# Abstracts

**On elicitable risk measures**

*Fabio Bellini (Department of Statistics and Quantitative Methods, Italy)*

Wednesday June 4, 16:00-16:30 | session P4 | Poster session | room lobby

A statistical functional is elicitable if it can be defined as the minimizer of a suitable expected scoring function (see Gneiting 2011, Ziegel 2013, and the references therein). With financial applications in view, we suggest a slightly more restrictive definition than Gneiting (2011), and we derive several necessary conditions. For monetary risk measures, we show that elicitability leads to a subclass of the shortfall risk measures introduced by Foellmer and Schied (2002). In the coherent case, we show that the only elicitable risk measures are the expectiles. Further, we provide an alternative proof of the result in Ziegel (2013) that the only coherent comonotone elicitable risk measure is the expected loss. A preliminary version of the paper can be downloaded on SSRN: http://dx.doi.org/10.2139/ssrn.2334746

**Double-barrier first-passage times of jump-diffusion processes**

*Lexuri Fernandez (Technische Universität München, Germany)*

Wednesday June 4, 16:00-16:30 | session P4 | Poster session | room lobby

Required in a wide range of applications in, e.g., finance, engineering, and physics, first-passage time problems have attracted considerable interest over the past decades. Since analytical solutions often do not exist, one strand of research focuses on fast and accurate numerical techniques. In this paper, we present an efficient and unbiased Monte-Carlo simulation to obtain double-barrier first-passage time probabilities of a jump-diffusion process with arbitrary jump size distribution.

We rely on numerical schemes and provide an unbiased, fast, and accurate Monte Carlo simulation based on the so-called ``Brownian bridge technique''. It proceeds as follows: First, the jump-instants of the process in consideration as well as the process immediately before and after the jump times are simulated. In between these generated points, one has a pure diffusion with fixed endpoints. Here, the so-called Brownian bridge probabilities provide an analytical expression for the first- passage time on a given threshold.

This simulation technique turns out to be (1) unbiased and (2) significantly faster than the standard Monte-Carlo simulation. We show how the Brownian bridge technique can be adapted to a large variety of exotic double barrier products. Those products are very flexible and thus allow investors to adapt to their specific hedging needs or speculative views. However, those contracts can hardly be traded without a fast and reliable pricing technique. Analytical solutions reach their limitations as they are often not flexible enough to adapt to complicated payoff streams and/or jump size distributions. To provide this flexibility for the Brownian bridge technique, we extend the existing algorithms and (1) allow to price double barrier derivatives that trigger different events depending on which barrier was hit first (a feature that is required to price, e.g., corridor bonus certificates) and (2) allow to evaluate payoff streams that depend on the first-passage time (a feature important in, e.g., structural credit risk models). Furthermore, (3) we show that time dependent barriers can easily be treated (a feature that is relevant for, e.g., window or step double barrier options). Finally, we discuss the implementation and show that -- in contrast to most alternative techniques -- the Brownian bridge algorithms are easy to understand and implement.

**Hydrodynamic limit of order book dynamics**

*Xuefeng Gao (The Chinese University of Hong Kong, Hong Kong)*

Wednesday June 4, 16:00-16:30 | session P4 | Poster session | room lobby

We study the temporal evolution of the limit order book shape on the “macroscopic” time scale (e.g. minutes). To gain useful analytical insights, we consider a scaling regime where the price tick size goes to. In this regime, we develop a first-order approximation for the sample path behavior of order book shape. Using a martingale method, we show that a pair of (scaled) measure-valued processes, representing the ``empirical sell-side shape'' and ``empirical buy-side shape'' of the order book, converges weakly to a pair of deterministic measure-valued processes in a certain Skorohod space. Moreover, the density profile of the limiting processes can be described by Ordinary Differential Equations whose coefficients are determined by the first-order statistics of order flows. We also perform empirical studies to test our theoretical model against historical order book data from NYSE Arca. The empirical results demonstrate the potential of our model in predicting the order book shape evolution for highly liquid stocks in a relatively stationary environment.

**Monetary Risk Measures on Orlicz Spaces produced by Set-Valued Risk Measures and Random Measures**

*Dimitrios Konstantinides (University of the Aegean, Greece)*

Wednesday June 4, 16:00-16:30 | session P4 | Poster session | room lobby

In this article we study the construction of coherent or convex risk measures defined either on a Orlicz heart, either on an Orlicz space, with respect to a Young loss function. The Orlicz heart is taken as a subset of $L^{0}(\omega, \mathcal{F}, \mu)$ endowed with the pointwise partial ordering. We define set-valued risk measures related to this partial ordering. We also derive monetary risk measures both by this class of set-valued risk measures and by the class of set-valued risk measures which has the properties mentioned in Jouini et al. (2004) and we compare their properties. We also use random measures related to heavy-tailed distributions in order to define monetary risk measures on Orlicz spaces, whose properties are also compared to the previous ones. The final part of the article is devoted to portfolio optimization problems which use these risk measures in practice and moreover with elements of their statistics.

**Statistically significant fits of Hawkes processes to FX data**

*Mehdi Lallouache (Ecole Centrale Paris, France)*

Wednesday June 4, 16:00-16:30 | session P4 | Poster session | room lobby

Most papers fitting Hawkes processes to financial data do not obtain statistically significant results. Significance on the activity of EBS limit order books cannot be achieved on mid-price changes. We argue that this is due to the limited time resolution (0.1s) of such data that prevents many events to be detected. This limitation can be self-consistently lifted for transactions on the bid and ask sides by using the volumes of transactions on both sides during a time slice. Fitting one hour of activity yields fits that pass Kolmogorov-Smirnov tests provided that at least two exponentials are used; all hours of the day have an endogeneity factor of about 0.7. We are able to fit accurately single days by accounting for the intra-day variability of activity, which however suggest larger endogeneity factor (0.8). We therefore argue that seasonalities and limited time resolution are major obstacles when fitting Hawkes processes to financial market data.

**Pricing and Hedging American and Hybrid Strangles with Finite Maturity**

*Souleymane Laminou Abdou (University of Rennes 1, France)*

Wednesday June 4, 16:00-16:30 | session P4 | Poster session | room lobby

This paper introduces variants of Strangles, called Euro-American or hybrid Strangles, and promotes a new numerical pricing technique. We highlight and compare properties of European, American and hybrid Strangles with pricing and hedging in mind. The new quadrature approach can deal with coupled integral equation systems locating early exercise boundaries of existing finite-lived contracts. The method is efficient, accurate and fast for pricing early exercisable Strangles. Other comparative advantages are that it avoids the non-monotonic gradient problem faced by precursors and allow users to control for errors. We then investigate the hedging of all Strangles, derive analytical expressions for some Greek parameters and finally stress how these parameters can differ (or not) one another.

**Pricing vulnerable option with credit default risk under stochastic volatility**

*Minku Lee (Sungkyunkwan University, South-Korea)*

Wednesday June 4, 16:00-16:30 | session P4 | Poster session | room lobby

Recently, the trade of financial derivatives in the over-the-counter markets has increased rapidly. Since there is no organized exchange to guarantee the promised payment in over-the-counter markets, the option holder is vulnerable to default risk. These options subject to the credit default risk are called vulnerable options. The value of a vulnerable option is less than a non-vulnerable option because of the possibility of default. Many researches subject to credit default risk have been studied extensively after Black and Scholes and Merton. Johnson and Stulz first proposed the option pricing formula for vulnerable European options, assuming that the option is the only liability of the counterparty. They assumed that an option holder receives all the assets of the option writer when the value of the option writer's assets is less than the value of the option. In the research of Hull and White, the other liabilities of the option writer were considered and the payment was determined by a proportion of the nominal claim when default occurs. But they did not consider the dependence between the value of the assets of the option writer and the asset underlying the option. Jarrow and Turnbull presented a new approach for pricing and hedging derivative securities with credit risk. Klein extended the study of Johnson and Stulz by allowing the counterparty to have other liabilities in the capital structure. Also, he obtained the formula under the assumption of recognizing the correlation between the option writer's asset and asset underlying the option, implying a payout ratio endogenous to the model is specified. He assumed that the option holder receives the proportion of nominal claim by the option writer in the event of financial distress and default boundary is fixed. In contrast to above assumptions, Klein considered the dependence of the total liabilities of the option writer on the value of the claim of the option holder. But the above papers assumed the volatility of the underlying asset is constant over the life of the vulnerable option. This simplified assumption is inappropriate to explain the volatility smile or skew of the implied volatility of the underlying asset. Hence a model must be developed to adapt to the real financial situation. Empirical evidence presents that volatility is a random process rather than a deterministic process. In the past two decades, many researches have been devoted to formulate these features of stochastic volatility models. For example, Heston model, the SABR model, the GARCH model and the Chen model. In this paper we consider the vulnerable option pricing with stochastic volatility extended the study of Klein to the case where the volatility of the underlying asset follows the mean-reverting OU process. The purpose of this paper is to provide asymptotic solutions of the vulnerable option pricing by applying singular perturbation method.

**Back to the Future, statistical data mining of Interest Rates Time-series**

*Michel Maignan (University of Lausanne, Switzerland)*

Wednesday June 4, 16:00-16:30 | session P4 | Poster session | room lobby

The centenial lowest interest rates and the amplitude of the most rapid descent since 15 september 2008 , as well as their crawling behaviour since 5 years seem to have created a surprise. The Bank Risk Manager aims to rely on fundamentals of interest modelling like « Economie et Intérêt, Allais, 1947 », the « John Taylor’s rule », 1993. The data mining of long-term time-series was performed on LIBOR and swap interest rates with daily data for a dozen of maturities with 20 years history for some major currencies : CHF ; USD ; Euro (10 Y), GBP, Yen, and with yearly or other frequencies for Swiss Francs mortgage rates and other rates since 1850. Findings of such data mining include the incontestable downward wavy trends since 20 years from now, which fit into the 160 years record of Swiss mortgage rates, showing now the third local long term minimum, in addition to those of 1890 and 1950. The date-maturity interest rates maps and their statistics depict the populations which vary according to maturity, from three different populations for LIBOR to one population for long term swaps. The bizarre moves on GBP can reveal the recent fraud on LIBOR. Adjustment to asymetric thick tails distributions could be achieved. In accordance with Bachelier’s finding, the daily yields (or moves) do not show time-memory with the autocorrelation computations, this is the contrary for the memory of the I.R. values themselves. The statistical autocorrelation of the signs and of the absolute daily moves reveal a one-week memory of the absolute values of change. The series of consecutive draw-up and draw-downs behave differently as medium terme memory was depicted by the statistics on the number of consecutive bullish or bearish days. Herealso the absolute daily moves show an autocorrelation structure. SOM Self-Organizing Maps maps confirm the clustering of time-periods , and ANN Artificial Neural Networks achieve an adapted fitting of trends and disturbancies, the MLP MultiLayer Perceptron stick on long-term trends during temptative forecasts. As a conclusion, Data Mining Methods with spatial statistics extract the usefull information for model development, out of the half-million figures analyzed.

**About decomposition of pricing formulas under stochastic volatility models**

*Raúl Merino (Universitat de Barcelona, Spain)*

Wednesday June 4, 16:00-16:30 | session P4 | Poster session | room lobby

We see that the classical Hull and White formula with stochastic volatility can be decomposed into the Black-Scholes formula plus other terms. We calculate the decomposition using three different methods: classical Itô calculus, functional Itô calculus and Malliavin calculus. We notice that solving the problem using classical Itô calculus is equivalent to use the functional Itô calculus. All the three methods are applied in the case that the price follows a more generic stochastic differential equation than the ones studied in others papers. In addition, we give an expression to the derivative of the implied volatility.

**The Volterra Integral Approach as a Useful Technique to Analytically Price Complex Options: The Example of Double-Barrier Options**

*Ahmed Loulit (Solvay Brussels School of Economics and Management ULB, Belgium)*

Wednesday June 4, 16:00-16:30 | session P4 | Poster session | room lobby

The quest for an analytical solution to complex derivatives such as double-barrier options has always been a challenge in the literature. Geman and Yor (1996) derive expressions for the Laplace transform of the double barrrier option price. However, they have to use a numerical inversion of the Laplace transform to obtain an approximate solution to option prices, thereby involving a tedious calculation. We study the price of complex derivatives, namely double barrier options proposing a new methodology that represents the solution of heat equations in the form of Volterra integral equations of the second kind. We provide an approximate solution whose error has an exponential decay allowing very rapid convergence with a few iterations only. Given the power of our approach we think that many more applications will follow, whilst Geman and Yor (1996) cover one type of barrier options at a time only: knock-out calls and puts. We provide some numerical applications and compare our results to German-Yor and to Kunitomo and Ikeda (2000) for double barrier knock-out options with no rebate and to the Crank-Nicholson finite difference and Kunitomo-Ikeda method for double barrier options that do pay a rebate at hit.