Universidad de Cartagena Cra. 6 No. 36-100, de la Universidad, Calle Cartagena,, Cartagena, Bolívar.

High-frequency statistics in Finance.

Jean Jacod (UPMC-Paris 6)

The aim of this course is to provide some basic facts about, and an overview of, statistics of processes which are observed at discrete times on a finite time interval. The domain of applications is primarily the study of observed stock prices.

After introducing the problem, we will explain which “parameters” of the model for the stock price or log-price can be identified, when it is observed at discrete times and when the frequency increases and eventually goes to infinity. The main parameters of this kind are the volatility and also the existence or not of jumps and their degree of activity when they are present. Then we will explain how it is possible to estimate these quantities, in a variety of settings (regular or irregular observation times, exact or noisy observation). If time permits, we will also mention some open Problems.

Stochastic control for insurers; what can we learn from finance, and what are the differences?.

Christian Hipp (Karlsruher Institute of Technology, Karlsruhe, Germany)

We give examples for stochastic control problems in insurance: optimal reinsurance (unlimited and limited excess of loss), optimal investment (without constraint: singularity, leverage, asymptotics), with constraints (no leverage, no shortselling and singularities caused by constraints), dividend optimisation and combinations. As methods for solution we discuss dynamic equations of Hamilton-Jacobi-Bellman type, the viscosity solution concept and a comparison argument for the insurance context. Emphasis is on numerical methods: we give an Euler type method which works in most cases and prove convergence.

Finally, we give a list of open problems together with heuristic solutions for a two objective problem: maximizing dividend payment under a ruin constraint.

Keywords: Stochastic Control, Viscosity Solutions, Euler type discretisations, Multi objective problem.

Stochastic control for insurers; what can we learn from finance, and what are the differences?.

Christian Hipp (Karlsruher Institute of Technology, Karlsruhe, Germany)

We give examples for stochastic control problems in insurance: optimal reinsurance (unlimited and limited excess of loss), optimal investment (without constraint: singularity, leverage, asymptotics), with constraints (no leverage, no shortselling and singularities caused by constraints), dividend optimisation and combinations. As methods for solution we discuss dynamic equations of Hamilton-Jacobi-Bellman type, the viscosity solution concept and a comparison argument for the insurance context. Emphasis is on numerical methods: we give an Euler type method which works in most cases and prove convergence.

Finally, we give a list of open problems together with heuristic solutions for a two objective problem: maximizing dividend payment under a ruin constraint.

Keywords: Stochastic Control, Viscosity Solutions, Euler type discretisations, Multi objective problem.

TUTORIAL ON STOCHASTIC PORTFOLIO THEORY.

IOANNIS KARATZAS

(Columbia University and INTECH Investment Management LLC)

The goal of this series of four one-hour lectures is to introduce Stochastic Portfolio Theory, a rich and flexible framework for analyzing portfolio behavior and equity market structure, and to explore some of its applications to portfolio management and performance measurement.

The first three lectures are intended for a general audience; very little background is assumed.The third lecture will be of particular interest to portfolio managers who wish to learn of new methods of measuring the impact of size on their equity portfolios. The last lecture is designed for a more mathematically advanced audience, although all relevant background materialwill be I ntroduced throughout.

Here is a detailed summary of the four one-hour lectures:

Lecture 1: The Model for a Stock PriceLogarithms; basic probability theory, random variables, mean and variance; random walks; Brownian motion; drift; stock price models; rates of return and growth rates.

Lecture 2: Introduction to Stochastic Portfolio TheoryMultiple stocks; portfolio rate of return and growth rate; Excess Growth Rate (EGR); relative growth rates and numeraire independence of EGR; the EGR of the market portfolio.

Lecture 3: Size and the Distribution of CapitalStability of markets; the size effe ct; market diversity and diversity-weighted portfolios; the distributional component of return.

Lecture 4: The Mathematics of Stochastic Portfolio TheoryDefinitions; why the EGR of an all-long portfolio is nonnegative; portfolio generating functions; diversity and relative arbitrage; local times and ranked generating functions.

References:

E.R. FERNHOLZ (2002) Stochastic Portfo

lio Theory. Springer Verlag, NY.

E.R. FERNHOLZ & I. KARATZAS (2009) Stochastic Portfolio Theory: An Overview.

Handbook of Numerical Analysis XV (A. Bensoussan & Q. Zhang, Editors). North-Holland, Amsterdam and Boston.

Short Course: TBA. Glenn Meyers. ISO Innovative Analytics.

The New Post-crisis Landscape of Derivatives and Fixed Income Activity under Regulatory Constraints on Credit risk, Liquidity risk, and Counterparty risk.

Nicole El Karoui, LPMA-UPMC, Paris

Introduction

The motivation for this course is to update academic community on the deep transformation after the financial 2008- crisis in the world of interest rates, and credit derivatives induced by the regulation. Liquidity risk, credit risk, counterparty risk have become more bulky over the recent years, maybe than the market risk, given the identified lack of transparence in the OTC Market.

These risks can be mitigated by the way trade and post-trade functions are structured. At trading level, risks can be reduced by improving operational efficiency, e.g. ensuring electronic trade execution, affirmation and confirmation.This would have the effect of making OTC trade execution more similar to the way transactions are handled on-exchange.

One way is to impose collateral and margin requirements. In the bilateral clearing, the two counterparties most often have collateral agreements in place that provide for regular monitoring of how the value of the contract evolves so as to manage their respective credit exposures to each other. In the Central Counter-party (CCP)clearing, the CCP acts as a counterparty to each side of a transaction. It makes collateral management simpler, as it is the CCP that collects and manages collateral.

Special attention is dedicated to reduce credit risks notably in Credit Default Swap (CDS) market, since CDS are particularly vulnerable on many respects. The risk they cover-the credit risk- is not immediately observable but requires specific information about the borrower, which typically only banks have had. Assessing the risk remains difficult, and amplified by the fact that the potential obligations that come with them are extreme.

It is of crucial importance in a derivative business at a aggregated level, to (i) measure counterparty exposure, (ii) compute capital requirements, and (iii) hedge counterparty risk. Measuring counterparty exposure is important for setting limits on the amount of business a firm is prepared to do with a given counterparty; hedging it gives a possibility of mitigating it and transferring risk; and from a regulatory perspective there is significant pressure on financial institutions to have the capability of producing accurate risk measures to compute capital. In addition, computing counterparty exposure can also give insights into prices of complex products in potential future scenarios.The Risk Control, function attracting relatively limited attention in the past, is now becoming a central activity of all major financial institutions, requiring significant resources from all parties.

The aim of the course is to provide a bridge between old and new practices including counterparty risk in fixed income and credit derivatives market, first at the level of the bilateral contract, second at the aggregated level. In particular, we try to make a rigorous formulation of the different problems

Outline

First talk

The first part is dedicated to the basic foundations of the interest rates derivatives in a perfect market, by making a clear distinction between the different notions of funding, risk-free rate, bond, and also the notions of forward curve and discounting curve. As a consequence, we deduced the standard HJM framework

on interest rates dynamics and the notion of forward neutral probability measure. In regard, we describe the standard contracts as forward or future contracts, swaps, and the associated derivatives.

The second part is an (non standard) introduction of the default derivative world, where the basic contact is the CDS, without specific mathematical tools. Default spreads and other similar quantities appear naturally. A general framework is then introduced. Examples of affine models. These tools are necessary to model the liquidity risk in the interbank market, and the multi-discounting curves. Different examples are developed.

Second talk

Pricing with collateral: some typical non-linear backward stochastic equation for pricing. Right-way/Wrong-way risk;

Hedging and Managing counterparty risk; aggregation and risk mitigation; stress testing.

Bibliography

Cesari, G., Aquilina, J., Charpillon, N., Filipovic, Z., Lee, G., & Manda, I. (2009). Modelling, pricing, and hedging counterparty credit exposure: A technical guide. Springer Science & Business Media.

Grbac, Z., & Runggaldier, W. J. (2015). Interest Rate Modeling: Post-Crisis Challenges and Approaches.

Henrard, M. (2013). Multi-curves framework with stochastic spread: A coherent approach to STIR futures and their options. OpenGamma Quantitative Research, (11).

Short Course: Using Bayesian MCMC Models for Stochastic Loss Reserving.

Glenn Meyers. ISO Innovative Analytics.

The course will open with an explanation of Bayesian MCMC models and the software used to implement these models. It will then walk through some of the details of the models discussed in the plenary talk. These will include:

The Correlated Chain Ladder (CCL) model for incurred loss triangles

The Changing Settlement Rate (CSR) model for paid loss triangles

A bivariate model for stochastic loss reserving of two lines of insurance

Using a Bayesian MCMC model to calculate cost of capital risk margins

The scripts that implement these models are written in the R programming language using the rstan package for Bayesian MCMC modeling. These scripts will be made available to course attendees upon request.

Keywords:Bayesian MCMC, Stochastic Loss Reserving, Dependencies, Risk Margins, Schedule P, R Programming Language, rstan

References

Meyers, Glenn G. 2015. “Stochastic Loss Reserving Using Bayesian MCMC Models” CAS Monograph Series, No. 1.

Meyers, Glenn G. 2016. “Dependencies in Stochastic Loss Reserve Models” Casualty Actuarial Society Forum (Winter)

To Borrow or Insure? Long Term Care Costs and the Impact of Housing.

Michael Sherris. (CEPAR, UNSW Business School)

Long term care costs are significant to individuals who survive to older ages. Many individuals own a house that not only provides consumption benefits but also can be used to offset long term care costs. Housing equity may also be a major component of any bequest wealth on death. Given an individual who owns a house and has to fund long term care costs at older ages, a natural question that arises is whether he/she should borrow with a reverse mortgage or purchase long term care insurance. We consider an individual’s optimal choice at retirement for consumption, reverse mortgage loans, and private long term care insurance in a discrete time life-cycle model that takes into account mortality risk, health shocks and house price risk. We quantify the extent to which the demand for private long term care insurance is reduced by high levels of home equity and the impact that the availability of a reverse mortgage has on the long term care insurance decision. We show that the welfare gain from long-term care insurance alone is marginal while that from a reverse mortgage is significant. The welfare gain is substantially increased when an individual is able to both borrow and purchase long term care insurance. We assess the sensitivity to these results and show that they are robust based on the life cycle model used.

Keywords: Long term care expenses; longevity risk; reverse mortgage; home equity; life-cycle model.

Riding the Bubble with Convex Incentives.

Fernando Zapatero* (USC, Los Angeles, CA, US) and Juan Sotes-Paladino (University of Melbourne).

Several empirical studies contradict the efficient markets contention that sophisticated investors like hedge funds should underweight overvalued assets in their portfolios. We rationalize this evidence within a dynamic model that accounts for hedge fund convex incentive fees. In response to these incentives, risk-averse hedge fund managers with superior information can aggressively overweight an overvalued asset with positive risk premium to beat a risk-less benchmark, even when they expect overpricing to fall in the short term. To secure outperformance, managers tilt their portfolios towards the risk-less benchmark and hold too much of a negative risk premium asset. This distortion can increase with managers’ information advantage over other market participants. The optimal investment strategy of managers exacerbates equilibrium mispricing (both over- and undervaluation) with respect to the case of no convex incentives.

5:00–5:30pm:

Heterogeneous Archimedean copula and t-copula in credit portfolio modeling

Ludger Overbeck*, University of Giessen

Besides its advantage in modelling tail-dependency, the main drawback of standard non-Gaussian copula is the homogeneity in the tail dependency. Several approaches to solve are meanwhile developed, hierachical copula, the grouped t-copula and the heterogeneous t-copula as recently described by Luo and Shevchenko. We will show results from a concrete implementation of a factor model using the later approach in a two step estimation procedure. In particular the effects on capital allocation will be highlighted. In the second part, we will also present how this can be extended to a wide class of Archimedean copula, in order to capture heterogeneous tail-dependencies and therefore tail-sensitive capital allocation in credit portfolio models. This first part is joint work in progress with Carsten Binnenhei, Melanie Frick and Benedikt Mankel (Deka Bank).

5:30pm–6:00pm

Option-Implied Objective Measures of Market Risk

Matthias Leiss*, ETHZ; Heinrich Nax, ETHZ

Foster and Hart (2009) introduce an objective measure of the riskiness of an asset that implies a bound on how much of one’s wealth is ‘safe’ to invest in the asset while (a.s.) guaranteeing no-bankruptcy in the long run. In this study, we translate the Foster-Hart measure from static and abstract gambles to dynamic and applied finance using nonparametric estimation of risk-neutral densities from S&P 500 call and put option prices covering 2003 to 2013. This exercise results in an option-implied market view of objective riskiness. The dynamics of the resulting ‘option-implied Foster-Hart bound’ are analyzed and assessed in light of well-known risk measures in- cluding value at risk, expected shortfall and risk-neutral volatility. The new measure is shown to be a significant predictor of ahead-return downturns. Furthermore, it is able to grasp more characteristics of the risk-neutral probability distributions than other measures, furthermore exhibiting predictive consistency. The robustness of the risk-neutral density estimation method is analyzed via Monte Carlo methods.

6:00pm–6:30pm

Counterparty Risk and Funding: Immersion and Beyond

Shiqi Song*, Université d’Evry ; Stéphane Crépey, University of Evry

In Cr’epey’s paper (textsc{Cr’epey, S.} (2015). Bilateral Counterparty risk under funding constraints. Part II: CVA. textit{Mathematical Finance} textbf{25}(1), 1-50.), a basic reduced-form counterparty risk modeling approach {was introduced} under a rather standard immersion hypothesis between a reference filtration and the filtration progressively enlarged by the default times of the two parties, also involving the continuity of some of the data at default time. This basic approach is too restrictive for application to credit derivatives, which are characterized by strong wrong-way risk, i.e.~adverse dependence between the exposure and the credit riskiness of the counterparties, and gap risk, i.e.~slippage between the portfolio and its collateral during the so-called cure period that separates default from liquidation.

{This paper} shows how a suitable extension of the basic approach can be devised so that it can be applied in dynamic copula models of counterparty risk on credit derivatives.

More generally, this extended approach is applicable in any marked default time intensity setup satisfying a suitable integrability condition. The integrability condition expresses that no mass is lost in a related measure change.

5:00pm–5:30pm:

Correlations between insurance lines of business: An illusion or a real phenomenon? Some methodological considerations

Greg Taylor*, UNSW Australia; Bernard Wong, UNSW Australia; Benjamin Avanzi, UNSW Australia

This paper is concerned with dependency between business segments in the non-life insurance industry. When considering the business of an insurance company at the aggregate level, dependence structures can have a major impact in several areas of Enterprise Risk Management, such as in claims reserving and capital modelling. The accurate estimation of the diversification benefits related to the dependence structures between lines of business (“LoBs”) is crucial for (i) capital efficiency, as one should avoid holding unnecessarily high levels of capital, and (ii) solvency of the insurance company, as an underestimation, on the other hand, may lead to insufficient capitalisation and safety.There seems to be a great deal of preconception as to how dependent insurance claims should be. Often, presence of dependence is taken as a given and rarely discussed or challenged, perhaps because of the lack of extensive data for public analysis. In this paper, we take a different approach, and consider how much correlation some real data sets actually display (the Meyers-Shi dataset from the USA, and the AUSI dataset from Australia). We develop a simple theoretical framework that enables us to explain how and why correlations can be illusory (and what we mean by that). We show with some real examples that, sometimes, most (if not all) of the correlation can be `explained’ by an appropriate methodology. Two major conclusions stem from our analysis:1. In any attempt to measure cross-LoB correlations, careful modelling of the data needs to be the order of the day. The exercise may not be well served by rough modelling, such as the use of simple chain ladders, and may indeed result in the prescription of excessive risk margins and/or capital margins.2. Such empirical evidence as examined in the paper reveals cross-LoB correlations that vary only in the range zero to very modest. There is little evidence in favour of the high correlation assumed in some jurisdictions. The evidence suggests that these assumptions derived from either poor modelling or a misconception of the cross-LoB dependencies relevant to the purpose to which they are applied.

5:30pm–6:00pm:

STATISTICAL TOOLS TO MANAGE LONGEVITY RISK

Ana Debón*, Universitat politècnica de val; Patricia Carracedo, Valencia International University

In recent decades the countries of the world have seen significant changes in the pattern of mortality has resulted in an increase in life expectancy. Although actuaries recognize that live longer is beneficial to all and not to anticipate these events is a problem for those involved to assess overall costs annuities a given portfolio and in particular, the sustainability of a system pensions. Therefore, we analyze the effect these changes have had on the development of models for predicting mortality and mortality indicators.In the view of much of authors who build mortality tables, the model must be adapted to the experience of the country. In particular, trends in mortality in Europe are decreasing, but there are big differences between Easter and Wester countries that is worth analyzing.Finally, it should be noted that the techniques used in practice often differ from the tools developed in academia. The experts on some of these reports tend to simplify the models, the number of indicators selected mortality and presentation of results. This approach, according to some authors and our results may lead to an underestimation of the projected life expectancy and dispersion that may have important implications for insurance companies and pension funds.

6:00pm–6:30pm:

Bridging risk measures and classical ruin theory

Jose Garrido*, Concordia University; Wenjun Jiang, Western University

Recent research has investigated possible bridges between ruin theory for the Cramer-Lundberg risk model with insurance risk management. Insurance risk models typically decompose into claim frequency and claim severity components, but also include other elements such as the premium loading. These proposed bridges are characterized by only some elements of the insurance risk process, typically the claim severity. Here we propose new risk measures based on solvency criteria that include all the insurance risk model components.An application to the optimal capital allocation problem serves as an illustrative use of these new risk measures.

5:00pm–5:30pm:

Cross-Dependent Volatility

Julien Guyon (Bloomberg L.P.)

Local volatilities in multi-asset models typically have no cross-asset dependency. In this talk, we propose a general framework for pricing and hedging derivatives in cross-dependent volatility (CDV) models, i.e., multi-asset models in which the volatility of each asset is a function of not only its current or past levels, but also those of the other assets. For instance, CDV models can capture that stock volatilities are driven by an index level, or recent index returns. We explain how to build all the CDV models that are calibrated to all the asset smiles, solving in particular the longstanding smiles calibration problem for the “cross-aware” multidimensional local volatility model. CDV models are rich enough to be simultaneously calibrated to other instruments, such as basket smiles, and we show that the model can fit a basket smile either by means of a correlation skew, like in the classical “cross-blind” multi-asset local volatility model, or using only the cross-dependency of volatilities itself, in a correlation-skew-free model, thus proving that steep basket skews are not necessarily a sign of correlation skew. We can even calibrate CDV models to basket smiles using correlation skews that are opposite to the ones generated by the classical cross-blind models, e.g., calibrate to large negative index skews while requiring that stocks are less correlated when the market is down. All the calibration procedures use the particle method; the calibration of the implied “local in basket” CDV uses a novel fixed point-compound particle method. Numerical results in the case of the FX smile triangle problem illustrate our results and the capabilities of CDV models.

Keywords: Option pricing, multi-asset models, cross-dependent volatility, correlation skew, smile calibration, basket options, particle method.

5:30pm–6:00pm:

Rough Volatility: From Microstructural Foundations to Smile

Mathieu Rosenbaum (Universite Pierre-et-Marie-Curie)

It has been recently shown that rough volatility models reproduce very well the statistical properties of low frequency financial data. In such models, the volatility process is driven by a fractional Brownian motion with Hurst parameter of order 0.1. The goal of this talk is first to explain how such fractional dynamics can be obtained from the behaviour of market participants at the microstructural scales. Using limit theorems for Hawkes processes, we show that a rough volatility naturally arises in the presence of high frequency trading combined with metaorders splitting. Then we will demonstrate that such result enables us to derive an efficient method to compute the smile in rough volatility models. This is joint work with Omar El Euch, Masaaki Fukasawa, Jim Gatheral and Thibault Jaisson.

6:00pm–6:30pm:

Hedging of covered options with linear price impact and gamma constraint

Bruno Bouchard (Universite Paris-Dauphine)

Within a financial model with linear price impact, we study the problem of hedging a covered European option under gamma constraint. Using stochastic target and partial differential equation smoothing techniques, we prove that the super-replication price is the viscosity solution of a fully non-linear parabolic equation. As a by-product, we show how ε-optimal strategies can be constructed. A numerical resolution scheme is proposed.

Open to all registered participants.

High-frequency statistics in Finance.

Jean Jacod (UPMC-Paris 6)

The aim of this course is to provide some basic facts about, and an overview of, statistics of processes which are observed at discrete times on a finite time interval. The domain of applications is primarily the study of observed stock prices.

After introducing the problem, we will explain which “parameters” of the model for the stock price or log-price can be identified, when it is observed at discrete times and when the frequency increases and eventually goes to infinity. The main parameters of this kind are the volatility and also the existence or not of jumps and their degree of activity when they are present. Then we will explain how it is possible to estimate these quantities, in a variety of settings (regular or irregular observation times, exact or noisy observation). If time permits, we will also mention some open Problems.

Stochastic control for insurers; what can we learn from finance, and what are the differences?.

Christian Hipp (Karlsruher Institute of Technology, Karlsruhe, Germany)

We give examples for stochastic control problems in insurance: optimal reinsurance (unlimited and limited excess of loss), optimal investment (without constraint: singularity, leverage, asymptotics), with constraints (no leverage, no shortselling and singularities caused by constraints), dividend optimisation and combinations. As methods for solution we discuss dynamic equations of Hamilton-Jacobi-Bellman type, the viscosity solution concept and a comparison argument for the insurance context. Emphasis is on numerical methods: we give an Euler type method which works in most cases and prove convergence.

Finally, we give a list of open problems together with heuristic solutions for a two objective problem: maximizing dividend payment under a ruin constraint.

Keywords: Stochastic Control, Viscosity Solutions, Euler type discretisations, Multi objective problem.