Pricing Asian Options in Affine Garch models

Lorenzo Mercuri

Dip. Metodi Quantitativi per le Scienze Economiche e Aziendali

Milano-Bicocca

28-29-30th of January, 2009.

X Workshop on Quantitative Finance

Outline

• Affine Garch models

• Geometric Asian options

• Arithmetic Asian options

• References

Affine Garch models

Affine Garch Models yield a closed form formula for option prices by means of Fourier

Transform

Stock price dynamics

where is an affine Garch process

1. The Heston and Nandi modelRef: Heston, Nandi (2000)

Log-returns dynamics

ttt XSS exp1

tX

111

211

10

1 ,0~

tt

ttt

ttt

ttt

hh

hh

hF

hrX

Affine Garch models

2. The Christoffersen, Heston and Jacobs modelRef: Christoffersen, Heston and Jacobs (2006)

Log-returns dynamic under real measure:

3. The Gamma model Ref: Bellini, Mercuri (2007)

Log-returns dynamic under real measure:

1

2

11110

21

1

~

t

tttt

ttttt

ttt

ttt

hhh

hwithIGFY

YF

hrX

11110

1

1

,~

ttt

ttt

ttt

ttt

hh

bahGaFY

Ya

bF

hrX

Affine Garch models

4. The Tempered Stable model Ref: Mercuri (2008)

Log-returns dynamic under real measure:

Tempered Stable distributionRef: Tweedie (1984), Hougaard (1986)

Let be the positively skewed stable density function with We say that is a Tempered Stable distribution with if its density is given by:

The characteristic function is given by:

11110

1

/)2(1

,,S~

)1(2

ttt

ttt

ttt

ttt

hh

bahTFY

ba

YF

hrX

,,,; cxs 1,0 baTS ,, 00,1,0 banda

xb

axsabbaxp

/1

2 2

1exp0,

sec2,1,;exp,,;

/1211exp biab

Affine Garch models:Tempered Stable distribution

Fig. 1. Behavior of the Tempered Stable density as function of b, from top to down

and 7.0,5.0 1,9.0 a

Affine Garch models:Tempered Stable distribution

Fig. 2. Convergence of the Tempered Stable density to the Gamma density (upper part) and to the Inverse Gaussian (lower part)

Affine Garch models:Change of measure

Conditional Esscher Transform Ref . Siu et Al. (2004), Buhlmann et Al. (1996), Gerber and Shiu (1994).

Definition Given a predictable stochastic process and an adapted process such that

We denote with

We define the stochastic process as

To price a contingent claim, we need an equivalent martingale measure Q such that

In order to obtain the martingale condition, we have to solve the following equation withrespect to

This equation is called Conditional Esscher Equation

Ntt 1tX

NkwithFXE kk ,...,1exp 1

1exp1

kkFX FXEMkk

t10

....,2,1

exp

11

NtwithM

Xt

k kFX

kkt

kk

rSFSE tttQ exp11

t

tFX

tFXr

M

M

tt

tt exp1

*

*

1

1

Geometric Asian options:Option pricing formula

We consider a geometric Asian call option with fixed strike where the underlying is

observed at equally-spaced times. The pay-off is given by:

Under Q measure, we price a geometric Asian call option with the formula:

In Affine Garch Models, it’s possible to compute the moment generating function (m.g.f)

of by a recursive procedure and then to price an option by the inversion of Fourier

transform.

1

1

0

0,max,

TT

ttT

T

SG

KGTKC

tTQt FKGEtTrC exp

tT FGln

Geometric Asian options: Recursive procedure for m.g.f.

We define:

We write the m.g.f. of in exponential form:

1. Heston and Nandi model

We get the following recursive relations:

2. The Christoffersen, Heston and Jacobs model

TT GY ln:

TY

tt

t

hhtT SuTtChuTtBuTtAS

T

uFuYEu ln,:,:,:ln

1exp:exp 1

1

0

uTtCT

uuTtC

uTtB

uTtCuTtCuTtBuTtB

uTtBuTtBuTtAruTtCuTtA

,:11

,:

,:121

,:12

1

2,:1,:1,:

,:121ln2

1,:1,:1,:1,:

1

22

1

10

uTtCT

uuTtC

TtBuTtCTtBuTtCh

uTtCuTtBuTtB

uTtBuTtBuTtAruTtCuTtA

t

,:11

,:

,:12,:121,:1,:1211,:1,:1,:

,:121ln2

1,:1,:1,:1,:

14

21

1

4

0

Geometric Asian options: Recursive procedure for m.g.f.

3. The Gamma model

4. The Tempered Stable model

With terminal conditions

1,:1,:

,:11ln,:1,:1,:

,:1,:1,:1,:

11

0

T

uuTtCuTtC

uTtBaa

uauTtCuTtBuTtB

uTtBuTtAruTtCuTtA

uTtCT

uuTtC

ba

uTtCuTtBabuTtCuTtBuTtB

uTtBuTtAruTtCuTtA

,:11

,:

1

,:1,:111,:1,:1,:

,:1,:1,:1,:

11

0

1,:

0,:

0,:

T

uuTTC

uTTB

uTTA

Geometric Asian options:Comparison between semianalytical price and Monte

Carlo simulations

Geometric Asian options:Comparison between semianalytical price and Monte

Carlo simulations

Arithmetic Asian options:Option pricing formula

We consider an arithmetic Asian call option with fixed strike where the underlying isobserved at equally-spaced times. The pay-off is given by:

Under Q measure, we need to evaluate the following expected value:

The distribution of the variable in unknown. Turnbull and Wakeman (1992) Approximate the true distribution with a more tractable distribution that matches the first fourth moments

0,1

1max,

0

KST

TKCT

tt

t

T

ttQt FKS

TEtTrC

01

1exp

T

ttST

A01

1:

Arithmetic Asian options:Recursive procedure for the moments

We define:

The nth-moment, given the information a time zero, is obtained by:

In Affine Garch models, it is possible to compute recursively the quantity:

Indeed

T

ttT SA

0

:

n

j

j

j

j

jTTQ

T

T

n

n

n

TQn

T

T

FXjXjEj

j

j

j

j

n

T

S

FAT

E

0 0 0011

1

2

1

1

0

0

1

1

2

1

...exp......1

1

1

tTTttQttTt FXjXjEXjXjjj ...exp...exp:,... 11111

111111 ,...,,:,...,,:...exp,... tTtTtttTt hjjTtBjjTtAXjXjjj

Arithmetic Asian options:Recursive procedure for the moments

1. The Heston and Nandi model

2. The Christoffersen Heston Jacobs model

Tt

t

tTtTt

TtTtTttTt

jjTtB

jjjjTtBjjTtB

jjTtBjjTtBjjTtArjjjTtA

,...,,:1212

1

2,...,,:1,...,,:

,...,,:121ln2

1,...,,:1,...,,:1,...,,:

21

212

1211

2120211

TttTttt

tTtTt

TtTtTttTt

jjTtBjjjTtBjh

jjjTtBjjTtB

jjTtBjjTtBjjTtArjjjTtA

,...,,:1221,...,,:1211,...,,:1,...,,:

,...,,:121ln2

1,...,,:1,...,,:1,...,,:

21124

121

1211

24

20211

Arithmetic Asian options:Recursive procedure for the moments

3. The Gamma model

4. The Tempered Stable model

TttTtTt

TtTttTt

jjTtBaa

uajjjTtBjjTtB

jjTtBjjTtArjjjTtA

,...,,:11ln,...,,,1,...,,:

,...,,:1,...,,:1,...,,:

21

1211

0211

2

1

,...,:111,...,:1,...,:

,...,:1,...,:1,...,:

1211211

20211

ba

jjjTtBabjjjTtBjjTtB

jjTtBjjTtArjjjTtA

tTttTtTt

TtTttTt

Arithmetic Asian options: Approximation formula

Rif. Lévy (1991) Turnbull and Wakeman (1992)

Let the lognormal density function where the parameters match the mean and variance of the

We can approximate the true distribution by using the fourth-order Edgeworth series

Where

With

Therefore the approximate Asian option price is given by:

2log ,; vmyf

TA

0022

02

0

log2log

log2

1log2

FAEFAEv

FAEFAEm

TQTQ

TQTQ

4

1

2log2

log2

,;

!,;,;

ii

i

iedg ye

y

vmyf

i

kvmyfvmyf

2log ,; vmyffk iii

fFfAEf

FfAEf

FfAEf

FAEf

TQ

TQ

TQ

TQ

204

14

03

13

02

12

01

3

2

2log

2

4

2log300

0

0

2

0,;

!4

,;

!31

log1

log

10, 2

2

y

vmyfk

y

vmyfk

T

Se

v

S

nKm

KNS

T

v

S

nKvm

NeT

SeKC rTmrT

edg

v

Arithmetic Asian options:Comparison between approximate formula and Monte

Carlo simulations

Arithmetic Asian options:Comparison between approximate formula and Monte

Carlo simulations

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Measure and Conditional Esscher Transform" CWI Quarterly 9(4) (1996) pp. 291-317.Carr, P. Madan, D. B. (1999). “Option Valuation using the Fast Fourier Transform” Journal

of Computational Finance 2 (4), 61-73.Christoffersen, P. Heston, S.L. Jacobs, C (2006). "Option valuation with conditional

skewness" Journal of Econometrics, 131, 253-284.Fusai, G. Meucci, A. (2008). "Pricing discretely monitored Asian options under Levy

processes," Journal of Banking & Finance, 32 (10), pp. 2076-2088. Fusai, G. Roncoroni, A. (2008). “Asian Options: An Average Problem, in Problem Solving in

Quantitative Finance: A Case-Study Approach” Gerber, H.U. Shiu, E.S.W (1994) "Option pricing by Esscher transforms“ Transactions of the Society

of Actuaries 46 (1994) pp. 99-191.Heston, S.L. Nandi, S. (2000). "A closed form option pricing model" Review of financial

studies 13,3, pp. 585-562.Hougaard, P. (1986) "Survival models for heterogeneous populations derived from stable

distributions" Biometrika 73, pp.387-396.Lévy, E., (1992). “Pricing European Average Rate Currency Options” Journal of

International Money and Finance 11, 474-491.Mercuri, L. (2008). “Option Pricing in a Garch Model with Tempered Stable Innovations“

Finance Research Letters 5, pp.172-182.Siu, T.K. Tong, H. Yang, H. (2004)"On pricing derivatives under Garch models: a dynamic

Gerber-Shiu approach" North American Actuarial Journal 8(3) pp. 17-31.Tweedie, M. C. K. (1984). “An Index wich Distinguishes between some important

exponential families” Statistics: Applications and New Directions: Proc. Indian Statistical Institute Golden Jubilee International Conference (ed. J. Ghosh and J. Roy), pp.579-604.