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Imposing No-Arbitrage Conditions In Implied Volatility Surfaces Using Constrained Smoothing Splines
Márcio Poletti Laurini
Insper Working PaperWPE: 096/2007
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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 marciopl@isp.edu.br.1
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.
IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 3
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.
IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 4
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
IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 5
(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
IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 6
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
IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 7
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:
IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 8
(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:
IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 9
(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.
IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 10
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.
IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 11
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.
IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 12
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.
IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 13
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
Implied Volatility 0.1
0.2
0.3
0.4
No Arbitrage Implied Volatility Surface for DAX Call Contract
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
IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 14
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
Implied Volatility 0.2
0.3
0.4
No Arbitrage Implied Volatility Surface for DAX Call Contract
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,
IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 15
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
IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 16
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
Implied Volatility and Local Volatility
Implied VolatilityDupire formulaDupire equation (Using Call Prices)Dupire equation (Using Implied Volatility
(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
Implied Volatility and Local Volatility
Implied VolatilityDupire formulaDupire equation (Using Call Prices)Dupire equation (Using Implied Volatility
(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].
IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 17
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
IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 18
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
IMPOSING NO-ARBITRAGE CONDITIONS IN IMPLIED VOLATILITY SURFACES USING CONSTRAINED SMOOTHING SPLINES 19
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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