on the calibration of distortion risk measures to bid-ask...
TRANSCRIPT
On the calibration of distortion risk measuresto bid-ask prices
Karl Friedrich Bannör
Chair of Mathematical Finance,Technische Universität München,
Parkring 11, 85748 Garching-Hochbrück, Germany,email: [email protected],
phone: +49 89 289 17415.
Matthias Scherer
Chair of Mathematical Finance,Technische Universität München,
Parkring 11, 85748 Garching-Hochbrück, Germany,email: [email protected],
phone: +49 89 289 17402.
Abstract
We investigate the calibration of a non-linear pricing model to quoted bid-ask pricesand show the existence of a solution in a broad class of distortion risk measures, fol-lowing the frameworks of [Cherny and Madan 2010] and [Bannör and Scherer 2011a].We present an approximation of distortion risk measures by a piecewise linearapproximation of concave distortions. This is used to construct a tractable non-parametric calibration procedure to bid-ask prices based on piecewise linear con-cave distortion functions. For an empirical proof of concept, we calibrate quotedbid-ask prices non-parametrically and w.r.t. parametric families and obtain a jump-linear structure. To conclude, we suggest using the parametric family of ess sup-expectation convex combinations as a suitable family of distortion functions, allow-ing for fast and efficient calibration.
Keywords
Distorted probabilities, distortion function, non-parametric calibration, conic finance,convex risk measure, spectral risk measure, bid-ask spread
1
1 Introduction
1 Introduction
Since the rise of risk-neutral valuation techniques, approaches to bid-ask price model-ing enjoy increasing attention by the derivatives community. Classical results interpretbid-ask prices as a consequence of incomplete markets, interpreting the ask (resp. bid)price as the supremum (resp. infimum) of the expectation w.r.t. all equivalent martin-gale measures. Meanwhile, the seminal paper [Artzner et. al. 1999] established the fruit-ful theory of convex risk measures (a standard reference is [Föllmer and Schied 2004]),which turned out to be interesting from both a mathematical and a practical point ofview. Since the supremum w.r.t. measures which are absolutely continuous w.r.t. thereal-world measure can be interpreted as convex risk measures, several papers endorseto calculate bid-ask prices by the usage of convex risk measures (as a natural gener-alization to incomplete markets pricing), either in an incomplete markets setting (see[Carr et. al. 2001, Xu 2006]), due to abstract considerations (see [Bion-Nadal 2009]), orin a model/parameter uncertainty setting (see [Cont 2006, Bannör and Scherer 2011a]).Unfortunately, a concrete implementation of the developed theory immediately invokesthe question which risk measure to choose for calculating bid-ask prices. Thus, in mostpapers, the choice of risk measure is regarded as somewhat subjective.
A more “objective” way of choosing the risk measure is obtained by calibrating modelbid-ask prices to given market prices, similar to the standard calibration to mid marketprices. [Cherny and Madan 2010] use some parametric calibration scheme to calibratetheir conic finance framework to bid-ask prices, focusing on parametric classes of distor-tion functions. However, restricting bid-ask pricing to a narrow class of parameterizeddistortion risk measures (characterized by a class of concave distortion functions) mayexclude better fitting alternatives, since the shape of the distortion functions is predefinedby its parametrization.
In this paper, we close this gap and discuss how one can re-engineer distortion risk mea-sures from quoted bid-ask prices. Thus, we first provide existence theorems, ensuringthat the bid-ask calibration problem can be solved in proper domains of distortion func-tions. In contrast to [Cherny and Madan 2010], we suggest a non-parametric approachfor obtaining the market-implied distortion risk measure for pricing, not restricting our-selves to a specific parametric shape of the distortion function. This ansatz is based on apiecewise linear approximation. Our non-parametric approach provides more flexibilityfor solving the bid-ask calibration problem and is simple to implement. Furthermore, itenables us to empirically observe the shape of possible market-implied distortion func-tions. We present an empirical analysis in the parameter risk setting established in[Bannör and Scherer 2011a, Bannör and Scherer 2011b], comparing our non-parametriccalibration to a calibration within the AVaR- and minmaxvar -families. Interestingly, inthe non-parametric approach we obtain a characteristic pattern with a jump close to zeroand a linear behavior afterwards. This gives rise to the introduction of an alternativeparametric class of distortion functions, the ess sup-expectation convex combinations,which allow for much faster calibration than the AVaR-/minmaxvar -families. Further-
2
2 Background and notation
more, we enhance the existence theorems to larger families of distortion functions withdiscontinuities at zero, containing the newly introduced family.
Our paper is organized as follows: First, we recall the necessary background concerningdistortion functions and Choquet integrals and summarize some suggestions to bid-askpricing employing distortion functions. Second, we state the bid-ask calibration prob-lem and proof the existence of a solution in the set of distortion risk measures (undermild technical restrictions, e.g. Lipschitzianity of the corresponding distortion function).Third, we present a numerical ansatz for the calibration problem based on a piecewiselinear approximation of distortion functions, enabling us to reduce the bid-ask calibrationproblem to a constrained optimization problem on the unit cuboid. Finally, we comparethe calibration performance of our non-parametric approach to a parametric calibrationà la [Cherny and Madan 2010] and introduce a new family of distortion functions basedon our calibration result.
2 Background and notation
2.1 Preliminaries from the theory of distortion risk measures
In this section, we recall the crucial definitions and theorems related to distortion riskmeasures. A standard reference for convex risk measures is [Föllmer and Schied 2004],for details about Choquet integration we refer to [Denneberg 1994].
Definition 2.1 (Distortion function)Let γ : [0, 1] → [0, 1]. γ is called a distortion function iff γ is monotone, γ(0) = 0, andγ(1) = 1.
The interpretation of distortion functions is as follows: Given a probability space (Ω,F , P ),instead of measuring the probability of a set A classically via P (A), we alternativelyconsider the distorted probability γ(P (A)) w.r.t. some distortion function γ. Obvi-ously, the set function γ P : F → [0, 1] is not a probability measure any more forgeneral γ, but preserves monotonicity w.r.t. the set order: A ⊂ B, A,B ∈ F impliesγ(P (A)) ≤ γ(P (B)). This type of set functions is often referred to as “capacities” (cf.[Föllmer and Schied 2004]). The ordinary Lebesgue integral w.r.t. probability measurescan be extended to an integral w.r.t. monotone set functions, the Choquet integral.
Definition 2.2 (Choquet integral w.r.t. distorted probabilities)Let (Ω,F , P ) be a probability space and γ a distortion function. Let furthermore X :Ω → R be a bounded random variable. The Choquet integral of X w.r.t. γ P is thendefined as
∫
ΩX d(γ P ) =
∫ 0
−∞γ(P (X > x)) − 1 dx+
∫ ∞
0γ(P (X > x)) dx.
3
2.1 Preliminaries from the theory of distortion risk measures
The Choquet integral is a monotone, translation invariant, positively homogeneous, andcomonotonic additive functional and coincides with the Lebesgue integral w.r.t. P for γ =id[0,1]. Furthermore, it can be shown (cf. [Denneberg 1994]) that if γ is concave, the Cho-quet integral is a subadditive functional. Thus, Choquet integrals w.r.t. concave distortedprobabilities fulfill the axioms of convex risk measures (even coherent risk measures)and are often referred to as distortion risk measures. Some authors (e.g. [Acerbi 2002])treat so-called spectral risk measures instead, which is a somewhat equivalent approachto distortion risk measures (cf. [Föllmer and Schied 2004, Gzyl and Mayoral 2008]). Inthis work we always use the notion of distortion risk measures, due to the convenientcomputability of the Choquet integral.
One of the well-known examples for a distortion risk measure is the celebrated riskmeasure Average-Value-at-Risk (AVaR), treated, e.g., in [Acerbi and Tasche 2002].
Example 2.3 (Average-Value-at-Risk)For a probability space (Ω,F , P ), a given significance level α ∈ (0, 1], and an integrableX ∈ L1(P ), the α-Value-at-Risk is defined as the upper α-quantile VaRα(X) := qP
X(1−α)and the α-Average-Value-at-Risk is defined by
AVaRα(X) :=1
α
∫ α
0VaRβ(X) dβ.
It can easily be shown that the AVaRα can be represented as a Choquet integral w.r.t. adistorted probability. The corresponding distortion function is given by
γα(u) =
uα, u ∈ [0, α]
1, otherwise
and γα is obviously concave.
As a more elaborate example, [Cherny and Madan 2009] introduce several parametricfamilies of concave distortion functions and calculate positions w.r.t. the induced distor-tion risk measures. The following minmaxvar -family of concave distortions is introducedin [Cherny and Madan 2009] and successfully employed in [Cherny and Madan 2010].
Example 2.4 (minmaxvar-family of concave distortions)Let ψx(y) : [0, 1] → [0, 1], x ∈ R≥0, defined by1
ψx(y) := 1 −(
1 − y1
x+1
)x+1.
It can easily be verified that ψx is a concave distortion function.
1Choosing x ∈ N allows for the following interpretation of the minmaxvar -distortion functions: If arandom variable X ∈ L1(P ) has the same distribution as minZ1, . . . , Zx+1, Z1, . . . , Zx+1 i.i.d.,then
∫
X d(ψx P ) =
∫
maxZ1, . . . , Zx+1 dP
follows.
4
2.2 Bid-ask pricing
2.2 Bid-ask pricing
Several authors (see, e.g., [Carr et. al. 2001, Xu 2006, Bion-Nadal 2009, Cherny and Madan 2010,Bannör and Scherer 2011a]) have suggested modeling bid-ask prices with convex riskmeasures. All approaches have in common that a certain convex risk measure Γ := Γask
is employed for the calculation of ask prices. Furthermore, the dual functional Γbid,defined by Γbid(X) := −Γask(−X), is usually used for calculating bid prices. In our case,we stick to this kind of “bid-ask symmetry”.
[Carr et. al. 2001] and [Cherny and Madan 2010] generalize the classical results from in-complete markets and deliver an explanation for bid-ask spreads. In incomplete marketswith real-world measure P , a set of “stress-test measures” Q is selected from the set of allequivalent martingale measures and the ask price of a contingent claim X is determinedas the supremum of the expectations w.r.t. the stress-test measures2:
Γ(X) := supQ∈Q
EQ[X].
In particular, [Cherny and Madan 2010] suggest choosing the set of stress test measuresby an acceptability index. Their result consists of using coherent risk measures inducedby a parametric family of concave distortion functions (γλ)λ∈Λ with
Γλ(X) :=
∫
X d(γλ P ).
This approach is described in [Cherny and Madan 2010] as “conic finance” and we willadopt this name.
A different approach to bid-ask prices, also relying on convex risk measures, is discussedin [Bannör and Scherer 2011a]: Here, bid-ask spreads are explained in the context of pa-rameter uncertainty of a parametric family of martingale measures (Qθ)θ∈Θ. In presenceof a distribution R on the parameter space Θ, one can define for contingent claims Xand law-invariant, normalized convex risk measures ρ the parameter risk-captured price,which may serve as an ask price, by
Γ(X) := ρ(θ 7→ Eθ[X]),
denoting the risk-neutral price w.r.t. a fixed parameter θ ∈ Θ by Eθ[X].
Obviously, we can restrict the choice of risk measures to distortion risk measures, whichare clearly law-invariant and normalized. Thus, using a concave distortion γ, the askprice turns into
∫
ΘEθ[X] (γ R)(dθ).
2[Cont 2006] discuss a similar supremum-based approach not restricting to an incomplete marketssetting.
5
3 The bid-ask calibration problem
In what follows, we discuss the calibration to bid-ask prices that are created in a dis-tortion risk measure-driven environment. Interestingly, it can be shown that the conicfinance approach of [Cherny and Madan 2010] can be embedded into the parameter riskframework, since it restricts parameter risk to the market price of risk. To cover bothideas in a unified way, we formulate the framework in a general manner and write f(X)(f being a linear map w.r.t. some matching vector spaces) as a symbol for the integrandunder the Choquet integral w.r.t. the distorted probability and R as symbol for theprobability measure. Hence, the generic ask pricing formula is given by
∫
f(X) d(γ R). (1)
In the parameter risk-capturing framework described in [Bannör and Scherer 2011a],f(X) = Eθ[X] and R is the probability measure on the parameter space Θ. In the conicfinance environment as described in [Cherny and Madan 2009], we identify f(X) = Xand R is simply the real-world measure P .3
3 The bid-ask calibration problem
For classical linear pricing systems, the calibration of liquid instruments C1, . . . , CM totheir quoted mid prices Cmid
1 , . . . , CmidM is a common way of obtaining the model’s un-
observable parameters; thus obtaining a market-implied distribution (from a previouslyselected family) for the expectation. Consequently, when extending to non-linear pricingsystems as briefly summarized in Section 2.2, it is natural to think about obtaining amarket-implied convex risk measure from quoted bid-ask prices. As a starting point,[Cherny and Madan 2010] develop closed-form solutions for the calculation of bid andask prices in their conic finance approach. Using these closed-form representations, theycalibrate to given bid-ask prices employing the minmaxvar -family of distortion func-tions.
In this section, we restate the bid-ask calibration problem in a formal way, using thegeneric language of (1), and show that there exists a solution in a mildly restrictedclass of distortion functions, not restricting to a fixed parametric shape of the distortionfunction. As a corollary, we show the existence of a solution of the bid-ask calibrationproblem in the AVaR- and minmaxvar -family of distortion functions.
We start with a general formulation of the bid-ask calibration problem.
Problem 3.1 (Bid-ask calibration problem)Let C1, . . . , CM be contingent claims (e.g. vanillas) with given market bid-ask quotes
(Cbid1 , Cask
1 ), . . . , (CbidM , Cask
M ). Let furthermore η : R2M≥0 → R≥0 be an error function4
measuring the pricing error between model prices and market quotes.
3Due to the general formulation, we omit the area of integration in the generic ask pricing formula (1),one could write dom(f(X)).
4We call η an error function (in the spirit of [Bannör and Scherer 2011b]) iff η(0, . . . , 0) = 0 and η iscomponentwise non-decreasing.
6
3 The bid-ask calibration problem
A convex risk measure Γ solves the (symmetric) bid-ask calibration problem on a domainG, if Γ minimizes the function
Γ 7→ η(
| − Γ(−C1) − Cbid1 |, . . . , | − Γ(−CM ) − Cbid
M |,
|Γ(C1) − Cask1 |, . . . , |Γ(CM ) − Cask
M |)
over the set of admissible functionals G.
The formulation of the bid-ask calibration (Problem 3.1) is quite general. Since theclass of convex risk measure functionals can be very large and a numerical solutionmay be difficult to obtain, we restrict ourselves to the set of distortion risk measures. Asdescribed in the previous section, they represent a very tractable and rich class of convexrisk measures. The tractability of distortion risk measures stems from the convenientrepresentation as a Choquet integral w.r.t. a distorted probability. We formulate ourresults in the generic language of (1).
The first theorem provides the existence of a best fit to market prices in broadly definedsubclasses of distortion risk measures. In short, Problem 3.1 is solvable.
Theorem 3.2 (Existence of a solution to the bid-ask calibration problem)Let K > 0, η be a continuous error function, f(X) be R-a.s. bounded and define GK :=γ : [0, 1] → [0, 1] : γ Lipschitz, concave distortion function with Lipschitz constant K.Furthermore, write Γγ(X) :=
∫
f(X) d(γ R) for γ ∈ GK . Then the bid-ask calibrationproblem has a solution in GK , i.e. there exists a minimizing distortion function γ ∈ GK
s.t. the function
η : γ 7→ η(
| − Γγ(−C1) − Cbid1 |, . . . , | − Γγ(−CM ) − Cbid
M |,
|Γγ(C1) − Cask1 |, . . . , |Γγ(CM ) − Cask
M |)
is minimized in γ in the following sense:
η(γ) ≤ η(γ) for all γ ∈ GK .
ProofStep 1: GK is compact in the uniform topology.
Let (γN )N∈N be a uniformly convergent sequence in GK with γ := limN→∞ γN . It is easyto see that γ is monotone and concave. Furthermore, γ(0) = 0 and γ(1) = 1. Thus, γ isa concave distortion function. It is also Lipschitz with Lipschitz constant K > 0: Chooseε > 0, then there exists some N ∈ N s.t. ‖γ − γN‖∞ < ε/2. Thus, by γN ∈ GK and thetriangular inequality,
|γ(x) − γ(y)| ≤ |γ(x) − γN (x)| + |γN (x) − γN (y)| + |γN (y) − γ(y)| < K|x− y| + ε
for all x, y ∈ [0, 1]. Hence, GK is closed. Since GK is uniformly bounded and equicon-tinuous by construction, the Arzelà-Ascoli theorem yields that GK is compact.
7
3 The bid-ask calibration problem
Step 2: The map γ 7→∫
f(X) d(γ R) is ‖ · ‖∞-continuous.
Let γN → γ and recall the definition of the Choquet integral (e.g. [Denneberg 1994]).Dominated convergence (applicable due to the boundedness of f(X)) immediately delivers
∫
f(X) d(γN R) =
∫ 0
−∞γN (R(f(X) > s)) − 1 ds+
∫ ∞
0γN (R(f(X) > s)) ds
→
∫ 0
−∞γ(R(f(X) > s)) − 1 ds+
∫ ∞
0γ(R(f(X) > s)) ds
=
∫
f(X) d(γ R),
hence γ 7→∫
f(X) d(γ R) is continuous.
Since η is a continuous error function and GK is compact, there exists a γ ∈ GK s.t.η(γ) ≤ η(γ) for all γ ∈ GK .
The assumption of the distortion functions being Lipschitz is not too restrictive forpractical purposes: Every concave distortion function on [0, 1] is Lipschitz on [ε, 1] withconstant 1/ε for every ε ∈ (0, 1) (furthermore, it can only have a jump in zero and iscontinuous on (0, 1]). In case of a numerical treatment, it is therefore sufficient to focuson Lipschitz functions.
Remark 3.31. One can show that the AVaR- and minmaxvar-families described in Examples 2.3
and 2.4 are Lipschitz when the inducing parameter set is mildly restricted. Thus,these examples are captured by our calibration framework.
2. If we scrutinize the proof, in the second part we have only used that γN → γpointwise, in particular, γ 7→
∫
X d(γ R) is sequentially continuous w.r.t. thepointwise topology.
We can immediately conclude the following corollary.
Corollary 3.4Let G ⊂ GK for some K > 0 (GK as above) be ‖ · ‖∞-closed. The bid-ask calibrationproblem then has a solution in G.
ProofSince G is ‖ · ‖∞-closed and GK is ‖ · ‖∞-compact, G is ‖ · ‖∞-compact. Thus, the samearguments as above provide the existence of a minimizing element γ ∈ G.
In particular, Corollary 3.4 can be applied to ensure the existence of a solution of thebid-ask calibration problem in some parametric family. Here, we show – under mildtechnical restrictions – the existence of a solution in the popular AVaR- and minmaxvar -families.
8
4 A non-parametric calibration scheme for bid-ask price
Example 3.51. Let ε > 0 and define the ε-bounded AVaR-family
Gε :=
γ : [0, 1] → [0, 1] : ∃α ∈ [ε, 1] : γ|[0,α](y) =y
αand γ|[α,1](y) = 1
.
Obviously, γN → γ implies αN → α, thus γN ∈ Gε for all N ∈ N implies γ ∈ N,thus Gε is closed. Corollary 3.4 shows that the bid-ask calibration problem has asolution in Gε.
2. Let K ∈ [0,∞) and define
GK :=
γ : [0, 1] → [0, 1] : ∃L ∈ [0,K] : γ(y) = 1 − (1 − y1
L+1 )L+1
,
i.e. γ ∈ GK is a minmaxvar-type distortion function with parameter L ∈ [0,K].If (γN )N∈N is a sequence in GK that converges uniformly, it is easy to show thatLN → L, in particular, limN→∞ γN ∈ GK . Thus, GK is ‖ · ‖∞-closed, applyingCorollary 3.4 shows that the bid-ask calibration problem has a solution in GK .
4 A non-parametric calibration scheme for bid-ask price
In this section, we present an alternative to the calibration of parametric families à la[Cherny and Madan 2010]. We estimate the distortion function in a non-parametric way,using a piecewise linear approximation which eventually reduces the bid-ask calibrationproblem to a constrained optimization problem on the unit cuboid. We argue that a non-parametric calibration scheme may be useful to obtain the shape of a market-implieddistortion function: Empirical results on market-implied distortion functions are veryrare, yet. Thus, a parametric approach may not capture the distortions appropriatelythat are quoted on the market. Furthermore, our approach can be used to find andjustify some parametric shape of the market-implied distortion function from empiricalobservations.
4.1 General results
We start with the following lemma, ensuring that a piecewise linear approximation ofa distortion function is a sensible approximation and does not produce much differentdistorted expectations.
Lemma 4.1 (Piecewise linear approximation)Let γ : [0, 1] → [0, 1] be a continuous concave distortion function. Let N ∈ N and choose0 =: y0 < y1 < · · · < yN := 1. Denote n(x) := maxn ∈ 0, . . . , N : x ≥ yn anddefine
γN (x) :=
γ(yn(x)) +γ(yn(x)+1)−γ(yn(x))
yn(x)+1−yn(x)(x− yn(x)) , x ∈ [0, 1)
1 , x = 1.(2)
9
4.1 General results
Then γN is a piecewise linear, concave distortion function and γN → γ uniformly, if themesh of the decompositions YN := y0, . . . , yN converge to zero, i.e. if maxn∈1,...,Nyn−yn−1 → 0 for N → ∞.
ProofLet N ∈ N. Obviously, γN is piecewise linear, monotone, and γN (0) = 0, γN (1) = 1.Thus, γN is a piecewise linear distortion function. Furthermore, γN is concave, sincethe a.e. defined derivative is decreasing. The convergence property follows from the proofthat the set of piecewise linear functions is uniformly dense in the space of continuousfunctions on the unit interval C[0, 1] (cf. [Shilov 1996, Section 1.23]).
In particular, for all K > 0: GlinK := γ ∈ GK : γ is piecewise linear is dense in GK .
Thus, approximation in GlinK is a sensible way for any distortion function in GK .
Furthermore, the continuity of the error function assures that integrals w.r.t. piecewiselinear concave distortion functions converge as well, so optimizing in the piecewise con-cave distortion functions may not provide significant disadvantages in accuracy comparedto optimizing in GK .
As a second result, we note that piecewise linear concave distortion functions are fullycharacterized by an N -tupel of real numbers (with constraints). Considering the de-composition 0 = y0 < y1 < · · · < yN = 1 and piecewise linearity on [yn−1, yn] for alln = 1, . . . , N , a piecewise concave distortion function γ is fully characterized by thedifference vector ∆γ ∈ R
N , where
∆γn := γ(yn) − γ(yn−1), n = 1, . . . , N .
Monotonicity, concavity, and γ(0) = 0, γ(1) = 1 immediately deliver for the discretederivatives
∆γ1
y1 − y0≥
∆γ2
y2 − y1≥ · · · ≥
∆γN
yN − yN−1≥ 0 and
N∑
n=1
∆γn = 1. (3)
In particular, solving the bid-ask calibration problem can be traced back to findingthe best matching (∆γ1, . . . ,∆γN ) fulfilling the conditions in (3). The key result forcalculating distorted expectations is presented in the following theorem.
Theorem 4.2 (Expectation w.r.t. a piecewise linear concave distorted probability)Let N ∈ N, γN be a piecewise linear concave distortion function with decomposition0 = y0 < y1 < · · · < yN = 1, and difference vector (∆γ1, . . . ,∆γN ) ∈ [0, 1]N . Letfurthermore f(X) ∈ L∞(R) and y0, . . . , yN are points of continuity of Ff(X). Then thedistorted expectation of f(X) w.r.t. γN R can be calculated via
∫
X d(γN R) =N∑
n=1
∆γnE
[
f(X)∣
∣
∣f(X) ∈
[
VaRyn(f(X)),VaRyn−1(f(X))]
]
.
10
4.1 General results
ProofLet N ∈ N. Define γn := ∆γn/(yn − yn−1) and fn(y) := minyn,maxyn−1, y − yn−1
for n ∈ 1, . . . , N. With some elementary substitutions it can be verified that γN as in(2) can be represented as
γN (y) =N∑
n=1
γnfn(y).
In particular, we obtain for X ∈ L∞(R) by applying linearity of Lebesgue-Stieltjes inte-grals in the integrators
∫
f(X) d(γN R) = −
∫ ∞
−∞x (γN F−f(X))(dx)
= −
∫ ∞
−∞x
(
N∑
n=1
γn(fn F−f(X))
)
(dx)
= −N∑
n=1
γn
∫ ∞
−∞x (fn F−f(X))(dx).
Scrutinizing fnF−f(X), one observes that fnF−f(X) is constant on [−∞, qyn−1(−f(X))]∪[qyn(−f(X)),∞], denoting by qα(−f(X)) the lower α-quantile of −f(X), due to the def-inition of fn. Therefore, the integration can be restricted to [qyn−1 , qyn ]. Hence,
−1
yn − yn−1
∫ ∞
−∞x (fn F−f(X))(dx)
= −1
yn − yn−1
∫ qyn
qyn−1
xF−f(X)(dx)
= −E
[
− f(X)∣
∣
∣− f(X) ∈
[
qyn−1(−f(X)), qyn(−f(X))]
]
= E
[
f(X)∣
∣
∣f(X) ∈
[
VaRyn(f(X)),VaRyn−1(f(X))]
]
delivers the desired result, since qα(−f(X)) = −VaRα(f(X)).
The Choquet integral of f(X) w.r.t. the distorted probability γN R can be used tocalculate ask prices. To get a similarly convenient result for bid prices, we find for−f(X):
−
(∫
−f(X) d(γN R)
)
=N∑
n=1
∆γnE
[
f(X)∣
∣
∣f(X) ∈
[
qyn−1(f(X)), qyn(f(X))]
]
.
In particular, the above formulae yield a tractable setting for solving the bid-ask calibra-tion problem within the piecewise linear concave distortion functions, given a decompo-sition Y = y0, . . . , yN : 0 = y0 < · · · < yN = 1 of the unit interval. Solving the bid-ask
11
4.1 General results
calibration problem corresponds to finding a vector ∆γ ∈ RN satisfying the constraints
in (3) and minimizing some distance between model bid-ask prices and quoted marketbid-ask prices.
Algorithm 4.3 (Bid-ask calibration to piecewise concave distortions on a fixed grid)Let (Cbid
1 , Cask1 ), . . . , (Cbid
M , CaskM ) be given bid-ask market quotes of contingent claims
C1, . . . , CM and η : R2M≥0 → R≥0 an error function. The bid-ask calibration problem
can be solved by the following algorithm:
1. Choose N ∈ N.
2. Choose a decomposition Y = y0, . . . , yN : 0 = y0 < · · · < yN = 1 of the unitinterval.
3. CalculateE
[
f(Cj)∣
∣
∣f(Cj) ∈
[
qyn−1(f(Cj)), qyn(f(Cj))]
]
andE
[
f(Cj)∣
∣
∣f(Cj) ∈
[
VaRyn(f(Cj)),VaRyn−1(f(Cj))]
]
for n = 1, . . . , N , j = 1, . . . ,M .
4. Solve the constrained optimization problem
min∆γ∈RN
≥0
η
(∣
∣
∣
∣
∣
N∑
n=1
∆γnE[f(C1)|f(C1) ∈ [qyn−1 , qyn ]] − Cbid1
∣
∣
∣
∣
∣
, . . . ,
∣
∣
∣
∣
∣
N∑
n=1
∆γnE[f(CM )|f(CM ) ∈ [qyn−1 , qyn ]] − CbidM
∣
∣
∣
∣
∣
,
∣
∣
∣
∣
∣
N∑
n=1
∆γnE[f(C1)|f(C1) ∈ [VaRyn−1 ,VaRyn ]] − Cask1
∣
∣
∣
∣
∣
, . . . ,
∣
∣
∣
∣
∣
N∑
n=1
∆γnE[f(CM )|f(CM ) ∈ [VaRyn−1 ,VaRyn ]] − CaskM
∣
∣
∣
∣
∣
)
subject to
N∑
n=1
∆γn = 1, ∆γ ≥ 0,
(
∆γ2
y2 − y1−
∆γ1
y1 − y0, . . . ,
∆γN
yN − yN−1−
∆γN−1
yN−1 − yN−2
)
=: D(∆γ) ≤ 0.
Thus, the core of the bid-ask calibration problem is reduced to a non-linear constrainedoptimization problem in the compact convex space
G :=
∆γ ∈ RN : ∆γ ≥ 0, D(∆γ) ≤ 0,
N∑
n=1
∆γn = 1
.
12
4.1 General results
Algorithm 4.3 treats the optimization on a fixed decomposition Y = y0, . . . , yN ofthe unit interval. Obviously, the methodology can be enhanced by varying over thedecompositions as well, which also delivers a constrained optimization problem. Theproblem accompanying optimization over decompositions is performance: Varying de-compositions considerably slow down the optimization procedure, since all conditionalexpectations
E
[
f(Cj)∣
∣
∣f(Cj) ∈
[
qyn−1(f(Cj)), qyn(f(Cj))]
]
andE
[
f(Cj)∣
∣
∣f(Cj) ∈
[
VaRyn(f(Cj)),VaRyn−1(f(Cj))]
]
have to be recalculated in every optimization step, while they only have to be calculatedonce when fixing the decomposition Y = y0, . . . , yN : 0 = y0 < · · · < yN.
One possibility to accelerate and simplify the bid-ask calibration is equidistant spacingof y0 < · · · < yN :5 Step 3 of Algorithm 4.3 then reduces to the calculation of
E
[
f(Cj)∣
∣
∣f(Cj) ∈
[
qyn−1(f(Cj)), qyn(f(Cj))]
]
for n = 1, . . . N , j = 1, . . . ,M and the concavity constraint D(∆γ) ≤ 0 simplifies to
∆2γ := (∆γ2 − ∆γ1, . . . ,∆γN − ∆γN−1) ≤ 0.
Furthermore, a closer look on the optimization problem in Algorithm 4.3 suggests thefollowing strategies:
Remark 4.4 (Numerical treatment)1. Note that if the objective function of the optimization problem is continuously dif-
ferentiable in ∆γ, the bid-ask calibration to piecewise linear concave distortionsfulfills the Karush-Kuhn-Tucker conditions for nonlinear optimization (see, e.g.,[Boyd and Vandenberghe 2004]). Thus, Algorithm 4.3 can be solved by using La-grange multipliers.
2. If we use the Euclidean distance (or some strictly monotone transformation ofit) as an error function (e.g. the popular RMSE error function), the bid-ask cal-ibration to piecewise linear concave distortions reduces to a linearly constrainedleast-squares optimization problem which is well treated in literature (see, e.g.,[Hanson and Haskell 1982, Hanson 1986]).
5Actually, this can be relaxed further: If y0 < · · · < yN are symmetrically spaced around 0.5, the sameacceleration argument can be done.
13
4.2 Application to parameter risk-captured prices
4.2 Application to parameter risk-captured prices
A framework for the calculation of parameter risk-captured prices is developed in [Bannör and Scherer 2011a].The central assumption in this framework is the presence of a distribution R on theparameter space Θ. In some cases of historical estimation of parameters (e.g. corre-lation estimation as in [Bannör and Scherer 2011a]), the distribution R on Θ is givenby the distribution of the parameter’s estimator. In other cases, e.g. the calibrationto market prices, the distribution R may be recovered from the calibration to midprices by an algorithm as described in [Bannör and Scherer 2011b]. While in the papers[Bannör and Scherer 2011a, Bannör and Scherer 2011b] the choice of risk measure wasregarded to be subjective, Algorithm 4.3 allows us to calibrate parameter risk-capturedprices to bid-ask prices using a broad and flexible class of risk measures, represented bypiecewise linear concave distortion functions.
Using the suggested procedure for obtaining a distribution R on the parameters, whencalibrating to market prices of vanillas, the result is a three-step calibration scheme:
First, we calibrate to mid market prices and obtain a parameter θ0 ∈ Θ. Second, usingthe parameter θ0, we construct the distribution R on the parameter space Θ as suggestedin [Bannör and Scherer 2011b].6 Finally, we calibrate to bid-ask prices using Algorithm4.3 to obtain the best matching concave distortion function γ0.
The choice of the error function is also somewhat delicate: [Detlefsen and Härdle 2007]observe that different choices of error functions deliver different calibration results. Fur-thermore, [Guillaume and Schoutens 2011] scrutinize different calibration methods (his-torical calibration, calibration to vanillas, etc.), restricted to the Heston model. We omitfurther interdependencies to those issues in the current investigation and concentrate onone error function and one calibration method.
5 Application to data
In the previous section, we have presented and discussed a non-parametric calibrationscheme based on piecewise linear concave distortion functions to bid-ask prices. In thissection, we apply our piecewise linear calibration scheme and compare it to a paramet-ric calibration scheme à la [Cherny and Madan 2010], using the introduced parametricfamilies of AVaR- and minmaxvar -type distortion functions. Therefore, we calibrate a
6In [Bannör and Scherer 2011b], the distribution R is constructed by transforming the error functionto mid prices ηmid via a decreasing function h s.t.
∫
Θh(ηmid(θ)) dθ = 1. The choice of h gauges the
weight being assigned to the parameters - the lower the pricing error, the higher the weight. Usingour solution of the bid-ask calibration problem, we obtain different market-implied concave distortionfunctions γ for different transformation functions h. We argue that fixing the transformation functionh (e.g. due to experience of traders) and solving for the market-implied concave distortion may bea sensible way of treating the bid-ask calibration problem in our parameter risk-capturing setting.Otherwise (solving for the transformation function and the concave distortion) our results maybecome difficult to interpret, due to the partial exchangeability of h and γ.
14
5.1 Parameter uncertainty setup and calibration
Barndorff-Nielsen–Shephard model with parameter risk (cf. [Bannör and Scherer 2011b])to quoted bid-ask prices of vanillas. As a result, we obtain a very characteristic shapeof the piecewise linear approximation. From our empirical results, we consequentlysuggest using a different parametric family of concave distortion functions: The ess sup-expectation convex combinations, simply interpolating between the essential supremumand the vanilla expectation w.r.t. the parameter distribution R. This family turns outto match the results from the piecewise linear calibration approach and allows for a fastand efficient calibration.
Our set of data is the DAX option surface, consisting of 501 bid and ask vanilla pricesas of December 2, 2011, with different maturities and strikes. For simplicity, we assumeno bid-ask spreads in the DAX spot price and EUR interest rates.
We use the Barndorff-Nielsen–Shephard model, for details see [Barndorff-Nielsen and Shephard 2001,Cont 2004]. The risk-neutral dynamics of the log-index price (Xt)t≥0 in the Barndorff-Nielsen–Shephard model are given by
dXt =
(
r −σ2
t
2+
λcρ
α+ ρ
)
dt+ σt dWt − ρ dZλt,
dσ2t = −λσ2
t dt+ dZλt,
with parameters r, S0, σ20, λ, c, ρ, α > 0, (Wt)t≥0 is a Brownian motion, and (Zt)t≥0 is a
compound Poisson process with exponentially distributed jump size, i.e. Zt =∑Nt
j=1 Uj
with a Poisson process (Nt)t≥0 with intensity c > 0 and (Uj)j∈N are exponentially i.i.d.with parameter α > 0. (Wt)t≥0 and (Zt)t≥0 are independent. Since we assume the DAXspot price S0 and risk-free rate r to be given, the unspecified parameters with exposureto parameter risk are gathered in the quintuple (σ2
0, c, α, λ, ρ).
For calculating vanilla prices, we use the Fourier pricing method of [Carr and Madan 1999,Raible 2000] and calculate call prices for the moneyness dimension simultaneously viaFFT. As an error function for both the initial calibration to mid prices and the calibra-tion to bid-ask prices, we use the popular RMSE without any weighting (for the impactof using different error functions, we refer to [Detlefsen and Härdle 2007]), standardizedby the mean option price, similar to the setting in [Bannör and Scherer 2011b]. Thus,our error function is
η(θ) =
√
1M
∑Mm=1(Eθ[Cm] − Cmid
m )2
1M
∑Mm=1 C
midm
.
5.1 Parameter uncertainty setup and calibration
We follow [Bannör and Scherer 2011b] and obtain a parameter risk distribution R onΘ by discretizing Θ and weighting the parameters with their error to market prices,transformed and normed by a decreasing function h s.t. the sum of all parameters equalsone. We hereby incorporate all parameters (on a discrete grid) up to an aggregate
15
5.2 Results
market error of 3.5% and weight them by transforming the error function with thenormal transformation function
hNλ (t) := c · exp
(
−(t− t0)
2
2λ2
)
,
where λ = 0.005, c > 0 matching, and t0 = 1.63% denoting the minimal aggregatemarket error. Doing so, we obtain a discrete distribution on Θ with a support of 9 430parameter vectors with an aggregate market error of less than 3.5%.
With a distribution R on Θ at hand, we calibrate to bid and ask prices in variousways: First, we calibrate to bid-ask prices with our non-parametric piecewise linearapproximation scheme described in Algorithm 4.3. As a unit interval decomposition,we use 100 and 1 000 equidistant nodes; for optimization purposes, we use again themean-standardized RMSE as in the mid prices calibration. Second, we compare ourresult to a parametric calibration using parametric families of distortion functions. Wehereby employ the popular AVaR- and minmaxvar -families for calibration and observedifferences in calibration performance.
5.2 Results
As a first result, we obtain that the calibration performances of the piecewise linearand the parametric approaches do not differ significantly, the standardized RMSEs of allapproaches are close to each other:
Distortion framework RMSE/mean to bid-ask prices CPU time
piecewise linear 1 000 nodes 1.64% 301.11 secpiecewise linear 100 nodes 1.64% 4.26 secminmaxvar -family 1.65% 3.17 secAVaR-family 1.64% 3.73 sec
Actually, the distortion functions that result from the calibration differ in shape (cf.Figure 2), in case of the AVaR-calibration due to its specific shape. The piecewise linearcalibration result is clearly the most flexible method and does not exhibit too strongperformance drawbacks compared to parametric calibration when choosing a reasonablenumber of nodes.
As a very remarkable result, one observes that both the 100 and 1 000 nodes approx-imation follow the same pattern: After a sharp increase close to zero, we can observelinear growth in the argument of the distortion function. This observation is quite ro-bust: When using different transformation functions for creating the distribution on Θand incorporating more parameters, this pattern remains the same. Even when usinga different model (e.g. the Heston model), a similarly shaped distortion function is ob-tained. Furthermore, the result is stable w.r.t. different choices of starting vectors in the
16
5.2 Results
optimization procedure, so it is also unlikely that it results from numerical instabilities.Hence, our results motivate to introduce another parametric family of concave distortionfunctions, see Definition 5.1.
Definition 5.1 (ess sup-expectation convex combinations)Let λ ∈ [0, 1]. The distortion function
γλ(u) :=
0, u = 0
λ+ (1 − λ)u, u ∈ (0, 1]
is called the ess sup-expectation convex combination risk measure with weight λ on theessential supremum.
The name of this family stems from the behavior of the Choquet integral w.r.t. a γλ-distorted probability: If we have a probability measure P and a bounded random variableX, the Choquet integral w.r.t. γλ P is a convex combination of the essential supremumof X and the ordinary expectation w.r.t. P , weighting the essential supremum with λand the expectation with 1 − λ. We can easily show the more general result:
Proposition 5.2Let (Ω,F , P ) be a probability space, γ a concave distortion function with a jump inzero, X an integrable random variable that is P -a.s. bounded from above, and let λ =limv↓0 γ(v) be the height of the jump. Then the function
γcont(u) =limv↓u γ(v) − λ
1 − λ
is a continuous concave distortion function and
∫
ΩX d(γ P ) = λ ess supX + (1 − λ)
∫
ΩX d(γcont P ).
In particular, for every concave distortion function with a jump in zero the Choquet inte-gral w.r.t. γ P is representable as a convex combination of the ess sup and a continuousconcave distortion function γcont.
We call γcont the continuous part of γ and λ the jump part of γ. It may be notedthat the continuous part γcont is not unique, but we denote with it the construction inProposition 5.2. For λ = 1, we trivially set γcont = 0.
Proof (of Proposition 5.2)It is easy to show that γcont is a continuous and concave distortion function (if λ = 1,the proposition is trivially true). Abbreviate x0 := ess supX (first assuming x0 ≥ 0),
17
5.2 Results
then the definition of the Choquet integral immediately yields
∫
ΩX d(γ P ) =
∫ 0
−∞γ(P (X > x)) − 1 dx+
∫ ∞
0γ(P (X > x)) dx
=
∫ 0
−∞γ(P (X > x)) − 1 dx+
∫ x0
0γ(P (X > x)) dx
=
∫ 0
−∞(1 − λ)γcont(P (X > x)) + λ− 1 dx+
∫ x0
0(1 − λ)γcont(P (X > x)) + λ dx
= λx0 + (1 − λ)
∫
ΩX d(γcont P ).
Similar for x0 < 0.
In particular, if we use γλ from Definition 5.1 as distortion function, we are interpolatingbetween the essential supremum and the expectation (since the continuous part of γλ
is just the identity function on [0, 1]) when calculating the Choquet integral. Since ourpiecewise linear calibration results in Figure 1 look similar to an ess sup-expectation con-vex combination, they deliver an appealing interpretation for trading: When setting thebid-ask prices by means of the calibration risk framework of [Bannör and Scherer 2011b],we get bid-ask spreads that are fairly in line with the market when calculating the worstcase of all parameters, the expectation w.r.t. the delivered distribution, and simply inter-polating between them, using the weight of the worst case as a “risk-aversion parameter”.Strikingly, this simple approach ties with the more sophisticated AVaR- and minmaxvar -methodologies in calibration performance and is much easier and faster to implement: Inour setting, we just have to calculate the supremum, the infimum, and the expectation,which can efficiently be done using vectorized programming systems (e.g. MATLAB).Afterwards, the optimization procedure only incorporates three values per vanilla optionand is much faster compared to other parametric distortion function types or the 100nodes piecewise linear function:
Distortion framework RMSE/mean to bid-ask prices CPU time
piecewise linear 100 nodes 1.64% 4.26 secminmaxvar -family 1.65% 3.17 secAVaR-family 1.64% 3.73 secess sup-exp-family 1.64% 0.21 sec
Furthermore, we obtain a firm economic interpretation of the expectation w.r.t. thedistribution: The expectation can be interpreted as the “true mid price” (which is notobservable in the market, since we only get bid and ask quotes), with a (non-symmetric)risk premium relative to the parameter risk which is expressed by the essential supre-mum (for ask prices) and infimum (for bid prices). This is in line with [Cont 2006] and
18
6 Conclusion
[Lindström 2010], who argue seperately for the supremum and the expectation incorpo-rating model/parameter risk.
Since our empirical observations motivate the introduction of discontinuous distortionfunctions, we finally generalize the existence theorem for a solution to the bid-ask cali-bration problem to discontinuous distortion functions.
Theorem 5.3Let f(X) be bounded, η a continuous error function, K > 0, ε ∈ (0, 1), and define
Gjump,εK := γ : [0, 1] → [0, 1] : γ is a distortion function with
jump height ≤ 1 − ε and γcont ∈ GK.
Then there is a solution of the bid-ask calibration problem in Gjump,εK .
ProofLet γ ∈ Gjump
K and take the representation γ = (λ, γcont) from Proposition 5.2. Themapping
(λ, γcont) 7→ λ ess sup f(X) + (1 − λ)
∫
f(X) d(γcont P )
is continuous w.r.t. the product topology on [0, 1 − ε] × GK . Furthermore, Tychonoff’stheorem ensures that [0, 1 − ε] ×GK is compact. Hence, a minimizing element exists.
6 Conclusion
In this paper, we state the bid-ask calibration problem in a generic manner and pro-vide existence theorems for its solution in the set of distortion risk measures; mildlyrestricted to certain concave distortions. Our results apply to both the conic financeapproach presented in [Cherny and Madan 2010] and the parameter risk-captured priceframework of [Bannör and Scherer 2011a]. Furthermore, we provide a non-parametriccalibration scheme to bid-ask prices based on a piecewise linear approximation of concavedistortion functions. This scheme turns out to be efficient to implement and reduces inprominent special cases to the well-known linearly constrained linear least-squares prob-lem. Applying our piecewise linear calibration scheme in the parameter risk-capturedprice framework from [Bannör and Scherer 2011a, Bannör and Scherer 2011b], we com-pare our model bid-ask spreads to bid-ask spreads provided by parametric bid-ask cal-ibration results as in [Cherny and Madan 2010] and obtain a new parametric family ofconcave distortions. The parametric family of ess sup-expectation convex combinationsmatches well with our observed non-parametric calibration results and allows for anefficient and accurate calibration.
19
References
References
[Acerbi and Tasche 2002] C. Acerbi, D. Tasche, On the coherence of expected short-fall, Journal of Banking and Finance 26:7 (2002) pp. 1487–1503.
[Acerbi 2002] C. Acerbi, Spectral measures of risk: A coherent representation of sub-jective risk aversion, Journal of Banking and Finance 26:7 (2002) pp. 1505-1518.
[Artzner et. al. 1999] P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, Coherentmeasures of risk, Wiley: Mathematical Finance 9:3 (1999) pp. 203–228.
[Barndorff-Nielsen and Shephard 2001] O.E. Barndorff-Nielsen, N. Shephard,Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financialeconomics, Journal of the Royal Statistical Society Series B, 63:2 (2001) pp. 167–241.
[Bannör and Scherer 2011a] K.F. Bannör, M. Scherer, Capturing parameter uncer-tainty by convex risk measures, working paper (2011).
[Bannör and Scherer 2011b] K.F. Bannör, M. Scherer, Incorporating calibration riskinto derivative prices: A model comparison regarding parameter risk, working paper(2011).
[Bion-Nadal 2009] J. Bion-Nadal, Bid-ask dynamic pricing in financial markets withtransaction costs and liquidity risk, Journal of Mathematical Economics 45:11 (2009)pp. 738-750.
[Boyd and Vandenberghe 2004] S. Boyd, L. Vandenberghe, Convex optimization,Cambridge University Press, Cambridge, 2004.
[Carr and Madan 1999] P. Carr, D. Madan, Option valuation using the fast Fouriertransform, Journal of Computational Finance 2 (1999) pp. 61–73.
[Carr et. al. 2001] P. Carr, H. Geman, D. Madan, Pricing and hedging in incompletemarkets, Journal of Financial Economics 62 (2001) pp. 131–167.
[Cherny and Madan 2009] A. Cherny, D. Madan, New Measures for PerformanceEvaluation, Review of Financial Studies 22 (2009) pp. 2571–2606.
[Cherny and Madan 2010] A. Cherny, D. Madan, Markets as a counterparty: Anintroduction to conic finance, International Journal of Theoretical and Applied Finance13:8 (2010) pp. 1149–1177.
[Cont 2004] R. Cont, P. Tankov, Financial modelling with jump processes, Chapman& Hall, 2004.
[Cont 2006] R. Cont, Model uncertainty and its impact on the pricing of derivativeinstruments, Wiley: Mathematical Finance 16:3 (2006) pp. 519–547.
[Denneberg 1994] D. Denneberg, Non-additive measure and integral, Kluwer AcademicPublishers, Dordrecht, 1994.
20
References
[Detlefsen and Härdle 2007] K. Detlefsen, W. Härdle, Calibration risk for exoticoptions, The Journal of Derivatives 14:4 (2007) pp. 47–63.
[Föllmer and Schied 2004] H. Föllmer, A. Schied, Stochastic Finance 2nd Edition,De Gruyter, Berlin, 2004.
[Guillaume and Schoutens 2011] F. Guillaume, W. Schoutens, Calibration risk: Il-lustrating the impact of calibration risk under the Heston model, Review of DerivativesResearch (2011) DOI 10.1007/s11147-011-9069-2.
[Gzyl and Mayoral 2008] H. Gzyl, S. Mayoral, On a relationship between distortedand spectral risk measures, Revista de economía financiera 15 (2008) pp. 8–21.
[Hanson and Haskell 1982] R.J. Hanson, K.H. Haskell, Algorithm 587: Two algo-rithms for the linearly constrained least squares problem, ACM Transactions on Math-ematical Software 8:3 (1982) pp. 323–333.
[Hanson 1986] R.J. Hanson, Linear least squares with bounds and linear constraints,SIAM J. on Scientific Computing 7:3 (1986) pp. 826–834.
[Lindström 2010] E. Lindström, Implication of parameter uncertainty on option prices,Advances in Decision Sciences 2010 (2010) Article ID 598103, 15 pages.
[Raible 2000] S. Raible, Lévy Processes in Finance: Theory, Numerics, and EmpiricalFacts, PhD thesis, Freiburg, 2000.
[Shilov 1996] G.E. Shilov, Elementary functional analysis, Dover Publications, 1996.
[Xu 2006] M. Xu, Risk measure pricing and hedging in incomplete markets, Annals ofFinance, 2 (2006) pp. 51–71.
21
References
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Probability
Dis
torte
d pr
obab
ility
Distortion function calibration results
Pcw linear 1000 nodesPcw linear 100 nodesminmaxvarAVaR
Figure 1 Different calibration results for a (non-parametric) piecewise linear calibra-tion with 1 000 and 100 nodes, a parametric calibration to the AVaR-distortionfunction family, and a parametric calibration to the minmaxvar -family. Whilethe piecewise linear calibration results with different nodes are very similarand have a characteristic jump close to zero and afterwards a linear behavior,we get very different results for the minmaxvar - and AVaR-results. In the up-per probability regions, the minmaxvar -result allocates more probability thanthe piecewise linear result, while the AVaR allocates even more probability.In the lower probability region, we get more similarity for minmaxvar - andAVaR-results, but a very different pattern to the jump in the piecewise linearcalibration results. Since the calibration results seem to be pretty similar, thehigher probability allocation to high-probability events like in the AVaR- andminmaxvar -case does not seem to impact the calibration result too much.
22
References
3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 80000
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Strike
Impli
ed vo
latilit
y spr
eads
(impli
ed as
k vol
− im
plied
bid v
ol)
marketnonparametricAVaRminmaxvar
3000 4000 5000 6000 7000 8000 9000 100000
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Strike
Impli
ed vo
latilit
y spr
eads
(impli
ed as
k vol
− im
plied
bid v
ol)
market spreadsnonparametricAVaRminmaxvar
Figure 2 Bid-ask spreads for differently calibrated distortions vs. real-world bid-askspreads, expressed in implicit volatilities for a short-term and a long-termmaturity. Since the BNS model with its downward jumps emphasizes the putside of the smile, we exhibit more parameter risk on the left wing and thoselarger bid-ask spreads can be captured well by all models. For the same reason,the right wing vol spread is matched less accurate, in particular for short-term maturities, but AVaR- and piecewise linear distortions match it moreefficiently than the calibrated minmaxvar -distortion. One has to encounterthat parameter risk may not be the main driver for far-OTM call prices, othereffects like illiquidity seem to be more predominant.
23