Models with time- -dependent parameters dependent parameters - - PowerPoint PPT Presentation

Models with time Models with time-

• dependent parameters

dependent parameters using transform methods: using transform methods: Application to Heston Application to Heston’ ’s model s model

Alberto Elices Alberto Elices

Model Validation Group Model Validation Group Risk Division Risk Division Grupo Santander Grupo Santander

Kick Kick-

• off
• ff-
• Workshop

Workshop Special Semester on Stochastics with empphasis on Finance Special Semester on Stochastics with empphasis on Finance J Johann

• hann Radon Institute for Computational and Applied

Radon Institute for Computational and Applied Mathematics Mathematics (RICAM) (RICAM) September 8 September 8-

• 12th 2008, Linz, Austria

12th 2008, Linz, Austria

2

Outline of the presentation

Introduction. Characteristic functions of models with time-dependent parameters. Application to Heston’s model. Case study: Calibration to Eurostoxx 50. Application to Forward start options. Forward skew of Heston’s model. Conclusions.

3

Introduction

Exotic valuation: usually carried out with Monte Carlo. Calibration: fast analytic models are needed for valuation of vanilla products. Analytic models depend on just a few parameters which cannot fit the whole set of market parameters. More degrees of freedom are needed in order to calibrate the market across all maturities. The most natural way of introducing more parameters is to let them depend on time.

4

Characteristic functions of models with time-dependent parameters

Characteristic function methods:

Useful when the characteristic function is analytic. The Inversion of the characteristic function is carried out

through the inverse Fourier transform.

Characteristic function:

( ) ∫

⋅ ⋅

= =

N v v

v u uv i i u uv

d f e e

R x X x X

x x X E x X ) ( ) ( ϕ

5

Introduction

Family of characteristic functions for which the methodology can be applied: The method proposed introduces time-dependent parameters for a wide variety of models which admit analytic characteristic function:

Merton jump model.

: sum of all Poisson distributed jumps up to time .

( )

u uv u uv

iGg G C g G + = ) ( exp ) ( ϕ

u

g

u

t

( )

u uv uv u uv

C x X D X x X ⋅ + = ) ( ) ( exp ) ( ϕ

)) ( , ), ( ( ) (

, 1 ,

X X X D

N uv uv uv

D D L = ) , , (

1 N

X X L = X )) ( , ), ( ( ) (

1

t x t x t

N

L = x

6

Introduction

Cox Ingersoll Ross model.

: short rate interest rate at time .

Heston stochastic volatility model.

: logarithm of underlying. : variance process.

Hybrids with jumps, stochastic interest rates and volatility.

( )

u uv uv u uv

r R iD R C r R ) ( ) ( exp ) ( + = ϕ

( )

u u uv uv u u uv

iXx v V X D V X C v x V X + + = ) , ( ) , ( exp ) , , ( ϕ

u

r

u

t

u

x

u

v

u u u uv u uv uv

iGg iXx v D r D C u u u u uv

e g r v x G R V X

+ + + +

=

1 , 2 ,

) , , , , , , ( ϕ

7

Characteristic functions of models with time-dependent parameters

All relevant information of a Markov process with independent increments at an instant is given by the joint probability distribution: Objective: Find in terms of

u

t

v

t

( )

x X

u

ϕ

( )

u uv

x X ϕ

u u

t = τ

u v uv

t t − = τ ) ( x X

v

ϕ ) ( x X

u

ϕ ) (

u uv

x X ϕ ) ( x X

v

ϕ

v

t

8

Characteristic functions of models with time-dependent parameters

Characteristic function under search: Joint density in terms of densities and (independent increments): Subtituting in :

( )

∫

⋅

=

N v

v v i v v

f e d

R x X

x x x x X ) ( ϕ

∫

=

N

u v uv u u u v v

f f d f

R

x x x x x x x ) ( ) ( ) (

( ) ( )

4 4 4 3 4 4 4 2 1

u uv uv u uv N v N

C u v uv i v u u u v

f e d f d

x X D X x X R x X R

x x x x x x x X

⋅ + = ⋅

∫ ∫

=

) ( ) ( exp /

) ( ) ( ) (

ϕ

ϕ ) ( x xv

v

f

( )

x X

v

ϕ

v

t t →

u

t t →

v u

t t →

9

Characteristic functions of models with time-dependent parameters

After substituting :

( )

( )

( )

( )

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⋅ + + =

− − 1 1

) ( ) ( ) ( exp x X D D X D X

X X

4 4 3 4 4 2 1 4 4 4 4 3 4 4 4 4 2 1

v v

D uv u C uv u uv

i i C C

( )

∫

⋅ + =

N

u uv uv u u u v

C f d

R

x X D X x x x x X ) ( ) ( exp ) ( ) ( ϕ

( )

u uv

x X ϕ

( )

( )

( )

4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 2 1

1

) ( 1

)) ( ( exp ) ( ) ( exp

x X D R

x X D x x x X

uv u N

i u uv u u u uv

i i f d C

−

⋅ =

−

∫

ϕ

10

Characteristic functions of models with time-dependent parameters

Identifying terms:

( ) ( )

( )

( )

( )

⎪ ⎩ ⎪ ⎨ ⎧ = + =

− −

) ( ) (

1 1

X D D X D X D X X

uv u v uv u uv v

i i C C C

( )

) ( ) ( exp ) ( x X D X x X ⋅ + =

v v v

C ϕ

1

t

M

t

01 01,D

C

01

ϕ

2

t

1 − M

t

12 12 ,D

C

M M M M

C

, 1 , 1

,

− −

D

2 1

ϕ

M M , 1 −

ϕ L L

11

Application to Heston´s model

Heston process: The two state variables for Heston’s process are the logarithm of the stock price and the variance process : These two state variables translate into and for the characteristic function:

) , ( V X = X )) ( ), ( (

u u u

t t x ν = x

( )

dt Y W d dY dt d dW S dt S dS

t t t t t t t t t t t

ρ ν σ ν θ κ ν ν μ = 〉 〈 ⎪ ⎩ ⎪ ⎨ ⎧ + − = + = , ) log(

t t

S x =

t

v X V

12

Application to Heston´s model

Joint characteristic function for Heston process:

( )

) ( ) , ( ) ( ) , ( ) , (

1 , 2 ,

) ( ), ( ,

u uv u uv uv

t x V X D t V X D V X C u u uv

e t v t x V X

+ +

=

ν

ϕ

( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − + − =

− − 2 τ τ

σ ρσ κ

d d uv

e g e g g d Xi V X D ~ 1 ~ ,

2 ,

( ) ( ) ⎟

⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − − + =

−

τ ρσ κ σ κθ τ μ

τ

d Xi g e X i V X C

d uv

~ 1 g ~ 1 ln 2 ,

2

( ) ( )

X i X Xi d d Xi d Xi g iV d Xi iV d Xi g + + − = + − − − = − + − − − − =

2 2 2

σ ρσ κ ρσ κ ρσ κ σ ρσ κ σ ρσ κ

2

~

( )

iX V X Duv = ,

1 ,

13

Application to Heston´s model

Characteristic function with time-dependent parameters at maturity :

( ) ( )

( )

( )

( )

⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = = + =

− −

iX V X D V X D i X D V X D V X D i X C V X C V X C

v uv u v uv u uv v

) , ( ) , ( , , ) , ( , , ,

1 , 2 , 1 2 , 2 , 2 , 1

( )

) ( ) , ( ) ( ) , ( ) , (

1 , 2 ,

) ( ), ( ,

t x V X D t V X D V X C v

uv v v

e t v t x V X

+ +

=

ν

ϕ

u

t

v

t

( )

x X

u

ϕ

( )

u uv

x X ϕ

u u

t = τ

u v uv

t t − = τ

v

t

14

Application to Heston´s model

Valuation of vanilla options: Characteristic function for cash or nothing option: Inversion formula: cummulative density in terms of characteristic function.

( )

{ }

( )

{ }

( )

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − = − =

> > +

4 3 4 2 1 4 4 3 4 4 2 1

nothing

• r

Cash ln nothing

• r

Asset ln

) (

K x K x x T T T

T T T

K e DF K S DF C 1 E 1 E E

T T T T T x iX T T

dv dx v x f e X X

T

∫

⋅

= =

R CN

x x x ) , ( ) , ( ) ( ϕ ϕ

∫

∞ −

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − + = > ) ( ) ( 1 2 1 2 1 ) ( dX e X e X iX a x P

iXa iXa

ϕ ϕ π

15

Application to Heston´s model

Characteristic function for asset or nothing option: Final expression of vanillas:

( ) ( )

) ( ) ( ) , ( ) , ( ) (

) ( T x iXx x x i X i T T T

S e e e e i i X X

T T T T

E E E E x x x

AN

= = − − =

−

ϕ ϕ ϕ

( )

( )

∫ ∫

=

R R T v x v x f T x T T T T iXx T

dv S e v x v x f dx e v x X

T T T T T

4 4 4 4 8 4 4 4 4 7 6

, ,

) ( , , ) , (

AN

E

AN

ϕ

( )

( )

) ln ( ) ln ( K x KP K x P S DF C

T T T T

> − > =

CN AN

E

16

Application to Heston´s model

Valuation of FX quanto options ( in USD per EUR): Characteristic function for asset^2 or nothing option: Final expression for FX quanto vanillas:

( )

{ }

( )

{ }

( )

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − = − =

> > +

4 4 3 4 4 2 1 4 4 3 4 4 2 1

nothing

• r

Asset ln nothing

• r

Asset^2 ln 2 $ $

) (

K x x K x x T T T T

T T T T

e K e DF S K S DF C 1 E 1 E E

( ) ( )

) ( ) ( ) 2 ( ) 2 ( ) (

2 2 2 ) 2 ( T x iXx x x i X i T T T

S e e e e i i X X

T T T T

E E E E x x x

N A2

= = − − =

−

ϕ ϕ ϕ

( ) ( )

( )

K x P S K K x P S DF C

T T T T T

ln ) ( ln ) (

2 $

> − > =

AN N A

E E

2

T

S

17

Calibration

A bootstrapping algorithm is proposed:

Periods in between vanilla maturities are chosen to let

parameters change.

• 1. n = 1
• 2. Search model parameters ( ) from to

to fit vanillas at minimizing the following objective function:

N.B. chosen to give more weight to options closer to ATM.

• 3. The parameters up to are fixed
• 4. n = n + 1
• 5. Return to step 2

ρ σ κ θ , , ,

( )

∑∑

=

− =

M i market i model i j i

price price w w FO

1 2

i

w

1 − i

T

i

T

i

T

i

T

18

Case study: Calibration to Eurostoxx 50.

K \ Mat 1m 3m 6m 9m 1y 2y 3y 4y 5y 10y 0.85 23.0 18.7 18.5 18.6 19.1 19.7 20.6 21.5 22.2 25.8 0.90 18.9 16.7 17.0 17.2 17.8 18.8 19.8 20.8 21.5 25.3 0.95 15.2 14.7 15.5 16.0 16.6 17.8 19.0 20.0 20.8 24.7 1.00 12.2 13.2 14.1 14.8 15.5 16.9 18.2 19.3 20.2 24.2 1.05 11.6 12.3 13.1 13.9 14.4 16.1 17.5 18.7 19.5 23.7 1.10 13.3 12.3 12.6 13.2 13.7 15.4 16.9 18.1 19.0 23.2 1.15 15.6 12.9 12.4 12.7 13.2 14.8 16.3 17.5 18.5 22.7

v0 θ κ σ ρ v0 θ κ σ ρ max 1 1 20 1.5 1 100 100 100 100 1 min

• 1
• 1

σ κ

Time dependent Heston model is calibrated to the following Eurostoxx 50 volatility surface: To avoid problems with discrete dividend payments, what is calibrated is the forward delivered at the last maturity rather than the underlying itself. Two calibrations are carried out:

Left: constrained calibration (esp. with respect to and ). Right: unconstrained calibration

19

Case study: Calibration to Eurostoxx 50.

Maximum error for both calibrations: 8 bp for most OTM options. Both calibrations are equivalent from a qualitative point of view:

Market is pricing in increasing uncertainty of volatility: is constant

while reduces (left) vs is constant while increases (right).

Market is pricing in increasing volatility (from 11% to around 45% at

10y) and increasing skew.

2 4 6 8 10 −1 1 2 3 4 5 6 7 8 Calibration of STOXX50E: ATM + 3 vanillas around. var

0 = 0.0174

Maturity in years (Moneyness weights: ATM: 100; Rest: 45, 35, 5) Values theta kappa sigma rho 2 4 6 8 10 −1 1 2 3 4 5 6 7 8 Calibration of STOXX50E: ATM + 3 vanillas around. var

0 = 0.0175

Maturity in years (Moneyness weights: ATM: 100; Rest: 45, 35, 5) Values theta kappa sigma rho

σ κ σ κ

20

Application to Forward start options.

Forward start option: Applying the tower law: The expectation E can be calculated integrating over the state variables and at times and .

( ) ( )

+ − +

− = − = ) ( ) , ( ) ( ) , ( K e e t P Ke e t P p

u v u u v

x x x v x x v

E E

[ ]

( )

E t P K e e t P p

v u x x x v

u v u

) , ( ) ( ) , ( = − =

+ −

x E E

( )

( )

( )

∫ ∫

+ −

− =

2 2

R R v u v uv x x u x u u u u

d f K e F e f d F E

u v u

x x x x x x

u

x

v

x

u

t

v

t

21

Application to Forward start options.

The increment depends on but not on : Doing the change of variable : Exchanging the order of integration, the expectation E can be calculated as a regular vanilla with respect to a new measure .

u

x

u

ν ) , ~ ( ) , / , ( ) , / , (

u v uv u v u v uv u u v v uv

f x x f x x f ν ν ν ν ν x = − =

( )

( )

( )

∫ ∫

+

− =

2 2

~ , ~

~ R R

x x x x x

v u v uv x u x u u u u

d v f K e F e f d F E

v u

( )

( )

( )

+ +

− = − =

∫

) ( ~ ~ ~ ~

~ ~

2

K e F f K e d F E

v v

x u v x v u

E x x x

R

( ) ( ) ( )

∫

=

2

, ~ ~ ~

R

x x x x x x

u u v uv u u u x v

d v f f F e f

u

u v v

x x x − = ~

u v v

x x x − = ~ f ~

22

Application to Forward start options.

Definition of the characteristic function of : Substituting and exchanging the order of integration: Replacing the definition of and reordering:

f ~

∫

⋅

=

2

~ ) ~ ( ~ ) ( ~

~ R X x

x x x x X

v v i

d f e

v

ϕ f ~

( ) ( )

( )

∫ ∫

+ = ⋅

=

2 2 , 2

) ( ) ( exp ) , ( ~

~ , ~ ) ( ~

R X X X R X x

x x x x x x X 4 4 4 4 3 4 4 4 4 2 1

u uv uv u uv v u

v D C v v u v uv i u u u x u

d v f e f F e d

ϕ

ϕ

( ) ( )

( )

( ) ( ) ( ) ( )

4 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 2 1

2 , 2 , 2 , 2 , 2

) ( , ) ( , exp ) ( , 2 , ) (

) ) ( ( ) ( exp ~

ν ϕ

ν ϕ

X X x X R X

X x x x x X

uv u uv u uv u uv

iD i D x iD i C iD i u uv u u u u u C

iD i x i i f d F e

− − + + − − = − −

∫

− + − = ) , (

u uv

v X ϕ

23

Application to Forward start options.

Final expression obtained for : The marginal distribution of the underlying is

• btained by setting in .

The final forward start option price is:

( ) ( ) ( )

( )

~ ~ exp ~ ν ϕ X X x X D x C + + =

( ) ( ) ( )

( )

( )

( )

⎪ ⎩ ⎪ ⎨ ⎧ − − = − − + + − = ) ( , ~ ) ( , ln ~

2 , 2 , 2 ,

X X X X X

uv u uv u uv u

iD i D D iD i C C F C

( )

~ x X ϕ

( )

+

− = ) ( ~ ) , (

~

K e F t P p

v

x u v

E

t

x ~ = V ) , ( V X = X

24

Forward skew of Heston’s model.

Consider the price of the forward start option when the underlying is driven by BS process with constant vol: It is understood by forward skew the implied volatility surface that results when the forward start option price above, equals the price of the same forward start option when is a Heston process.

( ) ( ) ( )

+ +

− = − = ) ( ) (

~

K e E e E DF Ke e E DF p

v u v u v v

x x t x x t t

x ~

t BS BS t

dW dt x σ σ μ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 2 1 − =

2

~

t

x ~

25

Forward skew of Heston’s model.

Lower maturity options are more sensitive to the variance distribution as the forward start term increases. Constrained calibration (left) seems a lot more reasonable than unconstrained calibration (right).

0.85 0.9 0.95 1 1.05 1.1 1.15 0.1 0.15 0.2 0.25 0.3 0.35 3m option for different forward start terms (constrained calib). Moneyness Implied Volatility in per unit 5y 4y 3y 2y 1y 6m 3m spot 0.85 0.9 0.95 1 1.05 1.1 1.15 0.1 0.15 0.2 0.25 0.3 0.35 3m option for different forward start terms (unconstrained calib). Moneyness Implied Volatility in per unit 5y 4y 3y 2y 1y 6m 3m spot

26

Forward skew of Heston’s model.

Longer maturity options are less sensitive to the variance distribution as the forward start term increases. Constrained and unconstrained calibrations seem to agree a lot more for longer maturity options.

0.85 0.9 0.95 1 1.05 1.1 1.15 0.1 0.15 0.2 0.25 0.3 0.35 5y option for different forward start terms (constrained calib). Moneyness Implied Volatility in per unit 5y 4y 3y 2y 1y 6m 3m spot 0.85 0.9 0.95 1 1.05 1.1 1.15 0.1 0.15 0.2 0.25 0.3 0.35 5y option for different forward start terms (unconstrained calib). Moneyness Implied Volatility in per unit 5y 4y 3y 2y 1y 6m 3m spot

27

Forward skew of Heston’s model.

Between both calibrations: big difference for short maturity forward start options.

Both calibrations fit the marginal distribution of the underlying but, the variance distribution is not specifically calibrated in either case.

Market volatility surface:

Gives info about the

marginal distribution of the underlying.

No info is given about the

distribution of the variance (this info could be given by forward start or cliquet

• ption quotes).

0.85 0.9 0.95 1 1.05 1.1 1.15 −20 20 40 60 80 100 3m option for different forward starting terms Moneyness Price difference in basis points 5y 4y 3y 2y 9m 6m 3m spot

28

Forward skew of Heston’s model.

What´s different from both calibrations? Consider the instantaneous volatility (obtained applying Itô): Calibrations with greater than one can lead to very negative drift:

Constrained: cannot be higher than 1.5. Unconstrained: far exceeds 1 at higher maturities:

– Variance is biased towards values near zero at long maturities. – Forward implied volatility is lower and forward skew does not make sense.

t t t t

dY dt d 2 1 ~ ~ 8 4 ~ σ σ κ σ σ κθ σ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ 2 − − =

2 t t

v = σ ~ σ σ σ

29

Forward skew of Heston’s model.

Calibration of the uncertainty of volatility ( and ):

Left: constraining to moderate values and calibrating

seems to provide a better forward skew.

Right: fixing and calibrating may bias the forward

skew towards artificially lower implied volatilities.

2 4 6 8 10 −1 1 2 3 4 5 6 7 8 Calibration of STOXX50E: ATM + 3 vanillas around. var

0 = 0.0175

Maturity in years (Moneyness weights: ATM: 100; Rest: 45, 35, 5) Values theta kappa sigma rho

σ σ κ κ σ κ

2 4 6 8 10 −1 1 2 3 4 5 6 7 8 Calibration of STOXX50E: ATM + 3 vanillas around. var

0 = 0.0174

Maturity in years (Moneyness weights: ATM: 100; Rest: 45, 35, 5) Values theta kappa sigma rho

30

Conclusions

A new method to introduce piecewise constant time- dependent parameters using transform methods is presented:

The characteristic function of the underlying for a time

horizon is calculated in terms of the characteristic functions

• f the sub-periods where the parameters change.

Analytic tractability is preserved for a wide family of models

such as hybrids with stochastic vol, interest rates an jumps.

The method has been applied to Heston’s model. Two calibrations were carried out on the Eurostoxx 50. The method has also been applied to valuation of forward start options. The forward skew of both calibrations is explored.