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## Rare events simulation and tail analysis

As expained in the previous sections, the computation of risk measures requires the evaluation of the low percentiles of the portfolio distribution and hence the estimation of probabilities of rare events. In this thesis, we propose to rely on (i) the Monte Carlo simulation with variance reduction allowing to estimate the probabilities of rare events efficiently and on (ii) asymptotic methods based either on the regular variation theory or the large deviations approach.

Rare events and Monte Carlo simulation The pricing of certain sophisticated financial instruments may require the simulation of rare events. For example, in order to price barrier options, one needs to compute the potentially very small probability that the stock price exceeds a certain level. In this context, the crude Monte Carlo method can be inaccurate as presented thereafter. Description of the problem. Let X be a random variable and assume that we need to estimate the probability p = P[X < x], and that p is small. Using a sample of size n, (X1, . . . ,Xn), we can estimate p by the empirical mean: ˆp = 1 n Xn i=1 1Xi≤0.

### Optimal importance sampling for L´evy processes

In this part we develop effective and easy to implement importance sampling estimators of expectations of functionals of L´evy processes, corresponding to option prices in exponential L´evy models. To model a financial market with a L´evy process, we assume that the market consists of a risk-free asset S0 t ≡ 1 and n risky assets S1, . . . , Sn where Si t = Si 0eXi t , and (X1, . . . ,Xn) is a L´evy process, such that Si is a martingale for each i, under the risk-neutral probability P. We fix a time horizon T < ∞ and consider a derivative written on (Si)1≤i≤n with a nonnegative pay-off P(S) which depends of the entire trajectory of the stocks up to time T. We are interested in computing the price of this derivative, given by the risk-neutral expectation E [P(S)].

The standard Monte Carlo estimator of E [P(S)] is defined as the empirical mean b PN := 1 N XN j=1 P(S(j)), where S(j), j = 1, . . . ,N are i.i.d. samples with the same law as S. The standard estimator often converges too slowly, and we therefore propose to use an importance sampling estimator based on the path-dependent Esscher transform.

#### Tail asymptotics of log-normal mixture portfolios

In this part we consider the tail behavior of the sum of n dependent positive random variables X = Xn i=1 Xi.

In financial mathematics, X may represent the value of a long-only portfolio of n assets, and understanding the tail behavior of X is important for risk management applications, such as computing the Value at Risk, evaluating tail event probabilities or designing efficient simulation algorithms for tail events. In particular, stress test scenarios may be constructed in a systematic manner by simulating the values of the components X1, . . . ,Xn conditionnally on the event that X takes a given small value.

This problem has received considerable attention in the literature, but mainly in the insurance context, where the random variables X1, . . . ,Xn represent losses from individual claims, and one is interested in the right tail asymptotics of X, so as to estimate the probability of having a very large aggregate loss. In this setting, provided the variables X1, . . . ,Xn are sufficiently fat-tailed (subexponential), under various assumptions on the dependence structure, it can be shown that the right tail behavior of X is determined by the single variable with the fattest tail. In our work, we focus on the context of financial risk management where the extreme event of interest corresponds to a small value of the random variable X. In this context, to estimate the probability of a large loss, one needs to focus on the left tail asymptotics of X. Owing to the positivity of the variables X1, . . . ,Xn, the asymptotic behavior of the left tail of X turns out to be very different from that of the right tail. In this work, we compute sharp asymptotics of the distribution function and the density of X in the left tail, and discuss the relevant risk management applications, under the assumption that X1, . . . ,Xn follow a log-normal mixture distribution. That is, we assume that for i = 1, . . . , n, Xi = eYi , where the vector Y = (Y1, . . . , Yn) follows a Gaussian variance-mean mixture distribution.

**Asymptotics of heavy-tailed risks with Gaussian copula dependence**

In this part, we show that a multivariate distribution with heavy-tailed (regularly varying) univariate margins and Gaussian copula dependence exhibits multivariate regular variation on the cone (0,∞)n even if the margins are not identically distributed, and the distribution is not multivariate regularly varying on [0,∞)n. This enables us to compute sharp tail asymptotics of certain functionals, such as the sum of components, of heavy-tailed random vectors with Gaussian copula dependence, and develop a number of applications such as the computation of risk measures for a portfolio of options in the Black-Scholes model, and of the probability distribution of the aggregate production of a wind farm.

**Table of contents :**

**1 Introduction (en francais) **

1 Les d´efis et m´ethodes de gestion des risques financiers

1.1 Valorisation d’instruments d´eriv´es

1.2 Les mesures de risque

1.3 Simulation d’´ev`enements rares et analyse asymptotique

2 Principales contributions de cette th`ese

2.1 Echantillonage d’importance dans le contexte de processus de L´evy

2.2 Analyse asymptotique des portefeuilles suivants des m´elanges log-normaux

2.3 Asymptotiques de distributions `a queues ´epaisses et structure de d´ependance `a copule Gaussienne

**2 Introduction **

1 Challenges and methods of financial risk management

1.1 Pricing of derivative instruments

1.2 Measuring the risk

1.3 Rare events simulation and tail analysis

2 Main contributions of this thesis

2.1 Optimal importance sampling for L´evy processes

2.2 Tail asymptotics of log-normal mixture portfolios

2.3 Asymptotics of heavy-tailed risks with Gaussian copula dependence

**3 Optimal importance sampling for L´evy processes **

1 Introduction

2 Pathwise large deviations for L´evy processes

3 Main results

4 Examples

5 Numerical illustrations

**4 Tail asymptotics of log-normal mixture portfolios **

1 Introduction

2 Notation and preliminaries

3 Asymptotic behavior of the left tail

4 Examples

4.1 Variance Gamma process

4.2 Heston model

5 Variance Reduction of Monte Carlo estimates

6 Proof of Theorem 4.3

**5 Asymptotics of heavy-tailed risks with Gaussian copula dependence **

1 Introduction

2 Main results

3 Examples

3.1 Portfolio of options in the Black-Scholes model

3.2 Aggregate production of a wind farm