EXTREME MEASURES IN CONTINUOUS TIME CONIC FINANCE YOSHIHIRO SHIRAI University of Maryland College Park Department of Mathematics

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EXTREME MEASURES IN CONTINUOUS TIME CONIC FINANCE

YOSHIHIRO SHIRAI

University of Maryland, College Park, Department of Mathematics

Abstract. Dynamic spectral risk measures deﬁne a claim’s valuation bounds as supremum

and inﬁmum of expectations of the claim’s payoﬀ over a dominated set of measures. The

measures at which such extrema are attained are called extreme measures. We determine

explicit expressions for their Radon-Nykodim derivatives with respect to the common dom-

inating measure. Based on the formulas found, we estimate the extreme measures in two

cases. First, the dominating measure is calibrated to mid prices of options and valuation

bounds are given by options bid and ask prices. Second, the dominating measure is esti-

mated from historical mid equity prices and valuation bounds are given by historical 5-day

high and low prices. In both experiments, we ﬁnd that the market determines upper bounds

by testing scenarios in which losses are signiﬁcantly lower than expected under the domi-

nating measure, while lower bounds by ones in which gains are only slightly lower than in

the base case.

1. Introduction

Much of the existing literature on continuous time valuation bounds deﬁnes upper and

lower prices as suprema and inﬁma of conditional expectations of discounted payoﬀs over a

certain set Mof measures. When Mis weakly compact and the bounds are time consistent,

the extrema are attained at the same measure at diﬀerent points in time (Delbaen (2006)).

We call such measures extreme for analogy with those analyzed in Cherny (2008) in a static

setting. The main mathematical contribution of this paper is to construct a pair of extreme

measures for the continuous time Conic Finance bounds deﬁned in Madan et al. (2017).1

The fundamental assumption of Conic Finance, introduced in Madan & Cherny (2010),

is that risks cannot be fully hedged and so the set of trades deemed acceptable by market

operators must strictly contain that of arbitrages. Acceptability is deﬁned in Conic Finance

by assuming that the market chooses a reference probability space (Ω,F,Q) and a set of test

measures Mdominated by Q, so that a payoﬀ is acceptable if its expected value under any

test measure is nonnegative. Other deﬁnitions include those of Ledoit (1995), Cochrane &

Sa`a-Requejo (2000), Cherny & Hodges (2000) and Bernardo & Ledoit (2000). Furthermore,

valuation bounds can also be deﬁned based on indiﬀerence pricing (see Carmona (2008)) and

model-free hedging (Hobson (1998)).

E-mail address:yshirai@umd.edu.

Date: October 23, 2023.

2020 Mathematics Subject Classiﬁcation. 60H10, 60G51, 91G20, 91G80.

Key words and phrases. Backward Stochastic Diﬀerential Equations, Convex Analysis, Conic Finance,

Dynamic Spectral Risk Measures, Empirical Analysis of Bid-Ask Spreads.

1The point of view in Madan et al. (2017) is that of risk measures, whereas here it is on valuation

bounds. The mathematical aspects of the two theories are the same (see e.g. Jashcke & Kuchler (2001)).

arXiv:2210.13671v2 [q-fin.RM] 20 Oct 2023

2 EXTREME MEASURES IN CONTINUOUS TIME CONIC FINANCE

Each of these alternatives presents its own advantages. As explained in Madan & Cherny

(2010), Conic Finance valuations are independent from agents’ preferences and initial wealth

and they are robust to model misspeciﬁcations. In addition, generating upper and lower

prices in Conic Finance does not require the existence of underlying liquid securities.

From a mathematical perspective, upper and lower valuations are deﬁned in continuous

time Conic Finance as the unique solutions of respective upper and lower backward stochas-

tic diﬀerential equations (BSDEs) driven by a Choquet integral and with pure jump Levy

generator X(see Madan et al. (2017)). In this case, and to the author’s knowledge, a full

proof for general formulas identifying the Radon-Nikodym derivatives of a pair of extreme

measures Qand Qwith respect to Qis absent from the literature. This paper’s mathemat-

ical contribution is then Theorem 3.3, in which such an explicit expression is provided in

terms of the level sets of the control processes Zand Zof the upper and lower BSDEs.

By the formulas obtained, the bounds deﬁned by continuous time Conic Finance are no

longer linear, as in static Conic Finance, over a pair of comonotonic payoﬀs. The requirement

for linearity is now that the control processes associated to the value of the two claims be

comonotonic for all t∈[0, T ], where Tis the expiration date, and we show in Remark

4.3 that this is indeed more restrictive than just comonotonicity of payoﬀs. This is no

surprise: by Theorem 7.1 in Delbaen (2021), a comonotonic and time consistent dynamic

expectation on the set of bounded random variables must be a conditional expectation.

Another interesting consequence of our formulas is that, diﬀerently from the static case

(Kusuoka (2001)), continuous time Conic Finance valuations are not law invariant. More

precisely, as observed in Remark 4.4, equivalence of the valuation bounds of two claims

requires the L´evy measure of Xunder Qof the α-level sets of Ztand Ztto be the same

for each level αand all t∈[0, T ]. Such result is in line with the one proved in Kupper

& Schachermeyer (2009) that the only time consistent, law invariant, dynamic nonlinear

expectation is the entropic risk measure.

In our L´evy setting, the existence of the extreme measures implies that, as shown in Barles

et al. (1996), there are functions u, ℓ : [0, T ]×RD\{0} → Rsuch that u(t, Xt) and ℓ(t, Xt)

are the upper and lower valuations for each t∈[0, T ]. Furthermore, as shown in Denneberg

(1994), the bounds are Malliavin diﬀerentiable and so the control processes satisfy

(1.1) Zt(y) = u(t, Xt−+y)−u(t, Xt−), Zt(y) = ℓ(t, Xt−+y)−ℓ(t, Xt−)

for every t∈[0, T ]. Based on (1.1), we show in Theorem 4.2 that if Xhas dimension 1 and f

is monotone, the level sets of the control processes are time independent and deterministic,

and so Xis a L´evy process under Qand Q.

This result paves the way for two empirical studies that constitute the second and third

contributions of this paper. In the ﬁrst such study, the law of Xunder the reference measure

Qis obtained by calibrating the bilateral gamma (BG) process introduced in Kuchler &

Tappe (2008) to mid prices of options on the SPY exchange trade fund (SPY). Since options’

payoﬀs are monotonic, Xis a L´evy process under the extreme measures associated to options’

valuation bounds, and the respective upper and lower L´evy measures can be calibrated to

bid and ask prices of options via FFT (see Carr & Madan (1998)). Our goals are:

1. to assess if continuous time Conic Finance bounds can match relative bid-ask spreads

across strikes; and

2. to infer, based on the diﬀerent relative bid-ask spreads across strikes, which events

the market is more uncertain about.

EXTREME MEASURES IN CONTINUOUS TIME CONIC FINANCE 3

The importance of 1. is that the calibration exercise determines the set Mof test measures.

If the relative options bid-ask spreads are matched, the same set Mcan be used to generate

quotes for certain derivatives traded out of the counter, such as straddles and, more in

general, combo options. This is an important issue for market makers as they need to quickly

provide quotes that are cheap but accurate enough. With this consideration in mind, we also

ﬁnd conditions on the set Mfor Xto be a BG process under both extreme measures and

assuming it is a BG process under Q. The conditions obtained resemble the Wang transform

(Wang (2000), Wang (2002)), with the BG distribution replacing the Gaussian. When they

are enforced, calibration is further facilitated, although the error is higher (see Remark 5.5).

The importance of 2. is that the ability to explain why some events are more uncertain than

others based on market data is one way to assess the calibrated set M. We set X=G−L,

where Gand Lare positive gamma processes referred to as gains and losses (see Kuchler &

Tappe (2008) for their existence). Then, we ﬁnd that bounds are determined by distorting

the loss process Lfor the ask price of calls and the bid price of puts, and the process Gfor the

bid price of calls and ask price of puts. That is, a call’s ask and a put’s bid are determined

by testing their payoﬀ under scenarios in which there is a high chance that LTis lower than

expected under Q. For a put’s ask or a call’s bid, instead, the payoﬀ is tested against the

event that GTmay be higher than expected. We then ﬁnd that the stress on the loss process

is higher than that on the gain process. This is related to the investors’ need to hedge against

economic downturns, which makes the out of the money (OTM) puts market more liquid

than that of OTM calls. Only few existing empirical studies are performed on options bid

and ask prices, so this contribution is quite unique in the empirical ﬁnance literature.

For the second empirical study of this paper we assume that, under the reference measure,

the law of Xis again BG but this time estimated from daily closing mid prices of the SPY.

Furthermore, the observed upper and lower valuations are deﬁned by the SPY 5-day high

and low prices. This is based on the fact that large trades put in place by institutional

investors are often executed over a few days at least. The payoﬀ of owning the SPY is

deﬁned by Y0eXT, where Y0is the current value of the SPY and Tis 5 days. Then Xis

again a L´evy process under Qand Qand its L´evy measure can be expressed in terms of an

integral. Hence, the probability density of XTcan be calculated by Fourier inversion, and

estimated by matching its tail to that of the empirical distribution (as in Madan (2015)).

The resulting estimator is called the digital moment estimator (DM). Our goals are:

1. to compare the estimates we obtain from DM with the one obtained through the

generalized method of moment (GMM); and

2. to infer, based on historical equity prices, which events market operators are more

uncertain about.

The importance of 1. is to determine the usefulness of our formulas for the Radon-Nykodim

derivatives for estimation purposes. In fact, computation of the tail measure is a demanding

task without knowledge of the probability density of XT, prone to numerical error and

approximations. Hence, without the formulas developed in this paper, one is forced to use

methods, such as the GMM, that are not designed to incorporate in their estimates events

that occur less frequently. The importance of 2. is as in the study on options. We ﬁnd that:

1. with DM estimators, and as found in our ﬁrst study on options data, upper valuations

are determined by uncertainty in the loss process and lower valuations by uncertainty

in the gain process; GMM estimators are, instead, more balanced;

4 EXTREME MEASURES IN CONTINUOUS TIME CONIC FINANCE

2. the drifts of the lower valuation process estimated by the two methods are similar;

however, the DM drift of the upper valuation is lower than the GMM one;

4. the correlation between the DM estimated upper drivers and upper returns is similar

to that between lower driver and lower returns; for GMM, it is substantially higher;

3. as in the case of calibration to options, the higher DM estimated upper driver implies

that the market tests scenarios in which losses are much lower than expected, and

gains only a bit higher; this appears related to the monetary authority’s support to

the ﬁnancial sector and the real economy during the 2010-2020 decade;

5. the standard deviation of the upper valuation implied by DM estimators is higher,

on average, than that of the density of the lower valuation; they are similar for GMM

estimated densities.

From these observations we argue that market operators were more uncertain over the

period considered about the SPY’s loss process than its gain process. Furthermore, the

GMM fails to incorporate in its estimates extreme realizations of upper and lower returns.

The rest of the paper is organized as follows. In section 2 we review Conic Finance valua-

tion bounds and prove preliminary results. The formula for the Radon-Nikodym derivatives

dQ/dQand dQ/dQis given in section 3. Section 4 considers the case of monotonic claims.

Results on numerical experiments are reported in Section 5 and 6. Section 7 concludes.

2. Assumptions and Preliminary Results

2.1. Assumptions. Unless otherwise speciﬁed, the following assumptions, most of which

are as in Madan et al. (2017), hold throughout the rest of the paper.

(i) Given a topological space (X, τ), B(X) denotes the Borel σ-algebra on X.

(ii) Given a measurable space (Ω,F), random processes Xi={Xi

t}t≥0on (Ω,F) and

constants Yi

0,i= 1, ..., D, and T > 0, we consider a market composed of a risk-free

asset with rate r≥0 and Drisky assets with payoﬀ Yi

T:= Yi

0eXi

T. To simplify

notation, the rate ris normalized to zero, except in the empirical studies (Sections

5 and 6). Furthermore, we set X:= (X1, ..., Xn) and {Ft}t≥0denotes the right

continuous, completed ﬁltration generated by X.

(iii) There is a probability measure Qon (Ω,F) such that, for each i,Yi

T∈L2(Ω,F,Q)

and the discounted process Yi={Yi

t}t≥0deﬁned by Yi

t:= Yi

0e−rt+Xtis a martingale

under Q. This is a condition necessary to avoid arbitrages (Jouini & Kallal (1995),

Theorem 2.1).

(iv) The process Xsatisﬁes, for t≥0 and d∈RD\ {0},

Xt=dt+Z[0,t]×RD\{0}

y˜

N(dy, ds).

where ˜

N(dy, ds) := N(dy, ds)−ν(dy)ds and Nis a Poisson random measure with

Q-compensator ν. It is assumed that νis σ-ﬁnite, has no atoms and, for some ε > 0,

ZRD\{0}

|y|2+εν(dy)∈RD\{0}.

(v) {Xt,x

s}s≥tis deﬁned for s≥t≥0 and x∈RDby

Xt,x

s=x+d(s−t) + Z[t,s]×RD\{0}

y˜

N(dy, ds).

EXTREME MEASURES IN CONTINUOUS TIME CONIC FINANCE 5

(vi) Γ = (Γ+,Γ−) is a pair of measure distortions, i.e. Γ+,Γ−: [0, ν(R)) →R+are

bounded, concave and satisfy Γ−(x)≤xand

Z(0,ν(R))

Γ±(y)

2y3/2dy < ∞.(2.1)

(vii) L2(ν) := L2(R\{0},B(R\{0}), ν) and g:L2(ν)→Ris speciﬁed for z∈L2(ν) by

g(z) := Z∞

Γ+(ν(z+> a))da +Z∞

Γ−(ν(z−> a))da.(2.2)

Example 2.1. An example of measure distortions is the pair Λ = (Λ+,Λ−)deﬁned by

(2.3) Λ+(x) := a1−e−cx1/(1+γ),Λ−(x) := b

c1−e−cx,

where x≥0,0< γ < 1,0< b ≤1and a, c > 0. These distortions are obtained in Eberlein

et al. (2013) by composing common probability distortions with the change of variable x→

1−e−cx. Note that if γ≥1then assumption 2.1 does not hold, and if b > 1there is x > 0

such that Γ−(x)> x. The requirement γ > 0ensures that the associated probability distortion

is strictly concave. See Eberlein et al. (2014) for a description of the parameters a, b, c, γ.

2.2. Notation. In addition to the assumptions above, the following notation will be used

throughout the paper.

i. For p∈[1,∞], Lp:= Lp(Ω,FT,Q); recall that L1⊃L2⊃... ⊃L∞.

ii. L2(ν) is endowed with the Borel σ-algebra generated by the L2(ν)-norm topology.

iii. For an L2(ν)-valued process {Vt}t≥0and y∈RD\{0}, we often write Vt(y) for Vt(ω, y).

iv. Edenotes the Doleans-Dade exponential.

v. Pdenotes the predictable σ-algebra on [0, T ]×Ω.

vi. We often identify the set of test measures Mwith the subset of L1of their Radon

Nikodim derivatives. A weak* topology on Mcan be deﬁned as in Corollary 14.11

of Aliprantis & Border (2006). This topology is equivalent to the weak topology on

L1, i.e. the topology such that if {χn}n∈N⊂ M, then χn→χ∈ M if and only if

ZΩ

C(ω)χn(dω)→ZΩ

C(ω)χ(dω),∀C∈L∞.(2.4)

If M ⊂ L2, we also have the weak topology in L2, i.e. (2.4) must hold for all

C∈L2. Recall also that, by the Eberlein-Smulian theorem, weak compactness in Lp

is equivalent to weak sequential compactness in Lp, 1 ≤p≤ ∞.

vii. C(Γ) denotes the set of functions q∈L2(ν) s.t., for A∈ B(RD\{0}) with ν(A)<∞,

−Γ−(ν(A)) ≤ZA

q(y)ν(dy)≤Γ+(ν(A)).(2.5)

viii For Γ+,Γ−diﬀerentiable and D= 1, we set

(2.6) ψΓ(y) := Γ′

+(ν([y, ∞))) 11{y>0}−Γ′

−(ν((−∞, y]))) 11{y<0},

ψΓ(y) := −Γ′

−(ν([y, ∞))) 11{y>0}+ Γ′

+(ν((−∞, y]))) 11{y<0}.

ix For Γ+,Γ−diﬀerentiable and D= 1, Q(Γ) and Q(Γ) denote test measures under

which the compensator of Nis respectively given by

(1 + ψΓ(y))ν(dy),(1 + ψΓ(y))ν(dy)

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EXTREMEMEASURESINCONTINUOUSTIMECONICFINANCEYOSHIHIROSHIRAIUniversityofMaryland,CollegePark,DepartmentofMathematicsAbstract.Dynamicspectralriskmeasuresdefineaclaim’svaluationboundsassupremumandinfimumofexpectationsoftheclaim’spayoffoveradominatedsetofmeasures.Themeasuresatwhichsuchextremaareattaineda...

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EXTREME MEASURES IN CONTINUOUS TIME CONIC FINANCE YOSHIHIRO SHIRAI University of Maryland College Park Department of Mathematics

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