GARCH AND STOCHASTIC VOLATILITY OPTION PRICING

(1)

GARCH AND STOCHASTIC VOLATILITY OPTION PRICING

Jin-Chuan Duan Department of Finance

Hong Kong University of Science & Technology

March 2000

Correspondence to:

Jin-Chuan Duan Department of Finance

Hong Kong University of Science & Technology Clear Water Bay, Kowloon

Hong Kong

Tel: (852) 2358 7671 Fax: (852) 2358 1749 E-mail: jcduan@ust.hk

Web: http://www.bm.ust.hk/~jcduan

(2)

Outline

1. Black-Scholes model

• Implied volatility vs. historical volatility

• Volatility smile

2. The GARCH option pricing model

• Data generating vs. risk-neutral price dynamics

• Foreign currency option pricing

3. Numerical methods for the GARCH option pricing model

• Monte Carlo simulations

• Markov chain approximation

• Lattice construction

• Analytical approximation

• Neural network approximation

4. Tackling volatility smile using GARCH

• Put-call parity regression

• FTSE 100 index options

5. The general properties of the GARCH option pricing model

• Skewness and kurtosis

• Risk premium and stationary volatility 6. Utilize volatility smile in applications

• Pricing exotic options

• GARCH delta and vega hedging

• Risk-neutral probabilities

7. Option pricing under stochastic volatility

• Diffusion limit of the GARCH model

• Diffusion limit of the GARCH option pricing model

• Numerical performance

(3)

1. Black-Scholes Model (1973)

• Asset price process

d ln( ) S _t = + ( r λσ σ − ² ) dt + σ dW _t 2

• Risk-neutralized asset price process

d ln( S _t ) = − ( r σ ² ) dt + σ dW _t ^* 2

• Pricing formula

For a European call option payoff at time T, Max S (

_T

− K , ) 0

its time-0 value is by the closed-form solution C S K T r

S N d Ke

^rT

N d T ( ; , , , )

( ) ( )

0 0

σ

= −

⁻

−

where

d

S

K r T

= ln ⁰ + + ( T ) 2 2 σ σ

• Implied Volatility

Find σ

^*

( K T

_i

,

_j

) to solve

C ^mkt ( K T _i , _j ) = C S ( ₀ ; K T r _i , _j , , σ ^* ( K T _i , _j ))

(4)

• Implied volatility vs. historical volatility

If the Black-Scholes model works well, the implied volatility should be roughly the same as the historical volatility. (March 11, 1998, South China Morning Post;

Historical volatility (250 days) equals 46%.)

(5)

• Volatility Smile

If the Black-Scholes model works well, implied

volatility σ ^* ( K T _i , _j ) should be independent of K

i

and T

j

. In reality, σ ^* ( K T _i , _j ) varies systematically with K

i

and T

j

.

Example 1: FTSE 100 index options

Implied Volatilities of FTSE 100 Index Calls (Euro style) on March 31, 1995

0.12 0.13 0.14 0.15 0.16 0.17 0.18

2975 3025 3075 3125 3175 3225 3275 3325 Strike Price

Implied Volatility

April May June September December

(6)

Implied Volatilities of FTSE 100 Index Calls (Euro style) on April 7, 1995

0.12 0.13 0.14 0.15 0.16 0.17 0.18

3025 3075 3125 3175 3225 3275 3325 3375 Strike Price

Example 2: Hang Seng index options

Implied Volatilities of HS Index Calls on Feb 28, 1997

0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25

12400 12600 12800 13000 13200 13400 13600 13800 14000 14200Strike Price

28 60 90

(7)

Implied Volatilities of HS Index Calls on March 7, 1997

0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25

12400 12600 12800 13000 13200 13400 13600 13800 14000 14200Strike Price

21 53 83

2. GARCH Option Pricing Model (Duan, 1995)

• Asset price dynamics modeled by a non-linear asymmetric GARCH (NGARCH)-in-mean process

ln

( )

| ~ ( , ) S

S r

F N

t t

t t t t

t t

P

+ + +

+ +

= + − +

= + + −

1 1 1

2

1 1

1 2

0 1

2 2

1

2 0 1

λσ σ σ ε

σ β β σ β σ ε θ

ε

(8)

• Locally risk-neutralized asset price process

ln

( )

| ~ ( , )

*

S

S r

F N

t t

t t t

t t t t

t t

Q

+ +

= − +

= + + − −

1 1

2

1 1

1 2

0 1 2

2 2 2

1

2 0 1

σ σ ε

σ β β σ β σ ε θ λ ε

This implies

S

_T

S Tr

_s

s T

s s

T

=  − ∑ + ∑

s







= =

0

2

1 1

1 exp 2 σ σ ε

^*

• Option pricing

For a European call option with a payoff at time T:

Max S (

_T

− K , ) 0 its time-0 value is

C S ( ₀ , σ ₁ ; K T r , , , β β β θ λ ₀ , ₁ , ₂ , + ) = e ⁻ ^rT E ₀ ^Q { Max S ( _T − K , )} 0

• Forex option pricing under GARCH

Risk-neutralized exchange rate process under Black- Scholes

d ln( ) e

_t

= − − ( r r

_f

σ

²

) dt + σ dW

_t^*

2

(9)

Garman and Kohlhagen (1983) formula C e K T r r

e e N d Ke N d T

f

r T

f

rT

( ; , , , , )

( ) ( )

0 0 0

σ

= ⁻ − ⁻ −

where

d

e

K r r T

T

= ln ⁰ + − ( f + ) 2 2 σ σ

Locally risk-neutralized exchange rate process under GARCH

ln

( )

| ~ ( , )

*

* e

e r r

F N t

t f t

t t

t t t t

t t

Q

+ +

= − − +

= + + − −

1 2 1

1 1

2 1

0 1 2

2 2 2

1

2 0 1

σ σ ε

σ β β σ β σ ε θ λ ε

Forex option price under GARCH

C e ( ₀ , σ ₁ ; , , , K T r r _f , β β β θ λ ₀ , ₁ , ₂ , + ) = e ⁻ ^rT E ₀ ^Q { Max e ( _T − K , )} 0

Note:

1. For quanto option pricing under GARCH, please see Duan and Wei (1999).

2. An arbitrage-free proof of the GARCH option pricing model can be found in Kallsen and Taqqu (1998). It is accomplished by using the geometric Brownian motion to connect the

discrete-time GARCH model.

(10)

3. Numerical methods

• Monte Carlo simulations

1) Standard Monte Carlo simulation S i S r t Z i

Z i Z i i

t t

t t t t t

( ) exp[( ) ] ( )

( ) ( ) exp[ . ( )]

= −

= ₋ − +

0 1 2

0 5 δ

σ σ ε

2) Empirical martingale simulation (EMS) (Duan &

Simonato, 1998)

S i S r t Z i

Z i Z i i

n Z i i

t t

t t t t t

t t t t

i n

( ) exp[( ) ] ( )

( ) ( ) exp[ . ( )]

( ) exp[ . ( )]

= −

= − +

− +

∑

−

= − 0

1 2

1 0 5

δ

σ σ ε σ σ ε

Thus,

1 1 0

n r t S i _t S

i n

exp[ ( − − ) ] ( )

∑ =

= δ

(11)

Worksheet: Standard Monte Carlo Simulation for valuing call option under GARCH

Current stock price = 51

Initial conditional s.d. (annualized) = 0.2 Interest rate (annualized) = 0.05

Number of sample paths = 10

Maturity (days) = 2 Strike price (X) = 50

GARCH parameters:

Beta0 = 1E-05

Beta1 = 0.8

Beta2 = 0.1

Theta = 0.5

Lambda = 0.3

Stationary standard deviations (annualized) implied by the GARCH parameters:

Data generating = 0.2206 Risk neutral = 0.3184

std. normal std. normal

-0.8131 0.7647

-0.5470 0.5537

0.4109 0.0835

0.4370 -0.6313

0.5413 -0.1772

-1.0472 2.4048

0.3697 0.0706

-2.0435 -1.4961

-0.2428 -1.3760

0.3091 0.3845

S(0) sd(1) S(1) sd(2) S(2) max(S(2)-X,0) 51 0.200 50.572 0.215 51.012 1.012 51 0.200 50.713 0.207 51.022 1.022 51 0.200 51.224 0.190 51.271 1.271 51 0.200 51.238 0.190 50.921 0.921 51 0.200 51.294 0.190 51.208 1.208 51 0.200 50.448 0.222 51.881 1.881 51 0.200 51.202 0.191 51.243 1.243 51 0.200 49.925 0.261 48.918 0.000 51 0.200 50.875 0.200 50.151 0.151 51 0.200 51.169 0.191 51.371 1.371 Discounted average (or Monte Carlo price) = 1.0079

(12)

Worksheet: Empirical Martingale Simulation for valuing call option under GARCH

Maturity (days) = 2 Strike price (X) = 50

GARCH parameters:

Beta0 = 1E-05

Beta1 = 0.8

Beta2 = 0.1

Theta = 0.5

Lambda = 0.3

-0.8131 0.7647

-0.5470 0.5537

0.4109 0.0835

0.4370 -0.6313

0.5413 -0.1772

-1.0472 2.4048

0.3697 0.0706

-2.0435 -1.4961

-0.2428 -1.3760

0.3091 0.3845

S(0) sd(1) S(1) S*(1) sd(2) S(2) S*(2) max(S*(2)-X,0) 51 0.200 50.572 50.712 0.215 51.012 51.126 1.126 51 0.200 50.713 50.854 0.207 51.022 51.137 1.137 51 0.200 51.224 51.366 0.190 51.271 51.386 1.386 51 0.200 51.238 51.380 0.190 50.921 51.036 1.036 51 0.200 51.294 51.436 0.190 51.208 51.323 1.323 51 0.200 50.448 50.588 0.222 51.881 51.998 1.998 51 0.200 51.202 51.344 0.191 51.243 51.357 1.357 51 0.200 49.925 50.063 0.261 48.918 49.027 0.000 51 0.200 50.875 51.016 0.200 50.151 50.264 0.264 51 0.200 51.169 51.311 0.191 51.371 51.486 1.486

Average 50.866 50.900

Discounted average (or Monte Carlo price) = 1.1109

(13)

A comparison: Black-Scholes option pricing model (batch size = 1000; S

₀

=100, r=0.1, δ =0, σ =0.2, T=1 month) At-the-money European call option price estimate

2.60 2.64 2.68 2.72 2.76

1 11 21 31 41 51 61 71 81 91

Crude Batc h EMS

Standard deviation of the price estimate

(at-the-money European call option; two hundred repetitions)

0.00 0.04 0.08 0.12

1 11 21 31 41 51 61 71 81 91

Crude Batch EMS

(14)

• Markov chain approximation (Duan & Simonato, 1999) Discretize the underlying asset prices

1 ) 1 ln(

) ln(

* ) 2 ( 1

= + +

+

−

= h t q t

S t t

h t r

p

where

] ) (

1 [ 1

*

2 2 1

0 λ θ β

β

+ +

−

= − h

Example: m=3, n=3 (p(1), q(1)) (p(2), q(1)) (p(3), q(1))

(p(1), q(1)) (p(2), q(1)) (p(3), q(1)) (p(1), q(2))

(p(2), q(2)) (p(3), q(2))

(p(1), q(2)) (p(2), q(2)) (p(3), q(2)) (p(1), q(3))

(p(2), q(3)) (p(3), q(3))

(p(1), q(3))

(p(2), q(3))

(p(3), q(3))

(15)

Transition probability matrix

Π =

π π π π π

π π π π

( , ; , ) ( , ; , ) ( , ; , ) ( , ; , ) ( , ; , ) ( , ; , ) ( , ; , ) ( , ; , ) ( , ; , ) ( , ; , ) ( , ; , ) ( , ; , ) ( , ; , ) ( , ; ,

1 11 1 1 1 2 1 1 1 1 1 11 2 1 1

2 11 1 2 1 2 1 2 1 1 2 11 2 2 1

11 1 1 2 1 1 1 11 2

! !

" " # " " # "

!

m m n

m m m m m ) ! ( , ; , )

" " # " " # "

π 1 2 m n





 





 

Since

1 2 1 2 1 1 0 2

1 1 1 1

) (

2 ln 1

λ θ ε

β β

β

ε

−

− +

+

=

+

−

=

+ + +

+

+ + + +

t t t

t

t t t t

t

h h

h

h h

S r S

q

_t+2

is a deterministic function of q

_t+1

, p

_t+1

, p

_t

; that is, q

_t+2

= Φ ( q

_t+1

, p

_t+1

, p

_t

)

{ } ( )



 

 ∈ = = Φ ∈

=

+ +

otherwise 0

) ( )

( ), ( ), ( if

) ( ),

(

| ) ( Pr

) ,

; , (

1 1 C k p p i q q j q j p k p i D l

p l k j i

t t

Q t

π

(16)

American option prices in the GARCH framework

Price vector

[ ⁽ ¹ ⁾ ⁽ ² ⁾ ⁽ ⁾ ⁽ ¹ ⁾ ⁽ ² ⁾ ⁽ ⁾ ]

' p p p m p p p m

P = ! ! !

American option price computation

{ ⁽ ^, ^, ⁾ ^, ⁽ ^, ¹ ⁾ }

max )

,

( P t = g P K t e ⁻ Π V P t +

V ^r

where ) , ( P t

V : American option’s price K : Strike price

) , , ( P K t

g : European option’s payoff function )

, , ( ) ,

( P T g P K T

V =

(17)

Density of the transition probability matrix (GARCH)

m=101, n=11,

2 . 0 ,

1 . 0 ,

8 . 0 ,

00001 .

0 ,

05 . 0 ,

50

₀ ₁ ₂

0

= ^r = β = β = β = λ =

S

(18)

• Lattice construction (Ritchken & Trevor, 1999)

(The following three figures are taken from Ritchken & Trevor, 1999)

Note: The maximum and minimum conditional volatilities are given in

the boxes. These figures are multiplied by 10

⁵

. S

0

=1000, r=0,

λ =0, β

0

=0.0000065, β

1

=0.9, β

2

=0.04,c=0.

(19)

(20)

(21)

• Analytical approximation (Duan, Gauthier & Simonato, 1999)

For a European call option payoff at time T, Max S (

_T

− K , ) 0

its time-0 value is by the approximate closed-form solution C _app = + C κ _{3 3} A + ( κ ₄ − 3 ) A ₄

where

C S N d Ke ^rT N d

= ₀ ( ~ − ⁻ −

T

) ( ~

σ _ρ )

~

ln( ) d d

d

S

K rT

rT

T T

T

= +

= + +

= − +

δ

σ σ

δ µ σ

σ

ρ ρ

ρ

0 2

2 1 2 1 2

A S d n d N d

T T T

3 1 0 2

3 2

= − −

! [( ~

) ( ~

σ _ρ σ _ρ σ _ρ )]

A S d d n d N d

T T T T

4 1 0 2 3

4 1 3

= − − − −

! [( ~

( ~

)) ( ~

) ( ~

σ _ρ σ _ρ σ _ρ σ _ρ )]

Note: µ _ρ

_T

and σ _ρ

_T

are the mean and standard deviation of the cumulative return, i.e., ln S

S T

0 , conditional on time- 0 information; κ ₃ and κ ₄ are the skewness and

kurtosis coefficients of the standardized cumulative

return, conditional on time-0 information.

(22)

Conditional moments of the cumulative return

Figure 1: Standardized first moment of the cumulative return

(GARCH parameter values : β₀ =0.00001,β₁ =0.7,β₂ = 0.1,λ+θ =1.0,h₁ =h^*)

0 50 100 150 200 250 300

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4x 10^-4

Number of Days

analytical moment 95% confidence interval monte Carlo estimate

(23)

Figure 2 : Standardized second moment of the cumulative return

0 50 100 150 200 250 300

0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035x 10^-4

Number of Days

(24)

Figure 3 : Skewness of the standardized cumulative return

0 50 100 150 200 250 300

-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1

Number of Days

(25)

Figure 4 : Kurtosis of the standardized cumulative return

0 50 100 150 200 250 300

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

Number of Days

(26)

• Neural network approximation (Hanke, 1997)

Euro Call Option under GARCH

S= 50.5

X= 50

Mat = 25 days

Lambda c-sig/sig S/X Mat (yrs) sigma beta2 beta1 r

Inputs 0 1 1.01 0.0685 0.2 0.1 0.8 0.05

Weights 0.214 0.417 1.713 -2.851 -1.799 -0.683 -0.467 -1.011 (layer 1) -0.033 0.161 0.415 0.360 -0.358 0.394 -0.504 -0.244 -0.005 0.452 1.979 0.314 0.077 -0.177 0.005 0.134 -0.502 -0.059 -2.166 -0.796 -0.144 -0.040 0.059 0.213 -0.726 0.097 0.192 1.500 -0.147 -0.275 0.231 -1.078 -0.016 -0.025 3.292 3.340 -0.432 0.022 0.034 0.266 -0.443 -0.093 1.353 0.587 0.071 0.020 0.129 0.170 -0.209 0.041 -14.132 0.345 0.361 -0.040 -0.017 -0.164 -0.087 -0.077 0.316 3.570 -0.994 0.068 -0.062 0.356 0.289 -0.025 3.680 3.649 -0.740 0.034 0.032 0.268 -0.296 -0.165 5.843 -5.310 0.006 -1.393 -0.376 -2.993 0.051 -0.771 2.812 -0.163 0.612 1.049 1.080 -0.197 0.136 0.065 2.504 2.733 -1.459 -0.225 -0.255 0.623 0.130 -0.860 -1.722 -0.063 0.206 0.339 1.247 -0.367 0.632 0.022 -14.877 -1.672 -0.158 0.014 -0.008 -0.116 0.034 0.440 -1.198 0.658 1.195 1.419 -0.667 0.044 -0.045 0.751 -1.830 -0.414 -0.589 -0.987 -1.009 0.300 0.420 -0.309 -1.273 4.204 1.422 0.453 0.302 0.631 -0.800 -0.051 3.899 0.316 -1.470 0.150 0.093 0.255 0.372 0.132 -1.663 2.097 1.040 -0.253 -0.155 -0.221 0.159 -0.070 -3.087 2.330 1.479 0.054 0.071 -0.617 -0.034 0.006 6.028 2.103 0.046 -0.045 -0.052 0.183 0.522 -0.050 -2.949 -3.214 0.021 0.036 0.063 0.430 0.623 0.024 -1.092 -1.101 -0.301 -0.332 -0.121 -0.447

(27)

Product Biases tanh(P+B) Weights (layer 2) Product

1.100 -0.236 0.698 0.094 0.066

0.157 -0.895 -0.628 0.003 -0.002

2.481 -1.694 0.656 -0.047 -0.031

-2.276 4.288 0.965 -1.717 -1.657

0.468 -0.949 -0.447 0.008 -0.003

3.485 -3.034 0.422 -1.651 -0.697

1.443 -1.464 -0.022 0.683 -0.015

-14.162 13.952 -0.207 0.131 -0.027

0.263 1.526 0.946 0.387 0.366

3.836 -3.283 0.503 1.323 0.666

4.784 -6.767 -0.963 0.016 -0.015

3.140 -5.557 -0.984 0.509 -0.501

2.294 -3.253 -0.744 0.185 -0.138

-1.549 1.970 0.398 0.001 0.000

-15.161 15.023 -0.137 -0.120 0.017

-0.876 0.771 -0.104 0.004 0.000

-2.135 4.251 0.971 0.524 0.509

-0.704 1.525 0.675 -0.170 -0.115

3.716 -3.130 0.527 -0.252 -0.133

-1.356 -1.377 -0.992 1.632 -1.619

-2.701 3.921 0.840 0.202 0.169

6.210 -6.547 -0.324 0.126 -0.041

-3.169 5.366 0.976 1.064 1.038

-1.366 1.301 -0.066 -0.328 0.022

Sum -2.1422

Bias (2nd layer) + 2.1702

Option price (before adjustment) 0.0280

Option price (after adjustment) 1.3996

Note: The above spreadsheet is based on the network constructed by

Hanke with one hidden layer of 24 neurons. This network is

limited to a range of parameters as follows: 0≤λ≤0.001,

0.8≤σ

1

/σ≤1.2, 0.7≤S/X≤1.3, 1/365≤τ≤30/365, 0.1≤σ≤0.4,

0.5≤β

1

≤0.8, 0.1≤β

2

≤0.3, 0≤r≤0.1.

(28)

4. Tackling volatility smile using GARCH

• Put-call parity regression

[ ]

C ( , ) τ X − P ( , ) τ X = S ( ) τ − X exp − τ τ r ( ) + ε τ ( , ) X

This regression can be run for every τ to obtain an intercept and slope. The intercept is S( τ ) and the slope is -exp[- τ ^r( τ ^)],

which implies r( τ ^).

Constrained put-call parity regression:

Make certain that S( τ ) is a non-increasing function of τ ^. Let τ

1

be the smallest maturity and use the following

formulation:

[ ]

0 ) ( and 0 ) ( where

, , , for

) , ( ) ( exp

) ( ) ( ) , ( ) , (

1 2 1

1 =

≥

=

+

−

=

−

τ τ

τ ε τ τ τ

τ τ

τ

a a

X r

X a

S X P X C

! n

(29)

FTSE 100 index options (Euro-style) on March 26, 1997

Maturity 23 23 51 51 86 86 177 177 268 268

Strike price Call Put Call Put Call Put Call Put Call Put

4125 179.5 11.5 217.5 38.0 245.5 58.5 302.5 94.5 364 119.5 4175 136.0 17.0 179.0 49.0 209.5 71.0

4225 96.0 27.0 143.0 63.0 174.5 85.5 235.5 124.5 297 148.5 4275 62.5 43.0 111.0 80.0 142.0 102.5

4325 36.0 66.5 83.0 102.0 113.0 123.0 175.0 161.5 236.5 184 4375 18.0 98.5 59.0 127.5 88.0 147.0

4425 8.0 137.5 40.0 158.5 68.5 177.0 126.0 210.0 183 226.5 4475 3.0 182.5 26.0 193.5 49.0 207.0

Maturity 23 51 86 177 268

Strike price C-P C-P C-P C-P C-P

4125 168.0 179.5 187.0 208.0 244.5

4175 119.0 130.0 138.5

4225 69.0 80.0 89.0 111.0 148.5

4275 19.5 31.0 39.5

4325 -30.5 -19.0 -10.0 13.5 52.5

4375 -80.5 -68.5 -59.0

4425 -129.5 -118.5 -108.5 -84.0 -43.5

4475 -179.5 -167.5 -158.0

Intercept 4267.3 4272.1 4257.0 4223.8 4204.5

Slope -0.9937 -0.9921 -0.9865 -0.9735 -0.96

R square 1 1 1 1 1

Imp. index 4267.3 4272.1 4257.0 4223.8 4204.5

Int. rate 10.04% 5.65% 5.75% 5.54% 5.56%

Constrained regressions

Imp. index 4269.7 4269.7 4257.0 4223.9 4204.5

Int. rate 9.16% 6.05% 5.75% 5.54% 5.56%

(30)

• FTSE 100 index options

A) Calibrate the GARCH option pricing model using the market prices of traded options in three steps.

Step 1: Estimate implied interest rates and index values using the put-call parity regression for March 26, 1997 Implied interest rates:

0.091591 0.060473 0.057472 0.055374 0.055604 Implied index spots:

4269.69 4269.69 4256.98 4223.86 4204.48

Step 2: Compute market implied volatilities for calls Maturities

23 51 86 177 268

Strike Price

4125 0.148192 0.167101 0.162538 0.156996 0.158193 4175 0.138595 0.161283 0.158904

4225 0.129007 0.154893 0.153415 0.150791 0.152135 4275 0.122565 0.149574 0.147791

4325 0.115908 0.144424 0.142836 0.143619 0.146566 4375 0.110632 0.138826 0.138783

4425 0.108071 0.134058 0.137396 0.138915 0.141300

4475 0.105673 0.130516 0.131567

(31)

Step 3: Calibrate the GARCH model to minimize the difference between the market implied volatility and the model implied volatility.

Estimated parameter values:

β

0

= 0.00000429 β

1

= 0.72507034 β

2

= 0.07560027 θ+λ = 1.35643575 σ

1

= 0.09889376

Imp. Risk-Neutral Stationary S.D. = 16.12%

Implied volatilities derived from the GARCH option pricing model Maturities

23 51 86 177 268

Strike Price

4125 0.144981 0.153777 0.153322 0.157534 0.158991

4175 0.139704 0.149455 0.150253 0.155788 0.157867

4225 0.134155 0.145439 0.147458 0.154096 0.156758

4275 0.127844 0.141323 0.144724 0.152474 0.155685

4325 0.121891 0.137218 0.142327 0.150752 0.154627

4375 0.116696 0.133375 0.140043 0.148963 0.153705

4425 0.112325 0.129775 0.137806 0.147262 0.152766

4475 0.108033 0.126464 0.135494 0.145531 0.151839

Root Mean Squared Error = 0.00643679

(32)

0.10 0.12 0.14 0.16 0.18 0.20

4125 4175 4225 4275 4325 4375 4425 4475

FTSE 100 Index Options on March 26, 97

Strike Price Imp. vol.

-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015

4125 4175 4225 4275 4325 4375 4425 4475

FTSE 100 Index Options on March 26, 97

Strike Price Imp. vol. Diff.

(33)

B) Out-sample performance on April 2, 1997 (1 week later) Step 1: Estimate implied interest rates and index values using

the put-call parity regression Implied interest rates:

0.087787 0.055221 0.053111 0.054358 0.058546 Implied index spots:

4215.80 4215.80 4204.43 4170.63 4140.97

Step 2: Compute market implied volatilities for calls Maturities

16 44 79 170 261

Strike Price

4075 0.185401 0.184434 0.176989

4125 0.171461 0.177604 0.170641 0.164747 0.163311 4175 0.160283 0.170957 0.165924

4225 0.151814 0.165289 0.160954 0.158433 0.157141 4275 0.142793 0.158884 0.156038

4325 0.137634 0.153740 0.151518 0.151709 0.151619 4375 0.131314 0.148823 0.147214

4425 0.130085 0.143318 0.143092 0.144766 0.147019 Step 3: Calibrate the GARCH model to minimize the difference

between the market implied volatility and the model implied volatility.

Pre-set parameter values:

β

0

= 0.00000429 β

1

= 0.72507034 β

2

= 0.07560027 θ+λ = 1.35643575 Estimated parameter values:

σ

1

= 0.16876672

Imp. Risk-Neutral Stationary S.D. = 16.12% (unchanged)

(34)

Implied volatilities derived from the GARCH option pricing model Maturities

16 44 79 170 261

Strike Price

4075 0.182192 0.170503 0.164460 0.163236 0.163042 4125 0.175398 0.165596 0.161139 0.161253 0.161672 4175 0.168165 0.160894 0.157961 0.159455 0.160368 4225 0.160163 0.156524 0.155017 0.157713 0.159062 4275 0.152872 0.152044 0.152459 0.155927 0.157798 4325 0.145817 0.147760 0.150011 0.154032 0.156564 4375 0.139310 0.142900 0.147438 0.152296 0.155454 4425 0.134518 0.138626 0.145090 0.150521 0.154428 Root Mean Squared Error = 0.00699941

0.10 0.12 0.14 0.16 0.18 0.20

4075 4125 4175 4225 4275 4325 4375 4425

FTSE 100 Index Options on April 2, 97

(35)

-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015

4075 4125 4175 4225 4275 4325 4375 4425

C) Out-sample performance on April 9, 1997 (2 weeks later) Pre-set parameter values:

β

0

= 0.00000429 β

1

= 0.72507034 β

2

= 0.07560027 θ+λ = 1.35643575 Estimated parameter values:

σ

1

= 0.15313167

Imp. Risk-Neutral Stationary S.D. = 16.12% (unchanged)

Root Mean Squared Error = 0.00697208

(36)

0.10 0.12 0.14 0.16 0.18 0.20

4125 4175 4225 4275 4325 4375 4425 4475

-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015

4125 4175 4225 4275 4325 4375 4425 4475

(37)

5. The general properties of the GARCH option pricing model

• Leptokurtic (fat-tailed) and skewed distributions Under the data generating probability

E v

u Cov

t

t t

( )

( , )

( )

ε

ε σ θβ β

β β θ

4 2

1

2 0 2

1 2

2

3 1

1 3

2 1 1

= −

− >

= − − − +

+

where u

v

= + + + + +

= + +

β θ θ β β θ β

β θ β

2

2 2 4

1 2 2

12 2

2

1

3 6 2 1

1 ( ) ( )

( )

Under the risk-neutralized probability

E v

u Cov

Q t

Q

t t

( )

( , ) ( )

[ ( ) ]

*4 *2

*

ε

ε σ θ λ β β

β β θ λ

= −

− >

= − +

− − + +

+

3 1

1 3

2 1 1

1

2 0 2

1 2

2

u v

*

[ ( ) ( ) ] [ ( ) ]

[ ( ) ]

= + + + + + + + +

= + + +

β θ λ θ λ β β θ λ β

β θ λ β

2

2 2 4

1 2

2

1 2

2 2

1

3 6 2 1

1 Note:

1. θ is expected to be positive for equity returns

2. λ is expected to be positive for equity returns

(38)

Conclusions:

1) Under the data generating probability

• ln S S

T 0

0 is negatively skewed if θ >

• ln S S

T 0

is leptokurtic

2) Under the risk-neutral probability

• ln S S

T 0

0 is negatively skewed if θ λ + >

• ln S S

T 0

is leptokurtic

Prediction for implied volatilities (or option prices):

A downward skewed implied volatility smile

• Risk premium and its effect on stationary (long-run) volatility

Physical long-run volatility:

σ β

β β θ

= − − +

0

1 2

1 ( 1

2

)

Risk-neutralized long-run volatility:

σ β

β β θ λ

*

[ ( ) ]

= − − + +

0

1 2 2

1 1

Note: σ

^*

is greater than σ if λ and θ share the same sign.

Prediction for implied volatilities (option prices):

Implied volatility should be higher than “realized” or

“historical” volatility.

(39)

6. Utilize the volatility smile in applications

• Pricing exotic options

On March 26, 1997, one can price exotic options, say a lookback call, using the estimated parameter values:

β

0

= 0.00000429 β

1

= 0.72507034 β

2

= 0.07560027 θ+λ = 1.35643575 σ

1

= 0.09889376

Worksheet: EMS for valuing a lookback call option under GARCH

Maturity (days) = 2 GARCH parameters:

Beta0 = 4E-06

Beta1 = 0.7251

Beta2 = 0.0756

Theta = 1.3564

Lambda = 0

-0.8131 0.7647

-0.5470 0.5537

0.4109 0.0835

0.4370 -0.6313

0.5413 -0.1772

-1.0472 2.4048

0.3697 0.0706

-2.0435 -1.4961

-0.2428 -1.3760

0.3091 0.3845

(40)

S(0) sd(1) S(1) S*(1) sd(2) S(2) S*(2)^Min(S*(t)) P 51 0.099 50.792 50.861 0.110 51.023 51.078 50.861 0.216 51 0.099 50.862 50.932 0.106 51.025 51.080 50.932 0.149 51 0.099 51.115 51.185 0.097 51.143 51.198 51.000 0.198 51 0.099 51.122 51.192 0.096 50.966 51.021 51.000 0.021 51 0.099 51.149 51.219 0.096 51.110 51.166 51.000 0.166 51 0.099 50.731 50.800 0.114 51.468 51.523 50.800 0.724 51 0.099 51.104 51.174 0.097 51.129 51.184 51.000 0.184 51 0.099 50.470 50.538 0.131 49.960 50.013 50.013 0.000 51 0.099 50.942 51.012 0.103 50.573 50.627 50.627 0.000 51 0.099 51.088 51.158 0.097 51.194 51.250 51.000 0.250

Average 50.937 50.959

Discounted average (or Monte Carlo price) = 0.1906

Note: P = max[S*(2)-min(S*(t);t=0,1,2),0]

• GARCH delta and vega hedging Static call option delta (see Duan, 1995):

δ ∂ σ

∂

χ

t

t t t r T t

t Q T

t S X

C S T X r S

e E S

S

^T

=

= 

  

  +

− −

≥ ( , ; , , , )

( )

{ }

1 2 Φ

Call option vega:

( )

Λ Φ

t t t

t r T t

t Q

T S X t t

t T

t k t k

C S T X r

e E S G

G G G

T

=

=  ∑ −

  

 

= + − − =

+ +

− −

≥ + − −

= +

−

∂ σ

∂σ

χ ε

σ

β β ε θ λ

τ

τ τ

τ

τ ( , ; , , , )

( )

{ }

*

,

, , *

2 1 2 1

1 1

1 1 1 2 2

0 1

2 1

1 where

and

(41)

Dynamic call option delta:

δ δ β ε _τ θ λ

t t t

S t

* *

( )

= + − −

2 Λ ² because

∆ Φ ∆ Λ ∆

Λ ∆

C S T X r S

S S

t t t t t t

t t t

t t

( , ; , , , )

( ^* )

σ δ σ

δ ∂σ

∂

δ β ε _τ θ λ

+ +

+

≅ +

≅  +

  

 

=  + − −

  

 

2 1

2 1 2 1

2 2

• Risk-neutral probabilities

( )

Pr { _t ^Q S _T x } E _t ^Q _{ _S _x _}

≤ = χ

T

_≤

The risk-neutral probability can be easily evaluated using the EMS.

One can thus use the estimated GARCH parameter values to

generate a risk-neutral probability distribution. This is a better

approach than other schemes, such as Shimko (1993) and

Rubinstein (1994), for obtaining the risk-neutral probability

distribution.

(42)

7. Option pricing under stochastic volatility

• Data generating system

Divide [0,T] into nT subintervals of length

s = n 1 . Let

! , 2 , 1 , k =

ε

k

be a sequence of i.i.d. standard normal random variables.

s s s

s s

r S

S

n k ks

n ks n

ks n

s k

n k ks n

ks n

s k n

ks

)]

1 ( )

( [

] 1 ) 1

( [

2 ) ( 1

ln ln

2 2 2 2

2 ) (

2 2 2 1

) 0 (

2 ) ( 2

) (

) 1 (

) ( 2

) ( )

( )

( ) 1 ( )

(

θ β

θ ε β σ

θ β

β σ

σ

ε σ σ

λσ

+

−

− +

+ +

=

−

+

− +

=

− +

−

Note that if s = 1, we have the NGARCH(1,1)-mean model.

As n goes to infinity, the approximating model becomes

( )

t t

t

dW dW

dt d

dW dt

r S

d

, 2 2 2 ,

2 1 2

2 2 2

1 2 0

, 2 1

2 2

] 1 ) 1

( [

2 ) ( 1

ln

σ β σ

θβ

σ θ

β β β

σ

σ σ

λσ

+

−

− +

+ +

=

+

− +

=

where W

1,t

and W

2,t

are two independent standard Brownian

motions.

(43)

• Locally risk-neutralized pricing system

Denote the risk-neutralized probability law by Q. Let

! , 2 , 1 for

* = _k + s , k =

k ε λ

ε . They form a sequence of i.i.d.

standard normal random variables with respect to Q. The pricing system becomes

s s

r S

S

n k ks

n ks n

ks n

s k

n k ks n

ks n

s k n

ks

)]

1 ( )

( [

] 1 ) 1

( [

2 ) ( 1 ln

ln

2 2 2 2 *

2 ) (

2 2 2 1

) 0 (

2 ) ( 2

) (

) 1 (

* ) ( 2

) ( )

( ) 1 ( )

(

θ β

λ θ ε β σ

θ β

β σ

σ

ε σ σ

+

−

− +

+ +

=

−

+

−

=

− +

−

Note that if s = 1, we have the NGARCH(1,1) option pricing model.

As n goes to infinity, the approximating model becomes

( )

* , 2 2

* 2 , 2 1 2

2 2 2 2

1 2 0

* , 2 1

2 2

] 2

1 ) 1

( [

2 ) ( 1 ln

t t

dW dW

dt d

dW dt

r S

d

σ β σ

θβ

σ λ θβ θ

β β β

σ

σ σ

+

−

+

− +

+ +

=

+

−

=

where W ₁ ^* _, _t and W ₂ ^* _, _t are two independent standard Brownian motions under Q.

Note that this is the pricing result directly deduced from the

GARCH option pricing theory.

(44)

• Numerical performance of the GARCH approximation Note: The following two tables are taken from Ritchken and

Trevor (1999), “Pricing Options under Generalized GARCH and Stochastic Volatility Processess.”

Comparison using Monte Carlo simulations

(45)

Convergence speed of Ritchken and Trevor’s lattice algorithm

(46)

References

Black, F. and M. Scholes, 1973, “The Pricing of Options and Corporate Liabilities,”

Journal of Political Economy 81, 637-659.

Duan, J.-C., 1995, “The GARCH Option Pricing Model,” Mathematical Finance 5, 13-32.

Duan, J.-C., 1996, “Cracking the Smile,” Risk (December), 55-59.

Duan, J.-C. and J.-G. Simonato, 1998, “Empirical Martingale Simulation for Asset Prices,” Management Science 44, 1218-1233.

Duan, J.-C. and J.-G. Simonato, 1999, “American Option Pricing under GARCH by a Markov Chain Approximation,” Journal of Economic Dynamics and

Control, forthcoming.

Duan, J.-C. and J. Wei, 1999, “Pricing Foreign Currency and Cross-Currency Options under GARCH,” Journal of Derivatives 7, 51-63.

Duan, J.-C., G. Gauthier, J.-G. Simonato, 1999, “An Analytical Approximation for the GARCH Option Pricing Model,” Journal of Computational Finance 2, 75- 116.

Garman, M. and S. Kohlhagen, 1983, “Foreign Currency Option Values,” Journal of International Money and Finance 2, 231-253.

Hanke, M., 1997, “Neural Network Approximation of Option Pricing Formulas for Analytically Intractable Option Pricing Models,” Journal of Computational Intelligence in Finance 5, 20-27.

Kallsen, J. and M. Taqqu, 1998, “Option Pricing in ARCH-Type Models,”

Mathematical Finance 8, 13-26.

Ritchken, P. and R. Trevor, 1999, “Pricing Option under Generalized GARCH and Stochastic Volatility Processes,” Journal of Finance 54, 377-402.

Rubinstein, M., 1994, “Implied Binomial Trees,” Journal of Finance 49, 771-818.

Shimko, D., 1993, “Bounds of Probability,” Risk (April), 33-37.

GARCH AND STOCHASTIC VOLATILITY OPTION PRICING