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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
In this model, the price process is given by
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
In this model, the price process is given by
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
Figure 12: 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 4-11
0.848
0.9 1 1.1 1.2 1.3 1.4
moneyness
510
0.2+
iv*E
-2
Figure 13: 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 4-12
1.648
0.9 1 1.1 1.2 1.3 1.4moneyness
24
68
10
0.2+
iv*E
-2
Figure 14: 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 4-13
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
cumulative hedging error
010
2030
40
Merton
0 0.1 0.2
cumulative hedging error
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
cumulative hedging error*E-2
05
1015
Merton
-15 -10 -5 0
cumulative hedging error*E-2
05
1015
20
Figure 25: Hedging results for 2y dop.
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hedging 6-4
Bates
-5 0
cumulative hedging error*E-2
010
2030
Heston
-5 0
cumulative hedging error*E-2
05
1015
2025
30
Merton
-6 -4 -2 0 2
cumulative hedging error*E-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
cumulative hedging error
05
10
Heston
-0.1 0 0.1
cumulative hedging error
05
10
Merton
-15 -10 -5 0 5 10 15
cumulative hedging error*E-2
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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