auto call able s

On the Pricing of Auto-Callable Equity Structures in the Presence of Stochastic Volatility and Stochastic Interest Rates

Alexander Giese

2Outline

1. Introduction to Auto-Callable Structures

2. Requirements for the Pricing Model

3. Black-Scholes & Vasicek++ Model

4. Heston & Vasicek++ and Heston & CIR++ Model

5. Pricing Example


Characterization

An auto-callable equity structure is a structured product which depending on the path of the equity underlying is automatically called and redeemed early on pre-prescribed dates known as the auto-call dates.

Auto-callable structures are also called auto-trigger or express structures.

4Euro Stoxx 50 = S0Start Date

End of 1st year Euro Stoxx 50 S0?

Euro Stoxx 50 S0?End of 2nd year

Euro Stoxx 50 S0?End of 3rd year

No

Early Redemption:107% of Notional

Yes


End of 7th year Euro Stoxx 50 S0?Yes

No

Yes

Yes


Early Redemption:113% of NotionalNo

No

Redemption:125% of Notional


. . .. . .


Some Figures:

There are currently more than 235 auto-callable equity certificates (express certificates) in the German certificate market.

The open interest (units x market price) in these products is about 4.7 billion which corresponds to a 9% share of the overall German certificate market.

HVB issued recently its 100th auto-callable equity certificate.

Source: Derivate Forum, Der Deutsche Markt fr Derivative Produkte, Januar 2006.


Closed-form Solution in the Black-Scholes Model with time-dep Volatility:

( ) ( )[ ]( ) ( )( )( ) ( ) ( )[ ]( ) ( ) ( )( )( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )7721777

22122

21*

22

111

1*

11

8

1

,,...,,,,100,,...,,,,125

...,,,,110

,110,107

107where

Structure Call-Auto of Price

==

=


Note that for the computation of the n-dimensional normal integrals we can

apply the following reduction formula (Schroder (1989)):

( ) ( ) ( )( ) { }

( ) ( ) ( ) dyyfbbNbbNbbN

ji

.fn-s

dyyfbbNbbNbbN

b

jij

i

b

nssnssnn

s

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

:exampleFor

. with for

structure following thehas matrix n correlatio theifdensity normal standard univariate thedenotes and 12 where

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

4''''

7''

43''

3'

13717

ji,

ji,

''''''1

''1

'111

+

=


Is the Black Scholes model with time-dep volatility

the right model for pricing auto-callable structures?

What are the requirements for the pricing model in

order to correctly price these structures?


Structures typically include multiple digital risks.

1. Pricing model should incorporate the implied volatility skew.

The redemption time of the structures is a stopping time and depends on the path of the equity underlying.

2. Pricing model should incorporate stochastic interest rates and correlation between interest rates and underlying.

Fast calibration to standard options in the equity and interest rate world (calls, puts, caps, swaptions) is required.

3. Pricing model should allow for the derivation of closed-form solutions for standard options.

10


Why is it important to incorporate

stochastic interest rates and

correlation between interest rates

and the equity underlying

into our pricing model?

11

Euro Stoxx 50 = S0Start Date

Euro Stoxx 50 S0?Early Redemption:100% of Notional

In order to simplify matters, consider the following elementary auto-callable structure:

End of 7th year

(Maturity)

Yes

No Redemption at Maturity:100% of Notional


End of 1st year

Assume that we delta hedge the auto-callable structure using the equity underlying, a 1 year zero bond and a 7 year zero bond.

12


1. Case: Equity underlying increases early redemption becomes more likely need to buy more of the 1y zero bond and to sell parts of the 7y zero bond position

If the correlation between equity and interest rates is positive, then interest rates willalso increase on average and the bonds will decrease. However, value of the 7y zerobond, which we have to sell, will decrease more than the value of the 1y zero bond,which we have to buy. Thus, we would make on average a net loss on the interest ratedelta rebalancing of the zero bond positions.

On the other hand, if correlation between equity and interest rates is negative, theninterest rates will decrease on average and bond will increase. Therefore, we wouldmake on average a net profit on the interest rate delta rebalancing of the zero bond positions since the value of the 7y zero bond will increase more than the value of the1y zero bond.

13


2. Case: Equity underlying decreases early redemption becomes less likely need to sell parts of the 1y zero bond position and to buy more of the 7y zero bond

If the correlation between equity and interest rates is positive, then interest rates willalso decrease on average. Thus, we would make again a net loss on the interest rate delta rebalancing of the zero bond positions since the value of the 7y zero bond, whichwe have to buy, will increase more than the value of the 1y zero bond. If the correlationbetween equity and interest rates is negative, then interest rate will increase onaverage and we would make on average a net profit on the interest rate deltarebalancing of the zero bond positions.

Consequently, we expect a higher price of the auto-callable structure if a positivecorrelation between equity and interest rates is assumed and a lower price if a negative correlation is assumed.

14

3. Black-Scholes & Vasicek++ Model Definition of the Model (Merton (1973), Brigo and Mercurio (2001)) :

- Equity part: Black-Scholes with time-dependent deterministic volatility function S(t)

- Interest rate part: instantaneous, mean reverting interest rate process (Vasicek)with deterministic shift function (t)

- Correlation: Brownian motions are correlated

( )[ ] ., ),()()()()()(

)()()())(()(

, dtWWd

tdWdttRtdR

ttRtrtdWtStS(t)dtdtrtdS

RStRS

RRRR

SS

=+=

+=+=

15

3. Black-Scholes & Vasicek++ Model Closed-from Solution for Standard Equity Options (Merton (1973)):

Price of a call option with strike K and maturity T:

Note that in comparison to the standard Black-Scholes formula only the volatility changes. The total volatility now depends on the equity and interest rate volatility aswell as on the correlation between equity and interest rates.

( )

( )

R

RRS

T

RSS

R

uuTv

TvTvKTFd

dKNdNTFTdfTKC

u-T-

22

0 ,22

22,1

21

e-1T)(u,

T)du (u,T)(u,)(2)()(

)(/)(21)(ln

)()()()(),(

=

++=

==

16

3. Black-Scholes & Vasicek++ Model Closed-from Solution for Zero Bond Options (Brigo and Mercurio (2001)):

Price of a call option with strike K and maturity T1 on a zero bond with maturity T2 :

Through the formula above we can also price caps and floors as well as swaptions. For details see Brigo and Mercurio (2001).

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( )( )( ) ( ) ( ) ( )R

-

R

22

2

22

R

2-

210,01

0,02

2/1

20,0

110,0

221

R

1R

12

1

2

12

122

0

e-1,B ;4

,5.0,exp,

2e-1, ;

2,0

,0ln1

,0,0),,,0(

tTR

R

RRR

T

RRTBdtt

RTB

RTBdttRTBdttCall

TtTtBtTTtBTtA

TTBvv

eTAKe

eTAv

h

hNeTAKehNeTAeKTTV

TT

TT

T

=

+=

=

=

=

17

3. Black-Scholes & Vasicek++ Model Calibration of the Model:Benchmark instruments: equity calls and puts, interest rate caps and floors, swaptions

Should we first calibrate to equity options and then try to best fit the interest ratebenchmark instruments, or vice versa, or should we simultaneously calibrate to allbenchmark products? We propose the following calibration procedure:

1. Calibrate the interest rate process to caps, floors, swaptions and the yield curve R, R,R and shift function determined

2. Fix the correlation between interest rates and equity S,R based on time series analysis or OTC market quotes

3. Calibrate the equity process to ATM standard call options holding the interest rate parameters constant.

time-dependent deterministic volatility function S determined

18


Pros:1. The model incorporates stochastic interest rates.

2. Direct correlation between stochastic interest rates and the equity underlying.

3. Closed-form solutions for standard call and put options as well as for caps and swaptions fast model calibration possible.

4. The Vasicek++ interest rate process produces a reasonably good fit to market prices of caps and swaptions.

5. The model is able to provide a perfect fit to all ATM call options.

19


Cons:

The model produces no implied volatility skew on the equity side.

Conclusion:

Since the implied volatility skew is an important factor for the price of any

structure with digital risk, we need to look for another model.

20


( )[ ] ,, ),()()()()()()())(()(

, dtWWd

tdWtvdttvtdv

tdWtStvS(t)dtdtrtdS

vStvS

vvvv

S

=+=+=

Definition of the Heston & Vasicek++ Model:- Equity part: mean reverting stochastic volatility process (Heston model)- Interest rate part: instantaneous, mean reverting interest rate process (Vasicek)

with deterministic shift function (t)

( )[ ][ ] .0, ,0,),()()(

),()()(

==

+=+=

tRv

tRS

RRRR

WWdWWd

tdWdttRtdRttRtr

( ) ),(, tdWtS RRS+

21


( )[ ] ,, ),()()()()()()())(()(

, dtWWd

tdWtvdttvtdv

tdWtStvS(t)dtdtrtdS

vStvS

vvvv

S

=+=+=

Definition of the Heston & CIR++ Model:- Equity part: mean reverting stochastic volatility process (Heston model)- Interest rate part: instantaneous, mean reverting interest rate process (CIR)

with deterministic shift function (t)

( )[ ][ ] .0, ,0,),()()()(

),()()(

==

+=+=

tRvt

RS

RRRR

WWd

WWd

tdWtRdttRtdR

ttRtr

( ) ),()(, tdWtStR RRS+

22

Closed-form Solution for Standard Equity Options (G. (2004)):Price of standard call with strike K and maturity T:

( ) ( ) ( )( ) ( )( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )TdfSiRTEvTDTC

SiRTEvTDTC

jKi

j

dT

efef

dife

RvSTKP

PTKdfPeSRvSTKC

ln0ln0,0,,2

0ln0,0,,1

0

ln

21

222

111

with

1,2.jfor Re1210,0,0,,

where,)(00,0,0,,

+++

+++

==

=

+=

=


23

Produced (stochastic) instantaneous correlation:

Isolating the coupling factor yields:

( )

4. Heston & Vasicek++ and Heston & CIR++ Model Link between coupling factor S,R and correlation

[ ][ ] [ ]

( )( ) ( )

( ) [ ][ ] [ ] ( ) )Vasicek&(Heston tv

rdSdrS,d

)CIR&(Heston tRtv

tRrdSd

rS,d

2RS,

RS,

t t

t,

2RS,

RS,

t t

t,

+++==

+++==

t

t

RS

RS

( ) ( )( ) ( )( ) ( )

( ) )Vasicek&(Heston 1tv

)CIR&(Heston 1tR

tv

2,

,RS,

2,

,RS,

++=

++=

t

t

t

t

RS

RS

RS

RS

24

If we want to know which value for S,R is approximately needed in order to produce anaverage instant. correlation of , we can use the following crude approximation:

4. Heston & Vasicek++ and Heston & CIR++ Model Link between coupling factor S,R and correlation

( )( )

( )

T.horizon timesomefor

)Vasicek&(Heston -1

dttvT1E

)CIR&(Heston -1dttR

T1E

dttvT1E

2,

T

0,

RS,

2,

T

0

T

0,

RS,

++

++

RS

RS

RS

RS

RS,

25

With

( ) ( ) ( ) ( )( ) ( )( )( ) ( ) ( )( )( )( ) ( )( ) ( ) ( ) ( )B

Aeefdf

v

0vdttvT1

0

2/3T

0

vvTT2T

vvTT2

23

T22vT

0

vvv

TT

0

23

T

0

T

0

T

0

T

0

T0

vv

v

v

E where/121dttv

T1E

ely,AlternativT2e3e4e1

0vT2ee12T2

edttvT1Var

0vT

e1dttvT1E

dttvT1E8dttv

T1Vardttv

T1Edttv

T1E

=

==

++++=

+=


26

Example (Heston & CIR++):

We would like to have an average instant. correlation of

Our approximation suggests to use

Average instantaneous correlation after 50.000 Monte Carlo runs:

( )( ) years 4T ,06.0 ,05.0 ,1.0 ,02.00R

7.0 ,5.0 ,09.0 ,4.0 ,0625.00v vS,=====

=====RRR

vvv

.35.0=RS,.54.0RS, =

.3473.0 RS, =


27

Calibration of the models

1. Calibrate the CIR++/ Vasicek++ interest rate process to the yield curve,

caps and swaptions.

2. Estimate the correlation between interest rates and underlying using

historical time series / OTC market quotes and estimate corresponding

coupling factor S,R.

3. Calibrate the stochastic variance process to the implied volatility surface

of the equity underlying holding all interest rate parameters and the

coupling factor constant.


28

Pros:

Both models incorporate stochastic interest rates and correlation between stochastic interest rates and equity underlying.

Closed-form solutions for standard call and put options as well as for caps and swaptions fast model calibration possible.

The CIR++ and the Vasicek++ interest rate processes produce a reasonably good fit to market prices of caps and swaptions.

Both models are able to provide a good fit to the whole implied volatility surface of the equity underlying.


29

Cons:

Correlation between interest rates and underlying is modeled only indirectly and extreme positive/negative correlations cannot be produced.

The models have problems to obtain a good fit to the implied volatility surface of the equity underlying for high positive correlations.

The interest rate part is modeled only by a short rate processes.


30

Euro Stoxx 50 = S0Start Date

End of 1st year Euro Stoxx 50 S0?

Euro Stoxx 50 S0?End of 2nd year

Euro Stoxx 50 S0?End of 3rd year

No


Yes

5. Pricing Example

End of 7th year

(Maturity)

Euro Stoxx 50 S0?Yes

No

Yes

Yes


Early Redemption:113% of NotionalNo

No



. . .. . .

We consider again the auto-callable equity structure from the introduction:

31

5. Pricing Example

Model / Correlation -40% -30% -20% -10% 0% 10% 20% 30% 40%

BlackScholes & Vasicek++

96.37

-0.47

96.49

-0.35

96.61

-0.23

96.74

-0.10

98.18

-0.14

96.84 0.0

98.22

-0.10

97.10

0.26

97.25

0.41

97.38

0.54

Heston & Vasicek++ 97.90-0.42

98.00

-0.32

98.09

-0.23

98.320.0

96.97

0.13

98.48

0.16

98.68

0.36

98.92

0.60

99.08

0.76

Heston & CIR++ 97.85-0.47

97.96

-0.36

98.07

-0.25

98.45

0.13

98.320.0

98.62

0.30

98.90

0.58

99.04

0.72

After the calibration of all 3 models to market data (Feb 2006) and for different values of the correlation between equity and interest rate, we obtain the following prices and price differences:

32

Conclusion:

Ignoring the implied volatility skew or the correlation between equity and interest rates leads to significant mispricing.

Heston & CIR++ and Heston & Vasicek++ produce similar results and their correlation exposure is comparable to the correlation exposure of the

Black-Scholes & Vasicek++ model for moderate correlation values.

5. Pricing Example

33

5. Pricing Example

EUSA5-SX5E

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

27-Jan-99

9-Apr-99

22-Jun-99

1-Sep-99

10-Nov-99

20-Jan-00

30-Mar-00

13-Jun-00

22-Aug-00

31-Oct-00

12-Jan-01

23-Mar-01

6-Jun-01

15-Aug-01

24-Oct-01

8-Jan-02

19-Mar-02

31-May-02

9-Aug-02

18-Oct-02

3-Jan-03

14-Mar-03

28-May-03

6-Aug-03

15-Oct-03

29-Dec-03

9-Mar-04

20-May-04

30-Jul-04

8-Oct-04

17-Dec-04

1-Mar-05

12-May-05

21-Jul-05

29-Sep-05

8-Dec-05

17-Feb-06

c

o

r

r

,

r

e

t

u

r

n

100 d corr 100 d corr

0.9 percentile corr 0.1 percentile corr

Remaining Question: Which correlation value should be used?

34

5. Pricing Example

Historical correlation between interest rates and equity varies considerably over time.

Inflation expectations, monetary policy and business cycle have impact on the equity-interest rate correlation.

1Y historical Correlation (1990-2002)

EuroStoxx 50 and

5y EUR Swap rate

S&P 500 and

5y USD Swap rate

Maximum 66% 70%

Average -10% -7%

Minimum -62% -58%

35

5. Pricing Example

OTC Quote (February 16, 2006):

1 year correlation swap* between 20 year USD swap rate and S&P 500:

-10 / +45

* based on daily realized log returns

36

5. Pricing Example

USSW20-SPX

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

22-May-98

4-Aug-98

14-Oct-98

24-Dec-98

10-Mar-99

20-May-99

2-Aug-99

12-Oct-99

23-Dec-99

7-Mar-00

17-May-00

28-Jul-00

10-Oct-00

20-Dec-00

6-Mar-01

16-May-01

27-Jul-01

12-Oct-01

24-Dec-01

8-Mar-02

20-May-02

31-Jul-02

10-Oct-02

20-Dec-02

6-Mar-03

16-May-03

29-Jul-03

8-Oct-03

18-Dec-03

3-Mar-04

13-May-04

27-Jul-04

6-Oct-04

16-Dec-04

1-Mar-05

11-May-05

22-Jul-05

3-Oct-05

13-Dec-05

27-Feb-06

c

o

r

r

,

r

e

t

u

r

n

100 d corr 100 d corr

0.9 percentile corr 0.1 percentile corr

37

ReferencesBakshi, Cao and Chen (1997): Empirical Performance of Alternative Option Pricing Models, Journal of

Finance, 52, 2003-2049.

Brigo and Mercurio (2001), On Deterministic-Shift Extensions of Short-Rate Models, Working Paper, Banca IMI.

Chang (2006), Hybrid Products, Derivatives Solutions, Credit Suisse.

Giese (2004), Closed-Form Solutions for Standard Calls in the Heston&Vasicek++ and the Heston&CIR++ Model, Working Paper, HypoVereinsbank.

Giese, Reder and Zagst (2004), Auto Trigger Structures: Closed-form Solutions and Applications, Working Paper, HypoVereinsbank.

Kruse (2002), Closed-form Solutions to Option Pricing under the Assumption of Stochastic Interest Rates, Working Paper, ITWM.

Merton (1973), Theory of Rational Option Pricing, Bell Journal of Economics and Management Science, 4, 141-183.

Schroder (1989), A Reduction Method Applicable to Compound Option Formulas, Management Science, 35, 823-827

Scott (1997), Pricing Stock Options in a Jump-Diffusion Model with Stochastic Volatility and Interest Rates: Application of Fourier Inversion Methods, Mathematical Finance, 7, 413-426.

On the Pricing of Auto-Callable Equity Structures in the Presence of Stochastic Volatility and Stochastic Interest RatesOutline2. Requirements for the Pricing Model2. Requirements for the Pricing Model2. Requirements for the Pricing Model2. Requirements for the Pricing Model2. Requirements for the Pricing Model2. Requirements for the Pricing Model2. Requirements for the Pricing Model2. Requirements for the Pricing Model3. Black-Scholes & Vasicek++ Model3. Black-Scholes & Vasicek++ Model3. Black-Scholes & Vasicek++ Model3. Black-Scholes & Vasicek++ Model3. Black-Scholes & Vasicek++ Model3. Black-Scholes & Vasicek++ Model4. Heston & Vasicek++ and Heston & CIR++ Model 4. Heston & Vasicek++ and Heston & CIR++ Model 4. Heston & Vasicek++ and Heston & CIR++ Model Link between coupling factor S,R and correlation4. Heston & Vasicek++ and Heston & CIR++ Model Link between coupling factor S,R and correlation5. Pricing Example5. Pricing Example5. Pricing Example5. Pricing Example5. Pricing ExampleReferences

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