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STOCHASTIC METHODS IN FINANCE
Torino, July 3–5, 2008
Torino, July 3–5, 2008
Stochastic Methods in Finance
Detailed Program
Thursday, July 3 – Lecture Room 1
13h30–14h20
Registration
Opening: 14h25–14h30
Session 1: 14h30–16h10
Chair: Wolfgang RUNGGALDIER (Università di Padova)
14h30–14h55
Cristina COSTANTINI (Università di Chieti - Pescara)
Existence, uniqueness and regularity of viscosity solutions for some degenerate valuation equations
14h55–15h20
Lucia CARAMELLINO (Università di Roma “Tor Vergata”)
Large deviation estimates of the crossing probability for pinned Gaussian processes
15h20–15h45
Emanuela ROSAZZA GIANIN (Università di Napoli “Federico II”)
Representation of the penalty term of dynamic concave utilities
15h45–16h10
Stefano DE MARCO (Scuola Normale Superiore, Pisa)
Smoothness of laws of diffusions with singular coefficients and applications to financial models
16h10–16h40
Coffee break
Session 2: 16h40–18h45
Chair: Carlo SEMPI (Università del Salento)
16h40–17h05
Markus FISCHER (Humboldt Universität Berlin & Universität Heidelberg)
On the numerical solution of high-dimensional optimal control problems: approximate dynamic
programming and Smolyak’s algorithm
17h05–17h30
Simone SCOTTI (Ecole Nationale des Ponts et Chaussées & Università di Torino)
Non-liquid assets and error theory
17h30–17h55
Stefano BACCARIN (Università di Torino)
Optimal impulse control to maximize lifetime utility from consumption of a geometric Brownian
motion
17h55–18h20
Mingxin XU (University of North Carolina at Charlotte)
Tradeable Measures of Risk
18h20–18h45
Gianfausto SALVADORI (Università del Salento)
E.V.T. Evidences against new Italian banking regulation
Friday, July 4 – Lecture Room 1
Session 3: 9h00–10h40
Chair: Franco PELLEREY (Politecnico di Torino)
09h00–09h25
Carlo SEMPI (Università del Salento)
Measures of non-exchangeability for bivariate random vectors
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Torino, July 3–5, 2008
Stochastic Methods in Finance
09h25–09h50
Fabrizio DURANTE (Johannes Kepler Universität, Linz)
Evolution of the dependence structure for tail events
09h50–10h15
Rachele FOSCHI (Università di Roma “La Sapienza”)
Evolution of survival copulas under longitudinal observation of dependent default times
Cecilia PROSDOCIMI (Università di Padova)
Partially exchangeable hidden Markov models
10h15–10h40
10h40–11h10
Coffee break
Session 4: 11h10–12h50
Chair: Cristina COSTANTINI (Università di Chieti - Pescara)
11h10–11h35
Tiziano VARGIOLU (Università di Padova)
Optimal prepayment rule for mortgage-backed securities
11h35–12h00
Giorgia CALLEGARO (Scuola Normale Superiore, Pisa & Université d’Evry Val d’Essonne)
Computing VaR and CVaR for energy derivatives
12h00–12h25
Claudio FONTANA (Università di Padova)
Credit risk and incomplete information: linear filtering and EM parameter estimation
12h25–12h50
Giacomo SCANDOLO (Università di Firenze)
A new approach for valuing a portfolio of illiquid assets
12h50–14h30
Lunch break
Session 5: 14h30–16h10
Chair: Paolo BALDI (Università di Roma “Tor Vergata”)
14h30–14h55
Massimo MARINACCI (Università di Torino)
Uncertainty averse preferences
14h55–15h20
Giovanni Luca TORRISI (Istituto per le Applicazioni del Calcolo - CNR, Roma)
A class of risk processes with delayed claims: ruin probabilities estimates under heavy-tail conditions
15h20–15h45
Pietro MILLOSSOVICH (Università di Trieste)
Regression-based algorithms for life insurance contracts with surrender guarantees
15h45–16h10
Claudio MACCI (Università di Roma “Tor Vergata”)
On the large deviations of a class of modulated additive processes
16h10–16h40
Coffee break
Session 6: 16h40–18h20
Chair: Michael MANIA (“A. Razmadze” Mathematical Institute)
16h40–17h05
Rita GIULIANO (Università di Pisa)
A strong “local” Law of Large Numbers and an almost sure “local” Limit Theorem
17h05–17h30
Mario ABUNDO (Università di Roma “Tor Vergata”)
First-passage problems for asymmetric diffusions
17h30–17h55
Ömer ÖNALAN (Marmara University, Istanbul)
Pure jump Lévy processes and self-decomposability in financial modelling
17h55–18h20
Patrizia SEMERARO (Università di Torino)
Extending time-changed Lévy asset models through multivariate subordinators
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Torino, July 3–5, 2008
Stochastic Methods in Finance
Saturday, July 5 – Sala d’onore
Session 7: 9h00–10h40
Chair: Maurizio PRATELLI (Università di Pisa)
09h00–09h50
Paolo GUASONI (Boston University)
Portfolios and Risk Premia for the Long Run
09h50–10h40
Michael MANIA (“A. Razmadze” Mathematical Institute, Tbilisi)
L2 -approximate pricing under restricted information
10h40–11h10
Coffee break
Session 8: 11h10–12h50
Chair: Marco FRITTELLI (Università di Milano)
11h10–12h00
Dirk BECHERER (Humboldt Universität Berlin)
Optimal portfolio liquidation in illiquid markets with finite resiliency
12h00–12h50
Vicky HENDERSON (University of Warwick)
Liquidation of option portfolios
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Torino, July 3–5, 2008
Stochastic Methods in Finance
Thursday
14h30–14h55 :
Existence, uniqueness and regularity of viscosity solutions for some degenerate valuation
equations
Cristina Costantini, Università di Chieti - Pescara
Abstract: In many recent models in finance, in particular involving interest rates, the price of a derivative is determined
by an integro-differential equation that is degenerate in various ways: the diffusion term vanishes somewhere or is identically zero in some directions, the coefficients are not Lipschitz continuous in the whole domain where the equation is
considered, boundary conditions are not specified. In this work we prove existence and uniqueness of a viscosity solution
for a (multidimensional) equation with all the above degeneracies, assuming locally Lipschitz continuous coefficients and
a Lyapunov type condition. In addition, we prove existence and uniqueness of a classical solution in the case when only
the integral part of the operator acts in some directions, and only the differential part acts in the remaining directions,
but the coefficients of the differential part depend on all directions; the differential part of the operator is assumed to be
strictly, but not uniformly, elliptic (in the directions on which it acts). Several examples from the finance literature are discussed.
Co-authors: Fernanda D’Ippoliti, Marco Papi
14h55–15h20 :
Large deviation estimates of the crossing probability for pinned Gaussian processes
Lucia Caramellino, Università di Roma “Tor Vergata”
Abstract: The talk deals with the asymptotic behavior of the bridge of a Gaussian process conditioned to stay in n fixed
points at n fixed past instants. In particular, functional large deviation results are stated for small time. Several examples
are considered: integrated or not fractional Brownian motion, m-fold integrated Brownian motion. As an application, the
asymptotic behavior of the exit probability is studied and used for the practical purpose of the numerical computation, via
Monte Carlo methods, of the hitting probability up to a given time of the unpinned process.
Co-author: Barbara Pacchiarotti
15h20–15h45 :
Representation of the penalty term of dynamic concave utilities
Emanuela Rosazza Gianin, Università di Napoli “Federico II”
Abstract: Starting from the well known representation of dynamic convex risk measures (or, equivalently, dynamic concave
utilities) (see [1], [3]), we will provide a characterization of the penalty functional in such a representation. More precisely,
such a characterization is deduced by applying the theory of Backward Stochastic Differential Equations and, in particular, of
the so called g-expectations (see [4], [5], [6] and [7]).
Co-authors: Freddy Delbaen, Shige Peng
[1] Bion-Nadal, J. (2006): “Time Consistent Dynamic Risk Processes Cadlag Modification”, Preprint available on arXiv
[2] Delbaen, F. (2006): “The structure of m-stable sets and in particular of the set of risk neutral measures”, in Memoriam
Paul-Andé Meyer, Lecture Notes in Mathematics 1874, 215-258.
[3] Detlefsen, K., Scandolo, G. (2005): “Conditional and dynamic convex risk measures”, Finance & Stochastics, 9/4, 539-561
[4] El Karoui, N., Peng, S., Quenez, M.C. (1997): “Backward stochastic differential equations in finance”, Mathematical
Finance 7/1, 1-71
[5] Pardoux, E., Peng, S. (1990): “Adapted solutions of a backward stochastic differential equation”, Systems & Control
Letters 14, 55-61
[6] Peng, S. (1997): “Backward SDE and related g-expectations”, in: Backward stochastic differential equations, N. El Karoui
and L. Mazliak eds., Pitman Res. Notes Math. Ser. Vol. 364, Longman, Harlow, 141-159
[7] Rosazza Gianin, E. (2006): “Risk measures via g-expectations”, Insurance: Mathematics and Economics 39, 19-34
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Torino, July 3–5, 2008
Stochastic Methods in Finance
15h45–16h10 :
Smoothness of laws of diffusions with singular coefficients and applications to financial
models
Stefano De Marco, Scuola Normale Superiore, Pisa
Abstract: One of the classical applications of Malliavin Calculus to diffusion processes is to establish some sufficient conditions
on the coefficients of a SDE so that the marginal laws of the solution admit a regular density w.r.t. the Lebesgue measure on
Rn . Using localization techniques and focusing on an analysis of the Fourier transform, we are able to show that a smooth
local density still exists for diffusions whose coefficients do have singular points. Our analysis hence includes financial models
that make use of diffusions with non-lipschitz coefficients to modelize interest rates or volatilities of stock prices, e.g. the
Heston model or other instances of stochastic volatility models.
Co-author: Vlad Bally
16h40–17h05 :
On the numerical solution of high-dimensional optimal control problems: approximate
dynamic programming and Smolyak’s algorithm
Markus Fischer, Humboldt Universität Berlin & Universität Heidelberg
Abstract: A procedure for numerically solving certain high-dimensional stochastic optimal control problems is presented. The
dynamics of the control problems are given by controlled SDEs, performance is measured in terms of expected costs over a
finite time horizon. The value functions of such problems can, in principle, be computed by first discretising the dynamics and
costs in time and space and then proceeding according to the principle of dynamic programming, see [1] and the references
therein. When the dimension of the state space is higher than two or three, a straight-forward discretisation in space leads
to tasks which are computationally too complex. A way out is to only discretise in time and to combine the dynamic
programming operator with an operator for function approximation; the two operators have to be compatible with each other
else the procedure may become unstable (cf. [2]). The functions to be approximated are related to the value function of the
original control problem. The value function is, in general, only Lipschitz continuous, and the task of recovering an unspecified
Lipschitz funtion is subject to a “curse of dimensionality” (e.g. [3]). While in many situations the value function is quite
smooth, this may be not easy to verify a priori. For this reason, interpolation methods based on a construction associated with
the name of S. Smolyak [4] look promising, as they “automatically” take advantage of higher than Lipschitz regularity (cf. [5]).
We will give an outline of Smolyak’s method and discuss its applicability in the context of approximate dynamic programming.
[1] Kushner, H.J., Dupuis, P. (2001): Numerical Methods for Stochastic Control Problems in Continuous Time. Springer,
New York, 2nd ed.
[2] Munos, R. (2007), Performance bounds in Lp norm for approximate value iteration, SIAM Control Optim. 46, 541-561
[3] Novak, E. (1988): Deterministic and Stochastic Error Bounds in Numerical Analysis. LNM 1349, Springer, New York
[4] Smolyak, S.A. (1963), Quadrature and interpolation formulas for tensor products of certain classes of functions, Sov.
Math., Dokl. 4, 240-243
[5] Barthelmann, V. et al. (2000), High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math. 12,
273-288
17h05–17h30 :
Non-liquid assets and error theory
Simone Scotti, Ecole Nationale des Ponts et Chaussées & Università di Torino
Abstract: In this article, we propose a methodology to price non-liquid assets when the underlying follows a local volatility
model. We start with a local volatility diffusion but we assume that the Brownian motion is uncertain, in a sense that we
explain. The uncertainty on the Brownian motion generates a noise on the trajectories of the underlying and we use this noise
to expound the presence of a bid-ask spread, besides we prove that this noise has an impact also on mid-price. We enrich our
analysis with a numerical simulation when the volatility is a power function of the asset price. Finally, we investigate the impact of this uncertainty on option prices defined on the non-liquid asset and we show an interesting application in power markets.
Co-author: Vathana Ly Vath
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Torino, July 3–5, 2008
Stochastic Methods in Finance
17h30–17h55 :
Optimal impulse control to maximize lifetime utility from consumption of a geometric
Brownian motion
Stefano Baccarin, Università di Torino
Abstract: We consider the problem of maximizing lifetime utility from consumption of a generalized geometric Brownian
motion in the presence of strictly positive intervention costs. Under general assumptions on the utility function and
on the controlling costs we show that, if the discount rate is large enough, there always exists an optimal impulse
policy for this problem. We characterize the value function as the minimum solution of a quasi-variational inequality in a weighted Sobolev space. The optimal policy is Markovian and it is characterized by an intervention region
where the optimal consumption is obtained by solving a static optimization problem. Our problem is essentially different from the impulse control problem studied in Bensoussan and Lions [1] and from most of the impulse control
problems dealt with in the literature (see, for instance, Harrison et al. [2], Bar-Ilan et al. [3], Amadori [4]). The
main difference is that in our objective function there is no additive separation between the utility from consumption
and the controlling costs and we consider a maximization problem where the value function is not necessarily finite.
[1] Bensoussan, A. and Lions, J.L. (1984) Impulse Control and Quasi-Variational Inequalities, Gauthiers-Villars, Paris.
[2] Harrison, J.M., T. Sellke and A. Taylor (1983): Impulse control of Brownian motion. Mathematics of Operations Research
8, 454-466
[3] Bar-Ilan,A., D. Perry and W. Stadje (2004) : A generalized impulse control model of cash management. Journal of
Economic Dynamics & Control 28, 1013-1033
[4] Amadori, A.L. (2004) : Quasi-variational inequalities with Dirichlet boundary condition related to exit time problems for
impulse control. SIAM J. Control Optim. 43, 2, 570-589
17h55–18h20 :
Tradeable Measures of Risk
Mingxin Xu, University of North Carolina at Charlotte
Abstract: We propose a new market where the tradable assets are risk measures. One function of this new market is to
provide a model independent market price of risk measures. We base the Tradable Risk Measures on Realized Risks over time
and use option pricing approach to derive the forward, call/put and swap contract prices on a particular Realized Risk named
Weighted Average of Ordered Returns. Then we give the convergence result which shows the forward price converges to the
theoretical law invariant coherent risk measure. Finally, we show that the dynamics of the forward price itself satisfies the
axioms of dynamic coherent risk measures.
18h20–18h45 :
E.V.T. Evidences against new Italian banking regulation
Gianfausto Salvadori, Università del Salento
Abstract: Our research finds out the potential competitive impact of the new Italian Banking capital regulation for operational
risks. Differently from the approach underlying the new discipline of the United States and many European countries,
the Italian regulation allows the access to the advanced measurement approaches (AMA) only to banks and financial
intermediaries whose size or specialization requirements meet predefined levels. In our research we compare the capital at risk
(estimated with a one year holding period and a 99.9th percentile confidence interval) and the capital charge calculated by
basic indicator methodology. Using operational loss data of one bank that does not meet regulation constraints, we show the
unfair penalization of the new supervisory regulation on capital requirements for a large group of intermediaries.
Co-authors: Simona Cosma, Giampaolo Gabbi
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Torino, July 3–5, 2008
Stochastic Methods in Finance
Friday
09h00–09h25 :
Measures of non-exchangeability for bivariate random vectors
Carlo Sempi, Università del Salento
Abstract: The exchangeability of a pair of random variables is reflected in the symmetry of their copula. Here we
introduce a set of axioms for measures of non-exchangeability for bivariate vectors of continuous and identically distributed
random variables and give some examples together with possible applications in statistical models based on the copula function.
Co-authors: Fabrizio Durante, Erich Peter Klement, Manuel Úbeda-Flores
09h25–09h50 :
Evolution of the dependence structure for tail events
Fabrizio Durante, Johannes Kepler Universität, Linz
Abstract: In financial and actuarial risk management, the construction of appropriate models for dependence between
risks is of obvious importance, due to the well recognized fact that neglecting dependence gives rise to a dramatic risk
underestimation. Copulas provide a widely accepted tool for building such dependence models. In particular, the need for
accurate modelling of extremal events requires a better understanding of the behaviour of copulas in the tails. Tail dependence
for bivariate copulas can be described using the concept of threshold copula. Specifically, given a random pair (U, V ) whose
distribution function is the copula C, the lower threshold copula Ct is defined as the copula of the conditional distribution of
(U, V ) given that U and V are under a given threshold t ∈ (0, 1]. In this talk, we investigate the family of lower threshold
copulas {Ct }t∈(0,1] associated with a copula C. In particular, we consider whether some positive dependence properties of C
are preserved by Ct , for t spanning (0, 1].
Co-authors: Rachele Foschi, Fabio Spizzichino
[1] Durante F., Foschi R., Spizzichino F. (2008), Threshold copulas and positive dependence, Statistics and Probability Letters,
in press.
09h50–10h15 :
Evolution of survival copulas under longitudinal observation of dependent default times
Rachele Foschi, Università di Roma “La Sapienza”
Abstract: We consider T1 , ..., Tn non-negative exchangeable r.v.’s and their survival copula Ĉ. Let T(1) , ..., T(n) be their order
statistics, we assume that the observation up to a time t > 0 is an event of the form {T(1) = t1 , ..., T(k) = tk , T(k+1) > t}.
Let Ft be the σ-algebra generated by such events. We consider the residual lifetimes, i.e. the r.v.’s Xt1 , ..., Xtn−k ,
t > 0, k = 1, ..., n − 1, defined as the exchangeable r.v.’s having order statistics T(k+1) − t|Ft , ..., T(n) − t|Ft , and the family
(n−k)
(2)
of their survival copulas {Ĉt
}. We concentrate attention on the family of copulas of two residual lifetimes {Ĉt }. We
(2)
analyze the behaviour of Ĉt at the instant of a default and in the interval between two subsequent defaults. Interpreting
T1 , ..., Tn as the times to default of n companies in a same market, our analysis aims to point out the connections between the
theme of tail dependence and the so-called phenomenon of default contagion. For the case when T1 , ..., Tn are conditionally
i.i.d. given an unobservable variable Θ, some specific results will be obtained in terms of the monotonicity behaviour of the
conditional hazard rate h(·|θ).
Co-author: Fabio Spizzichino
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Torino, July 3–5, 2008
10h15–10h40 :
Stochastic Methods in Finance
Partially exchangeable hidden Markov models
Cecilia Prosdocimi, Università di Padova
Abstract: Hidden Markov Models (HMM) have become increasingly popular in recent years in a wide range of applications,
including Market Models in Mathematical Finance. Special subclasses of HMM’s have been extensively studied in various
contexts. In particular the problem of estimating the memory of a mixture of Markov chains (Quintana [3]) has motivated
us to investigate the connection between hidden Markov and partially exchangeable stochastic sequences. In this work
we give a necessary and sufficient condition for a partially exchangeable sequences to be hidden Markov with countable
state space. More precisely we show that a partially exchangeable sequence is a HMM with a countable state space of the
underlying Markov chain if and only if it is a countable mixture of Markov chains. Our main theorem extends an old result
of Dharmadhikari [1] which proved that an exchangeable sequence is a HMM if and only if it is a countable mixture of i.i.d.
sequences. The main technical tool we use to generalize Dharmadhikari’s theorem is the Diaconis-Freedman [2] extension of
de Finetti’s theorem to partially exchangeable sequences.
Co-author: Lorenzo Finesso
[1] Dharmadhikari S. (1964), Exchangeable processes which are function of stationary Markov chains, The Annals of Mathematical Statistics 35,429-430
[2] Diaconis P., Freedman D. (1980), de Finetti’s theorem for Markov chains, The Annals of Probability 8, 115-130
[3] Quintana F.A., Newton M.A. (1998) Assessing the order of dependence for partially exchangeable binary data, JASA 93,
194-202
11h10–11h35 :
Optimal prepayment rule for mortgage-backed securities
Tiziano Vargiolu, Università di Padova
Abstract: We study the optimal stopping problem embedded in a typical mortgage. Despite a possible non-rational behaviour
of the typical borrower of a mortgage, the problem is worth to be solved for the lender to hedge against the prepayment
risk, and because many mortgage-backed securities pricing model incorporate this suboptimality via a so-called prepayment
function which can depend, at time t, on the fact that the prepayment is optimal or not. We state the pre-payment problem
in the context of the optimal stopping theory and present an algorithm to solve the problem via weak convergence. Numerical
results in the case of the Vasicek model and of the CIR model are also presented.
Co-author: Giulia De Rossi
11h35–12h00 :
Computing VaR and CVaR for energy derivatives
Giorgia Callegaro, Scuola Normale Superiore, Pisa & Université d’Evry Val d’Essonne
Abstract: Due to the peculiarity of energy markets (think for example of the spot price seasonality or of transport and
storability problems in the case of electricity and gas) many “variable volume” options have been introduced.
They are purchase or sale contracts which provide flexibility about the timing to delivery and about the overall minimum and
maximum take amounts, usually called “swing” of “take or pay” options. We focused our attention on the computation of
risk measures related to this kind of options, namely we considered the “Value at Risk” (VaR) and the “Conditional Value at
Risk” (CVaR). Our contribution is to present and compare three different solving procedures, based on stochastic recursive
algorithms of the Robbins-Monro type and made faster by the use of the importance sampling paradigm, and to show that
they are efficient, i.e., the corresponding algorithms converge to VaR and CVaR.
Co-authors: Olivier Bardou, Gilles Pagès
[1] Arouna B. (2003/04), Robbins-Monro algorithms and variance reduction in finance, The Journal of Computational Finance,
7(2).
[2] Arouna B. (2004), Adaptative Monte-Carlo Method, A Variance Reduction Technique, Monte-Carlo Methods and Applications, 10(1), 1-24.
[3] Bardou O., Bouthemy S., Pagès G. (2007), Optimal quantization for the pricing of swing options, preprint.
[4] Kushner H.J., Yin G.G. (2003), Stochastic Approximation and Recursive Algorithms and Applications, Springer.
[5] Rockafellar R.T., Uryasev S. (2000), Optimization of conditional value-at-risk, Journal of Risk, 2(3), 493-517.
9
Torino, July 3–5, 2008
12h00–12h25 :
Stochastic Methods in Finance
Credit risk and incomplete information: linear filtering and EM parameter estimation
Claudio Fontana, Università di Padova
Abstract: We consider a reduced-form credit risk model where default intensity and interest rate are linear functions of a not
fully observable Markovian factor process. We determine arbitrage-free prices of OTC products coherently with information
from the financial market, in particular yields and credit spreads and this can be accomplished via a linear filtering approach
coupled with a filter-based EM-algorithm for parameter estimation in lieu of the more traditional calibration. We furthermore
determine quantities related to risk management in accordance with information coming from both within and outside the
financial market such as the rating score.
Co-author: Wolfgang J. Runggaldier
12h25–12h50 :
A new approach for valuing a portfolio of illiquid assets
Giacomo Scandolo, Università di Firenze
Abstract: We present an hypotheses-free formalism for marking-to-market a portfolio in general illiquid markets. In this
formalism coherent measures of risk turn out to be appropriate to measure general portfolio risk including liquidity risk.
Coherent Risk maps and Value maps, defined on the space of portfolios, turn out to be convex and concave respectively,
displaying two distinct faces of the diversification principle, namely the traditional correlation benefit and a newly observed
granularity benefit. We show that the optimization problem implicit in the definition of the value of a portfolio is always a convex problem, ensuring straightforward industrial applicability of the method. Finally, some numerical applications are presented.
Co-author: Carlo Acerbi
14h30–14h55 :
Uncertainty averse preferences
Massimo Marinacci, Università di Torino
Abstract: We study uncertainty averse preferences, that is, complete and transitive preferences that are convex and monotone.
We establish a representation result, which is at same time general and rich in structure. Many objective functions commonly
used in applications are special cases of this representation.
14h55–15h20 :
A class of risk processes with delayed claims: ruin probabilities estimates under heavy-tail
conditions
Giovanni Luca Torrisi, Istituto per le Applicazioni del Calcolo - CNR, Roma
Abstract: We consider a class of risk processes with delayed claims, and we provide ruin probabilities estimates under
heavy-tail conditions on the claim size distribution.
Co-author: Ayalvadi Ganesh
15h20–15h45 :
Regression-based algorithms for life insurance contracts with surrender guarantees
Pietro Millossovich, Università di Trieste
Abstract: We present a general framework for pricing life insurance contracts embedding a surrender option. The model
allows for several sources of risk, such as uncertainty in mortality, interest rates and other financial factors. We describe
and compare two numerical schemes based on the Least Squares Monte Carlo method, emphasizing underlying modeling
assumptions and computational issues.
Co-authors: Annarita Bacinello, Enrico Biffis
10
Torino, July 3–5, 2008
15h45–16h10 :
Stochastic Methods in Finance
On the large deviations of a class of modulated additive processes
Claudio Macci, Università di Roma “Tor Vergata”
Abstract: We prove that the sample path large deviation principle holds for a class of processes that form a natural
generalization of semi-Markov additive processes. In the generalization, the sojourn times from which the phase process is
constructed need not be the points of a renewal process. Moreover the state selection process need not be independent
of the sojourn times and need not be a semi-Markov process. We assume that the phase process takes values in a finite
set and that the order in which elements in the set, called states, are visited is selected stochastically. The sojourn
times determine how long the phase process spends in a state once it has been selected. Based on assumed joint
sample path large deviation behavior of the state selection and sojourn processes, we prove that the empirical laws of the
phase process satisfy a sample path large deviation principle. From this large deviation principle, the large deviations of
modulated additive processes is deduced. Applications of the results are given to processes that arise in networking and finance.
Co-authors: Ken Duffy, Giovanni Luca Torrisi
16h40–17h05 :
A strong “local” Law of Large Numbers and an almost sure “local” Limit Theorem
Rita Giuliano, Università di Pisa
Abstract: The theory of the so called Almost Sure Central Limit Theorem has been developed starting from the classical
Central Limit Theorem. In this talk we develop a theory of the Almost Sure Local Limit Theorem, starting from the classical
Local Limit Theorem.
Co-authors: Andrei Volodin, Michel Weber
17h05–17h30 :
First-passage problems for asymmetric diffusions
Mario Abundo, Università di Roma “Tor Vergata”
Abstract: For a, b > 0, we consider a temporally homogeneous, one-dimensional diffusion process X(t) defined over
I = (−b, a), with infinitesimal parameters depending on the sign of X(t). We suppose that, when X(t) reaches the position
0, it is reflected rightward to δ with probability p > 0 and leftward to −δ with probability 1 − p, where δ > 0. Closed analytical
expressions are found for the mean exit time from the interval (−b, a), and for the probability of exit through the right end a,
in the limit δ → 0+ , generalizing the results of Lefebvre ([1]), holding for asymmetric Wiener process. As a generalization,
we could assume that the infinitesimal coefficients of X(t) change when the process crosses any given barrier, not necessarily
the origin. If X(t) is e.g. the price of a stock, it may happen that the volatility parameter and/or the drift undergo a sharp
variation when X(t) exceeds a certain threshold. In alternative to the heavy analytical calculations, a numerical method is
also presented to estimate approximately the quantities above ([2]).
[1] Lefebvre, M. (2006), First passage problems for asymmetric Wiener processes, J. Appl. Prob. 43, 175-184
[2] Abundo, M. (2007), On first-passage problems for asymmetric one-dimensional diffusions, Lecture Notes in Computer
Science, Computer Aided Systems Theory - EUROCAST 2007, vol. 4739, 179-186, Springer Berlin/ Heidelberg.
11
Torino, July 3–5, 2008
17h30–17h55 :
Stochastic Methods in Finance
Pure jump Lévy processes and self-decomposability in financial modelling
Ömer Önalan, Marmara University, Istanbul
Abstract: In this study first time, we review the connections between Lévy processes with jumps and then self-decomposable
laws. Self-decomposable laws arises as a limit laws of sequences of centered and normalized with general scaling constants
of sums independent random variables. Self-decomposable laws are sub class of infinitely divisible laws. Lévy processes and
additive processes can be related using selfsimilarity property. Lévy processes are related to the class of infinitely divisible
laws and self-similar additive processes are related to the class of self-decomposable laws. Self-decomposable distributions
occurs as limit law an Ornstein-Uhlenbeck type process associated with a Background Driving Lévy Process. We consider as a
model Normal inverse Gaussian process for asset returns. Finally, we use a nonparametric threshold estimator of the quadratic
variation which is proposed by [1] to test whether sufficient or not of these model for real financial data .Using these test
statistics, we research the presence of continuous component and whether the jump component has finite or infinite variation
in financial price process.
[1] Cont, R., and Mancini,C.,(2007) Nonparametric Tests for Analyzing the Fine Structure of Price Fluctations . Financial
Engineering Report no. 2007-13, Columbia University.
17h55–18h20 :
Extending time-changed Lévy asset models through multivariate subordinators
Patrizia Semeraro, Università di Torino
Abstract: The traditional multivariate Lévy process constructed by subordinating a Brownian motion through a univariate
subordinator presents a number of drawbacks, including the lack of independence and a limited range of dependence. In order
to face these, we investigate multivariate subordination (see Barndorff-Nielsen et al. [1]), with a common and an idiosyncratic
component (see Semeraro [5] for the variance gamma case). We introduce generalizations of some well known univariate
Lévy processes for financial applications: the multivariate compound Poisson, NIG (see [2] ), Variance Gamma (Madan and
Seneta [4]) and CGMY (Carr et al. [3]). In all these cases the extension is parsimonious, in that one additional parameter
only is needed. We characterize the subordinator, then the time changed processes via their Lévy measure and characteristic
exponent. We discuss their dependence features. We provide a calibration method and some examples of simulated trajectories,
scatter plots and linear dependence measures. The input data for these simulations are calibrated values of major stock indices.
Co-author: Elisa Luciano
[1] Barndorff-Nielsen, O.E., Pedersen, J. Sato, K.I. (2001). Multivariate Subordination, Self-Decomposability and Stability.
Adv. Appl. Prob. 33, 160-187.
[2] Barndorff-Nielsen, O.E.(1995). Normal inverse Gaussian distributions and the modeling of stock returns. Research report
no. 300, Department of Theoretical Statistics, Aarhus University.
[3] Carr, P. Geman, H. Madan, D. H., Yor, M. (2002) The fine structure of asset returns: an empirical investigation. Journal
of Business 75, 305-332.
[4] Madan, D. B., Seneta, E. (1990) The v.g. model for share market returns.Journal of Business 63, 511-524.
[5] Semeraro, P. (2008) A multivariate variance gamma model for financial application. Journal of Theoretical and Applied
Finance, 11, 1-18.
12
Torino, July 3–5, 2008
Stochastic Methods in Finance
Saturday
09h00–09h50 :
Portfolios and Risk Premia for the Long Run
Paolo Guasoni, Boston University
Abstract: This paper develops a method to derive optimal portfolios and risk-premia explicitly in a general diffusion model,
for an investor with power utility and in the limit of a long horizon. The market has several risky assets and is potentially
incomplete. Investment opportunities are driven by, and partially correlated with, state variables which follow an autonomous
diffusion. The framework nests models of stochastic interest rates, return predictability, stochastic volatility and correlation
risk. In models with several assets and a single state variable, long-run portfolios and risk-premia admit explicit formulas up
the solution of an ordinary differential equation, which characterizes the principal eigenvector and its corresponding eigenvalue
of a elliptic operator. Multiple state variables lead to a partial differential equation, which is solvable for most models of
interest. For each value of the relative risk aversion parameter, the paper derives the long-run portfolio, its implied risk-premia
and pricing measure, and their performance on a finite horizon.
Co-author: Scott Robertson
09h50–10h40 :
L2 -approximate pricing under restricted information
Michael Mania, “A. Razmadze” Mathematical Institute, Tbilisi
Abstract: We consider the mean-variance hedging problem under partial information in the case where the flow of observable
events does not contain the full information on the underlying asset price process. We introduce a certain type martingale
equation and characterize the optimal strategy in terms of the solution of this equation. We give relations between this
equation and backward stochastic differential equations for the value process of the problem. We examine particular cases of
diffusion market models, for which an explicit solution has been provided.
Co-authors: R. Tevzadze and T. Toronjadze
11h10–12h00 :
Optimal portfolio liquidation in illiquid markets with finite resiliency
Dirk Becherer, Humboldt Universität Berlin
Abstract: When liquidating a large portfolio position, a trader needs to balance two conflicting objectives. He is impatient
to realize the liquidation proceeds soon. But on the other side, to limit market impact costs he should not sell too quickly,
since large orders adversely affect the market prices against which they are executed. We present an extension of the Black
Scholes model, where an additional factor describes the market impact from previous transactions. We show how the optimal
liquidation strategy can be found explicitly, using classical calculus of variations.
12h00–12h50 :
Liquidation of Option Portfolios
Vicky Henderson, University of Warwick
Abstract: We consider the optimal liquidation problem of a risk averse agent with general utility who seeks to exercise a
portfolio of (perfectly divisible) American options. The optimal exercise strategy is of threshold form and can be characterized
explicitly as the solution of a calculus of variations problem. We consider a number of examples including one where the
exercising of options (or sale of stock) has an impact on the underlying price process.
13
Torino, July 3–5, 2008
Stochastic Methods in Finance
List of Participants 1/2
Full Name
Mario Abundo
Stefano Baccarin
Paolo Baldi
Dirk Becherer
Enea Bongiorno
Giorgia Callegaro
Antonella Calzolari
Lucia Caramellino
Cristina Costantini
Marzia De Donno
Stefano De Marco
Fabrizio Durante
Markus Fischer
Claudio Fontana
Rachele Foschi
Marco Frittelli
Maria Teresa Giraudo
Rita Giuliano
Paolo Guasoni
Vicky Henderson
Daniele Imparato
Simone Landini
Luana Lombardi
Maria Longobardi
Claudio Macci
Michael Mania
Massimo Marinacci
Francesco Martinelli
Pietro Millossovich
Omer Onalan
Franco Pellerey
Maurizio Pratelli
Cecilia Prosdocimi
Igor Pruenster
Luca Regis
Wolfgang Runggaldier
Emanuela Rosazza Gianin
Laura Sacerdote
Simon Salamon
Gianfausto Salvadori
Affiliation
Università di Roma “Tor Vergata”
Università di Torino
Università di Roma “Tor Vergata”
Humboldt Universität Berlin
Università di Milano
Scuola Normale Superiore - Pisa &
Université d’Evry Val d’Essonne
Università di Roma “Tor Vergata”
Università di Roma “Tor Vergata”
Università di Chieti - Pescara
Università Bocconi - Milano
Scuola Normale Superiore - Pisa
Johannes Kepler Universität - Linz
Universität Heidelberg & HumboldtUniversität Berlin
Università di Padova
Università di Roma “La Sapienza”
Università di Milano
Università di Torino
Università di Pisa
Boston University
University of Warwick
Politecnico di Torino
IRES Piemonte
Università dell’Aquila
Università di Napoli “Federico II”
Università di Roma “Tor Vergata”
“A. Razmadze” Mathematical Institute - Tbilisi
Università di Torino & Collegio Carlo
Alberto
UBI Banca
Università di Trieste
Marmara University - Istanbul
Politecnico di Torino
Università di Pisa
Università di Padova
Università di Torino
Università di Torino
Università di Padova
Università di Napoli “Federico II”
Università di Torino
Politecnico di Torino
Università del Salento
14
E-mail
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[email protected]
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[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
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Torino, July 3–5, 2008
Stochastic Methods in Finance
List of Participants 2/2
Full Name
Marina Santacroce
Giacomo Scandolo
Simone Scotti
Patrizia Semeraro
Carlo Sempi
Carlo Sgarra
Giovanni Luca Torrisi
Barbara Torti
Barbara Trivellato
Tiziano Vargiolu
Mingxin Xu
Affiliation
Politecnico di Torino
Università di Firenze
Ecole Nationale des Ponts et
Chaussées & Università di Torino
Università di Torino
Università del Salento
Politecnico di Milano
CNR - Istituto per le Applicazioni del
Calcolo
Università di Roma “Tor Vergata”
Politecnico di Torino
Università di Padova
University of North Carolina at Charlotte
15
E-mail
[email protected]
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[email protected]
[email protected]
[email protected]
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