Electronic copy available at: http://ssrn.com/abstract=2043503

Robust Calculation of MFIV from Calibrated

Surfaces

Philip Stahl∗and Philipp B. Rindler†

Working Paper — Feb 2012

Abstract

This paper proposes a new method to calculate model-free implied

volatility from a calibrated option price surface. This circumvents com-

mon interpolation/extrapolation problems found in established method-

ologies, where prices enter calculation directly, and is numerically more

stable. Areas outside the observable strikes are approximated better. Pre-

dictive regressions over 180 months based on this new method show that

the new method is indeed superior in most cases, even with only very few

observable strikes.

Keywords: Option pricing, model-free implied volatility, characteristic

functions

JEL-Classification: C61 C80 G13

1 Introduction

For financial institutions and investors, reliable ways of forecasting risk of in-

vestments is of utmost importance. Mostly reduced to the variation of returns,

∗mail@philipstahl.com†philipp.rindler@ebs.edu

1

Electronic copy available at: http://ssrn.com/abstract=2043503

active option trading allows to extract the views about return variation of the

average investor from option prices. Implied variance depends on the dynamics

of the underlying model, and are usually only computed for at-the-money op-

tions. Britten-Jones and Neuberger (2000) extended prior work by showing that,

for stochastic volatility diffusions, an infinite number of price processes can be

consistently fitted to observed option prices, thus deriving a model-free implied

variance (MFIV). This requires not only at-the-money option prices, but uses

the whole continuum of strikes, and levers available information more effective.

Jiang and Tian (2005) propose a procedure to deal with limited data observabil-

ity by interpolation and extrapolation of Black-Scholes implied volatility over

strikes. Their implementation is the established way of calculating MFIV nowa-

days, such as in Bollerslev et al. (2009) and Bollerslev et al. (2010). Also, the

CBOE volatility index VIX is based on this methodology.

Drechsler and Yaron (2011) show that the variance premium can give an

idea about long-run uncertainty aversion and may be helpful in predicting stock

market returns. Carr and Wu (2009) defined the variance premium as the

difference between realized variance under physical probability measure P and

the expected future realized variance under risk neutral measure Q. Under

no arbitrage, variance swaps represent the risk-neutral expectation of future

realized variance. They synthesize variance swaps from options to calculate

MFIV, and do so from a cross-section of options over strikes.

However, the cross-section of option prices is often very small. Only options

trading on the same stock and the same maturity can be considered, which

are rarely enough. Most studies limit their scope to indices with many active

options or sort out stocks with too few options. Furthermore, they employ the

methodology suggested by Jiang and Tian (2005), which relies on numerically

2

unstable interpolation with cubic splines and the assumption of constant option

prices outside the observable range. This introduces additional sources of error

in the estimation of MFIV and may give a false sense of confidence in the results.

In this paper we develop an indirect approach to calculate MFIV by cali-

brating a pre-specified model to fit option prices, and calculate MFIV from

a dense grid of theoretical option prices. The specification of the model can

be interpreted as sophisticated interpolation algorithm. Implied volatility is

still calculated from the cross section of option prices, but is less vulnerable to

option prices that behave different than expected (e.g., mispricing, or market

microstructure anomalies that may deform the surface). In regression analysis,

the proposed method has higher explanatory power for realized variance than

the Carr and Wu (2009)-implementation. The remainder of the paper is as fol-

lows: Section 2 gives an overview of the theory of MFIV and explains where

errors arise. Section 3 describes the methodology by Jiang and Tian (2005) as

well as the improved one in detail. Section 4 lays out regression results and

Section 5 concludes.

2 Model-Free Implied Variance

2.1 Synthetic variance swaps

Originally, Britten-Jones and Neuberger (2000) derived the MFIV only in

a diffusion setting. This was later extended by Jiang and Tian (2005), who

generalize the the underlying stochastic process to include jumps. Carr and Wu

(2009) show that the basic formula can be derived in a semi-martingale setup.

They quantify the return variance as variance swap that pays according to

realized variance and synthesize variance swap rates from option prices. Under

risk neutrality, the variance swap rate has to equal the expectation of the future

3

realized variance such that

SWt,T = EQt [RVt,T ] (2.1)

The following paragraphs redraw their derivation of future quadratic varia-

tion from option prices.

Let St denote the time t spot price of an asset, Ft its time t futures price

with maturity T > t. By no arbitrage, there exists a risk-neutral, filtered

probability space (Ω,F , (Ft),Q) on which the futures price Ft is a sum of a

purely continuous martingale and a purely discontinuous (jump) martingale:

dFt = Ft−σt−dWt +

∫R0

Ft−(ex − 1)[µ(dx, dt)− νt(x)dxdt] (2.2)

where Wt is a standard Brownian motion (under Q), R0 ≡ R\0, Ft− is the

left limit of the futures price at time t (the futures price prior to a jump at time

t), and µ(dx, dt) is the random counting measure which is nonzero for a given

value of x if and only if the futures price jumps from Ft− to Ft = Ft−ex at time

t. Finally, νt(x) is the compensator process corresponding to µ.1

Under the dynamics in (2.2), the quadratic variation on the futures return

from time t until maturity at T is given by:

Vt,T =

∫ T

t

σ2s−ds+

∫ T

t

∫R0

x2µ(dx, ds) (2.3)

The annualized return variance is then given by RVt,T = 1T−tVt,T . The idea

of MFIV is that the quadratic variation can be replicated by a static portfolio

1It is assumed that the jump process is of finite variation, i.e.∫R0 (|x| ∧ 1)νt(x)dx < ∞.

Moreover, σt− and νt(x) are both stochastic and predictable with respect to the filtration Ft

and restricted so that the futures price Ft is always positive.

4

of options with maturity T and a dynamic position in futures. When the un-

derlying process is a diffusion, this replication is exact. Assuming jumps in the

dynamics leads to higher-order correction terms as shown below.

Under Q, futures trading can be done at no cost, such that the risk-neutral

expected value of the quadratic variation can be approximated by the value of

the options in the static replicating portfolio. Carr and Wu (2009) specify the

following

Proposition 1. Under absence of arbitrage, the risk-neutral expected value of

return quadratic variation of an asset over horizon [t, T ] defined in (2.3) can

be approximated by the continuum of European out-of-the-money option prices

across all strikes K > 0 and at the same maturity date T

EQ[RVt,T ] =2

T − t

∫ ∞0

Θt(K,T )

Bt(T )K2dK + ε (2.4)

where Bt(T ) denotes the time-t price of a bond paying one dollar at T ,Θt(K,T )

denotesthe time-t value of an out-of-the-money option with strike price K > 0

and maturity T ≥ t (a call option when K > Ft and a put option when K ≤ Ft),

and ε denotes the approximation error, which is zero when the futures price

process is purely continuous.

For the remainder of the paper, equation (2.4) serves as definition of MFIV.

In the next section the calculation in practice is examined in detail.

2.2 Implementation issues

From equation (2.4) it is easy to see that calculation of MFIV requires nu-

merical integration over an infinite domain, i.e. an infinite amount of strikes

with infinitesimal distance ranging from K = 0 to K = ∞. In practice, these

requirements are not fulfilled. Strikes are often spaced several dollars from each

5

other, most options issued are somewhat close to being at-the-money (close in

relation to zero and infinity), and certainly only a finite number of options is

traded. Two sources of errors arise: Truncation and discretization.

Truncation refers to the reduction of the integration domain from [K0,K∞]

to [Kmin,Kmax], where Kmin < F0 < Kmax, thus strikes are observable around

the forward price. Jiang and Tian (2005) derive theoretical upper bounds for

truncation errors beyond the strike range.

Discretization stems from the distance between observable strikes ∆K > 0.

The strike surface is discrete and cannot be be integrated without interpolation.

Using approximations such as rectangle rule, trapezoidal rule, or Simpson’s rule

are interpolation methods. Interpolations induce an additional source of error to

the calculation, which can be severe under complicated or uncommon observed

price structures.

3 Methodology

3.1 Current methodology

Calculation of MFIV requires a continuum of strikes and in practice inter-

polation between strikes as well as extrapolation outside the observable region

is necessary. Jiang and Tian (2005) suggest to use natural cubic splines to

interpolate and to keep observed option prices constant outside [Kmin,Kmax].

According to Shimko (1993) and Ait-Sahalia and Lo (1998), the nonlinear re-

lationship of option and strike prices may induce an upward bias, and they

suggest using Black-Scholes implied volatility as symmetric transformation of

option prices to apply curve fitting methods that provide the required continuum

of strikes. Beyond the observable range of strikes, Jiang and Tian (2005) note

6

Figure 1: Extrapolation of strikes on a simulated implied volatility surface,based on parameter set 1 from Jiang and Tian (2005).

that other extrapolation methods than keeping the price constant may lead to

large estimation errors and negative implied volatilities. Figure 1 depicts differ-

ent extrapolation methods on an implied volatility surface of observable strikes.

The observed surface is calculated from a Heston (1993)-Model with Parameters

S = 100, r = 0, v0 = 0.04, ρ = −0.5, κ = 2, θ = 0.04, λ = 0, σvolatility = 0.225,

which are the same as in Jiang and Tian (2005, parameter set 1). The strikes

are equally spaced with ∆K = USD5 and symmetrically distributed around S.

The difficulty of curve-fitting is evident.

Jiang and Tian (2005) simulate a price surface with stochastic volatility and

jumps in the price process. From this surface, they estimate the effectiveness

of their curve-fitting method to assess its appropriateness to overcome limited

information by reducing the strikes that enter the MFIV calculation, but only

consider isolated situations where either truncation or discretization occur and

strikes are equidistant. In earlier work, we have extended this analysis and

considered random drafts from a simulated option price surface of decreasing

7

size to replicate the patterns observed in real data (Stahl (2010)). We conclude

that the method of Jiang and Tian (2005) is appropriate if the observed volatility

surface is very smooth, but may be far out if option prices exhibit deviations

from a smooth surface or very few prices are available.

Carr and Wu (2009) perform two interpolations: the first over strikes, the

second over maturities. On the strike dimension, they follow Jiang and Tian

(2005). Over maturities, they interpolate linear between two available maturi-

ties T1 and T2 to gain the variance between t to T by

SWt,T =1

T − t[SWt,T1

(T1 − t)(T2 − T ) + SWt,T2(T2 − t)(T − T1)

T2 − T1] (3.1)

which they compare against the corresponding ex-post realized variance. This

is given by

RVt,T =365

M

M∑i=1

(Ft+i,T − Ft+i−1,T

Ft+i−1,T

)2

(3.2)

where M is the maturity in days such that t+M = T . To ensure comparability

of results, M = 30. We refer to (3.1) as discretized MFIV and to (3.2) as realized

variance (RV).

3.2 MFIV using calibrated models

Current implementations of MFIV are inaccurate and numerically instable.

Strangely priced options are dangerous to the stability of direct interpolation

because they enter the calculation of MFIV directly and influence the inter-

polated neighborhood on the surface, multiplying their effects. Strikes outside

[Kmin,Kmax] may be mispriced as well. We propose to step away from cal-

culating MFIV from prices directly, rather use them to calibrate a model and

generate a large and dense volatility surface, which is translated into option

8

prices at a density unobserved in practice that serve as a basis for MFIV. This

has several advantages:

• Academics have a thorough understanding of the behavior of parametric

models. Curve-fitting methods based on splines behave unpredictable and

can generate extreme outliers. Option prices can easily become negative.

Additional procedures are necessary to make sure that the interpolation

stays within reasonable boundaries. A calibrated model fulfills these con-

ditions by design.

• Calibrated models replicate the behavior of option prices far out-of-the-

money better than constant extrapolation. MFIV considers a cross-section

of prices that are weighted by K2, thus strikes far from the forward price

are less important for the calculation, but in situations where all observ-

able strikes are truncated close to the forward price, truncation errors may

be considerable.

• If only one strike is observable (not uncommon – see table 1), the indi-

rect approach is still able to approximate the dynamics of the volatility

surface. Constant extrapolation would not only be as informationally inef-

ficient as common at-the-money implied volatility, but also introduce the

approximation as a new source of error.

• Parametric models follow from economic reasoning and empirical analy-

sis. Standard curve-fitting methods are not specifically designed for op-

tion markets. Economic reasoning may improve accuracy where educated

guesses are required by the real world.

Choosing the underlying model, we follow Gatheral (2006), who suggests using

a stochastic volatility model with jumps in the stock price process (SVJ model).

His reasoning is that while jumps are empirically necessary to explain volatility

9

surface behavior, estimation of simultaneously jumping prices and volatilities

(as with the SVJJ model) is computationally expensive and inaccurate, because

volatility cannot be observed directly and thus estimation of jumps makes fitting

the surface hard. Furthermore, Gatheral (2006) demonstrates that while the

SVJ model increases the fit compared to the Heston (1993) model, improvements

of the SVJJ model over the SVJ model are small2.

For every day and maturity, the specified model needs to be recalibrated.

Even though one would expect that it is sufficient to calibrate the model per

stock globally every once in a while and continue for the next dates with lo-

cal optimizations, errors grow fast and are cumulative. Every calibration is

therefore performed globally in a two step process: First, adaptive simulated

annealing (ASA) is performed on the SVJ model’s characteristic function to

find a set of parameters v0, κ, θ, η, ρ, α, δ, and λ that is likely to be close

to the global optimum. Second, pattern search is performed to find the local

optimal set, starting from the last set from ASA. By employing the second step,

the very slow transition from a good state to the best state of ASA can be

avoided and the resulting set is likely to fit better. Details and boundaries of

the optimization can be found in the appendix.

With a parameter set for a given t and T an indefinitely dense option price

surface can be calculated. Here, strikes are ranged from $1 to $5000 and spaced

by $2, so the interpolated option price surface contains 2500 prices. As the in-

terpolating SVJ model’s stochastic process is known by assumption, this can be

done computationally efficient through fourier inversion3. On this surface, op-

tion prices are sorted over strikes and the integral of weighted implied variances

2As characteristic functions are used for calibration, extension of results to the SVJJ orVariance-Gamma model is straightforward. These can be found in Gatheral (2006).

3See Gatheral (2006) or the appendix for details.

10

is approximated by trapezoidal numerical integration.

The weighting factor K2 pronounces strikes at-the-money, thus the most im-

portant part the model has is the interpolation. Extrapolation of far out-of-the-

money strikes has a relatively low impact. The underlying model is known, so

the behavior of the surface is stable. The calibration may correct false prices

and filters the effects of outliers on the estimation. However, only prices for a

specific date and maturity are considered, so in cases where only one option is

observable, the model may be vastly mis-specified.

4 Empirical Results

4.1 Data

The data to test the proposed methodology are daily historical option prices

from OptionMetrics, starting January 2, 2000, and ending December 31, 2010,

stocks where quotes that start later or end earlier are not considered. Maturity

of options is limited to be between 8 and 365 days to avoid anomalies of very long

term options or shortly before exercise date. All options are American options,

however OptionMetrics corrects these by subtracting an approximation of the

early exercise premium following a binomial tree approach. Even though this

might contain other biases, these are consistent over all analyzed stocks and do

not affect the main results. Furthermore, positive bid-ask spreads are required.

Stock prices, Black-Scholes implied volatilities, and the interest rate curve are

from WRDS and prices are dividend adjusted, but stock splits are not corrected.

From this data set we selected 15 stocks4 randomly. While other authors (for

example Carr and Wu (2009)) choose stocks according to availability of option

4The low number of stocks is mainly due to the time intensive computation. Tests will beextended in future versions of this paper.

11

Table 1: List of stocks in the sampleNo. Security ID Ticker Starting Date Sample size Min. Strikes Max. Strikes Name

1 100943 ASA 02-Jan-1996 9789 4 5 ASA GOLD AND PRECIOUS METALS LIMITED2 101172 AAI 02-Jan-1996 3480 1 5 AIRTRAN HLDGS INC3 101328 TWX 02-Jan-1996 16889 4 5 TIME WARNER INC4 101475 ABC 01-Aug-1998 8120 4 5 AMERISOURCEBERGEN CORP5 102322 BDN 14-Apr-1998 886 2 5 BRANDYWINE RLTY TR6 104939 F 02-Jan-1996 8013 1 6 FORD MTR CO DEL7 105964 IDA 28-Sep-1996 2987 1 4 IDACORP INC8 106652 KNDL 05-Dec-1998 3241 1 5 KENDLE INTERNATIONAL INC9 106776 LSI 02-Jan-1996 6428 1 5 LSI CORPORATION

10 106892 LM 29-Apr-1997 11302 4 5 LEGG MASON INC11 107015 L 02-Jan-1996 7044 4 5 LOEWS CORP12 109695 ROL 27-Oct-1998 333 1 4 ROLLINS INC13 110681 SYKE 02-Apr-1997 3477 1 4 SYKES ENTERPRISES INC14 110740 TE 13-Mar-1996 3965 3 5 TECO ENERGY INC15 112219 ZBRA 02-Jan-1996 5701 4 4 ZEBRA TECHNOLOGIES CORP

Table 1 summarizes the sample of stocks used in this paper. Quotes for calculations start 01-Jan-2000 and end 31-Dec-2010. Samplesize refers to the number of observations per stock, minimum and maximum strikes refer to the number of strikes observable for achosen maturity.

quotes, this shortage of data is what the proposed methodology is designed for

to overcome. A summary of the used data can be found in table 1.

4.2 Predictive regressions

To assess the forecast ability of both methods we follow Shu and Zhang (2002)

and run regressions on both the method as in Carr and Wu (2009) and the

indirect method on the future realized variance. The regression models are

RVt = α+ βMFIV CWt + εt (4.1)

RVt = α+ βMFIV CSt + εt (4.2)

where superscript CW denotes the Carr and Wu (2009)-Method, and CS de-

notes MFIV from calibrated surfaces. Table 2 summarizes the rsults of both

regressions on MFIV calculations for the first trading day of every month on 30

days in the future per security. In most cases, the new method’s R2 is higher

than before, and F -statistics as well as p-value are in favor of the MFIV from

calibrated surfaces. In absolute terms, R2 seems to be low, but there are two

possible explanations: First, the data is not adjusted for stock splits. This may

introduce an error in the calculation of the variance, depending on the exact

12

adjustment procedure of options. Second, the stocks are selected randomly from

all available stocks in the WRDS database, and may be illiquid or inactively

traded. Both apply to every stock in the data set, and do not affect the results

of the predictive regressions (4.1) or (4.2).e

5 Conclusion

In this paper, we develop a new method to calculate MFIV which does not

directly incorporates prices and is numerically more robust than prior methods.

It specifically addresses the interpolation/extrapolation method by Jiang and

Tian (2005), which relies on polynomial splines between observed option prices,

which are hard to control and may easily result in negative prices or volatilities.

Furthermore, the assumption of constant strike prices outside the observable

range is unrealistic. By calibrating a SVJ model to option prices, the price

surface can be approximated better, and will stay within boundaries that are

relatively well understood. Instead of taking the volatility from the model itself,

we calculate MFIV from the cross section of option prices, using all available

information. Predictive regressions confirm that the proposed method is indeed

better suited to extract future realized variance from option markets.

There are still issues with the calculation of MFIV from incomplete observa-

tions. First, for every considered date, the price surface has to be calibrated

globally. This is not only expensive in terms of computation, but it still requires

some data to calibrate on. In cases where only one or no option prices are ob-

servable, the model might be miscalibrated. One way to circumvent this could

be to analyze the stability of model parameters over time, and use them not only

as a starting set for the next fitting, but also to interpolate over time. Second,

empirical evidence points into the direction that volatility follows a continuous

13

Tab

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Reg

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stati

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rity

IDC

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and

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(200

9)-

Met

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1009

430.

0408

0.21

420.

0023

0.4

154

0.5

201

0.2

153

1.2

974

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101

0.0

918

17.9

925

0.0

000

0.1

960

1011

720.

2343

0.58

430.

0444

7.8

553

0.0

057

0.9

144

0.6

584

0.3

745

0.0

933

17.3

999

0.0

000

0.8

676

1013

280.

5246

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359

0.14

4330.0

086

0.0

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4.3

426

1.0

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0.2

668

0.0

355

6.5

586

0.0

113

4.8

944

1014

750.

0593

0.27

430.

0047

0.6

909

0.4

072

0.3

513

1.1

173

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0.1

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18.9

874

0.0

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0.3

126

1023

220.

5154

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952

0.48

4844.2

200

0.0

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0.7

490

2.7

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953

0.6

640

92.8

977

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0.4

884

1049

390.

1539

0.06

880.

2499

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5398

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7.1

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125.7

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0.0

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4378

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0.8

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319.6

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316.5

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1.3

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0.2

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638

Tab

le2

stat

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14

process. High frequency data, which has established to be standard for realized

variance estimations, should be analyzed in the context of of MFIV. This may

also improve ones ability to interpolate over time, as parameters are likely to

be more stable over 5 minutes than over 30 days.

It seems promising to apply the proposed methodology to estimate the vari-

ance risk premium. With a more accurate estimation, a part of what is un-

derstood as premium might be identified as measurement error. Further tests

could check wether classic risk factors influence the variance risk premium, or

wether current risk factors are brought into existence by variance risk premia.

A Calibration of SVJ model

A.1 Derivation of the Valuation Equation

In general, the price of a European call option can be computed as

C(Ft, τ) = e−rτEQ [max(FT −K, 0)] (A.1)

It turns out that it is quite straightforward to get option prices by inverting the

characteristic function of a given stochastic process if it is known in closed form:

C(Ft, τ) = Ft −√FtK

1

π

∫ ∞0

Re[eiuxtφτ (u− i

2 )]

u2 + 14

du (A.2)

where φT (u) is the characteristic function of S and xt = log(FtK

). In the fol-

lowing, we’ll follow the derivation of Carr and Madan (1999) and Lewis (2000)

to derive (A.2).

A covered call G(xt, τ) is a long position in S and a short position in a call

option C(S, τ) written on S with payoff G(k, 0) = min(ST ,K) = min(FT ,K).

15

To derive (A.2), consider the Fourier transform payoff of G(k, τ) with respect

to the log-strike k = log(KFt

)= −x defined by

G(u, τ) =

∫ ∞−∞

eiukG(k, τ)dk (A.3)

Then

1

FG(u, T − t) =

∫ ∞−∞

eiukE[min(exT , ek)|xt = 0

]dk

= E[∫ ∞−∞

eiuk min(exT , ek)dk∣∣∣xt = 0

]= E

[∫ xT

−∞eiukekdk +

∫ ∞xT

eiukexT dk∣∣∣xt = 0

]= E

[e(1+iu)xT

1 + iu− e(1+iu)xT

iu

∣∣∣xt = 0

]only if 0 < Im[u] < 1!

=1

u(u− i)E[e(1+iu)xT

∣∣xt = 0]

=1

u(u− i)φT (u− i)

To get the call option price in terms of the characteristic function, we express

the option in terms of the covered call position and invert the Fourier transform,

integrating along the line Im[u] = 12 .

C(F, τ) = F −G(F, τ)

= F − F 1

2π

∫ ∞+ i2

−∞+ i2

e−iuk

u(u− i)φT (u− i)du

= F − F 1

2π

∫ ∞−∞

e−i(u+i2 )k

(u+ i2 )(u− i

2 )φT (u− i

2)du

= F −√FK

1

π

∫ ∞0

Re[eiuxtφT (u− i

2 )]

u2 + 14

du

16

A.2 Used characteristic functions

As in Gatheral (2006, Chapter 5).

A.2.1 Black-Scholes GBM

In the Black-Scholes model, the dynamics are given by

dS = µSdt+ σSdW (A.4)

The characteristic function is given by

φT (u) = exp

iu(µ− 1

2σ2)T − 1

2u2σ2T

(A.5)

A.2.2 Merton Jump Diffusion

In the jump-diffusion model, the dynamics are assumed to be

dS = µSdt+ σSdW + (J − 1)Sdq (A.6)

where

dq =

0 with probability 1− λdt

1 with probability λdt

is a homogeneous Poisson process with arrival intensity λ and the jump size

J assumed to be log-normally distributed with mean log-jump α and standard

deviation δ. The characteristic function is given by

φT (u) = exp

(−1

2u(u+ i)σ2 − λ

[−(eiuα−

u2δ2

2 − 1)

+ iu(eα+

δ2

2 − 1)])

T

(A.7)

17

A.2.3 Heston Stochastic Volatility

The derivation of the characteristic function in the Heston model is somewhat

involved. It is given by

φT (u) = expC(u, τ)θ +D(u, τ)v0 (A.8)

where

C(u, τ) = κ

[r−τ −

2

η2log

(1− ge−dτ

1− g

)]D(u, τ) = r−

1− e−dτ

1− ge−dτ

g =r−r+

r± =b± dη2

d = d =√b2 − 4ac

c =η2

2

b = κ− ρηiu

a = − u2

2− iu

2

A.2.4 Stochastic Volatility with Jumps

To add jumps to the stochastic volatility Heston model, it is combined with the

jump part of the Merton model by multiplying the characteristic functions such

that

φT (u) = exp

C(u, τ)θ +D(u, τ)v0−(

λ[−(eiuα−

u2δ2

2 − 1)

+ iu(eα+

δ2

2 − 1)])

τ

(A.9)

18

Derivations of the characteristic functions above as well as for the SVJJ and

the Variance-Gamma model can be found in Gatheral (2006).

A.3 Adaptive simulated annealing

For the first step of the calibration, the last best set is used after either the

root mean squared error change over 500 iterations is smaller than 1e− 6 or the

algorithm has evaluated 24000 sets.

In the calibration of the SVJ model, the the parameters are limited to

0.01 ≤ v0 ≤ 1; 1 ≤ κ ≤ 20; 0.01 ≤ θ ≤ 1; 0.01 ≤ η ≤ 1;

−1 ≤ ρ ≤ 1;−1 ≤ α ≤ 0; 0 ≤ δ ≤ 1; 0 ≤ λ ≤ 1

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