capturing parameter risk with convex risk measures · schoutens et al (2004) examine diﬀerent...

European Actuarial Journal manuscript No.(will be inserted by the editor)

Capturing parameter risk with convex risk measures

Karl F. Bannör · Matthias Scherer

Received: date / Accepted: date

Abstract Adequately specifying the parameters of a financial or actuarial model is challenging.In case of historical estimation, uncertainty arises through the estimator’s volatility and possiblebias. In case of market implied parameters, the solution of a calibration to market data mightnot be unique or the numerical routine returns a local instead of a global minimum. This paperprovides a new method based on convex risk measures to quantify parameter risk and to translateit into prices, extending results in Cont (2006); Lindström (2010). We introduce the notion ofrisk-capturing functionals and prices, provided a distribution on the parameter (or model) set isavailable, and present explicit examples where the Average-Value-at-Risk and the entropic riskmeasure are used. For some classes of risk-capturing functionals, the risk-captured price preservesweak convergence of the distributions. In particular, the risk-captured price generated by thedistributions of a consistent sequence of estimators converges to the true price. For asymptoticallynormally distributed estimators we provide large sample approximations for risk-captured prices.Following Bion-Nadal (2009); Carr et al (2001); Cherny and Madan (2010); Xu (2006), we interpretthe risk-captured price as an ask price, reflecting aversion towards parameter risk. To acknowledgeparameter risk in case of calibration to market prices, we create a parameter distribution fromthe pricing error function, allowing us to compare the intrinsic parameter risk of the stochasticvolatility models of Heston and Barndorff-Nielsen and Shephard as well as the Variance Gammaoption pricing model by pricing different exotics.

Keywords Parameter risk · convex risk measure · risk-capturing functional · risk-captured price ·bid-ask spread · calibration risk.

1 Introduction

Crucial for the accurate valuation of contingent claims is the correct specification of the respec-tive model’s parameters. Some parameters are directly given from the market (like spot prices ofliquid stocks), others are the result of a calibration to market prices. Finally, some parameters areestimated from historic or forward looking data. With these techniques, we often do not obtain asingle parameter; exposing us to parameter uncertainty. In case one may assign probabilities forthe validity of each parameter, one exhibits parameter risk in the sense of Knight (1921). The mainobstacle is how to adequately incorporate parameter risk into a risk-captured price, consideringthat we are risk averse. This may be especially the case when estimating parameters where it is

Karl F. BannörParkring 1185478 Garching bei MünchenTel.: +49-89-289-17415E-mail: [email protected]

Matthias SchererParkring 1185478 Garching bei MünchenTel.: +49-89-289-17402E-mail: [email protected]

common practice to plug the calculated point estimate into the pricing functional. However, usinga point estimate bears the risk of estimation errors, since the estimator may have high variance (orskewness), which results in an exposure to estimation risk. Similarly, when parameters are obtainedfrom a calibration to given derivatives, the output of the minimizer might only be a local minimumor there might exist various parameter constellations with comparable pricing error.

In literature on model and parameter uncertainty (resp. risk), the approaches of Cont (2006) andLindström (2010) provide first ideas how to handle the above obstacle. Cont (2006) proposes a worstcase method based on convex risk measures to capture model uncertainty as well as a penalizedworst case approach, which was resummarized by Gupta et al (2010). Before, this worst-case pric-ing was established in incomplete markets (cf. the textbook Černy (2009)), which can be regardedas a special case of model uncertainty. Although Cont (2006) mainly focuses on comparing differ-ent models, he argues that parameter uncertainty can be translated into model uncertainty. Thespecial situation of uncertain volatility in a diffusion model is treated in Avellaneda et al (1995).However, handling parameter risk with a worst case approach leads to very conservative results.This may be suitable for quantifying the extremes of model uncertainty (the primary purpose ofCont (2006)), but leads to unrealistic results when used for pricing purposes. Furthermore, if thereis knowledge about the parameter’s distribution, this information is disregarded for the sake of con-servativeness. Lindström (2010) suggests to randomize the parameters of a financial model. Modelprices are then computed by integrating out the parameter distribution. Lindström (2010) primar-ily aims at explaining smiles and does not account for any aversion towards uncertainty aboutthe true parameter. In a calibration setting, Detlefsen and Härdle (2007) investigate calibrationw.r.t. different error functions yielding to differences in exotics prices. On a more macro level,Schoutens et al (2004) examine different models and their calibration results w.r.t. exotics pricingand discover that tremendous differences exist. Gupta and Reisinger (2012) suggest a Bayesianapproach to calibration with a focus on the local volatility model, using a Bayesian estimator forcalibration to noised data w.r.t. some prior distribution.

We provide a parameter risk framework that unifies and generalizes the proposals of Cont(2006); Gupta et al (2010); Lindström (2010) using the notion of convex risk measures1. Similarfunctionals to Cont (2006) are suggested in Gupta et al (2010), while Lindström (2010) advocatsrisk-neutral incorporation of parameter risk. Furthermore, it embeds the pricing suggestions forincomplete markets described in Carr et al (2001); Cherny and Madan (2010); Xu (2006). Ourapproach allows to capture parameter risk by explicitly incorporating the parameter’s distribution,such that a risk averse trader can acknowledge this risk.2 The settings described in Cont (2006)and Lindström (2010) are both extremal cases of an application of the Average-Value-at-Risk(AVaR) w.r.t. different significance levels α ∈ [0, 1] (a detailed examination of the AVaR is given inAcerbi and Tasche (2002)). We suggest to use risk-capturing functionals to determine parameterrisk-captured prices in face of a non-Dirac distribution on the parameter space (e.g. the pushforwardmeasure of an estimator). In line with the nonlinear pricing approaches of Bion-Nadal (2009);Carr et al (2001); Xu (2006); Cherny and Madan (2010), we propose to use the price deliveredby a risk-capturing functional as an ask price. The properties of the parameter’s distribution arereflected in the width of the bid-ask spread, and we give some examples how to calculate bid-askprices facing parameter risk.

A desirable feature of risk-captured prices is that weak convergence of the parameter distribu-tion implies convergence of the according risk-captured price to the risk-captured price computedwith the limit distribution. In case of a consistent estimator for the parameter, this implies thatthe risk-captured price converges to the price computed with the true parameter. This motivatesthat the charge for parameter risk is decreasing in the available amount of information that is usedto estimate the parameter. Under mild technical conditions, the risk-captured price arising fromspectral risk measures fulfills this convergence property. Meanwhile, Krätschmer et al (2012) havedeveloped extensions to these results applying techniques from functional analysis and possiblystronger topologies as the so-called ψ-weak topology from Weber (2006).

1 Convex risk measures are a fruitful topic in financial mathematics, treated in the special case of coherent risk mea-sures in the seminal paper Artzner et al (1999) and having been extended in many papers like, e.g., Kusuoka (2001);Föllmer and Schied (2002); Acerbi and Tasche (2002); Jouini et al (2006); Frittelli and Scandolo (2006); Krätschmer(2006). A standard reference for the theory of convex risk measures on the vector space of (a.s.) bounded measurablefunctions is the textbook Föllmer and Schied (2011).

2 Although we focus on applications in a financial context, our methodology can as well be used in an actuarialcontext to calculate risk-captured insurance premia.

2

2 Mathematical background and notation

When the parameters are obtained via calibration to market prices, a distribution on theparameter space does not arise naturally. We suggest a method employing a transformation ofthe distance between model and market prices to construct a distribution on the parameterspace. We exemplarily investigate the calibration risk w.r.t. three popular models: The stochas-tic volatility models of Heston (see Heston (1993)) and Barndorff-Nielsen and Shephard (seeBarndorff-Nielsen and Shephard (2001)) as well as the Variance Gamma model (see Madan et al(1998)). The discussed methodologies enable us to compare the exposure to parameter risk of thedifferent models and also of different exotic options.

2 Mathematical background and notation

In this section we define model and parameter uncertainty and risk and briefly recall the approachesof Cont (2006) and Lindström (2010) to capture these risks. Let throughout (Ω,F ,F) be a filteredmeasurable space and let (St)t≥0 denote a d-dimensional F-adapted stochastic process modellingthe basic instruments with St = (S

(1)t , . . . , S

(d)t ). To simplify notation, we assume without loss of

generality all claims being evaluated w.r.t. a martingale measure Q to be already discounted bythe associated numéraire process. Given a contingent claim X , we denote the evaluation mappingQ 7→ EQ[X ] (which maps a risk-neutral measure to the contingent claim price of X under Q) byεX , which is defined on some suitable set of risk-neutral measures.

Definition 1 (Model uncertainty) Let Q be a family of probability measures on (Ω,F) s.th.all stochastic processes modelling discounted basic instruments (S

(i)t )t≥0, i = 1, . . . , d, are Q-

martingales for all Q ∈ Q. The financial market model (Ω,F ,F, (St)t≥0,Q) faces model uncertaintyif |Q| > 1, where |Q| denotes the cardinality of Q. None

In particular, model uncertainty arises in incomplete markets with real-world measure P , since,in this case, more than one equivalent martingale measure Q ∼ P exists. In case that there is aconvenient parameterization of Q, one can interpret model uncertainty as parameter uncertainty.

Definition 2 (Parameter uncertainty) Let (Pθ)θ∈Θ be a family of pairwise different probabilitymeasures on (Ω,F) s.th. the models (Ω,F ,F, (St)t≥0, Pθ) are arbitrage-free for all θ ∈ Θ. Themodel faces parameter uncertainty if |Θ| > 1.3 None

Remark 1 (Interpreting parameter uncertainty as model uncertainty) Parameter uncertainty is aspecial case of model uncertainty, since every θ ∈ Θ induces a family of equivalent martingalemeasures Qθ. Therefore, exposure towards model risk exists with the model family Q =

⋃

θ∈Θ Qθ,as pointed out in Cont (2006). In case of incomplete models Pθ, we have uncertainty in two ways:The parameter uncertainty about the true real-world model Pθ and model uncertainty about theequivalent martingale measure Q ∈ Qθ := Q : Q ∼ Pθ obtained by a change of measure. None

An illustrating example for parameter uncertainty can be constructed in every model wherea hidden parameter vector θ has to be estimated and is not directly quoted by the market. Weconsider the Black-Scholes model and its volatility as the unknown parameter (which is a simpleexample for parameter uncertainty going beyond incomplete markets).

Example 1 (Parameter uncertainty in the Black-Scholes model) Examining the risk-neutral versionof the Black-Scholes model, the stock price is given by the SDE

dSt = rSt dt+ σSt dWt, S0 > 0, None

with (Wt)t≥0 being Brownian motion, r the risk-free interest rate, and σ > 0 the stock’s volatility.While the initial stock price S0 and the risk-free rate r are typically market-quoted quantities, onedoes not have direct information about the volatility σ. Hence, a priori every positive number canbe taken. Usually, one uses market data (e.g. historical estimation, calibration to market prices)to specify the volatility. None

3 As our notation in Definition 2 suggests, parameter uncertainty may arise from uncertainty of the real-worldmeasure and then transfers to uncertainty of the risk-neutral measure.

3

Cont (2006) (and Avellaneda et al (1995), in the case of uncertain volatility) proposes to handlemodel risk by a worst case approach: Given that the mapping Q 7→ EQ[X ] is bounded, one candefine an upper bound for the price of a contingent claim X ∈ C by Γ u(X) := supQ∈Q EQ[X ] andanalogously a lower bound Γ l(X) := −Γ u(−X) = infQ∈Q EQ[X ]. This definition clearly quantifiesthe extremes of model (resp. parameter) uncertainty, which is the main purpose of Cont (2006). Forpricing purposes, this approach might be too conservative in practice (see Figure 1): Using Γ u(X)may add a too large charge for model risk. In case of parameter uncertainty, if Q = Qθ : θ ∈ Θand Θ ⊂ R

m is a compact set and the map θ 7→ EQθ[X ] is continuous, the supremum and infimum

will be attained for certain parameters θu, θl ∈ Θ, so EQθu[X ] would be w.l.o.g. the largest sensible

price and EQθl[X ] would be the lowest sensible price. Therefore, a worst case approach might not

provide good practice in all situations to calculate an additional charge that may be accepted bythe market.4

In some situations, one might have additional information, e.g. a probability measure R on theset of models Q. This is the situation of model risk, following Knight (1921). In case of modelrisk, the ansatz of Cont (2006) disregards the additional information contained in the probabilitymeasure R.

Definition 3 (Model risk) Let (Ω,F ,F, (St)t≥0,Q) be a financial market model exhibitingmodel uncertainty, i.e. |Q| > 1. If there is a σ-algebra FQ on Q and a probability measureR : FQ → [0, 1] assigning a “likelihood of validity” to the models in doubt, the financial mar-ket model exhibits model risk. None

The situation of parameter risk is natural and arises when parameters are estimated. RecallingExample 1, there are situations where implied volatilities are not given (e.g. due to the lack ofliquid plain vanilla derivatives) and the volatility has to be estimated. In such a situation, we faceparameter risk concerning the Black-Scholes volatility.

Example 2 (Parameter risk from estimation of the Black-Scholes volatility) We consider a Black-Scholes setting, where the volatility σ is the key parameter for risk-neutral pricing. This parameteris not directly given by the market (different from the spot price S0 and the risk-free rate r). Hence,the determination of the volatility is a situation where one is exposed to parameter uncertainty (asdescribed in Avellaneda et al (1995)). If the whole stock price actually follows the Black-Scholesmodel, it may be a sensible idea to estimate the volatility from time series data. Given normallydistributed returns x1, . . . , xN , xj = logStj+∆t− logStj , j = 1, . . . , N , one may choose the classicalestimator for the variance, corrected for the frequency of the data ∆t, which results in the estimator

σ2N =

1

∆t(N − 1)

N∑

j=1

(xj − x)2, x =1

N

N∑

j=1

xj .

For independent normally distributed returns and a true variance σ20 > 0, this estimator is chi-

squared-distributed up to some scaling, i.e. σ2N ∼ σ2

0

∆t(N−1)χ2n−1, which is a Gamma distribution

with shape parameter (N − 1)/2 and scale parameter 2σ20/(∆t(N − 1)). Hence, the distribution

determining the parameter risk arising from the estimation of volatility (resp. variance) is essentiallydetermined by the chi-squared distribution, and the parameter risk triplet (Θ,FΘ , R) is given byΘ = R>0, FΘ = B(R>0) with B(R>0) denoting the Borel-σ-algebra w.r.t. R>0 and

R(dx) =(∆t(N − 1))

N−12

Γ(

N−12

)

(2σ20)

N−12

xN−3

2 exp

(

−x∆t(N − 1)

2σ20

)1x>0 dx.

Lindström (2010) explains the smile and term structure of implied volatilities by randomizingthe parameters of given models: In particular, he suggests substituting the volatility of the Black–Scholes model by a symmetrically distributed noise variable. Then, he calculates the option priceserially: First, the parameter θ ∈ Θ is fixed and EQ[X |θ = θ0] is calculated for each θ0 ∈ Θ.Afterwards, the function θ0 7→ EQ[X |θ = θ0] is integrated w.r.t. the parameter’s distribution. By

4 In incomplete markets, the sub-/superhedging prices (cf. Černy (2009)) are a natural worst-case ansatz fitting inthis framework. In incomplete markets, the problem of weaker variants of sub-/superhedging are also tackled fromthe hedging perspective by, e.g., mean-variance hedging (cf., e.g., Föllmer and Schweizer (1990); Schweizer (1991)),quantile hedging (cf. Föllmer and Leukert (1999)), and efficient hedging (cf. Föllmer and Leukert (2000)).

4

3 Constructing risk-capturing functionals from convex risk measures

defining Qθ0(A) := EQ[1A|θ = θ0] and considering the model family (Qθ0)θ0∈Θ, we can embedthis approach into our notion of parameter uncertainty. The ansatz of Lindström (2010) does notaccount for aversion towards parameter uncertainty, since no additional charge for parameter risk isstipulated. Beyond that, Jensen’s inequality shows that for concave functions θ0 7→ EQ[X |θ = θ0],adding a symmetric noise to the true parameter can even lower the price compared to using thesimple plug-in risk-neutral price.


In this section we introduce our concept of model (resp. parameter) risk-capturing functionals,extending the idea of using convex risk measures for pricing in an incomplete markets settingdescribed in Carr et al (2001); Xu (2006); Cherny and Madan (2010). Although we mainly focuson parameter risk, we introduce it in the more general setting of model risk.

We start by recalling the definition of convex risk measures on a vector space of measurablefunctions, as in, e.g., Föllmer and Schied (2011). We follow Frittelli and Scandolo (2006) in extend-ing the classical definition to translation invariance w.r.t. a linear form, since we want translationinvariance to hold not only for constants but also for all contingent claims without exposure towardsmodel (resp. parameter) risk.

Definition 4 (Convex risk measure) Let (Ω,F) be a measurable space and X ⊂ L0(Ω) bea vector space of measurable functions on Ω. Let Y ⊂ X be a sub-vector space and π ∈ Y⋆

(Y⋆ denotes the algebraic dual space of Y). ρ is called a convex risk measure with π-translationinvariance5 if it fulfills the following axioms:

1. ρ is monotone6: ∀X,Y ∈ X : X ≥ Y ⇒ ρ(X) ≥ ρ(Y ).2. ρ is convex: ∀X,Y ∈ X ∀λ ∈ [0, 1] : ρ(λX + (1− λ)Y ) ≤ λρ(X) + (1 − λ)ρ(Y ).3. ρ is π-translation invariant: ∀X ∈ X ∀Y ∈ Y : ρ(X + Y ) = ρ(X) + π(Y ).

Furthermore, ρ is called coherent if it is positively homogeneous, i.e. ρ(cX) = cρ(X) for allc > 0 and X ∈ X . ρ is called normalized, if ρ(0) = 0 holds. If P is a probability measure on (Ω,F),we call ρ P -law invariant, if ρ(X) = ρ(Y ) holds in case of PX = P Y . In canonical cases, we omitthe reference to the probability measure. None

We now state required properties of functionals to qualify for capturing model (resp. parameter)risk. These properties can be related to properties of convex risk measures, which we therefore applyto the current situation.Properties 5A functional Γ incorporating model risk w.r.t. the model family Q into contingent claim pricesshould fulfill the following properties:

1. Order preservation: If there exists a model-free order, it should be preserved when incorporatingmodel uncertainty, i.e. for contingent claims X,Y :

X(ω) ≥ Y (ω) for all ω ∈ Ω ⇒ Γ (X) ≥ Γ (Y ).

2. Diversification: Diversification of model uncertainty should not be penalized, i.e. a convex com-bination of two positions facing model uncertainty should not have a higher price than theconvex combination of the individual prices, i.e.

Γ (λX + (1− λ)Y ) ≤ λΓ (X) + (1− λ)Γ (Y )

should hold for contigent claims X,Y and λ ∈ [0, 1].3. Model independence consistency: If a contingent claim X is consistently priced under all models

(resp. parameters), no model uncertainty is present and the model risk-captured price agrees withthe risk-neutral price, i.e. no charge for model risk is added to the risk-neutral price:

Q 7→ EQ[X ] is constant on Q ⇒ Γ (X) = EQ[X ].

5 If we do not mention the translation invariance w.r.t. a specified linear form π and the sub-vector space Y , wealways assume that Y is the vector space of constant functions and π is the canonical linear form with π(1) = 1.

6 Most literature (e.g. Föllmer and Schied (2011); Artzner et al (1999); Krätschmer (2006); Frittelli and Scandolo(2006)) defines convex risk measures to be anti-monotone and anti-translation invariant. To match our purposesand for the sake of elegance, we follow Cont (2006) by employing ordinary monotonicity and translation invariance.

5

We now introduce the notion of a model risk-capturing functional and model risk-capturedprices. We treat the more general type of ambiguity, model risk, yet it can be applied in both casesof model risk and parameter risk. The basic idea of model risk-captured prices is that the modelrisk of the derivatives price can be measured by applying a convex risk measure to the evaluationmapping.

Definition 6 (Model risk-capturing functional) Let Q be a family of models and let R be aprobability measure on Q. Let A ⊂ L0(R) be a vector space of measureable functions containingthe constants and denote

CA :=

X ∈⋂

Q∈Q

L1(Q) : εX : Q 7→ EQ[X ] ∈ A

as the vector space of all A-regular claims being available for all models in the model familyQ. Intuitively speaking, CA consists of all contingent claims that are integrable w.r.t. all possibleprobability measuresQ ∈ Q and the evaluation mapping εX(Q) := EQ[X ] is in A.7 Let furthermoreρ : A → R be a normalized, law invariant convex risk measure. Then the mapping Γ : CA → R

defined by

Γ (X) := ρ(εX) = ρ(Q 7→ EQ[X ]) (1)

is called a (model) risk-capturing functional on the set of claims CA w.r.t. the distribution R. ρ iscalled the generator of Γ and Γ (X) is called the risk-captured (ask) price of X given the functionalΓ . None

The idea behind definition (1) is that a derivatives trader, facing model risk, should choose askprices that are high enough that some buffer prevents oneself from losses due to model risk, whichmay occur when selling too cheap. Conversely, one should account for model risk when buyingderivatives by setting bid prices low enough in order to prevent oneself from losses due to modelrisk. A consistent strategy is to regard the dual −Γ (−X) as the risk-captured bid price. Due to thedual nature of the bid and ask price, we use the name “risk-captured price” for the risk-capturedask price in the following.

Canonical choices for A are the Lebesgue spaces Lp(R) for p ∈ [1,∞], depending on the domainof ρ. The assumption that constants are included in A is important for the valuation of model-invariant payoffs. Furthermore, ρ being normalized is also natural from an economic point of view.

Obviously, parameter risk can be embedded into this framework: Given a set of equivalentmartingale measures (Qθ)θ∈Θ and a distribution R on Θ, the bijection ι : Θ → Q with Q := Qθ :θ ∈ Θ defines a distribution on Q. In cases of parameterized families, we speak about parameterrisk-capturing functionals. In case of parameter risk w.r.t a parameter space Θ, we abbreviateEθ[X ] := EQθ

[X ] for θ ∈ Θ.

Remark 2 It can be verified that if Γ : CA → R is a model risk-capturing functional with generatorρ, Γ is a normalized, law-invariant convex risk measure on CA which is translation invariant onthe subspace of model-invariant contingent claims X0 := X ∈ CA : Q 7→ EQ[X ] is constant w.r.t.the (well-defined) linear form π(X) = EQ[X ], X ∈ X0. It can be seen that Γ readily fulfills thedesirable properties from 5. None

The choice of ρ and therefore the choice of A completely determines the risk-capturing functionalΓ . This allows us to exploit the rich pool of convex risk measures fulfilling law-invariance andnormalization to calculate risk-captured prices.

7 This definition may look vast at first sight, but consists of mainly technical conditions that have to hold forwell-definedness. Essentially, we take all contingent claims X such that the price EQ[X] is defined for all Q ∈ Q

(that is the integrability condition) and the evaluation mapping εX : Q 7→ EQ[X] can be plugged in the risk measureρ (which is guaranteed by the second condition).

6


Examples: AVaR- and entropic-driven risk-captured prices

A popular convex risk measure is the Average-Value-at-Risk, abbreviated AVaR.8 Given a probabil-ity space (Ω,F , P ), we define for a given significance level α ∈ (0, 1] and an integrable X ∈ L1(P )the α-Value-at-Risk to be the upper α-quantile VaRα(X) := qPX(1 − α) and the α-Average-Value-at-Risk9 to be

AVaRα(X) :=1

α

∫ α

0

VaRβ(X) dβ.

The AVaR is a coherent law-invariant risk measure (see Acerbi and Tasche (2002), Föllmer and Schied(2011)) and therefore fulfills the requirements to use it for measuring model risk.

Example 3 (Average-Value-at-Risk-induced risk-capturing functional) Let Q be a family of martin-gale measures inducing model risk and let R be a distribution on Q. Consider the L1(R)-regularclaims and R ∗AVaRα : CL1(R) → R to be the risk-capturing functional generated by the coherentrisk measure AVaRα : L1(R) → R for a given confidence level α ∈ (0, 1], so

R ∗AVaRα(X) := AVaRα(Q 7→ EQ[X ]).

R ∗ AVaRα captures the model risk quantified by the distribution R, which is isolated from themodel-intrinsic risk within a specific model Q ∈ Q. R ∗AVaRα deals with the risk from the upperα-tail of the price distribution w.r.t. different parameters by averaging over the tail prices. It shouldnot be confused with the regular Average-Value-at-Risk of a single model, which captures the riskwithin a specific model.

The Average-Value-at-Risk-induced risk-captured prices generalize both ansatzes of Cont (2006)and Lindström (2010): Starting with Lindström (2010), the existence of a single martingale mea-sure Q is assumed. Furthermore, the parameter θ bearing risk follows a distribution Qθ. In ourterminology, Lindström (2010) proposes to calculate Qθ ∗AVaR1, which can be easily seen by

∫

Θ

EQ[X |θ = θ0]Qθ(dθ0) =

∫

Θ

EQθ0[X ]Qθ(dθ0) =

∫ 1

0

qQθ

EQ·[X](β) dβ

=

∫ 1

0

VaRβ(EQ·[X ]) dβ = AVaR1(EQ·

[X ]) = Qθ ∗AVaR1(X).

Furthermore, since the Average-Value-at-Risk can naturally be extended for bounded functionsf ∈ L∞(R) by setting AVaR0(f) := ess sup f (further details about this extension can be found inFöllmer and Schied (2011)), our approach of risk-captured prices also generalizes the suggestionsfrom Cont (2006). By choosing a discrete dummy distribution R on Q, the essential supremum turnsout to become a regular supremum and therefore R∗AVaR0(X) = supQ∈Q EQ[X ] holds. Therefore,when there is uncertainty about at most countably many models, the suggested worst-case ansatzfrom Cont (2006) agrees with R ∗AVaR0 as the risk-capturing functional.

This shows that Cont (2006) and Lindström (2010) use the extreme points of R ∗ AVaR, soR ∗ AVaR allows to interpolate between these approaches and provides prices with extra chargesfor uncertainty, being more conservative than an expected value and less conservative than asupremum.

Clearly, the Average-Value-at-Risk dominates the Value-at-Risk. Hence, when calculating arisk-captured ask (resp. bid) price with the R-Average-Value-at-Risk with significance level α > 0,the ask (resp. bid) price does not understate (overstate) the true price with a probability of α.None

In presence of a distribution R on Q, the R ∗ AVaRα for a given significance level providesan objective risk-captured price, since the only preference will be set by specifying the significancelevel. A risk neutral type of preference would be the expected value, the most conservative choice isthe (essential) supremum. The next example provides a more subjective view on capturing modelrisk.

8 A detailed overview on the AVaR is presented in Acerbi and Tasche (2002).9 According to different purposes (as, e.g., actuarial and financial ones), there can be found numerous definitions

for the Value-at-Risk and the Average-Value-at-Risk in literature. To be concise for the reader, we use the definitionas in Cont (2006).

7

4.1 Convergence results of risk-capturing functionals

Example 4 (Entropic-induced risk-capturing functional) If (Ω,F , P ) is a probability space andX ∈ L∞(P ), the entropic risk measure with risk aversion parameter λ ∈ (0,∞) is defined as

ρentλ (X) :=1

λlog(EP [exp(λX)]).

It is well known that the entropic risk measure is normalized and law-invariant (but not coherent),see the treatment in Föllmer and Schied (2011). We examine (in presence of a distribution R onQ) the resulting R-entropic risk-capturing functional generated by the entropic risk measure anddenote it by Γ ent

λ for λ ∈ (0,∞). To ensure that Γ entλ is well-defined, we restrict the set of evaluable

claims to CL∞

. None

Remark 3 (Properties of the entropic-induced risk-capturing functional)

– x 7→ exp(λx) being convex, Jensen’s inequality provides

Γ entλ (X) = ρent(E·[X ]) =

1

λlog(ER[exp(λE·[X ])])

≥ 1

λlog(exp(λER[E·[X ]])) = ER[E·[X ]],

i.e. the entropic risk-capturing functional is more conservative than the expected value. Fur-thermore, it is known that the limit cases for λց 0 and λր ∞ are the expected value and theessential supremum. Therefore, the entropic-driven risk-capturing functional is an alternativegeneralization of (and interpolation between) the approaches of Cont (2006) and Lindström(2010).

– Jensen’s inequality also sheds some light on the role of λ: Heuristically speaking, the higherλ ∈ (0,∞), the “more convex” the function x 7→ exp(λx) and therefore the more conservativethe bid-ask pricing will be.

– An interesting feature of the entropic risk measure is that it is not positively homogeneous (un-like AVaR). Therefore, it may account for the risk being associated with large trades, reflectingthat large trades may bear additional risk due to liquidity effects, risk management issues, orexternal regulatory constraints. None

4 Convergence and asymptotics of risk-captured prices

In case of a true (but unknown) parameter θ0, we typically use an estimator θN : ΩN → Θ (givenN ∈ N samples), inducing a distribution on Θ by the pushforward measure PN

θ0

θN . An exampleis given in Example 2. One desirable property of a risk-capturing functional, provided that theestimator is consistent and we enlarge the sample size, is convergence of the risk-captured priceto the price computed with the true parameter θ0. Furthermore, large sample approximations canbe helpful to simplify calculations when the estimator’s distribution is complicated or unknown,but the asymptotic distribution is at hand. In this section we provide convergence results forsome classes of risk-capturing functionals and some large sample approximations for the classes ofAVaR- and entropic-induced risk-capturing functionals. Meanwhile, since the first publication ofthese results in a working paper, similar results have been developed in a more general setting andfor possibly stronger topologies in the seminal paper Krätschmer et al (2012).10

4.1 Convergence results of risk-capturing functionals

As motivating example, we treat prices arising from distributions induced by a consistent se-quence of estimators. More generally, we can postulate the following convergence property forrisk-capturing functionals based on weak convergence of probability measures, making it useful forother applications beyond estimation.

10 In Krätschmer et al (2012), the so-called ψ-weak topology introduced by Weber (2006) is employed whichcoincides with the weak topology in many important cases.

8

4 Convergence and asymptotics of risk-captured prices

Definition 7 (Convergence property (CP)) Let Θ be a parameter space and RN → R0,N → ∞, be a sequence of weakly convergent distributions on Θ. Let A ⊂ L0(Θ) be a vectorspace of measurable functions and (ρN )N∈N be a sequence of convex risk measures on A s.th. ρNis RN -law invariant. The sequence of risk-capturing functionals ΓN : CA → R with

ΓN (X) := ρN(θ 7→ Eθ[X ])

is said to have the convergence property (CP) on A, if

limN→∞

ΓN (X) =: Γ0(X) = ρ0(θ 7→ Eθ[X ])

converges for every X ∈ CA and Γ0 is a risk-capturing functional for the distribution R0 withgenerator ρ0. None

Intuitively, (CP) describes that a sequence of risk-capturing functionals is consistent with ap-proximations of the parameter’s distribution in the sense of weak convergence. Typically, one em-ploys the same convex risk measure ρ (e.g. the AVaR), while varying the parameter distributions(RN )N∈N, eventually converging to a limit distribution R0.

Obviously, if Γ carries (CP) and θN is a consistent estimator for the true parameter θ0,limN→∞ ΓN (X) = Eθ0 [X ] holds, since the consistency of the estimator θN implies weak conver-gence of RN := PN

θ0

θN towards R0 := δθ0 . Practically, this means that reduction in the uncertaintyarising from estimation eventually reduces the risk-captured ask price. Under some mild technicalconditions (continuity, boundedness) we can prove that some families of risk-captured functionalsindeed fulfill (CP).

Proposition 1 (AV aR-induced risk-capturing functionals fulfill (CP)) Let Θ ⊂ Rm be a

Euclidean parameter space. Let RN → R0 be a weakly convergent sequence of probability distribu-tions on Θ and α ∈ (0, 1]. Then RN ∗ AVaRα(X) → R0 ∗ AVaRα(X), N → ∞, for all X s.th.θ 7→ Eθ[X ] is continuous and bounded, so the sequence (RN ∗ AVaRα)N∈N fulfills the convergenceproperty (CP) on Cb(Θ). None

Proof See the Appendix. None

The result for the AVaR-based functionals can easily be extended to all spectral risk measures.

Definition 8 (Spectral risk measure) Let (Ω,F , P ) be a probability space, X ∈ L∞(P ), andφ : (0, 1) → R≥0 be a decreasing and normed (i.e.

∫ 1

0φ(x) dx = 1) function. Then the functional

ρφ : L∞(P ) → R defined by

ρφ(X) :=

∫ 1

0

VaRα(X)φ(α) dα

is called the spectral risk measure with spectrum φ.11 None

Corollary 1 (Spectral-induced risk-capturing functionals fulfill (CP)) Let Θ ⊂ Rm be

a Euclidean parameter space. Let RN → R0 be a weakly convergent sequence of probability dis-tributions on Θ. Then for every spectrum φ : (0, 1) → R≥0 the spectral risk measure-inducedrisk-capturing functionals fulfill (CP) on Cb(Θ), i.e. if we define

ΓSφ (X) :=

∫ 1

0

VaRSα(E·[X ])φ(α) dα

for a probability distribution S on Θ, we obtain ΓRN

φ (X) → ΓR0

φ (X) for all X ∈ CCb(Θ), N →∞. None


11 A detailed discussion of spectral risk measures can be found in Acerbi (2002). In particular, it is shown thatspectral risk measures are a subclass of coherent risk measures.

9

4.2 Asymptotics of risk-capturing functionals

Remark 4 As a corollary, we obtain that in case of a compact parameter space, the assumptionof boundedness of a continuous function θ 7→ Eθ[X ] is fulfilled automatically and can therefore bedropped. Furthermore, Prokhorov’s theorem immediately yields that the spectral risk measures (asa function of the distribution with fixed contingent claim) are continuous w.r.t. the weak topologyon the distributions. None

The conditions in Proposition 1 and Corollary 1 are not too strict for practical use. To recall themotivation of developing (CP), in many cases the historical estimation of a parameter is necessary(e.g. because of a lack of liquid benchmark instruments), prices of contingent claims are continuous.Consistency is fulfilled by most estimators, e.g. Maximum Likelihood estimators fulfill it under mildtechnical assumptions.

The class of risk-capturing functionals generated by entropic risk measures fulfills (CP) forconsistent estimators as well, not being coherent and therefore not being covered by the abovepropositions.

Proposition 2 (Entropic-induced risk-capturing functions fulfill (CP)) Let Θ ⊂ Rm be a

Euclidean parameter space. Let (RN )N∈N be a sequence of probability distributions on Θ convergingweakly to a probability distribution R0. Let X be a contingent claim s.th. the mapping θ 7→ Eθ[X ]is bounded and continuous. Denote the entropic parameter risk-capturing functional w.r.t. RN

by Γ ent,Nλ and the one w.r.t. R0 by Γ ent,0

λ . It then follows Γ ent,Nλ (X)

N→∞−−−−→ Γ ent,0λ (X) for all

λ ∈ (0,∞). None


Having proved that convergence of prices holds, we apply the results to the Black-Scholessetting.

Example 5 Continuing with Example 2, it is a well-known fact that the estimator σ2N for the

variance with sample size N ∈ N is consistent (i.e. converges in probability to the true varianceparameter σ2

0) and follows a Gamma distribution. Unfortunately, the function assigning to everyvolatility σ the Black-Scholes prices BS(σ) for some option is continuous but not bounded. Hence,one cannot easily apply the above results. But, if we either restrict the available volatilities tosome bounded set Σ ⊂ R≥0 or cut the Black-Scholes price at some C > 0, i.e. regarding thefunction BScut(σ) := C ∧BS(σ), we obtain a bounded and continuous function in σ. In practice,similar restrictions are often done (e.g. in the uncertain volatility model of Avellaneda et al (1995)or for finite element PDE pricing). Hence, when applying these mild restrictions, the estimationrisk-captured Black-Scholes prices converge to the Black-Scholes price w.r.t. the true volatilityparameter σ0. None

However, property (CP) is not fulfilled by every risk-capturing functional. It can be seen that theess sup-driven risk-capturing functional does not provide convergence for bounded and continuousfunctions θ 7→ Eθ[X ].

Example 6 (Essential supremum does not fulfill (CP)) Consider the parameter space Θ = R andthe sequence of distributions RN ∼ N (0, 1/N). Let f ∈ Cb(R). Obviously, for N → ∞, RN → δ0weakly, but on the other hand we obtain ess supRN

f = supx∈R f(x) for all N ∈ N, since for everyx ∈ R and every ε > 0 the environment Bε(x) := y ∈ R : |x− y| < ε has positive measure underthe normal distribution, i.e. for all N ∈ N RN (Bε(x)) > 0 holds. Thus, limN→∞ ess supRN

f =supx∈R f(x) which, in general, does not coincide with ess supδ0 f = f(0). None

Further investigations about convergence of convex risk measures have been made in Krätschmer et al(2012). Our notion of weak convergence of measures is a too weak postulate in case the functionθ 7→ Eθ[X ] is not bounded or not continuous, e.g. θ 7→ Eθ[X ] ∈ Lp(Θ) for p ∈ [1,∞]. They intro-duce a stronger topology on the set of distributions, yielding similar convergence results in case ofdropping continuity and/or boundedness, which, however, goes beyond the scope of this paper.

4.2 Asymptotics of risk-capturing functionals

The calculation of risk-captured prices bears a substantial obstacle: For many estimators, the dis-tribution is not known in closed form. Fortunately, many estimators share the feature of asymptotic

10

5 Application: Bid-ask prices implied by parameter risk

normality. In this case, the delta method (see van der Vaart (2000)) provides convenient approxi-mations for large samples.

Remark 5 (Delta method) Let (θN )N∈N be an asymptotically normal sequence of estimators forthe true parameter θ0 ∈ Θ ⊂ R

m with positive definite covariance matrix Σ, i.e.√N(θN − θ0) →

Nm(0, Σ) weakly. Let furthermore X ∈ CL∞(Θ). If θ 7→ Eθ[X ] is continuously differentiable andthe gradient ∇Eθ0 [X ] 6= 0, then

√N(EθN [X ]− Eθ0 [X ])

N→∞−−−−→ N (0, (∇Eθ0 [X ])′ ·Σ · ∇Eθ0 [X ]) weakly.

With this approximation, the calculation of the AVaR- and entropic-induced risk-capturingfunctionals can be reduced to a simple symmetric interval around the evaluation at the true pa-rameter since they preserve weak convergence of the distributions. We start with the entropicrisk-capturing functional and approximate EθN [X ] by a random variable Y with

Y ∼ N(

Eθ0 [X ], N−1(∇Eθ0 [X ])′ ·Σ · ∇Eθ0 [X ])

.

Hence, we readily obtain

Γ ent,Nλ (X) ≈ 1

λlogE[exp(λY )] = Eθ0 [X ] +

λ

2N(∇Eθ0 [X ])′ ·Σ · ∇Eθ0 [X ],

since the inner term is the moment-generating function of a normal distribution. For θN∗AVaRα(X),α ∈ (0, 1), we use the calculation of (McNeil et al, 2005, p. 45) for the AVaR of a normal distributedvariable to obtain

θN ∗AVaRα(X) ≈ Eθ0 [X ] +ϕ(Φ−1(α))

α√N

√

(∇Eθ0 [X ])′ ·Σ · ∇Eθ0 [X ],

denoting by ϕ the density and by Φ the distribution function of the standard normal law.The normal approximation sheds some light on the convergence rate. The entropic risk measure

converges with O (1/N) to the risk-neutral price computed with true parameter θ0, while θN ∗AVaRα converges with O(1/

√N). This result may look surprising, since the extreme points of

both θN ∗ AVaRα and Γ ent,Nλ are the essential supremum and the expected value. Yet, on the

one hand, for the expected value the upper approximation does not hold, since for λ → 0 resp.α→ 1 the feasible approximation would simply be Eθ0 [X ]. On the other hand, property (CP) doesnot hold for the essential supremum (see Example 6). Thus, convergence is not guaranteed in theextremes.


Bion-Nadal (2009) proposes to use convex risk measures to model bid-ask prices. Similarly, in Cont(2006) it is mentioned that one can construct bid-ask intervals by the worst-case approach. Withour introduction of the notion of risk-capturing functionals, we fall into Bion-Nadal (2009)’s generalframework and additionally provide a possible explanation for bid-ask spreads of derivative prices.We provide an example coming from the historical estimation of a parameter, where a distributionnaturally arises. The fundamental idea for the application of parameter risk-capturing functionalsto derive bid-ask prices is that higher uncertainty about the true parameter should result in awider bid-ask spread. A trader who is unsure about the true value of a parameter usually adds apremium to the given theoretical price (the systematics behind this are often called edge rules).In case of a quantification of the uncertainty by a distribution, the risk-captured price provides asystematic approach for traders to account for parameter risk. The estimator’s distribution deliversuseful information how wide a reasonable bid-ask spread could be. The choice of risk-capturingfunctional reflects the subjective preferences of the trader (e.g. parameter risk aversion, aversionto large trades). When Γ : C → R denotes a risk-capturing functional on a suitable vector space ofcontingent claims C, we treat Γ (X) as the ask price and Γ (X) := −Γ (−X) as the bid price for thecontingent claim X . Note that due to convexity and normalization we always have Γ (X) ≤ Γ (X)(cf., e.g., Bion-Nadal (2009)).

11

Case study: Estimation of correlation risk for an exchange option

The correlation of two stocks is not a liquid asset class and therefore it is hard to find liquid pricesto calibrate the correlation to. But it may be estimated by historical correlation over a suitableperiod. We present a numerical example and a comparison between the different risk-capturedprices, dependent on the sample size used to estimate the correlation.

Example 7 (Exchange option with estimated correlation) Suppose we have a Black-Scholes modelwith three assets, the money market account S(0) and two stocks S(1) and S(2). Under the real-world measure P they follow the dynamics

dS(0)t = rS

(0)t dt, dS(i)

t = µiS(i)t dt+ σiS

(i)t dW

(i)t , i = 1, 2,

with (W(i)t )t∈[0,T ] being Wiener processes, µi ∈ R, r, σi ∈ R>0 for i = 1, 2. Furthermore, we assume

the covariation of the Wiener processes to be given by

dW(1)t dW

(2)t = ρ dt

for an unknown ρ ∈ [−1, 1]. The covariation between the stock prices is dS(1)t dS

(2)t = ρσ1σ2S

(1)t S

(2)t dt.

The risk neutral dynamics follow

dS(0)t = rS

(0)t dt, dS(i)

t = rS(i)t dt+ σiS

(i)t dW

(i)t , i = 1, 2,

and the covariation between the stocks remains dS1t dS

2t = ρσ1σ2S

1t S

2t dt.

We are interested in the fair value of a European exchange option, giving the holder the rightto exchange stock 1 into an equal amount of stock 2, so the payoff of the contingent claim withexercise date T > 0 is X = (S

(2)T −S(1)

T )+. In Margrabe (1975) a closed-form solution is calculated:

BS(S(1)0 , S

(2)0 , σ1, σ2, T, ρ) = S

(2)0 Φ(d1)− S

(1)0 Φ(d2),

with

d1 :=log(S

(2)0 )− log(S

(1)0 ) + 0.5T σ2

σ√T

, d2 := d1 − σ√T , σ =

√

σ21 + σ2

2 − 2σ1σ2ρ.

We assume that all parameters except for the correlation ρ are known. One possible approachto obtain the correlation is to estimate it via Pearson’s sample correlation coefficient

ρ(N) =N∑N

i=1 xiyi −(

∑Ni=1 xi

)(

∑Ni=1 yi

)

√

∑Ni=1 x

2i −

(

∑Ni=1 xi

)2√

∑Ni=1 y

2i −

(

∑Ni=1 yi

)2

with N ∈ N being the largest number of feasible samples, xi, yi denoting the log-returns of stock 1resp. 2 for i = 1, . . . , N . Since a change of measure to the equivalent risk neutral measure does notaffect the parameter ρ, it is reasonable to estimate the correlation ρ under the real-world measure.

Without accounting for the uncertainty of the parameter ρ, we would simply plug in the re-sulting point estimate of Pearson’s correlation estimator into the pricing formula. To calculate arisk-captured price, however, we first have to determine the distribution of the correlation in theparameter space [−1, 1] induced by the estimator ρ(N).

In a Black-Scholes world, log returns are independent over time and follow a bivariate normaldistribution with correlation ρ. Therefore, the Fisher transformation (see Fisher (1915)) can beapplied and the transformed distribution approximately fulfills

artanh(ρ(N)) ∼ N(

artanh(ρ0),1

N − 3

)

, N ≫ 3,

where ρ0 denotes the true correlation parameter. Pearson’s correlation estimator is consistentbut only asymptotically unbiased. To obtain an unbiased estimator, one can use the Olkin-Prattadjustment of the estimator as described in Olkin and Pratt (1958) at the cost of a much morecomplicated distribution of the estimator.

12


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 12

4

6

8

10

12

14

16

18

Correlation (ρ)

Pric

e

Black−Scholes pricesup f(ρ)

inf f(ρ)

Figure 1 Exchange option price as a function of the stocks’ correlation. In this example, the stock price is decreasingand concave in correlation.

We now calculate risk-captured prices. We start with the calculation of the entropic-drivenrisk-captured price: We fix λ ∈ (0,∞) and obtain

Γ entλ,N (X) =

1

λlog

(∫ 1

−1

exp(

λ ·BS(S(1)0 , S

(2)0 , σ1, σ2, T, ρ)

)

P ρ(N)

(dρ)

)

≈ 1

λlog

(∫ ∞

−∞

exp(

λ ·BS(S(1)0 , S

(2)0 , σ1, σ2, T, tanh(x

√N − 3 + ρ0))

)

ϕ(x) dx

)

.

This integral has to be evaluated numerically and defines the entropic-induced ask price.For an efficient calculation of ρ(N)∗AVaR, general theory of convex risk measures can help: Since

the distribution of ρ(N) is continuous, the characterization of AVaR as tail conditional expectationholds (compare Föllmer and Schied (2011)), by shortening we calculate

ρ(N) ∗AVaRα(X) = Eρ(N) [f |f ≥ VaRα(f)] ,

which may be better suited from a computational point of view. The conditional expectationEρ(N) [f |f ≥ VaRα(f)] can be estimated by Monte Carlo simulation given the distribution of ρ(N).In our example, we use the Fisher transformation approximation due to its simplicity.

In our numerical example we assume S(1)0 = 120, S(2)

0 = 110, σ1 = 0.16, σ2 = 0.32, and T = 1.The correlation ρ is supposed to be unknown, so we are exposed to parameter uncertainty w.r.t.ρ. As visualized in Figure 1, the correlation parameter has massive influence on the price of theexchange option. In case ρ = 1, the price is 3.39, while for ρ = −1 the price is 17.16. Furthermore,Figure 1 shows that in this case the upper and lower risk-captured prices provide very roughestimates for a spread price when a supremum-generated risk-capturing functional to determinerisk-captured prices is used, so the supremum/infimum approach may hardly be useful to calculatean acceptable bid-ask spread.

If the stocks’ correlation is estimated, we have more information about the parameter andcan apply the risk-capturing functionals. We assume to have estimated the correlation with Pear-son’s correlation estimator from bivariate normal distributed stock returns, the real correlation issupposed to be ρ0 = 0.4 and obtain the entropic risk-captured price and the AVaR-induced risk-captured prices dependent on the selection of λ ∈ (0,∞) resp. α ∈ (0, 1] and the sample size N .Since f is a continuous function on the compact interval [−1, 1], it fulfills the conditions to applythe convergence results of the previous section. Convergence can also be visualized by plotting ourrisk-captured prices against the sample size (see Figure 2).

13

0 100 200 300 400 500 600 700 800 900 10002

4

6

8

10

12

14

16

18

Number of samples

Pric

e

AVaR0.05

bid price

AVaR0.05

ask price

Entropic bid price λ=3

Entropic ask price λ=3real price

Figure 2 Different risk-captured prices as a function of the sample size. One can see the convergence of the risk-captured prices towards the risk-neutral price with the true parameter. Note that even for 250 samples (about oneyear of daily returns), the bid-ask spreads induced by parameter risk are still considerably large.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50002

4

6

8

10

12

14

16

18

Number of samples

Pric

e

AVaR0.01

bid price

AVaR0.01

ask price

AVaR0.1

bid price

AVaR0.1

ask price

AVaR0.5

bid price

AVaR0.5

ask price

expectation pricereal priceinfimum bid pricesupremum ask price

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50002

4

6

8

10

12

14

16

18

Number of samples

Pric

e


Entropic ask price λ=1


Entropic ask price λ=10


Entropic ask price λ=50real priceinfimum bid pricesupremum ask price

Figure 3 Left: The AVaR-induced bid-ask prices for different significance levels. The higher the significance levelα ∈ (0, 1] is chosen, the wider is the bid-ask spread. The extreme case of a supremum-induced ask price (whichagrees with ρ(N) ∗AVaR0) does not depend on the sample size and leads to a very conservative bid-ask spread whichmay not comply to bid-ask spreads being quoted in the market. The calculation of the expected value leads to pricesbeing very close to the plug-in price of the true parameter.Right: The entropic-induced bid-ask prices for different risk-aversion parameters. A higher risk aversion parameterλ ∈ (0,∞) results in wider bid-ask spreads, leading to supremum resp. infimum induced prices in the limit. Theentropic-induced bid-ask prices are more volatile with rising risk aversion parameter due to numerical issues.

First, we examine the AVaR-induced risk-captured price. The significance level α ∈ (0, 1] steersthe level of conservativeness. For α = 0 we obtain the supremum approach and therefore the askprice ρ(N) ∗ AVaR0(X) = 17.16; independent of the sample size. The higher the significance levelα ∈ (0, 1] is chosen, the narrower are bid-ask spreads. For illustration, we have plotted some spreadsfor different significance levels in Figure 3. As one can see, the expectation-induced price convergesvery fast to the real price, while convergence velocity shrinks for lower significance levels.

When regarding the entropic-induced prices, a higher risk-aversion parameter λ leads to widerbid-ask spreads. Due to the fact that the entropic risk measure is not positively homogeneous, ahigher quantity of the same claim leads to a higher price per notional in the same way a higherrisk-aversion parameter does, since for a quantity a ∈ R>0 the entropic-induced risk-captured price

14

6 Accounting for calibration risk using risk-capturing functionals

per notional w.r.t. a distribution S is calculated as

Γ entλ (aX)

a=

1

λlog(ES

[

exp(λE·[X ])]

) = Γ entλ

(X)

with λ := aλ. Due to this symmetric relationship between quantities and risk-aversion parameters,we have only visualized how the risk-captured prices evolve for different risk-aversion parameters(see Figure 3). None

6 Accounting for calibration risk using risk-capturing functionals

Necessary for using the machinery of risk-captured prices is the existence of a distribution on theparameter space. Unfortunately, this is not given in a standard calibration setting. In this section,we briefly summarize the key ingredients of a calibration to market prices and present possibleconstructions of a distribution on Θ, being consistent with the calibration result.

For a calibration to market prices it is crucial to have a concept to measure the distance betweenmodel and market prices. Therefore, we first define the notion of an error function, summarizingthe distance between model and market prices into a single number.

Definition 9 (Error function) Let Θ ⊂ Rk be a parameter space. We call εC1,...,CM

: Θ → R≥0

an error function if there is a componentwise monotone function ε : RM≥0 → R≥0 s.th. εC1,...,CM

(θ) =

ε(|Eθ [C1]− C⋆1 |, . . . , |Eθ[CM ]− C⋆

M |) with ε(0, . . . , 0) = 0.12 None

A popular example is the root mean square error

RMSEC1,...,CM(θ) :=

1

M

M∑

j=1

(Eθ[Cj ]− C⋆j )

2

12

.

A detailed discussion of the effect of error function choice on calibration results and exotics pricingcan be found in Detlefsen and Härdle (2007). Given an error function, the standard calibrationprocedure works as follows: A numerical optimization procedure determines the error minimizingparameter

θ0 = argminθ∈Θ

εC1,...,CM(θ).

Afterwards, the parameter θ0 is used as the model’s “true” parameter for pricing exotics. Note thatthe result of this procedure might be seen as a Dirac distribution on Θ with all probability massconcentrated on the parameter θ0.

Since a distribution on Θ does not arise naturally in a calibration framework, it has to beconstructed. The information used in a calibration stems from the error function ε, so a naturalstarting point is a suitable transformation of the error function to imply a Lebesgue density onthe parameter space Θ. Hence, we try to find a function h s.th. h ε is a proper Lebesgue densityon Θ; reflecting pricing errors in a consistent manner. We formally describe which properties atransformation h of the error function has to fulfill.

Properties 10 (Transformation function requirements)Let ε : Θ → R≥0 be an error function. A transformation function h : R≥0 → R≥0, creating ameaningful density from the error function ε, has to fulfill the following properties:

1. h is decreasing: This is important to guarantee that the density’s values are in concordance withthe error function’s results. Parameters with lower aggregate errors to market prices should beattributed with a higher likelihood.

2.∫

Θh(ε(θ)) dθ = 1: This normalization assures that hε indeed induces a distribution R on Θ via

R(dθ) = h(ε(θ)) dθ. This property can easily be obtained by scaling, as long as∫

Θh(ε(θ)) dθ <

∞.

12 For the sake of notation simplicity, we occasionally omit the dependence on the liquid contingent claimsC1, . . . , CM .

15

Transformation of measures yields∫

Θ

h(ε(θ)) dθ =

∫ ∞

0

h(t) (λ ε−1)(dt), (2)

which may be useful to exploit for considerations whether a function h is appropriate to induce adensity on Θ. A first observation reveals that if “too many parameters” (in the sense of Lebesguemeasure) with low pricing errors exist, a Lebesgue density may not be established on Θ. Clearly,in this case we are exposed to massive parameter risk and model prices should be considered withuttermost caution.

Remark 6 (Situations with extreme parameter risk) Let h : R≥0 → R≥0 be decreasing and δ ≥ 0s.th. h(δ) > 0. If ε−1([0, δ]) has infinite Lebesgue measure, then

∫

Θh(ε(θ)) dθ = ∞. None

Proof Let λ(ε−1([0, δ])) = ∞. Then we immediately obtain∫

Θ

h(ε(θ)) dθ =

∫ ∞

0

h(t) (λ ε−1)(dt)

≥∫ δ

0

h(t) (λ ε−1)(dt) ≥ h(δ) · λ(ε−1([0, δ])) = ∞. None

Geometrically speaking, this means that the error to market prices may not be too smallfor too large (in particular unbounded) sets of parameters.13 We consider such situations to bepathological, since we do not want to incorporate this huge amount of parameter uncertainty intoprices via a Lebesgue-a.c. distribution. Furthermore, it is shown that the choice of transformationfunction h may be limited. Some possible choices for the transformation function h are functionswith compact support or functions decreasing to zero fast enough.

Example 8 (Suitable transformation functions) Decreasing functions which may ensure small valuesfor parameters with large errors to market prices are the normal transformation function

hNλ (t) := c · exp(

−(

t

λ

)2)

, t ≥ 0

and the triangular transformation function

h∆λ (t) := c · 1t≤λ

(

− t

λ+ 1

)

, t ≥ 0,

each equipped with a scaling parameter λ > 0 and c > 0 chosen s.th. the function induces a density.The scaling parameter determines the amount of weight which is assigned to parameters with lowpricing error. None

The normal transformation function of Example 8 is a Schwartz function, i.e. for every α > 1and every k ∈ N the function t 7→ tα · ∂khλ(t)/∂tk is bounded. Thus, Schwartz functions decreasefaster to zero than every polynomial increases to infinity. We denote the set of all Schwartz functionson [0,∞) by S([0,∞)). A detailed discussion about Schwartz functions can be found in Werner(2007) and Hörmander (1990).

7 Application: Comparing the parameter risk of different models and exotic options

We now apply the techniques described above and calculate bid-ask spreads of different exoticoptions induced by calibration risk. We induce a distribution on the respective parameter set byevaluating the errors to market prices of a Heston model, a BNS model, and a Variance Gammamodel. This enables us to calculate parameter risk-captured prices for different exotic options(similar to Schoutens et al (2004) and Jessen and Poulsen (2010)), to compare parameter risk ofthe three models, and to profile exotics w.r.t. their parameter risk.

13 This would typically happen when the parameters are underspecified, e.g. when sophisticated models withseveral parameters meet few liquid market prices to calibrate to.

16


Our universe of liquid securities consists of 887 DAX plain vanilla call options with differentstrikes and maturities as of February 26, 2009, at a spot S0 = 3 942. We use the FFT method(see Carr and Madan (1999)) to calculate vanilla prices for a whole strike grid simultaneously. Forcalculating the distribution on the parameter space, we discretize the parameter space and re-evaluate the error function on the grid. Afterwards, we evaluate different exotic options (discretebarrier option, Asian option, discrete lookback option) via Monte Carlo simulation and calculateAVaR0.05-induced bid-ask prices implied by parameter risk for different densities.

We choose the relative root mean squared error ε(θ) = RMSEC1,...,CM(θ)/C⋆, i.e. the mean

deviation from mean market prices of the securities C1, . . . , CM with market prices C⋆1 , . . . , C

⋆M

as error function. This error function is comfortable to interpret and calibration is equivalent to acalibration with RMSE, since it is just a strictly monotone transformation of RMSE. We transformε by the normal transformation function

hNλ (t) = c · exp(

−(

t0 − t

λ

)2)

, t ≥ t0,

to a density with t0 := ε(θ0), denoting by θ0 the calibration parameter, scaling parameter λ > 0,and c > 0 matching s.th. a density is obtained. We correct the centered transformation functions forthe minimal encountered error to market prices t0. First, this corrects for advantages in calibrationperformance of one model compared to another. Second, the correction ensures that the induceddistributions Rλ(dθ) = hNλ (ε(θ)) dθ/

∫

ΘhNλ (ε(θ)) dθ converges weakly to δθ0 for λ → 0. Since the

AVaR-induced risk-capturing functionals fulfill (CP), we obtain the calibration plug-in price in thelimit λ→ 0.

7.1 The Heston model

Heston (1993) modifies the classical Black-Scholes model by modeling the square of Black-Scholesvolatility using a Cox-Ingersoll-Ross process. The risk-neutral dynamics of the Heston model followthe SDEs

dSt = rSt dt+ σtSt dW(1)t ,

dσ2t = κ(σ2

long − σ2t ) dt+ ξσt dW

(2)t ,

dW(1)t dW

(2)t = ρ dt,

(W(i)t )t≥0, i = 1, 2, being correlated Brownian motions and r, S0, σ

20 , κ, σ

2long, ξ > 0, ρ ∈ [−1, 1].

Assuming that the spot S0 and risk-free rate r are quoted in the market, parameter uncertaintyarises from the quintuple (σ2

0 , κ, σ2long, ξ, ρ) that has to be calibrated. To guarantee for strictly

positive volatilities, we require our parameters to fulfill the Feller condition ξ2 ≤ 2κσ2long. Hence,

our parameter space for the Heston model is

ΘHeston = (σ20 , κ, σ

2long, ξ, ρ) ∈ R

4>0 × [−1, 1] : ξ2 ≤ 2κσ2

long.

For FFT pricing, we need the characteristic function of the log-price process Xt := logSt, whichis, e.g., given in Schoutens et al (2004). It may be noted that there are two specifications forthe Heston characteristic function as pointed out in Albrecher et al (2007), we have chosen the(numerically) less pathological one.

7.2 The BNS model

Barndorff-Nielsen and Shephard (2001) developed a stochastic volatility model in which the vari-ance process is a subordinator-driven Ornstein-Uhlenbeck process. One of the most common spec-ifications for the subordinator is a compound Poisson process with exponentially distributed jump

17

7.4 Error function calculation

sizes as in Schoutens et al (2004). To account for the leverage effect, the stock price has a neg-ative jump whenever the volatility has an upward jump. The dynamics of the log-price processXt := log St are governed by the SDEs:

dXt =

(

r − σ2t

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 a Brownian motion and (Zt)t≥0 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. Given the observable spot price S0 and risk-free rate r, the unspecifiedparameters with exposure to uncertainty is the quintuple (σ2

0 , c, α, λ, ρ). Thus, without any furtherrestrictions, our parameter space is

ΘBNS = (σ20 , c, α, λ, ρ) : σ

20 , c, α, λ, ρ ∈ R>0 = R

5>0.

The characteristic function of the log-price process is given in, e.g., Schoutens et al (2004); Cont and Tankov(2004).

7.3 The Variance Gamma model

The variance gamma (VG) model is an exponential Lévy model, where the log-returns follow thevariance gamma process. This model was introduced in Madan and Senata (1990) for modelingreturns, Madan et al (1998) introduced an option pricing model based on the variance gammaprocess. The variance gamma process is a special case of a time-changed Brownian motion withdrift Zt = ϑΛt + σWΛt

with ϑ ∈ R, σ > 0, (Wt)t≥0 a Brownian motion, and (Λt)t≥0 a Lévysubordinator with Λ0 = 0. In case of the variance gamma process, the subordinator (Λt)t≥0 is aGamma process, i.e. for h > 0 and t ≥ 0 the increments Λt+h −Λt follow a Γ (h/κ, κ)-distribution.Thus, under a risk-neutral measure, the log-price process Xt is given (compare Madan et al (1998))by

dXt =

(

r +1

κlog

(

1− ϑκ− σ2κ

2

))

dt+ dZt.

Like the other two models, the variance gamma model can be regarded as an extension of theclassical Black-Scholes model. Instead of making volatility stochastic, time is made stochastic, as itwas first proposed in Clark (1973). The characteristic function of the log-price process is calculatedin Madan et al (1998).

7.4 Error function calculation

Since the characteristic function of the log-prices of all three models is known, we use Fourierpricing as described in Carr and Madan (1999). Provided that the βth moment, β > 0, of thelog-price exists, it is shown in Carr and Madan (1999) that the price of a plain vanilla call optionC(K) at maturity T > 0 with strike K > 0 is given by

C(K) =1

πKβ

∫ ∞

0

exp(− log(K)vi)ψT (v) dv, ψT (v) =exp(−rT )φT (v − (β + 1)i)

β2 + β − v2 + (2β + 1)vi,

denoting by φT the characteristic function of the log-price at time T . We choose (as in Schoutens et al(2004)) β = 0.75. The usage of FFT allows us to calculate call prices on a large strike grid simul-taneously. We interpolate between the given strikes to obtain prices for the strikes from our data.

18


7.5 A case study on parameter risk

We discretize the parameter space and evaluate the error function in a connected set where allparameters have a pricing error smaller than the smallest pricing error t0 plus 2%. This enablesus to compare the pure parameter risk effect w.r.t. the same transformation function. For theHeston and BNS models we have chosen not to observe parameter risk arising from the short-termvolatility σ2

0 (following, e.g., Guillaume and Schoutens (2011)) to reduce the parameter space tofour dimensions. The following table gathers the result of the algorithm and the minimal relativeRMSE to market prices, denoting by t0 the minimal error to market prices.

Model Discretization step vector ♯ of feasible parameters t0Heston model

(

0 0.07 0.1 0.07 0.02)

5 868 1.44%BNS model

(

0 0.1 0.5 0.15 0.2)

9 536 1.79%VG model

(

0.04 0.04 0.02)

6 705 5.41%

On the obtained parameter set, we evaluate three exotic options: An ITM discrete barrier calloption, an arithmetic Asian call option, and a discrete lookback call option. An ITM discretebarrier call option with maturity T > 0, strike K > 0, barrier B > 0, and observation points0 < t1 < · · · < tL =: T , L ∈ N, is given by the payoff ZBarrier = 1⋂

Ll=1Stl

<B(ST − K)+.An arithmetic Asian call option with maturity T > 0, strike K > 0, and observation points

0 < t1 < · · · < tL =: T , L ∈ N, is given by the payoff ZAsian =(

1L

∑Ll=1 Stl −K

)+

. Finally,the payoff of a discrete lookback call option with maturity T > 0, strike K > 0, and observationpoints 0 < t1 < · · · < tL =: T , L ∈ N, is given by ZLookback =

∨Ll=1 (Stl −K)+ . The payoff

always takes place at maturity. In our case, for all options we choose T = 1, K = 4 000, L = 24observation points and, in case of the barrier option, a barrier at B = 5 000. For simplicity, wedisregard dividends and assume the risk-free rate to be r = 0.03.

The contingent claims described above are path-dependent, we evaluate them by Monte Carlosimulation. We draw accordingly distributed random numbers with the same seed and rescale themwith the different parameters. This prevents us from mixing parameter risk with noise risk dueto differently drawn random numbers. In our Monte Carlo simulation, we evaluate using 10 000sample paths.

7.6 Parameter risk in different models

Having evaluated the above contingent claims in different models, we can empirically evaluatethe amount of model specific parameter risk. Our methodology to create a distribution on theparameter set allows us to scrutinize the pushforward distributions of the prices. The probabilitydistributions can be visualized in scatterplots. For a proper visual model comparison, we calculateand compare the resulting probability distribution functions of the option prices and compare theAVaR0.05-induced bid-ask spreads for the different models when varying the scaling parameter λ.The results are shown in Figures 4-6, together with an illustrating scatterplot for one selectedmodel. Statistical properties of the distributions are compared in Tables 1-3. We calculate themodus (which essentially is the plug-in calibration price), the expected value, the coefficient ofvariation, and the skewness. We have decided to apply the coefficient of variation for measuringdispersion due to its invariance w.r.t. multiplicative factors. Since we compare prices with differentlevels, the coefficient of variation enables us to directly compare their dispersion. The expectedvalue references to the AVaR1-induced parameter risk-captured price, which was suggested forincorporating parameter risk by Lindström (2010).

Results: The Heston model

The Heston model shows narrow-peaked price distributions for all options with little variance.The price distributions are quite symmetric (skewness is close to zero). Compared to the pricedistributions arising in the BNS model, the distribution has lower coefficients of variation for allexotics. This may be partly attributed to the better calibration performance, so values with worsecalibration performance are incorporated in the BNS model as well. Furthermore, the Hestonmodel, being completely driven by bivariate Brownian motion, has continuous paths, while the

19

7.6 Parameter risk in different models

Asian option Heston model BNS model VG modelModus (Calibration plug-in value) 311.8868 316.1214 274.9964Expected value(integrating out the distribution on Θ)

311.4427 316.0965 274.2531

Coefficient of variation 0.0125 0.0152 0.0348Skewness 0.2707 −0.3451 −0.2017

Table 1 The coefficient of variation is smallest for the Asian price distributions. In particular, the dispersion of theBNS model distribution is not much larger than the dispersion of the Heston model distribution. The VG modelimplies much different option prices (due to bad fit to ATM vanilla options) and considerable higher dispersion.Furthermore, the jump model price distributions are slightly skewed to the left, while the Heston model prices havea slight skew to the right.

Barrier option Heston model BNS model VG modelModus (Calibration plug-in value) 77.5529 97.9619 82.5580Expected value(integrating out the distribution on Θ)

75.0691 92.1595 98.8515

Coefficient of variation 0.0420 0.1257 0.2650Skewness −0.0156 0.2889 1.6040

Table 2 The dispersion of the barrier price distribution is a lot higher than for the Asian and lookback option dueto high sensitivity. In particular, the coefficients of variation are a lot higher in the BNS model than in the Hestonmodel and the dispersion of the VG model is by far the highest among all models. While the Heston model pricedistribution observes almost no skewness, the BNS model price distribution is slightly skewed to the right and theVG model price distribution has very strong skewness, compared to the other price distributions.

BNS model (in our specification) is also driven by a compound Poisson process and allows fordiscontinuities in the stock price process.

Results: The BNS model

The BNS model delivers distributions for all three options without a clear single peak and consid-erably larger dispersion. This means that many parameters deliver an equally good fit to marketprices as the parameter obtained from the standard calibration, but differ substantially in thecalculated prices for exotics. We attribute this result to the strong focus of the jump componentto replicate smile and term structure of given market prices. Only slightly different parametersconcerning the jump component may result in considerable price differences, especially concern-ing more sensitive contingent claims like lookback or barrier options. It is observed that the BNSmodel - considered isolated - bears substantial parameter risk, even if only parameters with verylow differences are incorporated. It is observed in (Cont and Tankov, 2004, pp. 488–490) that theBNS model has problems to fit to several smiles, thus it has some problems with calibration perfor-mance. This is verified by our observations. Furthermore, the calibration result seems to be quiteunstable in the sense that there are many parameters matching equally well. From a computationalpoint of view, this is a clear drawback of the BNS model. From an economic point of view, theBNS model contains a higher degree of parameter risk than the Heston model.

The most striking differences occur for the ITM barrier option. Not only is the price of the ITMbarrier option significantly higher in the BNS model than in the Heston model, which may be dueto the downward pressure of the downward-directed jumps. Furthermore, observed parameter riskin the BNS model is huge. Besides larger parameter risk, there are significant differences regardingthe lookback option: In the BNS model, the distribution is clearly skewed to the left. This may bethe result of the jump component allowing jumps only to move stock prices downwards, while inthe Heston model the distribution looks quite symmetric.

Results: The VG model

The VG model’s calibration performance is a lot weaker than the calibration performance of theHeston and the BNS model. This is consistent with the results in Jessen and Poulsen (2010) whichstate that the calibration performance of the VG model is inferior to that of the Heston modeland ties approximately with the classical Black-Scholes model. From this point of view, it playsin a different league than the Heston and the BNS model. The mediocre calibration performanceis reflected in the high degree of parameter risk. The price distributions for all three options are

20


Lookback option Heston model BNS model VG modelModus (Calibration plug-in value) 847.9149 837.7727 840.3980Expected value(integrating out the distribution on Θ)

852.1520 856.3299 824.4740

Coefficient of variation 0.0127 0.0380 0.0638Skewness 0.2621 −0.1539 −0.2031

Table 3 The coefficient of variation of the BNS model distribution is a lot higher than in the Heston model. Indeed,in the Heston model the lookback price distribution has comparable dispersion to the Asian price distribution, whilevariability is higher in the jump models. Furthermore, the sign of skewness in the Heston model is positive, while inboth jump models skewness is negative (probably due to jumping down). The VG model shows by far the highestcoefficient of variation.

tremendously wider than the price distributions for the BNS and the Heston model. Many observedparameter constellations that match the criterion for the VG model have quite poor performancein matching market prices, thus, in case of lookback and ITM barrier options, this mismatch mayproduce extreme results. In case of the barrier option price, it is heavily skewed to the right. Sincethe VG model did not match ATM options well, it is no surprise that the price of the Asian optiondeviates a lot from the prices in the BNS and the Heston model.

7.7 Results: Parameter risk profiles of different exotics

Comparing the parameter risk w.r.t. the different types of options, we mainly expect the parameterrisk profiles of exotics to follow their sensitivities: Contingent claims with high deltas and highBlack-Scholes vegas are supposed to show greater parameter risk, since stock price and volatilityare supposed to be captured by the models. Figure 7 compares the relative deviation of AVaR0.05

induced bid-ask prices to the plug-in prices for different options within the Heston model and theBNS model.

It can be easily identified that the parameter risk of the Asian option is quite small in all models.This may be attributed to the averaging taking place in Asian payoff profiles, so calculating priceswith slightly different parameters does not lead to much different results. On the other hand, therisk profile of an Asian option is close to that of a PV option, delta and vega tend to be lower onan absolute basis. Thus, since all parameter constellations on average match vanilla market pricesquite well, too large deviations from parameters are not expected.

For lookback options, the Heston model does not bear substantially higher parameter risk thanfor Asian options. Parameter risk is more skewed to the ask price, which can be explained by highermaximums being covered with the risk-captured prices. This is mainly due to the sensitivity ofthe maximum of the stock price to parameter changes. For the BNS model, the lookback optionsuffers from more parameter risk, especially on the ask side. The parameter risk of the VG modelis skewed a lot more. Bid prices deviate significantly more than the ones in the BNS model, whileask prices do not.

The ITM barrier option shows large variability in prices, mainly due to the knock-out feature.For some parameter constellations, the price becomes quite sensitive since a high probability ofstock prices hitting the barrier cause the option price to fall. Overall, we have an option with highlyvarying delta and vega. Compared to the other options, the parameter risk is substantially higherfor both the Heston and the BNS model. In the BNS model, incorporating parameter risk deliversa bid price of more than 30% below the plug-in price. Even worse deviations can be observed forthe VG model. While the bid price deviation can be compared to the one from the BNS model,the ask price has enormous parameter risk - there are values that are 150% higher than the plug-inprice.

In a nutshell, our observations meet our expectations concerning the degree of parameter risk.Options with low deltas and Black-Scholes vegas (e.g. Asian options) seem to be quite robust w.r.t.parameter risk. On the other hand, setting the price for lookback and barrier options classicallymay bear large amounts of parameter risk. Calculating risk-captured prices may give useful hintsfor setting bid-ask spreads, which should be narrow for good-natured exotics like Asians and widefor ITM barrier options.

21

References

8 Conclusion

We have introduced the notion of risk-capturing functionals and along with this definition a newframework to incorporate parameter risk into risk-neutral prices, provided a distribution on theparameter space is available. The works of Cont (2006) and Lindström (2010) are particular casesof risk-capturing functionals. Furthermore, for various risk-capturing functionals we have shownthat continuity w.r.t. the weak topology holds, especially in case of parameter estimation witha consistent estimator, the risk-captured price converges to the plug-in price computed with thetrue parameter. As an application, we have given a possible explanation for bid-ask spreads inconcordance with Bion-Nadal (2009). To apply the framework in a calibration setting, we describehow to induce distributions from the error function’s values based on theoretical and computationalconsiderations. This enables us to calculate parameter risk-induced bid-ask prices of exotic optionsand to compare the parameter risk of different models expressed. In a case study, we comparedthe parameter risk of the Heston model, the BNS model, and the VG model. The popular Hestonmodel is affected only little by parameter risk. The BNS model exhibits larger parameter risk forsensitive derivatives like lookback options and barrier options. The VG model does not fit marketprices well which is accompanied by very high parameter risk, especially for the ITM barrier option.

To sum up, the calculation of risk-captured prices may help an exotics trader to set bid-askspreads in line with parameter risk, in particular in a calibration setting. As we have shown in ourempirical results, parameter risk can be considerable for certain types of options. In this regardone is well adviced to incorporate parameter risk in quoted bid-ask prices.

Acknowledgements We thank C. Bluhm and J.-F. Mai for an initial discussion on parameter risk and uncer-tainty, A. Min for fruitful remarks on the delta method, A. Schied for helpful comments on a previous version ofthe manuscript, and V. Krätschmer for fruitful discussions about convergence properties of convex risk measures.Furthermore, we thank the TUM Graduate School for supporting these studies.

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A Appendix

Proof (of Proposition 1) Let X ∈ Cb(Θ) and α ∈ (0, 1] be arbitrary, abbreviate f(θ) := Eθ [X]. A property of weakconvergence is that it transfers to pushforward measures of continuous functions (cf. Bartoszynski (1961)), especiallyFf,RN

(x) → Ff,R0(x), N → ∞, with Ff,S(x) := S(f ≤ x) denoting the S-distribution function of f . Since the

quantile function is the quasi-inverse of the distribution function, it follows qRNf

(β) → qR0f

(β) Lebesgue-a.e. on(0, 1). Hence, by dominated convergence,

RN ∗AVaRα(X) =1

α

∫ α

0VaR

RNβ

(E·[X]) dβ =1

α

∫ α

0qRN

E·[X](1 − β) dβ

N→∞

−−−−→1

α

∫ α

0qR0E·[X]

(1 − β) dβ = R0 ∗AVaRα(X). None

Proof (of Corollary 1) In Acerbi (2002) it is pointed out that every spectral risk measure can be represented by aBorel measure µ on [0, 1] with zero mass in 0 such that

ρφ(X) =

∫ 1

0AVaRα(X)µ(dα),

so the AVaR w.r.t. different security level α ∈ (0, 1) are the “building blocks” of spectral risk measures. Hence, itfollows by Proposition 1 and dominated convergence

ΓRNφ

(X) =

∫ 1

0RN ∗AVaRα(X)µ(dα)

N→∞

−−−−→

∫ 1

0R0 ∗AVaRα(X)µ(dα) = Γ

R0φ

(X). None

Proof (of Proposition 2) Let λ ∈ (0,∞) be arbitrary but fix. Since f(θ) := Eθ [X] is assumed to be continuous andbounded, uλ f is continuous and bounded as well for uλ(x) := exp(λx). Since the expectation of uλ f w.r.t. themeasure RN is exactly the AVaR1 of uλ f , we obtain by Proposition 1 the convergence

∫

Θ

uλ(Eθ [X])RN (dθ)N→∞

−−−−→

∫

Θ

uλ(Eθ [X])R0(dθ)

and since u−1λ

(y) = λ−1 log(y) is continuous, also

u−1λ

(∫

Θ

uλ(Eθ [X])RN (dθ)

)

= Γent,Nλ

(X)N→∞

−−−−→ Γent,0λ

(X)

follows. None

24

A Appendix

240 250 260 270 280 290 300 310 320 3300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Cumulative distribution functions for Asian option price

Asian option price

Pro

babi

lity

of a

low

er/e

qual

opt

ion

pric

e

upper 5% quantilelower 5% quantileBNS price distributionHeston price distributionVG price distributionBNS calibration priceHeston calibration priceVG calibration price

290 295 300 305 310 315 320 325 330 335 3400

1

2

3

4

5

6

7

8

x 10−4

Asian option price

Pro

ba

bility

Barndorff−Nielsen−Shephard model

0 0.005 0.01 0.015240

250

260

270

280

290

300

310

320

330Bid ask prices − Asian option

Scaling parameter

Asia

n o

ptio

n p

rice

Heston bid priceHeston ask priceBNS bid priceBNS ask priceHeston plug−in priceBNS plug−in priceVG bid priceVG ask priceVG plug−in price

Figure 4 The cumulative distribution functions of Asian option prices induced by the normal transformationfunction with scaling parameter λ = 0.008. It is observed that the Heston price distribution observes slightly lowerdispersion than the BNS price distribution. Furthermore, the BNS price distribution is slightly skewed to the left.Overall, the Heston model is less exposed to parameter risk in this case. The VG price distribution is a lot moredispersed and the peak of the distribution is quite different. This is due to the large calibration error of the VGmodel, in particular when fitting to ATM prices. The bid-ask prices of the BNS and Heston models are quite close,while the VG model bid-ask spread is broader and has a different location.

25

60 80 100 120 140 160 180 200 2200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Cumulative distribution functions for Barrier option price

Barrier option price

Pro

babi

lity

of a

low

er/e

qual

opt

ion

pric

e


50 60 70 80 90 100 110 120 1300

1

2

3

4

5

6

7

8

x 10−4

Barrier option price

Pro

ba

bility

Heston model

0 0.005 0.01 0.015

60

80

100

120

140

160

180

200

220

Scaling parameter

Ba

rrie

r o

ptio

n p

rice

Bid ask prices − Barrier option


Figure 5 The cumulative distribution functions of ITM barrier option prices induced by the normal transformationfunction with scaling parameter λ = 0.008. Dispersion for ITM Barrier options is a lot higher for the BNS model thanfor the Heston model, thus parameter risk for barrier options is higher in the BNS model than in the Heston model.Furthermore, due to the discontinuity of the payoff function, prices in the BNS model jump significantly. Moreover,BNS model prices are considerably higher than Heston model prices. This may be due to downward jumps, makingstock prices not breaking the barrier. Jumps are also very likely to be the reason for the snatchy shape of the BNSdistribution function. The VG model prices are heavily skewed to the right and have much higher variability due tothe poor calibration performance. The bid-ask prices of the VG model clearly dominate the induced bid-ask pricesof the Heston and the BNS models.

26

A Appendix

700 750 800 850 900 9500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Cumulative distribution functions for Lookback option price

Lookback option price

Pro

babi

lity

of a

low

er/e

qual

opt

ion

pric

e


600 650 700 750 800 850 900 950 10000

1

2

3

4

5

6x 10

−4 Variance Gamma model

Lookback option price

Pro

ba

bility

0 0.005 0.01 0.015650

700

750

800

850

900

950

Scaling parameter

Lo

okb

ack o

ptio

n p

rice

Bid ask prices − Lookback option


Figure 6 The cumulative distribution functions of lookback option prices induced by the normal transformationfunction with scaling parameter λ = 0.008. For lookback options, the BNS model also shows considerably higherdispersion than the Heston model. The VG model observes much more variability compared to the other models.This may be directly related to the results of the barrier option since little stock price maximums make barriers notto break. The bid-ask prices of the VG model clearly dominate the induced bid-ask prices of the Heston and BNSmodels, although their plug-in prices are very close.

27

0 0.005 0.01 0.015−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Scaling parameter

Re

lative

de

via

tio

n to

plu

g−

in p

rice

Relative deviation − Heston model

Asian bid deviationAsian ask deviationLookback bid deviationLookback ask deviationBarrier bid deviationBarrier ask deviation

0 0.005 0.01 0.015−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Scaling parameter

Re

lative

de

via

tio

n to

plu

g−

in p

rice

Relative deviation − BNS model


0 0.005 0.01 0.015−0.5

0

0.5

1

1.5

2

Scaling parameter

Re

lative

de

via

tio

n to

plu

g−

in p

rice

Relative deviation − VG model


Figure 7 Relative deviations from the plug-in price compared among different option types for the Heston, BNS,and Variance Gamma models. In the Heston model, both Asian and lookback options observe low parameter risk. Forlookback options, parameter risk is skewed to the ask side. In the BNS model, Asian options observe low parameterrisk but lookbacks have considerable parameter risk. For both models, parameter risk is largest for the ITM barrieroption, the BNS model observes large deviations (> 20%) from the plug-in price. The Variance Gamma model hashigher deviations from the plug-in price. An extreme deviation can be observed for the ITM barrier option ask price.Here we have deviations of > 150% from the original price.

28

capturing parameter risk with convex risk measures · schoutens et al (2004) examine diﬀerent...

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