## CMAP

# Contributed talks

Each talk will be 25 minutes long, including questions.

## Tuesday 11.15- 13.00

**Correlation Calibration for the Multi-Asset Heston model Lokmane Abbas-Turki (Université Paris-Est, Marne la Vallée)**

This work deals with the calibration of the correlation parameters in the multi- asset Heston model. This includes both studying the monotony with respect to the correlation parameters and detailing the numerical procedure used to fix these parameters. The monotony is established for the bidimensional Heston model in two cases: First, for some values of the products {η

_{i}ρ

_{i}}

_{{i=1,2}}, and {η

_{i }- √ 1-ρ

_{i}

^{2}}

_{{i=1,2}}, where η

_{i}is the volatility of the volatility and ρ

_{i}is the asset/volatility correlation coefficient. Second, for European exchange options with short maturities, for which we also give an accurate expression of the derivative for maturities T ≤ 0.3. Then, we present some ideas of possible extensions to options written on more than two assets and on models that are derived from Heston model, like the double Heston model. The second part is devoted to the numerical considerations. By a large number of simulations, we first demonstrate the monotony of various contracts and which comforts the fact that we are generally satisfying the required conditions. Then, employing the established theoretical results, we present a calibration algorithm based on Monte Carlo that can be instantaneously executed using an Nvidia 480GTX GPU.

**Mean variance hedging under defaults risk Sébastien Choukroun (Université Paris Diderot)**

We solve a Mean Variance Hedging problem in an incomplete market where multiple defaults can appear. For this, we use a default-density modeling approach. The global market information is formulated as progressive enlargement of a default-free Brownian filtration and the dependence of default times is modeled by a conditional density hypothesis. We prove the quadratic form of each value process between consecutive defaults times and solve recursively systems of quadratic backward stochastic differential equations. Moreover, we obtain an explicit formula of the optimal trading strategy. We illustrate our results with some specific cases.

**A recursive solution for the density of the Parisian stopping time Jia Wei (London School of Economics)**

In this paper, we obtain the density function of the single barrier one-sided Parisian stopping time. The problem reduces to that of solving a Volterra integral equation of the first kind, where a recursive solution is consequently obtained. The advantage of this new method as compared to that in previous literature is that the recursions are easy to program as the resulting formula only involves a finite sum and does not require a numerical inversion of the Laplace transform. For long window period, an explicit formula for the density of the stopping time can be obtained. For shorter window length, we derive a recursive equation from which numerical results are computed. From these results, we compute the prices of one-sided Parisian options.

**Non-random overshoots and fluctuation theory for Lévy processes Matija Vidmar (The University of Warwick, Department of Statistics)**

Fluctuation theory for Lévy processes is particularly explicit in the spectrally negative case. Crucially for the latter, given any x ∈ (0,∞) the overshoots R

_{x}:= X(T

_{x}) - x are almost surely constant conditionally on the event {T

_{x}< ∞}, i.e. that the process ever goes above x (here T

_{x}is the first passage time to [x, ∞)). It is thus natural to ask which other (if any) Lévy processes exhibit this property of non-random overshoots. A precise characterization of this class is given. Moreover, the fluctuation theory in such an instance is essentially analogous to the spectrally negative case, even when the processes exhibit positive jumps. This is of interest in finance, and anywhere else where Lévy processes find wide application.

## Wednesday 11.15- 13.00

**A simple Price Action Model Han Wang (Imperial College)**

Parametric models can be quite useful in explaining the current price action state of a financial asset and predicting changes to it. In this work we apply statistical methods to learn and classify different price action states for a given asset, and filtering techniques to detect and predict changes in regime. Our model is trained and tested using intra-day foreign exchange data.

**Filtering and forecasting futures market prices under an HMM framework Anton Tenyakov (University of Western Ontario)**

We propose a model for the evolution of arbitrage-free futures prices under a regime-switching framework. The state variable for the spot prices follows an Ornstein-Uhlenbeck process with parameters modulated by a finite-state Markov chain in discrete time. The estimation of model parameters is carried out using hidden Markov model (HMM) filtering algorithms under a multivariate setting. A numerical implementation to a dataset compiled by DataStream is included. The forecasting performance of our approach through the analysis of predicted log-returns and futures prices as well as the issue of model adequacy is examined. This is joint work with Drs P. Date (Brunel University, West London, UK) and R. Mamon (University of Western Ontario, Canada)

**An optimal trading rule under switchable market Duy Nguyen ( Department of mathematics - University of Georgia)**

This work provides an optimal trading rule that allows buying and selling of an asset sequentially over time. The asset price follows a model that allows one switch between different models. The objective is to determine a sequence of trading times to maximize an overall return. The sequence of trading times can be given in terms of various threshold levels. Numerical examples are given to demonstrate the results.

**Running cost problem in the Wachter's market model Yaroslav Melnyk (Kaiserslautern University of Technology)**

When considering portfolio optimisation problems with fixed transaction costs one usually is able to prove verification theorems assuming existence of solutions to corresponding quasi-variational inequalities which satisfy certain differentiability properties and growth conditions. However, except for a few simple examples in one-dimensional models it is usually impossible to find a closed-form solution satisfying the required properties. Using the example of Wachter's market model we will demonstrate how to find weak value functions of some impulse control problems (for instance the running cost problem) and prove a version of Ito's formula, which allows us to construct optimal and suboptimal impulse control strategies. The used method provides also several ideas for numerical approximation of solutions to quasi- variational inequalities what is of great relevance for applications.

## Thursday 11.15- 13.00

**Model-independent no-arbitrage conditions on American Put options Christoph Hoeggerl (Department of Mathematical Sciences, University of Bath)**

Suppose European Put options with fixed maturity and for a finite number of strikes are traded in the market and these prices are consistent with no (model-independent) arbitrage. Using Skorokhod embedding techniques, we investigate necessary and sufficient conditions on the American Put option prices corresponding to the absence of arbitrage in the extended market, where not only European, but also American options can be traded.

**Pricing options with call prices Lauri Viitasaari (Aalto University School of Science and Technology)**

There exists several methods how more general European options can be priced with call prices. We extend these results to cover wider class of options and market models. In particular, we introduce a new pricing formula which can be used to price more general options if prices for call options and digital options are known for every strike prices. Similar results for barrier type options are introduced.

**Good-Deal Bounds via Correspondences and BSDEs Klébert Kentia Tonleu (Humboldt Universitaet zu Berlin, Department of Mathematics)**

In incomplete financial markets, the existence of infinitely many pricing measures typically results in arbitrage bounds being too large for practical use, yielding trading opportunities which are ''too good''. These kinds of opportunities are called good-deals and should be ruled out from the market. A recent approach to no-good-deals consists in tightening the arbitrage bounds by choosing as pricing measures only those equivalent martingale measures which satisfy a bound on the expected growth rates of returns on contingent claims. In this talk, we present a no-good-deal valuation and hedging approach using correspondences (multi-valued mappings) to describe the no-good-deal restriction. In a setting where asset prices are Ito processes, this approach naturally leads to a derivation of good-deal bounds and hedging strategies in terms of solutions to BSDEs. We present more concrete results in a particular case where the correspondences are given by ellipsoids. Finally for practical applications, we provide an example where explicit formulas for the good-deal valuation bound and hedging strategy of European options on a non-tradable asset are obtained

**Minimal Supersolutions of BSDEs with Lower Semicontinuous Generators Christoph Mainberger (Humboldt Universitaet zu Berlin, Department of Mathematics)**

We study minimal supersolutions of backward stochastic differential equations. We show the existence and uniqueness of the minimal supersolution, if the generator is jointly lower semicontinuous, bounded from below by an affine function of the control variable, and satisfies a specific normalization property. In particular, we do not need convexity of the generator. Semimartingale convergence is used to establish the main result.