hedging exotic options - hu-berlin.de fileintroduction 1-4 schoutens (2004) schoutens et al. studied...

Hedging Exotic Options

Kai Detlefsen

Wolfgang Hardle

Center forApplied Statistics and EconomicsHumboldt-Universitat zu BerlinGermany

introduction 1-1

Models

The Black Scholes model has some shortcomings:

- volatility is not constant

- returns are not normally distributed

Hence, alternative models have been considered.

Strengths and weaknesses of different models?


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introduction 1-2

Bakshi (1997)

Bakshi et al. compared stochastic volatility models with jumps andstochastic interest rates by

- in-sample fit

- stability of parameters

- hedging performance of European options


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introduction 1-3

Aims

- We repeat Bakshi’s analysis for a European data set from01/2000 to 06/2004.

- We consider exotic options for hedging.

Thus, we extend the analysis of Bakshi to exotic options andrepeat it with recent DAX data.


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introduction 1-4

Schoutens (2004)

Schoutens et al. studied modern option pricing models that

- all led to good in-sample fits

- but had different prices for exotic options

Additional aim:Does our study also lead to different prices for exoticoptions?


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introduction 1-5

Overview

1. introductionX

2. data

3. calibration

4. Monte Carlo simulation

5. hedging

6. conclusion


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data 2-1

Components of data set

Our data is a time series from January 2000 to June 2004 thatcontains for each trading day

- an implied volatility surface of settlement prices

- the value of the DAX

- the interest rate curve (EURIBOR).


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data 2-2

Components of data set

Our data is a time series from January 2000 to June 2004 thatcontains for each trading day

- an implied volatility surface of settlement prices

- the value of the DAX

- the interest rate curve (EURIBOR).


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data 2-3

Number of observations

moneyness summaturity 0.5− 0.9 0.9− 1.1 1.1− 1.5

1.0− 5.0 24476 18383 21353 642120.25− 1.0 37670 41047 38832 1175490.04− 0.25 31783 47574 29677 109034

sum 93929 107004 89862 290795


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data 2-4

2000 2001 2002 2003 2004

years

510

1520

2530

0.15

+iv

*E-2

Figure 1: Time series of mean implied volatilities for long maturities.(blue: in the money, green: at the money, red: out of the money)


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data 2-5

2000 2001 2002 2003 2004

years

0.2

0.3

0.4

0.5

iv

Figure 2: Time series of mean implied volatilities for mean maturities.(blue: in the money, green: at the money, red: out of the money)


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data 2-6

2000 2001 2002 2003 2004

years

0.2

0.3

0.4

0.5

0.6

0.7

iv

Figure 3: Time series of mean implied volatilities for short maturities.(blue: in the money, green: at the money, red: out of the money)


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data 2-7

2000 2001 2002 2003 2004

years

12

34

56

2000

+D

AX

*E3

Figure 4: DAX.


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data 2-8

0 1 2 3 4

years

510

1520

2530

35

0.01

5+ir

*E-3

Figure 5: Interest rates for maturity 1 year.


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data 2-9

Preprocessing

In order to delete arbitrage opportunities in the data we have used

- a method by Hafner, Wallmeier to correct tax effects

- a method by Fengler to smooth the whole ivs.


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models 3-1

The option pricing models

We consider

- the Merton model (jump diffusion/exponential Levy model)

- the Heston model (stochastic volatility model)

- the Bates model (stochastic volatility model with jumps)

The Bates model is the combination of the Merton and the Hestonmodel.


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models 3-2

The Merton model

In this model, the price process is given by

St = s0 exp(µt + σWt +Nt∑i=1

Yi ).

W is a Wiener process, N a Poisson process with intensity λ andthe jumps Yi are N(m, δ2) distributed.µ is the drift and σ the volatility.


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models 3-3

0.88

0.96 1.04

1.12 1.20 0.06

0.46 0.86

1.26 1.66

0.14

0.22

0.30

0.37

0.45

Figure 6: Implied volatility surface of the Merton model for µM =0.046, σ = 0.15, λ = 0.5, δ = 0.2 and m = −0.243.(Left axis: time to maturity, right axis: moneyness)


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models 3-4

The Heston model


dSt

St= µdt +

√VtdW

(1)t

and the volatility process is modelled by a square-root process:

dVt = ξ(η − Vt)dt + θ√

VtdW(2)t ,

where the Wiener processes W (1) and W (2) have correlation ρ.


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models 3-5

The Heston model II

The other parameters in the Heston model have the followingmeaning:

- µ drift of stock price

- ξ mean reversion speed of volatility

- η average volatility

- θ volatility of volatility

The volatility process stays positive if ξη > θ2

2 .


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models 3-6

0.88

0.96 1.04

1.12 1.20 0.06

0.46 0.86

1.26 1.66

0.23

0.26

0.29

0.32

0.36

Figure 7: Implied volatility surface of the Heston model for ξ =1.0, η = 0.15, ρ = −0.5, θ = 0.5 and v0 = 0.1.(Left axis: time to maturity, right axis: moneyness)


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models 3-7

The Bates model


dSt

St= µdt +

√VtdW

(1)t + dZt

dVt = ξ(η − Vt)dt + θ√

VtdW(2)t

where W (1) and W (2) are Wiener processes with correlation ρ andZ is a compound Poisson process with intensity λ and independentjumps J with ln(1 + J) ∼ N{ln(1 + k)− 1

2δ2, δ2}.

The meaning of the parameters is similar to the interpretations inthe Merton and the Heston model.


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models 3-8

0.88

0.96 1.04

1.12 1.20 0.06

0.46 0.86

1.26 1.66

0.28

0.32

0.37

0.42

0.46

Figure 8: Implied volatility surface of the Bates model for λ =0.5, δ = 0.2, k = −0.1, ξ = 1.0, η = 0.15, ρ = −0.5, θ = 0.5and v0 = 0.1. (Left axis: time to maturity, right axis: moneyness)


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calibration 4-1

FFT

For the calibration it is essential to have a fast algorithm forcalculating the prices/implied volatilities of plain vanilla options.

We have used the FFT based method by Carr and Madan whichuses the characteristic functions of the log price processes.


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calibration 4-2

Error functional

As measure for the errors we have used the squared distancebetween the observed iv σobs and the model iv σmod .

In order to give the at the money observations with long maturitiesmore importance we used vega weights V .

In order to make the errors on different days comparable weincluded additional weights.


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calibration 4-3

Error functional II

error(p)def=

∑τ

∑K

1

nτnS(τ)V (K , τ){σmod(K , τ, p)− σobs(K , τ)}2

where p is a vector of model parameters, nτ is the number of timesto maturity of the observed surface and nS(τ) is the number ofstrikes with time to maturity τ .


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calibration 4-4

Minimization algorithms

The calibration problem can be stated as

minp

error(p)

where the minimum is taken over all possible parameter vectors p.

For this optimization, we have considered

- Broyden-Flechter-Goldfarb-Shanno algorithm

- simulated annealing algorithm.


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calibration 4-5

Minimization algorithms II

These algorithms have been tested with fixed starting values,moving starting values and the problem has been regularized.

Simulated annealing with moving starting values withoutregularization seems to give the best results with respect tocomputation time and fit.


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calibration 4-6

Results: errors

The Bates model gives the smallest errors (median 0.7).

The errors in the Heston model are similar (median 1.0).

The Merton model performs worse than the other two models(median 3.9).


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calibration 4-7

2000 2001 2002 2003 2004

years

05

1015

2025

30

squa

red

erro

r

Figure 9: Error functional after calibration in the Bates model (blue),the Heston model (green) and the Merton model (red).


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calibration 4-8

2000 2001 2002 2003 2004

years

05

10

squa

red

erro

r

Figure 10: Error functional after calibration in the Bates model (blue)and the Heston model (green).Hedging Exotic Options

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calibration 4-9

0.068

0.6 0.8 1 1.2 1.4

moneyness

0.2

0.4

0.6

0.8

1

iv

Figure 11: Original iv (black) and calibrated iv in the Bates model(blue), the Heston model (green) and the Merton model (red).


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calibration 4-10

0.248

5 10 15 20 25

0.85+moneyness*E-2

510

0.2+

iv*E

-2



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calibration 4-11

0.848

0.9 1 1.1 1.2 1.3 1.4

moneyness

510

0.2+

iv*E

-2



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calibration 4-12

1.648

0.9 1 1.1 1.2 1.3 1.4moneyness

24

68

10

0.2+

iv*E

-2



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calibration 4-13

0.068

1 2 3 4 5 6 7

3000+moneyness*E3

05

1015

2025

30

pric

e*E

2

Figure 15: Original prices (black) and prices from iv calibration inthe Merton model (red).


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calibration 4-14

1.648

1 2 3 4 5

5000+moneyness*E3

510

15

pric

e*E

2

Figure 16: Original prices (black) and prices from iv calibration inthe Merton model (red).


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calibration 4-15

Results: parameters

But the parameters in the Bates model are unstable.

The parameters in the other two models are stable.


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options 5-1

Options

We have considered the following six types of barrier options:

- d&o put with maturity 1 year, 80% barrier and 110% strike

- d&o put with maturity 2 years, 60% barrier and 120% strike

- u&o call with maturity 1 year, 120% barrier and 90% strike

- u&o call with maturity 2 years, 140% barrier and 80% strike

- forward start (1 year) d&o put with maturity 1 year, 80%barrier and 110% strike

- forward start (1 year) u&o call with maturity 1 year, 120%barrier and 90% strike


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options 5-2

Monte Carlo simulation

We have computed the prices and greeks of these options byMonte Carlo simulations.

We have found that butterfly spreads give a good variancereduction as control variates.


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options 5-3

0.5 1 1.5 2

time to maturity

00.

20.

40.

60.

8

corr

elat

ion

Figure 17: Correlation of the 1 year d&o put barrier and control vari-ates: Black Scholes barrier (black), underlying (blue), European put(green), butterfly spread (red) and option with final barrier payoff(cyan).


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options 5-4

Prices

The prices of the puts differ across the models.

The prices of the calls on the other hand are similar for all models.

Hence, Schoutens’ results can be confirmed partly.But we conclude more precisely that there are also classes withoutsignificant price differences.


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options 5-5

1 year down-and-out put

2000 2001 2002 2003 2004years

12

34

5

pric

e pe

r no

tiona

l*E

-2

Figure 18: Prices of 1y dop in the Bates model (blue), the Hestonmodel (green) and the Merton model (red).


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options 5-6

2 years down-and-out put

2000 2001 2002 2003 2004years

24

68

10

0.04

+pr

ice

per

notio

nal*

E-2

Figure 19: Prices of 2y dop in the Bates model (blue), the Hestonmodel (green) and the Merton model (red).


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options 5-7

1 year up-and-out call

2000 2001 2002 2003 2004

years

24

6

pric

e pe

r no

tiona

l*E

-2

Figure 20: Prices of 1y uoc in the Bates model (blue), the Hestonmodel (green) and the Merton model (red).


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options 5-8

2 years up-and-out call

2000 2001 2002 2003 2004years

510

15

pric

e pe

r no

tiona

l*E

-2

Figure 21: Prices of 2y uoc in the Bates model (blue), the Hestonmodel (green) and the Merton model (red).


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options 5-9

forward start down-and-out put

2000 2001 2002 2003 2004years

12

34

0.01

+pr

ice

per

notio

nal*

E-2

Figure 22: Prices of fs dop in the Bates model (blue), the Hestonmodel (green) and the Merton model (red).


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options 5-10

forward start up-and-out call

2000 2001 2002 2003 2004

years

24

68

pric

e pe

r no

tiona

l*E

-2

Figure 23: Prices of fs uoc in the Bates model (blue), the Hestonmodel (green) and the Merton model (red).


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hedging 6-1

Hedging

We have considered three hedging methods:

- delta hedging

- vega hedging

- delta hedging with minimum variance


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hedging 6-2

Bates

0 0.1 0.2

cumulative hedging error

010

2030

40

Heston

0 0.1 0.2


010

2030

40

Merton

0 0.1 0.2


010

2030

40

Figure 24: Hedging results for 1y dop.


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hedging 6-3

Bates

-15 -10 -5 0

cumulative hedging error*E-2

05

10

Heston

-15 -10 -5 0


05

1015

Merton

-15 -10 -5 0


05

1015

20

Figure 25: Hedging results for 2y dop.


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hedging 6-4

Bates

-5 0


010

2030

Heston

-5 0


05

1015

2025

30

Merton

-6 -4 -2 0 2


05

1015

2025

30

Figure 26: Hedging results for 1y uoc.


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hedging 6-5

Bates

-0.1 0 0.1


05

10

Heston

-0.1 0 0.1


05

10

Merton

-15 -10 -5 0 5 10 15


05

1015

20

Figure 27: Hedging results for 2y uoc.


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conclusions 7-1

Conclusions

Bakshi: We have concluded that the Heston model gives the bestcalibration results with respect to fit and stability of parameters.Moreover, hedging in the Heston model does not perform worsethan in the other models. These findings correspond to Bakshi’sresults.


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conclusions 7-2

Conclusions II

Schoutens: We have found in our study that the prices of someexotic options differ among various models although the modelsare all calibrated to the same plain vanilla ivs. But we have alsoseen examples where these price differences are only small. Hence,we can not support Schoutens’ results fully. It seems that there areclasses with price differences and other classes without.


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bibliography 8-1

Reference

Bakshi, G., Cao, C. and Chen, Z.Empirical Performance of Alternative Pricing ModelsThe Journal of Finance, 1997, 5: 2003–2049.

Schoutens, W., Simons, E. and Tistaert, J.A Perfect Calibration!Wilmott magazine.


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hedging exotic options - hu-berlin.de fileintroduction 1-4 schoutens (2004) schoutens et al. studied...

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