefficient GARCH Model”,Econometric Theory.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH

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OutlineIntroductionFC-GARCH

Conditional Esscher transformParametric cases

Simulation ResultsConclusionsReferences

Thanks!!!

Thanks for the patience and attention of you all.

Renato Alencar Adelino da Costa, Alvaro Veiga, Tak Kuen Siu Option Pricing under a Nonlinear and Nonnormal GARCH