Index

absolute risk aversion 1

absolutely continuous measure 1

acceptance set 1, 2, 3, 4, 5, 6, 7

– for conditional risk measure 1

– one-step 1

act 1

adapted process 1

admissible strategy 1, 2

affine hull of a convex set 1

affine numerical representation 1

Allais paradox 1, 2

American claim

– arbitrage-free price of 1, 2

– attainable 1

– hedging strategy 1

American option 1

– call 1

– put 1, 2

arbitrage bounds 1

arbitrage opportunity 1, 2, 3, 4, 5

arbitrage-free price

– of a contingent claim 1

– of a European claim 1

– of American claim 1

– of an American claim 1

– universal bounds 1, 2

Archimedean axiom 1, 2

Arrow–Debreu equilibrium 1

– and interest rate 1

– w.r.t. general utility functionals 1

Arrow–Pratt coefficient 1

Asian option 1, 2

asymmetric relation 1

at-the-money option 1

atom of a probability space 1, 2, 3

atomless probability space 1, 2

attainable

– American claim 1

– European claim 1

– payoff 1

average

– price option 1

– strike option 1

Average Value at Risk 1

– comonotonicity 1, 2

– sensitivity 1

– time-inconsistency 1

backwards martingale 1

balayage order 1, 2, 3

Banach space 1

Banach–Alaoglu theorem 1

barrier option 1, 2

– Black–Scholes price of 1

– down-and-in put 1

– up-and-in call 1, 2

– up-and-out call 1, 2, 3

barycenter of a measure 1

basket option 1, 2

Bermuda option 1

Bernoulli, Daniel 1, 2

Bernoulli, Nicholas 1

binomial model 1, 2, 3

Black–Scholes formula 1, 2

Black–Scholes p.d.e. 1, 2

Black–Scholes price 1

– of a European call option 1, 2

Bolzano–Weierstraß theorem

– randomized version 1

boundedly supported measures 1

Brownian motion 1

butterfly spread 1

call option 1, 2, 3

– American 1, 2

– Asian 1

– Bermuda 1

– Black–Scholes price 1, 2

– European 1, 2, 3, 4

– universal bounds on price 1

–with barrier 1, 2

call-put parity 1

cap 1

CARA utility 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

cash additivity 1

cash invariance 1, 2

– conditional 1

catastrophe bond 1

central limit theorem 1, 2

– functional 1

–multiplicative 1

certainty equivalent 1, 2, 3, 4, 5

– as reservation price 1

– as risk measure 1

– cash-invariant 1

– quasi-convex 1

– relative 1

– translation property 1

certainty independence 1

–weak 1

Choquet boundary 1

Choquet integral 1, 2, 3, 4

coherent conditional risk measure 1

coherent measure of risk 1

– and superhedging 1

– representation theorem 1, 2

coherent monetary utility functional 1

coherent risk measure 1, 2

– conditional 1, 2

– dynamic 1

– dynamically consistent 1

– representation theorem 1

– time-consistent 1

comonotone random variables 1, 2, 3, 4, 5

comonotonic

– functional 1

– independence 1, 2

– risk measure 1

complete

–market 1, 2

– relation 1

compound lottery 1

concave distortion 1, 2

concave function 1

– proper 1

concave monetary utility functional 1

concave stochastic order 1

conditional Average Value at Risk

– time-inconsistency 1

conditional cash invariance 1

conditional coherent risk measure 1

conditional convexity 1

conditional covariance 1

conditional entropic risk measure 1

conditional expectation 1, 2

conditional mean-standard deviation risk measure 1

conditional relative entropy 1

conditional risk measure

– coherent 1

– relevant 1

– representation 1

– sensitive 1

– time consistent 1

Conditional Value at Risk 1

conditional Value at Risk 1

– time-inconsistency 1

conditional variance 1, 2

conjugate function 1, 2, 3

– of a convex risk measure 1, 2

connected topological space 1

constant absolute risk aversion 1, 2

constant relative risk aversion 1

contingent claim 1

– American 1

– arbitrage-free price of 1

– attainable 1

– European 1

– redundant 1

– replicable 1

continuation region 1

continuity axiom 1, 2

continuity from above 1, 2, 3, 4

continuity from below 1, 2, 3, 4

convex combination 1

convex function 1

– duality theory 1

– proper 1

convex hull 1

convex measure of risk 1

– continuous from above 1, 2

– continuous from below 1, 2

– in financial market 1, 2, 3

– law-invariant 1

– penalty function for 1, 2, 3, 4, 5

– relevant 1

– representation theorem 1, 2

– sensitive 1, 2

– tight 1

convex risk measure 1, 2

– conditional 1

– representation theorem 1

convex set 1

convex stochastic order 1, 2, 3

convexity

– conditional 1

core of a set function 1

cost of superhedging 1

cost process 1

countably convex set 1, 2

Cox-Ross-Rubinstein model 1, 2, 3

CRR model 1, 2, 3

CRRA utility 1

declining a favorable bet 1

Delta hedging 1, 2, 3

Delta of a call option 1

density function 1

density process 1, 2

derivative 1

derivative securities 1, 2, 3

digital option 1, 2

Dini’s lemma 1

Dirac measure 1

discount certificate 1

discounted American claim 1

discounted claim 1

discounting 1, 2

distortion function 1

distortion of a probability measure 1

distribution

– log-normal 1, 2

– normal 1

– Poisson 1

distribution function 1

– and stochastic dominance 1

– and stochastic order 1

– modified 1

divergence 1, 2

divergence risk measure 1

– sensitivity 1

divergence risk measures 1

diversification 1

Doléans–Dade exponential 1

Donsker’s invariance principle 1

Doob decomposition 1, 2, 3

– uniform 1, 2

Doob’s stopping theorem 1, 2

Doob’s systems theorem 1

down-and-in option 1, 2, 3

down-and-out option 1

dual space 1

duality theory of convex functions 1

Dunford–Pettis theorem 1

dynamic coherent risk measure 1

dynamic consistency of a risk measure 1

dynamic risk measure

– relevant 1

– sensitive 1

– time consistent 1

–weak time consistency 1

dynamically consistent 1

early exercise premium 1

Eberlein–Šmulian theorem 1

effective domain of a convex function 1

efficient market hypothesis 1

Ellsberg paradox 1

enlagement of filtration 1

entropic risk measure 1, 2, 3, 4

– conditional 1

– sensitivity 1

entropy of a measure 1

entropy-minimizing risk-neutral measure 1, 2

equivalent martingale measure 1, 2

equivalent measure 1

Esscher transform 1

essential infimum 1

– of a random variable 1

essential supremum 1

Euclidean inner product 1

Euclidean norm 1

European call option 1, 2, 3

– Black–Scholes price 1, 2

– efficient hedging of 1

– intrinsic value of 1

– time value of 1

European contingent claim 1

– arbitrage-free price 1

– attainable 1, 2

– redundant 1

– replicable 1

European put option 1

exercise strategy for American claim 1

Expected Shortfall 1

expected shortfall 1, 2

expected utility representation 1

expectile 1

expiration date of a contingent claim 1

exponential family 1

exponential utility function 1, 2, 3

extreme point 1

– of comonotonic convex risk measures 1

fair premium 1

fair price 1

Fatou property 1, 2, 3

–on Lp 1

favorable bet 1

feasible allocation 1

Fenchel–Legendre transform 1, 2, 3, 4, 5, 6

– of a convex risk measure 1, 2

filtered probability space 1

filtration 1

– enlargement of 1

financial position 1

finitely additive set function 1, 2, 3

first-order stochastic dominance 1, 2

fork convex sets 1

forward contract 1, 2

forward starting option 1

Fréchet bounds 1

FTAP 1, 2

– second 1, 2

– under portfolio constraints 1

–with contingent initial data 1

function

– concave 1

– convex 1

fundamental theorem of asset pricing

– for multi-period market 1

– geometric form 1

– in one period 1

– second 1, 2

– under portfolio constraints 1

–with contingent initial data 1

–without a priori measure 1

g-divergence 1, 2, 3

gains process 1, 2

Gamma of a call option 1

gauge function 1, 2, 3

generalized likelihood quotient test 1

generalized trading strategy 1

–mean self-financing 1

geometric Brownian motion 1

Gini coefficient 1

Gini’s mean difference 1

Girsanov formula 1

global quadratic risk 1

Greeks 1

– and Black–Scholes equation 1

– Delta 1

–Gamma 1

– Rho 1

– Theta 1

– Vanna 1

– Vega 1

– Volga 1

–Vomma 1

Hahn–Banach theorem 1

Halmos–Savage theorem 1

HARA utility 1, 2, 3, 4

Hardy–Littlewood inequality 1, 2

Hausdorff space 1

hedging strategy for American claim 1

Hoeffding–Fréchet bounds 1

hyperbolic absolute risk aversion 1

in-the-money option 1

Inada conditions 1

independence axiom 1, 2, 3, 4

–on 1

indifference relation 1

inequality

– Hardy–Littlewood 1, 2

inf-convolution of two risk measures 1

inner product

– Euclidean 1

insider information 1

insurance 1, 2, 3

interest rate 1

intrinsic value 1, 2

inverse function 1

– left-continuous 1

– right-continuous 1

Itô integral 1

Itô’s formula 1

Knightian uncertainty 1, 2, 3

knock-in option 1, 2

knock-out option 1, 2

Kolmogorov’s law of large numbers 1, 2, 3

Krein–Šmulian theorem 1

Kreps–Yan theorem 1, 2

Kunita–Watanabe decomposition 1

λ-efficient allocation 1

λ-quantile 1, 2

L2-admissible strategy 1

law of large numbers 1, 2, 3

law of one price 1, 2

least favorable measure 1

– and hypothesis testing 1

Lebesgue decomposition 1

Lebesgue property 1, 2, 3

left-continuous inverse 1

Legendre transform 1

leverage effect 1, 2

lexicographical order 1

likelihood quotient test 1

liquid option 1

local martingale 1

local risk process 1

locally convex space 1

locally risk-minimizing strategy 1

locally riskless bond 1

log-normal distribution 1, 2

– change of measure 1

lookback call 1

lookback option 1, 2

lookback put 1

loss function 1, 2

lottery 1, 2

– variance of 1

lower quantile function 1

lower Snell envelope 1, 2

m-stable sets 1

market clearing condition 1

market portfolio 1

Markov property 1

martingale 1, 2

– backwards 1

– convergence theorem 1

– local 1

– of successive densities 1, 2

– reversed 1

– strongly orthogonal 1

martingale measure 1, 2, 3

– as extension of a pricing rule 1

– minimal 1

martingale representation property 1

maturity of a contingent claim 1

MAXMINVAR 1

MAXVAR 1

mean self-financing strategy 1

mean-preserving spread 1, 2, 3

mean-standard deviation risk measure 1, 2

– conditional 1

mean-variance trade-off process 1, 2

measure

– absolutely continuous 1

– entropy of 1

– equivalent 1

– simple 1, 2

– support of 1

measure of risk

– acceptance set of 1

– cash invariant 1

– coherent 1, 2

– comonotonic 1

– continuous from above 1, 2

– continuous from below 1

– convex 1

– in financial market 1, 2, 3

– monetary 1

– monotone 1

– normalized 1

– penalty function for 1

– positive homogeneous 1

– relevant 1

– representation theorem 1

– sensitive 1, 2

– subadditive 1

– translation invariant 1

metric space 1

minimal martingale measure 1

Minkowski’s separating hyperplane theorem 1

MINMAXVAR 1

MINVAR 1, 2

mixture space 1

model uncertainty 1

modified distribution function 1

moment generating function 1

monetary conditional risk measure 1

monetary measure of risk 1

monetary risk measure 1

monetary utility functional

– coherent 1

– concave 1

monotone preference relation

– on assets 1

– on distributions 1

monotone set function 1

negative part 1

negative transitive relation 1

Neyman–Pearson lemma 1, 2, 3

– generalized 1

nonredundance condition 1, 2

norm

– Euclidean 1

normal distribution 1, 2, 3, 4

normalized set function 1

numéraire 1, 2, 3

– change of 1, 2

numerical representation 1, 2

– affine 1

– of robust Savage form 1

– of Savage form 1, 2

– of von Neumann–Morgenstern form 1, 2, 3, 4, 5

optimal stopping problem 1

– continuation region 1

– stopping region 1

option 1

– Black–Scholes price 1

optional decomposition 1

– theorem 1

– under constraints 1

optional sampling theorem 1, 2

order dense set 1

orthogonal decomposition

– of a contingent claim 1

out-of-the-money option 1

ψ-weak topology 1, 2, 3

P&L 1

paradox

– Allais 1, 2

– Ellsberg 1

– St. Petersburg 1

partial order 1

pasting of two probability measures 1, 2, 3

penalty function 1, 2, 3

– one-step 1

Poisson distribution 1, 2

Polish space 1

portfolio 1

portfolio insurance 1

portmanteau theorem 1

positive affine transformations 1

power of a statistical test 1, 2

predictable process 1

predictable representation property 1

predictably convex set 1

preference order 1

preference relation 1

– asymmetry of 1

– continuous 1

– continuous from above 1

– continuous from below 1

– lexicographical 1

– monotone 1, 2

– negative transitivity of 1

– numerical representation of 1, 2, 3, 4, 5, 6

– random 1

– risk averse 1

–weak 1

price density 1

price system 1

pricing measure 1

probability measure

– absolutely continuous 1

– equivalent 1

Prohorov’s theorem 1

– for ψ-weak topology 1

proper concave function 1

proper convex function 1

put option 1

– American 1

– Asian 1

– European 1, 2

– universal bounds on price 1

–with barrier 1

put-call parity 1, 2

Q-martingale 1

Q-submartingale 1

Q-supermartingale 1

QS-Snell envelope

– upper 1

quadratic risk

– global 1

– local 1

– remaining 1

quantile 1

quantile function 1, 2

– and stochastic order 1

– for nonadditive set functions 1

quantile hedging 1

quasi-convexity 1

Radon measure 1

Radon–Nikodym

– derivative 1

– theorem 1

random walk 1

randomized statistical test 1, 2, 3, 4

recursiveness 1

reference portfolio 1

reflection principle 1

regular conditional distribution 1, 2

relation

– antisymmetric 1

– asymmetric 1

– complete 1

– equivalence 1

– indifference 1

– negative transitive 1

– reflexive 1

– transitive 1, 2

relative entropy 1, 2, 3

– conditional 1

relative interior of a convex set 1

relative risk aversion 1

relevant risk measure 1, 2

remaining conditional risk 1

replicating

– portfolio 1

– strategy 1

representation theorem for risk measures 1

–on L∞ 1

reservation price 1

return 1

reverse convertible bond 1

reversed martingale 1

Rho of a call option 1

right-continuous inverse 1

risk 1

risk aversion 1, 2, 3, 4, 5

– absolute 1

– relative 1

risk measure 1

– acceptance set of 1

– cash invariant 1

– coherent 1, 2

– comonotonic 1

– conditional 1, 2

– continuous from above 1, 2

– continuous from below 1, 2

– convex 1, 2

– dynamic 1

– dynamically consistent 1

– entropic 1, 2, 3, 4

– in financial market 1, 2, 3

– monetary 1

– monotone 1

– normalized 1

– penalty function for 1

– positive homogeneous 1

– relevant 1, 2

– representation theorem 1

– sensitive 1, 2, 3

– subadditive 1

– tight 1

– time-consistent 1

– translation invariant 1

risk neutrality 1

risk premium 1

risk transfer 1

risk-neutral measure 1, 2

– entropy-minimizing 1, 2

robust representation

– of conditional risk measure 1

robust Savage representation 1, 2, 3

robust shortfall risk 1

running maximum 1, 2

Savage representation 1, 2

– robust 1, 2, 3

second fundamental theorem of asset pricing 1, 2

second-order stochastic dominance 1, 2, 3, 4

self-financing strategy 1

– in continuous time 1

sensitive risk measure 1, 2, 3

separable metric space 1

separating hyperplane theorem 1

separation theorems 1

set

– convex 1

set function

– core of 1

– finitely additive 1, 2, 3

– monotone 1, 2

– normalized 1, 2

– strongly additive 1

– strongly subadditive 1, 2, 3

– submodular 1, 2, 3, 4

Sharpe ratio 1

short sales 1

– constraints 1, 2

shortfall 1

shortfall risk 1, 2

– robust 1

significance level of a statistical test 1, 2

simple measure 1, 2

simple probability distribution 1

size of a statistical test 1

Snell envelope 1, 2, 3

– lower 1

– upper 1, 2, 3, 4

– upper QS- 1

space of reference portfolios 1

St. Petersburg paradox 1

stability under pasting 1

stable set of measures 1, 2

stable subspace of H 2 1

standard normal distribution 1

statistical test 1

– for composite hypothesis 1, 2

– randomized 1, 2, 3

stochastic differential equation 1

stochastic dominance 1

– first order 1, 2

– second order 1, 2, 3

stochastic kernel 1, 2

stochastic order

– concave 1, 2

– convex 1, 2

– monotone 1, 2

stochastic process 1

– adapted 1

– predictable 1

Stone vector lattice 1

stop-loss contract 1

stopping region 1

stopping theorem 1, 2

stopping time 1

straddle 1

strategy

–L2-admissible 1

– locally risk-minimizing 1

–mean self-financing 1

– self-financing 1

– self-financing in continuous time 1

stress test measures 1

strike price 1, 2

strong law of large numbers 1, 2, 3

strong orthogonality 1

strongly additive set function 1

strongly subadditive set function 1, 2, 3

submartingale 1

– local 1

submodular 1

submodular set function 1, 2, 3

substitution axiom 1

success ratio 1

success set 1

superhedging 1, 2

– duality 1, 2, 3

– strategy 1, 2

supermartingale 1

– local 1

superreplication strategy 1, 2

support of a measure 1

supremum norm 1

systems theorem 1

theorem

– Doob’s systems theorem 1

– Dunford–Pettis 1

– Eberlein–Šmulian 1

– Hahn–Banach 1

– Halmos–Savage 1

– Krein–Šmulian 1

– Kreps–Yan 1, 2

– optional decomposition 1

– optional sampling 1

– portmanteau 1

– Radon–Nikodym 1

– randomized Bolzano–Weierstraß 1

– separating hyperplane 1

– separation 1

– stopping 1, 2

– superhedging duality 1

Theta of a call option 1

tightness

– of a set of measures 1

– of risk measure 1

time consistency 1

–weak 1

time consistency of a risk measure 1

time value 1, 2

– of a European call option 1

topological space

– connected 1

– Hausdorff 1

topological vector space 1

– locally convex 1

topology

–weak 1

–weak∗ 1

total variation 1, 2

trading strategy 1

–L2-admissible 1

– generalized 1

– replicating 1

– self-financing 1

transitive relation 1

translation invariance 1

translation property 1

– of certainty equivalent 1

type 2 error 1

type 1 error 2

uncertainty 1

uncertainty aversion 1

underlying 1

uniform Doob decomposition 1

– under constraints 1

universal arbitrage bounds 1, 2

up-and-in call 1

up-and-in option 1, 2

– Black–Scholes price of 1

up-and-out call 1, 2

up-and-out option 1, 2, 3

upper QS-Snell envelope 1

– characterization of 1

upper hedging price 1

upper quantile function 1

upper Snell envelope 1, 2

– characterization of 1

– optional decomposition of 1

utility function 1, 2, 3

– CARA 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

– CRRA 1

– decreasingly risk averse 1

– exponential 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

– HARA 1, 2, 3

– random 1, 2

utility-based shortfall risk measure 1, 2

Value at Risk 1, 2, 3, 4, 5, 6, 7

– comonotonicity 1

– conditional 1

– time-inconsistency 1

value process 1

Vanna of a call option 1

variance 1

– conditional 1

variance-optimal strategy 1

vector lattice 1

Vega of a call option 1

volatility 1, 2

Volga of a call option 1

Vomma of a call option 1

von Neumann–Morgenstern representation 1, 2, 3, 4, 5

weak certainty independence 1

weak information 1

weak preference order 1

weak time consistency 1

weak topology

– on a Banach space 1

– on a locally convex space 1

weak topology for measures 1

weak∗ topology 1

weakly compact set 1

Wiener measure 1

Wiener process 1

worst conditional expectation 1, 2

worst-case risk measure 1

–on L∞ 1, 2