Probability

(exogenous liquidity risk) rather than as a result of an individual trader attempting to dispose of an unusually large position (endogenous liquidity risk). We define exogenous liquidity risk measurement in terms of a confidence interval or a tail probability. We define the exogenous cost of liquidity (COL) based on a certain average spread, S , plus a multiple of the ~ spread volatility, a⋅ σ , to cover most (say 99%) of the spread situations. COL = 1 2 ~ Pt S + aσ c h (4) where Pt is

10iJq ~O ,'Iffff ~(j .,ff;~/'rl? !;{l:1~11'1 fH L~)l"")l~ :>111 !:I(,rl1in ')jiU t".Jnianxt ""!;~!;Jr-rj .'z.' ';. :t .lloitG:>irlii';'lO ?~ri;ln rlrdqc,,;,';"'!,:' Jluon2 l:itv(,dJ;: 'jdi~';) ":-tQ~'.(~." '..,rlt 0, with probability 1. An arbitrage portfolio is thus basically a deterministic money making machine, and \e interpret the existence of an arbitrage portfolio as equivalent to a serious case of mispricing on the market. It is now natural to investigate when a given

Security. In an execution of P , we say an adversary A wins if it asks a single Test-query to a Fresh player U and correctly guesses the bit b used in the game Gameake (A, P ). We denote the AKE advantage as Adv ake (A) = 2 Pr[b = b ] − 1, P where the probability space is over all the random coins of the adversary and all the oracles. 3 Assumptions Before presenting the protocol, let us remind the algorithmic assumptions on which its security will be based on. These assumptions were shown in [9]

Backtesting Green zone - up to 4 exceptions OK Yellow zone - 5-9 exceptions increasing k Red zone - 10 exceptions or more intervention 27 Daniel HERLEMONT Probability of Multiple Exceptions Each period the probability of exception is 1%, then after 250 business days the probability that there will be 0 exceptions is 250! 0.010 ⋅ 0.99 250 = 0.081 0!⋅250! General formula of binomial distribution is n! p x ⋅ (1 − p ) n − x x!⋅(n − x)! 28 Daniel HERLEMONT

(14) 2.4.2 The Zhou volatility with covariance estimated on a longer sample The previous estimator can be changed by measuring the incoherent correction on a much longer time interval. This should reduce the variance, and therefore the probability to get negative values. In this direction, several variations of the Zhou estimator can be written. One possible estimator is the following: rk j r k j t ¡ ¡ ¢ where n ∆t t is the number of ticks between t ∆t and t. Essentially, the ratio Covariance

Probabilistic model............................................................................................................... 82 6.4 Confidence intervals for liquefaction-induced ground deformation .................................... 84 6.5 Probability calculation.......................................................................................................... 90 6.6 Future work ...........................................................................................................

counterparty risk We denote by τ1 the default time of the credit underlying the CDS, and by τ2 the default time of the counterparty. We assume the investor who is considering a transaction with the counterparty to be default-free. We place ourselves in a probability space (Ω, G, Gt , Q). The filtration (Gt )t models the flow of information of the whole market, including credit and defaults. Q is the risk neutral measure. This space is endowed also with a right-continuous and complete sub-filtration Ft representing

probabilities, Journal of Computational and Applied Mathematics, Vol. 230, No 1, 107-120 (2009). – [113] C. Lef`vre, S. Loisel, Finite-Time Ruin Probabilities for Discrete, e Possibly Dependent, Claim Severities, Methodology and Computing in Applied Probability, Vol. 11, No 3 (2009), 425-441. – ∗ [69] L. Devineau, S. Loisel, Construction d’un algorithme d’acc´l´ration ee de la m´thode des ”simulations dans les simulations” pour le calcul du e capital ´conomique Solvabilit´ II, Bulletin Fran¸ais d’Actuariat

........................... 164 4. Scenario Generation ............................................................................................. 165 4.1. Estimating the probability of a plant failure ..................................................................................... 167 4.2. Estimating the probability of a specific plant to fail .......................................................................... 167 4.3. Estimating the duration and start time of plant failures ....

Created more than a century ago by the actuaries to calculate the probability of bankrupt or the stocks of solvency, the value-at-risk is a widely used risk measure of the loss on a specific portfolio of financial assets. Until the late 80’s, the methods used to detect and manage risks were adapted to specific products; it was impossible to compare the risk measures between the different “tables” of the same trading floor. Thanks to the value-at-risk and its generalisation in the different markets