calibration risk for exotic optionsweb.math.ku.dk/~rolf/siven/jod_su_07_detlefsen.pdf · 2008. 2....

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SUMMER 2007 THE JOURNAL OF DERIVATIVES 47

Equity derivative pricing models are calibrated tomarket data of plain vanilla options by minimiza-tion of an error functional. From the economic view-point, there are several possibilities to measure theerror between the market and the model. These dif-ferent specifications of the error give rise to differentsets of calibrated model parameters and the resultingprices of exotic options vary significantly.

We provide evidence for this calibration riskin a time series of DAX implied volatility surfacesfrom April 2003 to March 2004. We analyze fac-tors influencing these price differences for exotic optionsin the Heston and in the Bates models and recom-mend an error functional. Moreover, we determinethe model risk of these two stochastic volatility modelsfor the time series and compare it to calibration risk.

Recently, there has been consider-able interest, both from a prac-tical and a theoretical point ofview, in the risks involved in

option pricing. Schoutens et al., [2004] haveanalyzed model risk in an empirical study andCont [2004] has put this risk into a theoret-ical framework. Another source of risk ishidden in the calibration of models to marketdata. This calibration risk is also fundamentalfor the banking industry because it significantlyinfluences the prices of exotic options. More-over, calibration risk exists even if an appro-priate model has been chosen and model riskdoes not exist anymore.

Calibration risk arises from the differentpossibilities to measure the error between theobservations on the market and the corre-sponding quantities in the model world. A nat-ural approach to specify this error is to considerthe absolute price (AP) differences. (See e.g.,Schoutens et al., [2004]). But the importance ofabsolute price differences depends on the mag-nitude of these prices. Hence, another usefulway for measuring the error is relative price(RP) differences. (See e.g., Mikhailov et al.,[2003]). As models are often judged by theircapability to reproduce implied volatility sur-faces, other measures can be defined in termsof implied volatilities. There are again the twopossibilities of absolute implied volatilities (AI)and relative implied volatilities (RI). We con-sider these four ways to measure the differ-ence between model and market data andexplore the implications for the pricing ofexotic options.

To this end, we focus on the stochasticvolatility model of Heston. In order to analyzethe influence of the goodness of fit on calibra-tion risk, we consider in addition the Batesmodel, which is an extension of the Hestonmodel with similar qualitative features. Thesetwo models are calibrated to the prices of plainvanilla options on the DAX. We use a time seriesof implied volatility surfaces from April 2003 toMarch 2004. As exotic options we considerdown and out puts, up and out calls, and cliquetoptions for one, two, or three years to maturity.

Calibration Risk for Exotic OptionsK. DETLEFSEN AND WOLFGANG K. HÄRDLE

K. DETLEFSEN

is a PhD candidate infinance at the Center forApplied Statistics and Eco-nomics, (CASE), Hum-boldt-Universität zu Berlinin Berlin, [email protected]

WOLFGANG

K. HÄRDLE

is professor of statistics anddirector of CASE at Humboldt-Universität zuBerlin in Berlin, [email protected]

Copyright © 2007

In this framework we determine the size of calibration riskand analyze factors influencing it.

Besides calibration risk, there is also model risk, whichrepresents wrong prices because a wrong parametric modelhas been chosen. We consider the model risk between theHeston and the Bates models and analyze the relationbetween the two forms of risk in pricing exotic options.

The next section introduces the models anddescribes their risk neutral dynamics that we use for optionpricing. Moreover, this section contains information aboutthe data used for the calibration. The following sectiondescribes the calibration method and defines the errorfunctionals analyzed in this work. The goodness of fit isshown by representative surfaces and statistics on the errors.We then present the exotic options that we consider forcalibration risk and price these products by simulation.In our next section we analyze the model risk for the twostochastic volatility models under the four error func-tionals. In the last section, we summarize the results anddraw our conclusions.

MODELS AND DATA

In this section, we describe briefly the Heston modeland the Bates model for which we are going to analyzecalibration risk. Moreover, we provide some descriptivestatistics of the implied volatility surfaces that we use asinput data for the calibration.

Heston Model

We consider the popular stochastic volatility modelof Heston [1993]:

(1)

where the volatility process is modelled by a square-rootprocess:

(2)

and W 1 and W 2 are Wiener processes with correlation ρ.The variance process (Vt) remains positive if its volatility

θ is small enough with respect to the product of the meanreversion speed ξ and the average variance level η:

dV V dt V dWt t t t= − +ξ η θ( ) 2

dS

Sdt V dWt

tt t= +µ 1

(3)

The dynamics of the price process are analyzed undera martingale measure under which the characteristic func-tion of (St) is given by:

(4)

where , see e.g., contet al. [2004].

Bates Model

Bates [1996] extended the Heston model by con-sidering jumps in the stock price process:

(5)

where Z is a compound Poisson process with intensity λand jumps k that have a lognormal distribution:

(6)

We analyze the dynamics of this model under a mar-tingale measure under which the characteristic functionof (St) is given by:

(7)

φ λ δ δtB z k zz t e( ) exp{ ( ){log( ) }= −− / + + −2 2 1

222 1 1i }}

exp( )

( )coth ( )×

− ++ −

z z V

z zz t

20

2

i

iγ ξ ρθγ

×+ − +−exp{ ( ) log( )}( )ξη ξ ρθ

θt z zt r k z Si i i2 0

((cosh sinh )( )( )

( )γ ξ ρθγ

γξη

θz t zz

z t2 2

22+ −i

log( ) ~ log( ) – ,1 12

22+ +

k kN

δδ

dS

Sdt V dW dZ

dV V dt V dW

t

tt t t

t t t t

= + +

= − +

µ

ξ η θ

1

2( )

γ θ ξ ρθ( ) ( ) ( )z z z z= + + −def 2 2 2i i

φγ ξ ρθγt

Hz t

zz z V

z z( ) exp

( )

( )coth ( )=

− ++ −

20

2

i

i

×+ +−exp{ log( )}( )ξη ξ ρθ

θt z ztr z Si i i2 0

((cosh sinh )( )( )

( )γ ξ ρθγ

γξη

θz t zz

z t2 2

22+ −i

ξη θ>

2

2

48 CALIBRATION RISK FOR EXOTIC OPTIONS SUMMER 2007

Copyright © 2007

where . (See e.g., Contet al. [2004].)

The Bates model has eight parameters while theHeston model has only five parameters. Because of thesethree additional parameters, the Bates model can betterfit an observed surface but parameter stability is more dif-ficult to achieve.

Data

Our data consists of EUREX-settlement impliedvolatilities on the DAX. Hence, these volatilities are givenby inversion of the Black-Scholes formula, i.e., if youplug these volatilities into the Black-Scholes formulatogether with the other corresponding parameters thenyou get the settlement prices of European options on theDAX. In this context, we approximate the risk-free interestrates by the EURIBOR. On each trading day we use theyields corresponding to the maturities of the impliedvolatility surface. As the DAX is a performance index, itis adjusted to dividend payments. Thus, we do not con-sider dividend payments explicitly.

We analyze the time period from April 2003 toMarch 2004. Since March 2003 the EUREX trades plainvanilla options with maturities up to five years. Until March2004, it has not changed its range of products. Hence, thedata is homogeneous in the sense that the implied volatilitysurfaces are derived from similar products.

From this time period, we analyze the surfaces fromall the Wednesdays when trading has taken place. We restrictourselves to these days because of the computationally

γ θ ξ ρθ( ) ( ) ( )z z z z= + + −def 2 2 2i i intense Monte Carlo simulations for the pricing. Thus, we

consider 51 implied volatility surfaces. We exclude obser-vations that are deep out of the money because of illiq-uidity of these products. More precisely, we consider onlyoptions with moneyness K/S0 ∈ [0.75, 1.35] for small timesto maturity T ≤ 1. As we analyze exotic options that expirein one, two, or three years we exclude also plain vanillaswith time to maturity less than three months.

Some information about the resulting impliedvolatility surfaces is summarized in Exhibit 1. The surfacescontain on average 140 prices and nine times to maturitywith a mean moneyness range of 65%.

The values of the underlying in the sample periodare shown in Exhibit 2. This exhibit also shows the (inter-polated) at the money implied volatilities for one year tomaturity. The market value of the DAX went up in thisperiod and accordingly the implied volatilities went downas Exhibit 2 shows.

CALIBRATION

In this section, we specify the calibration routineand describe the four error functionals. The calibrationresults illustrate how well the plain vanilla prices can bereplicated by the Heston and the Bates models.

Calibration Method

Carr and Madan [1999] found a representation ofthe price of a European call option by one integral for awhole class of option pricing models. Their method, that


E X H I B I T 1

Note: Description of the implied volatility surfaces.

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is applicable to the Heston and the Bates model, is basedon the characteristic function of the log stock price underthe risk neutral measure.

Carr and Madan showed that the price C(K,T) ofa European call option with strike K and maturity T isgiven by

(8)

for a (suitable) damping factor α > 0. The function ψTis given by

(9)

where φT is the characteristic function of log (ST); see theprior section on Models and Data.

For the difference between market and model, weconsider the following four objective functions based onthe root weighted square error:

(10)

AP

RP

def

def

= −( )

=

=

=

∑

∑

i

n

i imod

imar

i

n

iim

w P P

wP

1

2

1

oodimar

imar

i

n

i imod

P

P

w IV IV

−

= −=∑

2

1

AIdef

iimar

i

n

iimod

imar

imar

wIV IV

IV

( )

=−

=∑

2

1

RIdef

2

ψφ α

α α αTTv

rT v

v v( )

exp( ) { ( ) }

( )=

− − ++ − + +

1

2 12 2

i

i

C K TK

v K v dvT

( )exp{ ln( )}

exp{ ln( )} ( )

, =−

−+ ∞

απ

ψi0∫∫

where mod refers to a model quantity and mar to a quan-tity observed on the market, P to a price, and IV to animplied volatility. The index i runs over all nt observationsof the surface on day t. The weights wi are non negativewith Σiwi = 1. Hence, the objective functions can be inter-preted as mean average errors. While the error functionalsAP and RP measure the differences between option prices,the other two error measures focus on Black-Scholesimplied volatilities because these quantities are normallyused in reality for price quotations. The model impliedvolatilities IVmod are computed from the prices of Euro-pean options in the Heston and in the Bates models bynumerical inversion of the Black-Scholes formula.

We choose the weights in such a way that on eachday all maturities have the same influence on the objectivefunction. In order to make different surfaces comparable,each maturity gets the weight 1/nmat where nmat denotesthe number of maturities in this surface. Moreover, weassign the same weight to all points of the same maturity.This leads to the weights

(11)

where nistr denotes the number of strikes with the same

maturity as observation i. This weighting leads asymp-totically to a uniform density on each maturity.

Given these weights, the average time to maturityof an implied volatility surface can be measured by a mod-ified duration:

(12)i

ni i

in

i

w

w= =∑

1 1

τΣ

wn ni

mat stri

=def 1


E X H I B I T 2

Note: DAX and ATM implied volatility with 1 year to maturity on the trading days from 01 April 2003 to 31 March 2004.

Copyright © 2007

where τi is the time to maturity of the option i. The meanduration of the 51 surfaces is 2.02 and the minimal (max-imal) is 1.70 (2.30). Thus, the point of balance for thematurities lies around 2 for our time series of surfaces. Aswe analyze exotic options with one, two or three years tomaturity, this point of balance confirms a correct weightingfor our purposes.

We consider only out-of-the-money prices. Thus,we use call prices for strikes higher than the spot and putprices for strikes below the spot. This approach ensuresthat we compare only prices of similar magnitude. It hasno impact on the errors based on implied volatilities.Because of put-call parity, the use of OTM options has noimpact on the absolute price error (AP). But the relativeprices are weighted in such a way that the observationsaround the spot receive less weight. Hence, only the rela-tive price error (RP) is influenced by this choice of prices.

In order to estimate the model parameters, we applya stochastic global optimization routine and minimize theobjective functions with respect to the model parame-ters. In addition to some natural constraints on the rangeof the parameters, we have taken into account inequality (3)that ensures the positivity of the variance process.

Calibration Results

We consider 51 implied volatility surfaces betweenApril 2003 and March 2004. Each of these is calibratedwith respect to the four error functions described in ourprior sub-section. These calibrations are done for theHeston and the Bates models.

The resulting errors of these 408 calibrations havebeen summarized in Exhibit 3 for the Heston model andin Exhibit 4 for the Bates model. Descriptive statistics onthe calibrated parameters are given in Exhibit 5 for theHeston model and in Exhibit 6 for the Bates model.Exhibit 7 shows the fit of the implied volatility surface inthe Heston model on a day that is representative for theAI error.

Exhibit 3 reports in each line the means of the fourerrors when the objective function given in the leftcolumn is minimized. In the Heston model, we get amean absolute price error of 7.3 and a mean relative priceerror of 9.7% when we calibrate with respect to AP. Usingthe RP error functional we get the opposite result witha mean absolute price error of 11 and a mean relativeprice error of 6.1%. The errors based on implied volatil-ities are smaller for the RP objective function than for

the AP objective function. The results for the AI and RIobjective functionals differ only slightly: the mean absoluteimplied volatility error is about 0.68% and the mean rel-ative implied volatility error is about 2.5%. Moreover, theprice errors for these objective functions lie between theprice errors of the other two objective functions. Thecalibration with respect to RI gives the best overall fitbecause it has the smallest RI error and the second besterrors for the rest. The meaning of these numbers is illus-trated by Exhibit 7, which shows an implied volatility fitthat is representative for an AI error of 0.68%. In orderto make the AP errors comparable for different days (withdifferent values of the spot) we have computed the meanof AP/DAX as 0.21%, 0.34%, 0.27%, 0.25% for the fourerror functionals.

The calibrated parameters which are described byExhibit 5 form two groups because the parameters forthe RP, AI, and RI calibration are quite similar. The ini-tial variance V0 and the average variance level η are bothabout 0.07 for all objective functionals. For the AP cali-bration we get a reversion speed ξ = 0.9, a volatility ofvariance of θ = 0.34, and a correlation ρ = –0.82. Theother calibrations lead to similar parameters with a rever-sion speed ξ = 1.3, a volatility of volatility of θ = 0.44,and a correlation ρ = –0.75. As the correlations are sig-


E X H I B I T 3

Note: Mean calibration errors in the Heston model for 51 days. (AP = absolute price differences, RP = relative price differences, AI = absoluteimplied volatility differences, RI = relative implied volatility differences)

Copyright © 2007

nificantly greater than –1, the calibrated Heston modelshave really two stochastic factors.

The Bates model exhibits similar qualitative resultsas the Heston model: the AP and the RP calibrationsdiffer clearly while the AI and the RI calibrations lead tosimilar results. The Bates model can be regarded as anextension of the Heston model. The additional three para-meters for the jumps in the spot process lead to bettercalibration results for all error functionals: the AP erroris reduced (on average) by 4%, the RP error by 16%, theAI and the RI error both by 12%.

The calibrated parameters of the Bates model aregiven in Exhibit 6. As in the Heston model, they formtwo groups with the AP calibration on the one hand andthe RP, AI, and RI calibrations on the other hand. Theparameters ξ , η, θ, and V0 are similar to the calibrationsfor the Heston model. Only the correlation ρ rises to alevel of –0.93 for all objective functions. Hence, this cri-terion for distinguishing between the two groups disap-pears. It is replaced by the expected number of jumps peryear: for the AP calibration we expect (on average) a jumpevery three years while we expect a jump every two yearsfor the other calibrations. It is interesting that all calibra-tions lead to a mean jump up of about +8% for the returns.The expected jumps upwards correspond to the marketgoing up as shown in Exhibit 2.

Schoutens et al. [2004] found that the Heston andthe Bates option models can both be calibrated well to theEuroStoxx50. In summarizing the results of this section,we can say that DAX implied volatility surfaces can bereplicated by these models for different error functionalsas well as the EuroStoxx50 data in Schoutens et al. [2004].As in that work, we find that the Bates model gives onlyslightly better fits for the AP calibration. In addition wehave shown that it leads to a considerable improvementin the fit for the other objective functions.

EXOTIC OPTIONS

In this section, we analyze the price differences ofexotic options for calibrations with respect to differenterror measures. We consider barrier and cliquet options.The prices of these products are calculated by MonteCarlo simulations using Euler discretizations.

Simulation

We price all exotic options by Monte Carlo simu-lations. To this end, we use for each derivate one million


E X H I B I T 4

Note: Mean calibration errors in the Bates model for 51 days. (AP =absolute price differences, RP = relative price differences, AI = absoluteimplied volatility differences, RI = relative implied volatility differences)

E X H I B I T 5

Note: Mean parameters (std. dev.) in the Heston model for 51 days.

Copyright © 2007

paths generated by Euler discretization. (See e.g.,Glasserman [2004]). As we take into account the posi-tivity constraint (3) the square root process for the vari-ance can be simulated by truncation at zero

where ∆t is the time step and Zi are independent standardnormal variables. This simple scheme leads to an accept-able small bias when the positivity constraint is fulfilled.For each exotic option we consider three maturities: 1year, 2 years, and 3 years. We analyze three exotic options:up and out calls, down and out puts, and cliquet options.These products are described in the following sectionswhere remaining parameters are also specified.

The payoffs of barrier options depend only onwhether the underlying price process exceeded the barrierin some time interval. Hence, the value of barrier optionsdepends on the minimum or maximum of the underlyingprice process. We approximate such continuous extremaby discrete extrema using one observation for each tradingday. We use 252 time steps to simulate a process for a yearassuming 252 trading days a year.

V V V V Zt t t t t t ii i i i+= + − ∆ + ∆

+

1ξ η θ( )

The calibration results are presented in the followingsections together with a discussion of the options. Theaccuracy of the Monte Carlo results is given by the rela-tive standard error in Exhibit 8. This exhibit confirmsthat the estimators have sufficiently small variance after onemillion paths compared to the price differences that weobserve in Exhibits 10, 13, and 16.

Barrier Options

We consider two types of barrier options: up and outcalls and down and out puts. These options are quite pop-ular on the market. Down and out puts are, for example,sold together with zero-strike calls as “bonus certificates.”These structured products are actively traded in Germanyand also in many other markets.

Up and out call options The prices of up and out callswith strike K, barrier B, and maturity T on an underlying(St) are given by

exp(–rT ) E[1{MT<B}(ST–K )+] (13)

where


E X H I B I T 6

Note: Mean parameters (std. dev.) in the Bates model for 51 days.

Copyright © 2007


E X H I B I T 7

Note: Implied volatilities in the market and in the Heston model for the maturities 0.26, 0.52, 0.78, 1.04, 1.56, 2.08, 2.60, 3.12, 3.64, 4.70 (left toright, top to bottom) for AI parameters on 25/06/2003. Solid: model, dotted: market. X-axis: moneyness.

Copyright © 2007

(14)

We choose as strike K and barrier B

K = (1 – 0.1T )S0 (15)

B = (1 + 0.2T )S0 (16)

where T denotes time to maturity. Up and out calls withsuch strikes and barriers are widely traded on the Germanmarket.

Up and out call options have the payoff profile ofEuropean call options if the underlying has not exceededthe barrier. Otherwise their payoff is zero. Thus, up andout calls are path dependent exotic options.

We want to analyze the difference between the pricesof the exotic options when the underlying model hasbeen calibrated with respect to different error measures.To this end, we have calibrated the Heston and the Batesmodels to implied volatility or price data on each daywith respect to the four error functionals introduced inthe prior sub-section on Calibration Method. Hence, wehave four time series of calibrated models parameters thatare described in the sub-section on Calibration Results.By Monte Carlo simulations we calculate on each day theprices of up and out calls for the four sets of model para-

M ST t T t=≤ ≤

defmax0

meters. In this way, we get four time series of up and outcall prices corresponding to the four error measures forthe Heston model and four corresponding time series ofprices for the Bates model. We are interested in how theprices of exotic options differ when the four differenterror functionals are used. If the prices of up and out callsthat are computed from the AP parameters are denotedby PAP

UOC and the corresponding prices from the RP para-meters by PAP

UOC then we measure the difference betweenthese prices by the ratio PAP

UOC/PAPUOC. The other five price

differences are measured by corresponding price ratios.Hence, we observe on each day six price ratios thatdescribe the differences of the prices of exotic optionsresulting from different error measures.

The six time series of price ratios are summarizedin Exhibit 9 for up and out calls with three years to matu-rity in the Heston model. In the boxplots the central linegives the median and the box contains 50% of the obser-vations. Hence, the AP prices lie on average about 6%over the other prices and the AP prices are on 75% ofthe 51 days at least 4% higher than the other prices. TheRP prices are about 2% below the AI or RI prices, whichare very similar to each other.

We analyze the influence of time to maturity on theseprice ratios by also considering one year and two years tomaturity (and by adjusting the barrier and the strike appro-priately). The medians of the price ratios are presented inExhibit 10 for all three times to maturity. This exhibit


E X H I B I T 8

Note: Maximal ratio of standard error and price in Monte Carlo simulations. (Maximum over all time points and all objective functions.) Up and out calls withstrike 1 – 0.1T and barrier 1 + 0.2T relative to initial spot, down and out puts with strike 1 + 0.1T and barrier 1 – 0.2T relative to initial spot, cliquetswith reset times ti = Ti/3(i = 0, ..., 3) and local caps of 8% and local floors of –8%. (T denotes time to maturity).

Copyright © 2007

shows that the price differences become smaller for shortertimes to maturity for the AP prices. The other price ratiosremain almost constant. For one year to maturity, the pricedifferences are about 2% – 3% and the AP prices are lowerthan the other prices. For two years to maturity, the APprices are again higher than the other prices.

In order to analyze the influence of the goodness offit on the price ratios, we consider also the Bates model.The boxplots of the price ratios in this model are givenin Exhibit 11 for three years to maturity. Compared to theHeston boxplots, the boxes are longer in the Bates model.Thus there is more variation between the prices for dif-ferent error functionals. Moreover, the median ratiosbetween the AP prices and the other prices are biggerthan in the Heston model—especially for AP/AI andAP/RI. The ratios between RP, AI, and RI are similarto those in the Heston model. The corresponding resultsfor one year and two years to maturity are presented inExhibit 10. Qualitatively the situation is similar to theHeston model: For shorter times to maturity the pricedifferences decrease—especially for AP prices.

Thus, the price ratios in the Heston and in the Batesmodel are similar among the RP, AI, and RI prices whilethe AP price differences are bigger in the Bates model.Moreover, the variation of the price ratios is higher inthe Bates model.

Down and out p put options The prices of the downand out puts with strike K, barrier B, and maturity T onan underlying (St) are given by

exp(–rT ) E[1{mT>B}(K – ST)+] (17)

where

(18)

For our analysis, we use the strike K and the barrier B

K = (1 + 0.1T)S0 (19)

B = (1 – 0.2T)S0 (20)

where T denotes time to maturity. The strikes and barriersare set analogously to those for the up and out calls. Suchdown and out puts are often part of bonus certificates.1

Down and out put options have the payoff profileof European put options if the underlying has been abovethe barrier during the lifetime of the option. Otherwisetheir payoff is zero.

As described above, we calculate on each day theprices of the down and out puts for the four parametersets. The resulting six time series of price ratios are shownin the Exhibit 12 for three years to maturity in the Hestonmodel. The AP prices are (on average) about 3.5% smallerthan the other prices and 75% of the AP prices are at least2% smaller than the other prices. The RP prices lie abovethe prices from the calibrations to implied volatilities.These AI and RI prices are quite similar so that we canidentify again the two groups that we have alreadyobserved for the up and out calls.

Compared to the up and out calls, the price differ-ences are smaller for the down and out puts. This can beseen also from Exhibit 13 that reports the median of theprice ratios for one, two, and three years to maturity. Thisexhibit shows that the price ratios change for increasingtime to maturity: For one year to maturity, the AP priceslie above the other prices but with increasing time tomaturity, the AP prices become relatively smaller. TheRP and AI prices remain on a similar level for all timesto maturity and the RI prices tend to this level for longertimes to maturity.

The situation in the Bates model that gives a betterfit to the plain vanilla data is described in Exhibit 13 andExhibit 14. The AP prices lie about 7% below the otherprices. Thus this difference is bigger than in the Hestonmodel. The other price ratios still lie on average on thesame level but their variance has grown compared to theHeston model.

m ST t T t=≤ ≤

defmin0


E X H I B I T 9

Note: Relative prices of the up and out calls in the Heston model for 3years to maturity.

Copyright © 2007

The situation for the barrier options can be sum-marized as follows: the AP prices differ significantly fromthe other prices for both types of barrier options. Whilethe AP prices are higher for up and out calls, they arelower for down and out puts relative to the other prices.In this sense the situation is symmetrical. The differ-ences become bigger for longer times to maturity and

the better fit of the Bates model does not lead to smallerprice differences.

Cliquet Options

We consider cliquet options with prices

exp(–rT )E[H] (21)


E X H I B I T 1 1

Note: Relative prices of the up and out calls in the Bates model for 3 yearsto maturity.

E X H I B I T 1 2

Note: Relative prices of the down and out puts in the Heston model for 3 years to maturity.

E X H I B I T 1 0

Note: Median of price ratios of up and out calls.

Copyright © 2007

where the payoff H is given by

(22)

H c f

c fS

g gi

N

li

li

= , ,

, ,

=∑def

min max min

max

1

tt t

t

i i

i

S

S

−

−

−

1

1

Here cg (fg) is a global cap (floor) and cgi (fg

i) is a local cap(floor) for the period [ti–1, ti].

We consider three periods with and the caps and floors are given by

cg = ∞

fg = 0(23)

cli = 0.08, i = 1, 2, 3

fli = –0.08, i = 1, 2, 3

While barrier options are simple exotic options, cli-quet options are more difficult because of their forwardstructure. Moreover, there exist many different types cor-responding to the parameters. Hence, our specificationcannot give a representative picture of all the traded cli-quets. But these caps and floors are typical because theoption holder cannot lose money and the returns arebounded above only by the local return bounds.

Cliquet options pay out basically the sum of thereturns Ri =

def (Sti – Sti)/Sti–1. In order to reduce risk, local

and global floors f are introduced for the returns R. Inthe same way the returns are bounded from above by localand global caps c.

t i i …iT= = , ,3 0 3( )


E X H I B I T 1 3

Note: Median of price ratios of down and out puts.

E X H I B I T 1 4

Note: Relative prices of the down and out puts in the Bates model for 3 years to maturity.

Copyright © 2007

The distributions of the six time series of price ratiosfor cliquet options are described in Exhibit 15 for threeyears to maturity in the Heston model. The ratios arecloser to 1 than in the case of the barrier options. The APprices lie above the other prices but the difference is sig-nificant only for the AP and RP prices. The differencesbetween the other prices is also small. Thus, we cannotrecognize directly from this exhibit the two groups thatwe identified for the barrier options.

Exhibit 16 that reports the median price ratios forone, two, and three years to maturity gives some insightinto this situation: the AP prices are about 2% smallerthan the other prices for one year to maturity. Withincreasing time to maturity, the AP prices grow relativelyand are about 1.5% higher than the other prices for threeyears to maturity. As Exhibit 16 confirms, the other priceratios remain relatively constant for different times tomaturity. Thus there are again the two groups that wehave identified for the barrier options: The changing rel-ative AP prices on the one hand and the constant otherprice ratios on the other hand.

The relative prices of the cliquet options in theBates model are presented in Exhibit 17 for three yearsto maturity. Here we see that the AP prices are about7% smaller than the other prices. The RP prices lie about2% under the AI prices that are 3% higher than the RIprices. The RP and RI prices are similar. Thus, thereare quite big differences for the cliquet options in theBates model. Moreover, the variance is larger relative tothe Heston model. Exhibit 16 describes the situation of

different times to maturity and shows that the AP pricesgrow relatively with increasing time to maturity while theother price ratios remain rather constant for differenttimes to maturity.

Comparing the results for the two types of barrieroptions and the cliquet options we see in all cases twogroups, the AP prices and the other prices. The AP pricesdiffer a lot from the other prices and in addition this dif-ference changes for different times to maturity. Moreover,the variance of the price ratios with AP prices is biggerin general than for the other price ratios. The other groupof RP, AI, and RI prices shows similar prices and smallvariances. The Bates model that gives better fits to impliedvolatility surfaces has higher price differences (with highervariances).

MODEL RISK

In the last section, we have described price differ-ences that result from the calibration with respect to thefour error functionals. In this section, we consider modelrisk and its relation to calibration risk and compare ourresults with the findings of Schoutens et al. [2004]. Model risk is generally understood as the risk of “wrong” prices because an inappropriate parametric model hasbeen chosen for modelling the price process of the underlying.

In order to analyze this model risk for the two sto-chastic volatility models, we consider the ratios of theprices of the exotic options in the Bates model and thecorresponding prices in the Heston model. The distri-bution of these ratios for up and out calls with three yearsto maturity is described by Exhibit 18. The prices in theBates model lie below the prices in the Heston model forall four error functionals: The difference varies between2% for the AP prices and 6% for the RI prices. Thusmodel risk is not independent of the calibration method,i.e., calibration risk. The results for smaller times to matu-rity are given in Exhibit 19. The exhibit suggests thatmodel risk does not change significantly for differenttimes to maturity.

The model risk of down and out puts is shown inExhibit 20 for three years to maturity. The prices in theBates model lie below the prices in the Heston model forall error functionals. Compared to the up and out calls,the model risk is bigger for the down and out puts: itvaries between 9% for AP prices and 14% for RI prices.But again we observe the highest difference for RI prices


E X H I B I T 1 5

Note: Relative prices of the cliquet options in the Heston model for 3 yearsto maturity.

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and the smallest for AP prices. Moreover, the variance isbigger than for the up and out calls. Exhibit 19 that givesthe results for smaller times to maturity can be interpretedin such a way that the model risk becomes smaller forshorter times to maturity.

Finally, we consider the model risk of cliquet optionsin Exhibit 21. For these options the Bates prices lie above

the corresponding Heston prices for all calibrationmethods. The smallest price difference that appears forthe AP prices is about 8% while the biggest difference of16% is for RI prices. Exhibit 19 shows again smaller pricedifferences for shorter times to maturity.

The model risk between the Heston and the Batesmodel can be described for barrier and cliquet options as


E X H I B I T 1 6

Note: Median of price ratios of cliquet options.

E X H I B I T 1 7

Note: Relative prices of the cliquet options in the Bates model for 3 years tomaturity.

AP/RP AP/AI AP/RI RP/AI RP/RI AI/RI

0.7

0.8

0.9

1

1.1

1.2

1.3

cliquet options, T=3

E X H I B I T 1 8

Note: Ratio of Bates to Heston prices for up and out calls with 3 years tomaturity on 51 days.

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follows: model risk measured by the price ratios in the twomodels increases for longer times to maturity. Moreover,it is ordered with respect to the calibration method. Thecalibration with respect to implied volatilities leads tobigger price differences than the calibration with respectto prices. The model risk is smallest for the AP calibra-tion and bigger for the RP calibration. It is even bigger forthe AI calibration and the price differences are the biggestfor the RI calibrations. This emphasizes the importanceof the implied volatility surfaces and their calibration.Moreover, model risk differs across option types. Themore exotic cliquet options have a higher model risk thanthe barrier options.

Schoutens et al. [2004] consider up and out calls (withstrike equal to spot) and cliquet options with three years tomaturity. For a barrier 50% above the spot, they find a modelrisk for the up and out calls of about 14%. For the cliquetoptions, they do not find a significant model risk. Theseresults do not correspond completely to our AP results.There may be several reasons for these different results: while

we look at a time series of 51 implied volatility surfaces,they focus only on one day. Moreover, they have analyzedEuroStoxx50 prices and we use DAX data.

CONCLUSION

We have looked at the stochastic volatility model ofHeston and analyzed different calibration methods andtheir impact on the pricing of exotic options. Our analysishas been carried out for a time series of DAX impliedvolatility surfaces from April 2003 to March 2004.

We have shown that different ways to measure theerror between the model and the market in the calibra-tion routine lead to significant price differences for exoticoptions. We have considered the four error measures thatare defined by the root mean squared error of absolute orrelative differences between prices or implied volatilitiesin the market and in the model. Among these measureswe have identified two groups. Calibrations with respectto relative prices, absolute implied volatilities, or relative


E X H I B I T 1 9

Note: Median of ratios of Bates to Heston prices.

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implied volatilities lead to similar prices of exotic options.Calibrations with respect to absolute prices imply exoticsprices that are quite different from the prices of the firstgroup. As implied volatilities are of comparable size fordifferent strikes and different times to maturity, calibra-tions with respect to absolute or relative implied volatilitydifferences lead to similar prices of exotic options. Asthese prices are also rather similar to those computed fromcalibrations with respect to relative price differences, weconclude that all three of these error measures have a sim-ilar weighting of the errors. The price differences betweenthe described two groups of error measures increase forlonger times to maturity. Moreover, the differences donot decrease in the Bates model although it is an exten-sion of the Heston model with similar qualitative featuresand a better fit to plain vanilla data. The price differencesof exotic options differ also across option types and arebigger for barrier options than for cliquets.

The observed price differences of exotic optionsbetween the two groups of error measures can beexplained in the following way: as we use out of themoney options for the calibration, measuring the error byabsolute prices puts most weight at the money. The char-acteristic feature of implied volatility surfaces around themoney is the skew because the smile can be identifiedonly from deep out-of-the-money calls. Hence, errors interms of absolute prices focus on the skew. In the Hestonmodel, the skew is mainly controlled by the correlationbetween the Brownian motions of the stock and the vari-ance process. Our calibrated model parameters confirm

this because the calibrations with respect to absolute priceslead to higher (absolute) correlations between these twoprocesses. At the same time such calibrations imply smallervolatilities of variance which control the smile. These twoeffects make the distribution of the stock more rightskewed so that up and out calls (down and out puts) aremore (less) expensive in models that are calibrated toabsolute price differences.

Moreover, we have looked at the model risk betweenthe Heston and the Bates model. Model risk and cali-bration risk are not independent because model risk islowest for calibrations with respect to absolute prices andhighest for calibrations with respect to relative impliedvolatilities. As this holds for all considered options, modelrisk seems to be ordered with respect to the error mea-sure used in the calibration. In the analyzed time periodthe DAX experienced a strong upward trend. This is alsoreflected in the positive expected jumps in the Batesmodel. These jumps can explain why the barrier (cliquet)options are more (less) expensive in the Heston modelthan in the Bates model. Because of the jumps, the con-sidered down and out puts often finish out of the money.The up and out calls are knocked out more often becauseof the upward jumps. The higher prices of the cliquetoptions in the Bates model may be due to the higherreturns that result from the upward jumps. If the markethas a downward trend that is also reflected in negativeexpected jumps in the Bates model, then model riskshould have the same size in general but could have a dif-ferent sign, e.g., for barrier options.


E X H I B I T 2 0

Note: Ratio of Bates to Heston prices for down and out puts with 3 yearsto maturity on 51 days.

E X H I B I T 2 1

Note: Ratio of Bates to Heston prices for cliquet options with 3 years tomaturity on 51 days.

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As model risk is bigger than calibration risk, cali-brations should be carried out with respect to absoluteprices if the choice of an appropriate model is unclear. Butif a model has already been chosen we suggest measuringthe error between the model and the market in terms of(relative) implied volatilities, because this error measurebest reflects the characteristics of the model that are essen-tial for pricing exotic options.

ENDNOTE

This research was supported by Deutsche Forschungs-gemeinschaft through the SFB 649 “Economic Risk” and byBankhaus Sal. Oppenheim.

1A bonus certificate is a structured product incorporatinga zero-strike call and a down and out put option. The payoffstructure at expiry can be described as follows: if the value ofthe underlying at expiry is above the strike then the investorreceives the underlying. If the price of the underlying is belowthe barrier then the investor bears the full loss of the under-lying because she gets only the underlying. Otherwise theinvestor gets the underlying and in addition a bonus if the bar-rier has not been exceeded before expiration. Because of itspayoff profile, this product is constructed mainly for sidewaysmarkets.

REFERENCES

Bates, D. “Jump and Stochastic Volatility: Exchange RateProcesses Implicit in Deutsche Mark Options.” Review of Finan-cial Studies, 9 (1996), pp. 69–107.

Carr, P., and D. Madan. “Option Valuation Using the Fast FourierTransform.” Journal of Computational Finance, 2 (1999), pp. 61–73.

Cont, R. “Model Uncertainty and its Impact on the Pricingof Derivative Instruments.” Mathematical Finance, 16 (2005),pp. 519–542.

Cont, R., and P. Tankov. Financial Modelling With Jump Processes.Chapman and Hall/CRC, 2004.

Glasserman, P. Monte Carlo Methods in Financial Engineering.New York: Springer, 2004.

Heston, S. “A Closed-Form Solution for Options with Sto-chastic Volatility with Applications to Bond and CurrencyOptions.” Review of Financial Studies, 6 (1993), pp. 327–343.

Mikhailov, S., and U. Nögel. “Heston’s Stochastic VolatilityModel. Implementation, Calibration and Some Extensions.”Wilmott magazine (July 2003), pp. 74–94.

Schoutens, W., E. Simons, and J. Tistaert. “A Perfect Calibra-tion! Now What?” Wilmott magazine (March 2004), pp. 66–78.

To order reprints of this article, please contact Dewey Palmieri [email protected] or 212-224-3675


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