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Imposing No-Arbitrage Conditions In Implied Volatility Surfaces Using Constrained Smoothing Splines

Márcio Poletti Laurini

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IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USINGCONSTRAINED SMOOTHING SPLINES

MÁRCIO POLETTI LAURINI

ABSTRACT. We apply the constrained smoothing b-splines introduced by [He & Ng, 1999] to the construction

of arbitrage-free implied volatility surfaces extracted from option price data. The constrained smoothing b-

splines permits to impose the constraints of monotonicity and convexity given by the option pricing equation and

are related to no-arbitrage conditions in the constructionof smoothed implied volatility surfaces. The methodol-

ogy share the robustness properties of quantile regressionmethods, as the method formulates the b-spline using

LP projections. We illustrate the methodology through the calculation of implied volatility surfaces free of ar-

bitrage, and also for the calculation of local volatilitiesand risk neutral densities, showing that the methodology

also can be used as a pre-processing tool for general treatment of option data.

1. INTRODUCTION

The estimation of measures of volatility is of fundamental importance in financial engineering. Asset

pricing, dynamic hedging strategies, risk management and asset allocation are directly based on the volatility

of the underlying assets. Volatility is a key component in precification of virtually any risk asset, from simple

vanilla options to complex equity, interest rates and FX instruments.

Measurement of volatility can be divided on three general methodologies. The first is the called his-

torical volatility, where the volatility is estimated using past returns of underlying asset. We can put in

this class parametric models for modelling the volatility as GARCH ([Engle, 1982]) and Stochastic Volatil-

ity ([Taylor, 1986]) models, as the estimation is based on past squared returns. The second methodology

are the measures knowed as realized volatility estimators,and are of special importance on option pricing,

since these measures are based on the concept of quadratic variation, given by the quadratic variation in the

high frequency intra-daily returns (e.g. [Andersenet al. , 2003], [Barndorff-Nielsen & Shephard, 2002] and

[Barndorff-Nielsen & Shephard, 2004]).

The third class of volatility measures contains the measures known as implied volatility, and they are the

motivation of this article.The measures of implied volatility differ from the two previous ones for the fact of

that they are not based on the variation of the returns of the underlying asset, but calculated through the data

of options on these assets.

The implied volatility is the volatility which placed inside a choosed option pricing formula, it takes

to an price equal to the observed one in the market for an option with the same characteristics. The

implied volatility usually is quoted in terms of the volatility extracted using the Black-Scholes model

Ibmec São Paulo e Departamento de Estatística - Imecc-Unicamp. email de contato [email protected]

IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 2

FIGURE 1. Call Prices and Strikes

6400 6600 6800 7000

010

020

030

040

050

0

strikes

call p

rice

Call Prices for Dax Contract

([Black & Scholes, 1973]), by the inversion, using a numerical method for root finding, of the Black-Scholes

formula in terms of the observed call price, spot price, timeto maturity, risk free interest and dividend rates.

The implied volatilitysurface, defined as the set of volatilities obtained by the inversionof simple Black-

Scholes prices for distinct strikes and maturities, is a keyinput on many financial applications. As examples

we have the pricing of volatility derivatives, the extraction of risk neutral density implicit on option prices

[Shimko, 1993], the construction of local volatility models ([Dupire, 1994] and [Derman & Kani, 1998]),

non-parametric option pricing ([Ait-Sahalia, 1996]) and extraction of market expectations from option data

([Svensson & Soderling, 1997]).

But the correct construction of implied volatility surfaceis a practical problem, not solved in a consensual

way - quoting [Gatheral, 2006]:

“The problem is that we don´t have a complete implied volatility surface, we only have a few bids and

offers per expiration. To apply a parametric method, we needto interpolate and extrapolate the know implied

volatilities.

Its very hard to do this without introducing arbitrage”

The Figure 1 shows the usual situation, using as example aggregated call prices on DAX Index. Call

options are not traded for all possible strikes and expirations, and in the case of DAX Index strikes, strikes

are traded in multiples of 50. Is many financial applications, is necessary to have a near continuous number

of strikes for the numerical methods work properly, and for obtain the not observable strikes, we need to use

some method of data interpolation and extrapolation.


The practical problem is that the usual methods of interpolation do not guarantee that the interpolated

points, and same the points observed in the curve, are free ofarbitrage. This represents a problem of

basic importance thus, since the great majority of the applications is derivated with basis on no-arbitrage

conditions, the presence of arbitrage points leads to invalidation of many of these methods.

We propose a non-parametric method, based on constrained smoothing b-splines under restrictions of

monotonicity and convexity. The restrictions are necessary to impose no-arbitrage in interpolated curves,

and they guarantee that the constructed implied volatilitysurface is arbitrage free.

The article is constructed with the following structure: inSection 2 we revise the necessary conditions

for no-arbitrage in options price data, and we revise some sources of arbitrage on option data. Section 3

shows a compact review of the theory of constrained smoothing splines under restrictions. Section 4 shows

the applications of the methodology for the construction ofimplied volatility surfaces, in comparison with

the other methodologies used in this literature. In this Section we also shows also two derived applications,

the first one the use of constrained smoothing splines in local volatility estimation and second applications

the estimation of risk neutral density implied in option prices. The final conclusions are in Section 5.

2. NO-ARBITRAGE IN OPTION PRICING

2.1. Necessary Conditions for No-Arbitrage. In the classic option pricing model of [Black & Scholes, 1973]

the no-arbitrage conditions are equivalent to the existence of a risk neutral density (state price density), who

gives the price of base asset in every possible state of the nature in the risk neutral measure. Fundamental

conditions for the imposition of no-arbitrage are given by the relationships between the option price and the

strike price.

The option price for a call option is1:

(1) C(St, E, τ, rt,τ , σ2) = e−rt,ττ (max(St − E, 0)p(St|St, E, τ, rt,τ , σ2)dSt

whereSt is the price of the asset at time t, E is the strike price,τ is the time until the expiration of the

contract,rt,τ is the risk free interest rate andp(St|St, E, τ, rt,τ ) is the risk neutral density.

The conditions that guarantee the existence of an risk neutral density can be summarized (following

[Rebonato, 2004]):

(1) Market Conditions - The market is complete, frictionless, there are not exists bid-ask spreads, short-

sales are allowed and there are not taxes.

(2) Traded Instruments - Is this economy are traded the underlying asset and plain-vanilla calls and puts

options for all maturities and strikes. There also exists deterministic bonds whose income is given

by a risk free interest rate, and the payoffs of derivative instruments depends on the history of the

underlying asset until the expiration date.

(3) Probability Spaces - Information set in given by a filtered probability space (Ω, Ft, Q), whereΩ is

the state space,Ft, is the filtration andQ is the probability measure on the risk neutral world. State

1The same conditions can be derived for a put option using the put-call parity.


space contains all present and possible future values of underlying asset and derivative options, and

Ft is the natural filtration generated by history of prices of the underlying and the options for a finite

but large number of dates.

(4) Pricing Conditions - Using the notationC(St, E, τ, rt,τ , σ2) for the price of call options, we require

that exists a measureQ satisfying:

(2) EQmax(St − E, 0)|Ft = C(St, E, τ, rt,τ , σ2)

The relationship between the option priceC(.) and the strike can be viewed by the derivative of the option

price with respect to the strike E:

(3)∂C(St, E, τ, rt,τ , σ2)

∂E= e−rt,ττ

∫∞

E

p(St|St, E, τ, rt,τ , σ2)dSt

We can also to check that given two strikesE1andE2 we have :

(4)∂C(St, E2, τ, rt,τ , σ2)

∂E−

∂C(St, E1, τ, rt,τ , σ2)

∂E=

(5) e−rt,τ τ

∫ E2

E1

p(St|St, E, τ, rt,τ , σ2)dSt

The no-arbitrage conditions in the Black-Scholes model canbe extracted of equations 3 and 5. Equation

3 implies that we must have a monotonically decreasing relationship between the option price and the strike,

and Equation 5 shows that price function must be a convex function of the strike price E. Any points violating

this restrictions are arbitrage conditions.

Under this set of relationships, we define aadmissibleimplied volatility surface (e.g. [Rebonato, 2004])

by the following set of conditions:

(6)∂C(St, E, τ, rt,τ )

∂E< 0

(7)∂2C(St, E, τ, rt,τ )

∂E2> 0

(8)∂C(St, E, τ, rt,τ )

∂T> 0

(9) Call(St, E, τ, rt,τ )|E=0 = St


(10) limE→∞

Call(St, E, τ, rt,τ ) = 0

The practical problem is how to construct interpolated or smoothed call prices curves in function of strikes

respecting conditions (6,7,8,9 and 10). In the following sub-section we argue about the practical aspects of

this problem. A quotation from [Rebonato, 2004] shows the importance of this question:

“So, the ’arbitrage-free’ label has become thesine-qua-nonbadge of acceptability of a model, but, as

with so many labels, it is often forgotten how little it actually guarantees in practice”.

2.2. No-Arbitrage in Practice. Exists two important literatures two important literatures who utilizes the

relationships given by equations 3and 5, and helps to consolidate the importance of the no-arbitrage con-

ditions. The first one is the use of option prices to extract the risk neutral density implied in option prices

observed in market. In this literature the scaled risk neutral density is extracted from option prices when we

calculate the second derivative of option price in relationto the strike price:

(11)∂2C(St, E, τ, rt,τ )

∂E2= e−rt,τ τp(St|St, E, τ, rt,τ )

The second literature, who we pursue in this article, is the extraction of the volatility implied in the option

prices observed in the market. Given the observed priceC andSt, E, τ, rt,τ of the contract, we can invert

the Black-Scholes model to recuperate the volatilityσ implicit in the traded option. Note that for each strike

observed in market we have a implied volatility, the impliedvolatility curve for the observed strikes.

Although the Black-Scholes model assumes a constant volatility, a stylized fact is the existence of the

“smile” effect, which reflects the stochastic volatility present in the returns of assetS .The implied volatility

surface is the estimation of implied volatility curve for each distinct time to maturityτ for contracts for a

same expiration date, which is nominated the term structureof volatility. A problem is that usually exist

few observed options with different strikes in a same trading day, and the implied volatility curve has to be

interpolated or smoothed to the permits the construction ofimplied volatility or the risk neutral extraction.

We can have two situations where the constraints of no-arbitrage, related to the conditions of monotonicity

and convexity in Equations 3 and 5 can be violated. The first situation in when the observed call data

already contains arbitrage conditions, that is, exists call prices increasing or not convex in function of strikes.

[Henstchel, 2003] reports that sources of potential errorsin data includes bid-ask bounce, asynchronous

pricing and finite quote precision in option prices.

The second situation is when the interpolating scheme or thesmoothing method generates points which

violates the monotonicity and convexity constraints. The more frequent situation is a mixture of this two

situations - the presence of arbitrage points generates interpolating and smoothing curves contaminated with

arbitrage situations.

It can be argued that if arbitrage situations exist, they notbe removed from data, and imposing the

monotonicity and convexity constraints we introduce distortions in the observed process. But exist two


situations where is necessary to work with data free of arbitrage. The first one is already cited method of

extraction of risk neutral densities. In this methodology is crucial that the data are arbitrage free so that

the extraction of the density is correct. [Ait-Sahalia & Duarte, 2003] use constrained least squares for the

construction of risk neutral density. The other situation is the use of scheme know as local volatility models,

as the Smile model of [Dupire, 1994].

Arbitrage conditions (real arbitrage situations or “pseudo arbitrage” induced by prices measured with

errors and created by the interpolation scheme) can be of substantial impact, given the nonlinear transforma-

tions applied in calls prices. In risk neutral density extraction the arbitrage conditions in call prices can lead

to bad properties of extracted densities, given that risk neutral is related to differentiate the data two times,

introducing large fluctuations. Arbitrage conditions can lead to the presence of negative probability points

and multimodality in the extracted risk neutral density.

In local volatility models, the arbitrage conditions can affect severely the local volatility, since estimation

of the local volatility surface can be based directly on the call prices or in the implied volatility surface

under no-arbitrage condtions. The presence of arbitrage also affects the stability properties of the numerical

methods used in the resolution of partial diferential equations present in local volatility equations.

There are a large range of methods used in interpolation and smoothing of implied volatility surface. As

examples of the parametric methods employed we have the quadratic specification used in [Shimko, 1993],

and the least squares kernel smoother in [Gourierouxet al. , 1994] and [Fengleret al. , 2003]. Non and

semi-parametric methods includes the use of Nadaraya-Watson regression (e.g. [Ait-Sahalia & Lo, 1998],

[Rosenberg, 2000] and [Cont & da Fonseca, 2002]) and Local Polynomial Smoothing ([Rookley, 1997] and

[Ait-Sahaliaet al. , 2001a] ). But in all these methods, is not guaranteed that the surface is free of arbitrage.

Our contribution to this literature is the use of constrained smoothing b-splines incorporating the restric-

tions of monotonicity and convexity in the process of smoothing of the implied volatility surface. Our work

is related to [Wanget al. , 2004] and more closely to [Fengler, 2005]. In [Wanget al. , 2004] the problem

is the interpolation of option price data, and the shape restrictions in interpolated curves is imposed by the

use of semi-smooth equations minimizing the distance between the implied risk neutral density and a prior

approximation based onL2 norm. [Fengler, 2005] is, like us work, a smoothing problem,but is based on the

theory of natural smoothing splines under shape constraints, where the constraints are imposed via a initial

pre-smoothing procedures and after via recursive quadratic programming methods.

3. CONSTRAINED SMOOTHING B-SPLINES

Our method is based on the constrained smoothing b-splines introduced by [He & Ng, 1999].The con-

strained smoothing b-splines permits to impose monotonicity and convexity in the smoothed curve, and also

additional pointwise constraints. The methodology also share the robustness properties of quantile regres-

sion ([Koenker & Basset, 1978]) methods, as the method formulates the b-spline usingLP projections, and

is less sensible to outliers in reduced samples that the methods of smoothing splines and other interpolation


schemes, and this property is attractive when used in intra-day data and in the case of markets with low

liquidity, what normally it occurs in emerging markets.

The method of smoothing splines are extended by [Boschet al. , 1995] to the problem of estimating a

quantile smoothing spline, i.e. estimating a conditional quantile function specified by the choice of quantile

τ :

(12) ming∈R

n∑

i=1

ρτ (yi − g(Xi))2

+ λ

∫(g,,(x))

2dx

[Koenkeret al. , 1994] consider this problem a special case inLp fitting, in specialL1andL∞ , in the

form:

(13) J(g) = ||g,,||p =

∫(g,,(x)p)1/p

The methodology of [He & Ng, 1999] can be viewed as a special case of 12, again formulating the

smoothing problem using a conditional quantile functiongτ (x), which it is a function ofx such asP (Y <

gτ (x)|X = x) = τ . Sorting the observations(xi, yi)ni=1 with a = x0 < x1 < ... < xn < xn+1 = b , can

be defined a smooth functiong and a indicator functionρτ (u) = 2[τ−I(u < 0)u = [1+(2τ−1)sgn(u)]|u.

Using the concept of fidelity in the form:

(14) fidelity =

n∑

i=1

ρτ (yi − g(xi))

[He & Ng, 1999] utilizes theLp quantile smoothing spline of [Koenkeret al. , 1994] gτLp(x) as the so-

lution of the problem:

(15) ming

fidelity + λLproughness

The roughness measure can be related toL1andL∞ problems as:

(16) L1roughness = V (g′) =

n−2∑

i=1

|g′(x+i+1) − g′(x+

i )|

(17) L∞roughness = V (g′) = ||g′′||∞ = maxxg′′(x)]

and the fidelity measures as:


(18) fidelity =

n∑

i=1

|yi − g(xi)|

(19) s(x) =

N+m∑

j=1

ajBj(x)

[He & Ng, 1999] notes that this problem can be formulated as:

(20) minθ∈RN+⋗

N+m∑

i=1

|yi − xiθ

yi =

(y

0

)andX =

[B

λC

]

whereθ = (a1, a2, ..., aN+m) are the parameters at knotxi..

TheB matrix is given by:

|

(21) B =

B1(x1) . . . BN+m(x1)... . . .

...

B1(xn) . . . BN+m(xn)

and theC matrix by the expression:

(22) C =

B′1(tm+1) − B′

1(tm) . . . B′N (tm+1) − B′

N (tm)... . . .

...

B′1(tN+m) − B′

1(tN+m−1) . . . B′N (tN+m) − B′

N(tN+m−1)

The estimation is based on applying linear programming (interior point method) in

(23) min1′(u + v)|yi − xiθ = u − vi, (u′, v′) ∈ R2(n+M)

The attractive feature of the method for the use in implied volatility construction is the possibility of

incorporate general constraints of monotonicity. The constraints are imposed constructing a matrixH in

the form:


(24) H =

B′1(tm) . . . B′

N+m(tm)... . . .

...

B′1(tN+m+!) . . . B′

N+m(tN+m+1)

The monotonicity can be imposed for decreasing functions making Hθ ≤ 0 for decreasing functions

andHθ ≥ 0 for increasing functions. Convexity constraints also can be imposed, in the case ofmL1 the

convexity is imposed makingCθ ≥ 0 and for the case of and formL∞ trough the use of[

D 0]θ ≥ 0,

and concavity is obtained reverting the signals.

It´s possible to incorporate pointwise constraints:

(25) g(x) = yi

(26) g(x) ≥ yi

(27) g(x) ≤ yi

(28) g′(x) = y

as additional constraints in the linear programming problem.

The knot selection and the smoothing parameterλ in the constrained smoothing method of [He & Ng, 1999]

can be made using the Akaike Information Criteria (AIC) and the Schwartz Information Criteria (SIC). The

AIC and SIC in constrained smoothing splines of [He & Ng, 1999] are given by:

(29) SIC(λ) = log(1

nρτ (yi − mλ))) +

1

2pλlog(n)/n

(30) AIC(λ) = log(1

nρτ (yi − mλ))) + 2(N + m)/n

To construct aadmissibleimplied volatility surface, we impose conditions given by the 6,7,8, equivalent

to the restrictions of a decreasing and convex curve in the matrices B (Eq. 21) and C (Eq. 22), and the

conditions 9 and 10 in the form pointwise contraints. The condition 9 is not observed in data, and can safely

be ignored. Condition 10 is important, and is related to the more out-of-money options, and can be imposed

using a conditionC(St, E, τ, rt,τ ) ≥ 0 in the constrained smoothing spline formulation.


FIGURE 2. Observed Call Prices and Strikes

Strik

es

6400

6600

6800

7000

Time To Maturity

0.05

0.10

0.15

0.20

Observed C

alls 100

200

300

400

500

Observed Call Prices for Dax Contract

4. EMPIRICAL APPLICATIONS

To illustrate the practical use of the methodology, we will go to demonstrate its use in a series of situations

frequently found in the treatment of options data. The first one is the construction of an smoothed implied

volatility surface free of arbitrage using all maturities for a given expiration os the DAX Index, in Section

4.1.

The second example, in Section 4.2, is the interpolation of acalls-strikes curve using all intra-day data

in a given maturity, and comparing with alternative methodologies. In Section 4.3 we will show the use of

the methodology in the construction of local volatility, interms of call prices and the implied volatilities,

when the curve of strikes is contaminated by arbitrage. And in Section 4.4 we will show an application of

the methodology in the risk neutral densities in the presence of arbitrage prices.

4.1. Smoothed Implied Volatility Surface - DAX Index. We demonstrate the methodology for the con-

struction of implied volatility surfaces using aggregatedcall options on DAX Index with expiration date on

01/19/2007. Figure2 shows observed calls prices for all maturities in this contract. The curve has a typical

behaviour, with a large number of calls in more distant maturities, and contains a large number of strikes

with no transactions in some days.


FIGURE 3. Call Prices and Implied Volatility - DAX Call Option 56 Days to Expiration

O

O

O

O

O

O

O

O

O

OO

O

0.98 1.00 1.02 1.04 1.06

5010

015

020

025

030

0

Moneyness

Cal

l Pric

e

*************************************************************************************************************************

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

O*+

Observed DataConstrained SplineSmoothing Spline

0.98 1.00 1.02 1.04 1.06

0.14

0.16

0.18

0.20

0.22

0.24

Moneyness

Impl

ied

Vol

atili

ty

Observed DataConstrained SplineSmoothing Spline

In Figure 3 we focus on the calls prices traded in 56 days (11/24/2006) before expiration. The first graphic

shows the observed calls prices for each strike in this day. Note that we have a evident arbitrage point in for

the moneyness2 of 1.05. Is clear that this point generates a locally crescent and non-convex curve.

We interpolate the call prices using a normal smoothing spline and the constrained smoothing spline, and

with these curves we calculate the implied volatility. The smoothed curve obtained using the smoothing

spline generates a non convex and crescent curve, and this effect generates a deformated implied volatility.

Using the constrained smoothing spline, the smoothed curveis not affected by the arbitrage point, and

clearly respects the no-arbitrage conditions given by a decreasing and convex curve. The implied volatility

surface obtained using the constrained spline shows the usual smile effect, evidencing that the method of

constrained smoothing spline is a effective method to deal with arbitrage-situations. Figure 4 shows the

smoothed curve in call price direction for all observed strikes in this contract. In each day the implied

volatility is by construction free of arbitrage.

We also use the method for generating extrapolation points,that is, to obtain call prices and related

implied volatilities for strikes not observed in this contract. Figure 5 shows the implied volatility surface

using interpolation and extrapolation by the constrained smoothing spline. Again in this figure we interpolate

only in call-strikes direction.

2Moneyness is defined by the ratio between the strike and the forward price for the underlying asset.


FIGURE 4. Implied Volatility Surface - DAX Call Options - Interpolation in Strikes Direction

Strik

es

6400

6600

6800

7000

Time To Maturity

0.10

0.15

0.20

Implied Volatility 0.1

0.2

0.3

0.4

No Arbitrage Implied Volatility Surface for DAX Call Contract

Figure 6 shows the smoothed implied volatility surface using interpolation and extrapolation in strikes

direction, and we impose the restriction∂C(St,E,τ,rt,τ)∂T > 0, using the constrained smoothing spline in the

maturity direction. The result is a fully smoothed curve. This type of smoothed curve is a input on some

models as the model for Forward-staring options presented by [Samuel, 2002] and general models assuming

deterministic smiles. Examples of models generating deterministic smiles are the geometric diffusion as-

sumed in the Black-Scholes model, jump-diffusions with constant ot time-dependent coefficients, displaced

difusions ([Rubinstein, 1983]) and generalizations as displaced jump-diffusions, Derman-Kani restricted-

stochastic volatility model and variance-gamma process.

4.2. Intra-day Interpolation . Another frequent situation in the construction of a call-strike curve using

intra-day data in option prices. In intra-day data we have many quoted call prices for each strike, and the

call-strike curve can be very problematic by the presence ofaberrant call prices in some strikes. Is this

situation we have the necessity of a outliers resistant method of smoothing or interpolation for generate a

well behaved curve.

The constrained smoothing spline method of [He & Ng, 1999] isspecially attractive is this situations,

given the connection of this method with the quantile regression methodology [Koenker & Basset, 1978].

The advantage can be viewed by the robustness to outliers of the median in relation to the mean, and we use

the .5 quantile (the median) as the choice in the Eq. 12.


FIGURE 5. Implied Volatility Surface - DAX Call Options - Interpolation and Extrapola-tion in Strike Direction

Strik

es

6400

6600

6800

7000

Time To Maturity

0.10

0.15

0.20


0.2

0.3

0.4


To verify this property, we construct a call prices-strike curve using the constrained smoothing spline

and compare with the 3 more used methodologies - smoothing spline, Local Polynomial Smoothing and

Nadaraya-Watson Regression. Figure 7 shows the result of this methods for intra-day data on the DAX-Index

for day 19/03/2007. The result shows that Smoothing Spline generates a crescente and non-convex curve,

but also the Nadaraya-Watson and Local Polynomial are affected by the more extreme points, generating

slightly non convex curves. Another interesting fact is that the curves of constrained smoothing spline is

concentred in the region with more observations, while the curves constructed by the other methodologies

in some strikes (moneyness betwenn .96 and .98) are in regions with few points, as the result of influence of

outliers in call prices and can be not be representative of the observed prices in this market.

4.3. Local Volatility Calculation. Local volatility, also knowed as forward volatility, was introduced by

[Dupire, 1994], and can be viewed as the market consensus of instantaneous volatility for strike E for some

future date t. Local volatility is defined as the risk neutralexpectation onQ measure of squared instantenous

volatility conditional onSt = E for a given information set , defined by a filtrationFt :

(31) σ2E,t = EQσ2(St, E, τ, rt,τ )|St = E, Ft


FIGURE 6. Implied Volatility Surface - DAX Call Options- Interpolation in Strikes andand Maturity Direction

Strik

es

6400

6600

6800

7000

Time To Maturity

0.10

0.15

0.20


0.3

0.4


The Dupire formula permits to estimate the local volatilityin terms of observed calls prices and their

derivatives, and is given by:

(32) σ2E,t(St, t) = 2

∂C(St,E,τ,rt,τ)∂t − δC(St, E, τ, rt,τ ) + (r − δ)E

∂C(St,E,τ,rt,τ )∂E

E∂C2(St,E,τ,rt,τ)

∂E2

whereδis the Dirac delta function.

Another used formula, of special interest in this article, of asymptotic equivalence to 32, is the local

volatility in terms of implied volatility and its derivatives (e.g. [Andersen & Brotherton-Ratcliffe, 1997] and

[Dempster & Richards, 2000]):

(33) σ2E,t(St, t) =

bστ + 2∂bσ

∂t + 2E(rt − δ) ∂bσ∂E

E2

1K2 bσt

+ 2 d1Ebσ√τ

∂bσ∂E + d1d2bσ (

∂bσ∂E

)2+ ∂2bσ

∂E2

To verify the properties of constrained smoothing splines in the calculation of local volatility, we construct

a experiment using call prices and implied volatilities to construct local volatility. The data contains calls

prices in function of strikes mimicking call data observed in market for a fixed maturity. The original data,


FIGURE 7. Intraday Interpolation and Strikes

0.94 0.96 0.98 1.00 1.02 1.04 1.06

010

020

030

040

0

Moneyness

Cal

l Pric

e

Constrained SplineSmoothing Spline SplineNadaraya−WatsonLocal Polynomial

marked by the ’o’ symbol in subfigure a) of Figure 4.3 respect the no-arbitrage conditions. We replace the

call price for strike 2950 for a point violating the no arbitrage conditions, generating a contaminated curve.

We construct the local volatility using three approachs - the first solving numerically the PDE generated

by the Dupire equation, second using the local volatility interms of call prices (Eq. 32) and the third using

the local volatility in terms of implied volatilities (Eq. 33).

The sub-figures b), c) and d) shows respectly the local volatilities for the original and arbitrage-free data,

contaminated data and the local volatility using contaminated data treated by the constrained smoothing

splines. The presence of a single arbitrage point affects severaly the local volatility estimation, for all three

methods of calculating the local volatility.

The local volatility calculated in terms of implied volatility is affected in the same direction as the implied

volatility, being dislocated for top of the original curve.The local volatility in terms of Dupire PDE and call

prices are more affected, converging to a zero local volatility in strike 2950. These effects shows the extreme

dependence on conditions of no-arbitrage conditions in calculation of local volatility.

The local volatility calculated using the contaminated data, but treated by the constrained smoothing

spline shows (sub-figure c) )that the methodology it allows to recover a local volatility curve with the same

characteristics of the true curve. The curve calculated solving the PDE is more affected, but retains the


FIGURE 8. Local Volatility

2650 2700 2750 2800 2850 2900 2950 3000

05

01

00

15

02

00

strikes

Ca

ll P

rice

s

+

(a) Original and Contaminated Data

2700 2750 2800 2850 2900 29500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.1

66

66

7

Implied Volatility and Local Volatility

Implied VolatilityDupire formulaDupire equation (Using Call Prices)Dupire equation (Using Implied Volatility

(b) Local Volatility for No-ArbitrageCurve

2700 2750 2800 2850 2900 29500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.1

66

66

7



(c) Local Volatility for ContaminatedCurve

2700 2750 2800 2850 2900 29500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.1

66

66

7



(d) Local Volatility Using ConstrainedSmoothing Spline

general shape of true curve, while the curves calculated in terms of call prices and implied volatility are less

affected and resembles more closely the true local volatility curve.

4.4. Risk Neutral Density Extraction. The methodology also can be used as a pre-processing technique

in situations indirectly related to volatility, as the construction of risk neutral densities using option data.

As example, we use the methodology of fitting the risk neutralestimation by the mixture of log-normals as

[Bahra, 1996], [Melick & P., 1997] and [Ritchey, 1990].


The risk neutral estimator by the mixture of-log normals is based on fitting a mixture of log-normals to

the second diference of the observed curve of call prices prices in functions of strike. Ignoring discounting,

the estimatorp for the risk-neutral density is given by the relationship:

(34) p(St|St, E, τ, rt,τ ) =∂2C(St, E, τ, rt,τ )

∂E,2

whereC andE are the observed calls and strikes. Remember that a simple way of calculating a second

derivative is differenciate the data two times, but this estimation is too irregular and render a poor description

of the risk neutral expectations (this approach was introduced by [Breeden & Litzenberger, 1978]). To obtain

a more precise estimation of the risk neutral density, is fitted a log-normal distribution or a mixture of log-

normal distribution for the differenced data, and is this way is obtained a density with the usual properties.

But this simple method is not imunne to violations of no-arbitrage conditions. To exemplify, we utilize

the data on LIFFE Bund Options analized in [Svensson & Soderling, 1997]. We replace the call data on

strike 99.50 by a point violating the decreasing condition of no-arbitrage, as showed in sub-figure a) of

Figure 4.4. We extract the risk neutral extraction using thesame methodology of mixture of log-normals

used in [Svensson & Soderling, 1997], for the contaminated and constrained smoothed data. The original

risk neutral density estimations are in [Svensson & Soderling, 1997].

The risk neutral density using the contaminated data generates a bimodal risk neutral density (sub fig-

ure b), Figure 4.4) when using only one-log normal in approximation. The mixture of log-normals is more

robust to this arbitrage point, but the variance of log-normal mixture is slight bigger that the original with-

out contamination. When we treat the data using the constrained smoothing spline, both the risk neutral

densitities, using one and two normal densities, are very close to the original risk neutral density without

the contamination for the arbitrage point, and confirming that the methodology can be used as a tool for

pre-processing options data for financial analysis that need the no-arbitrage conditions.

.

5. CONCLUSIONS

In this article we introduce the use of constrained smoothing splines for the treatment of arbitrage situa-

tions present in options data, with special reference to theconstruction of implied volatility surfaces.

The methodology it showed to be efficient in some practical problems, as the construction of smoothed

and free of arbitrage implied volatility surfaces, and alsoon other applications derived from call prices as the

construction of local volatility using [Dupire, 1994] methodology, and the extraction of risk neutral densities.

The methodology has some apparent advantages on competing methodologies. It allows to impose directly

the shape restrictions of no-arbitrage in the format of the curve, and is robust the aberrant observations.

An additional advantage is that this methodology is of general purpose in the situations that need free

of arbitrage data, and a possible implementation would be asa pre-processing tool for options data before


FIGURE 9. Risk Neutral Estimation

92 94 96 98 100 102 104−1

0

1

2

3

4

5

6Contaminated CurveCOBS Interpolated Curve

(a) Original and Contaminated Data

4.45 4.5 4.55 4.6 4.650

2

4

6

8

10

12

14

16

18Pdf from LIFFE Bunds option data, 6 April 1994

log Bund price

two N()one N()

(b) Risk Neutral Extraction - ContaminatedData

4.45 4.5 4.55 4.6 4.650

2

4

6

8

10

12

14

16

18Pdf from LIFFE Bunds option data, 6 April 1994

log Bund price

two N()one N()

(c) Risk Neutral Extraction - ConstrainedSmoothed Data


the use in other applications, as we illustrate in the empirical section.. Examples are the principal compo-

nent analysis and the fitting of dynamics of implied volatility surfaces (e.g. [Cont & da Fonseca, 2002] and

[Fengleret al. , 2003])

Although exists alternative methodologies for some specific applications, in special extraction of risk

neutral densities, these tools are not of ample use due the specific nature of the problems for which they had

been developed, as example [Bondarenko, 2003].

The methodology presented here can be useful for an ample class of applications which uses data derived

from options prices and implied volatility, answering to a practical market demand.

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