## Abstracts

Title

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A heat kernel approah to interest rate models.
(joint work with Y. Hishida, J.Teichmann, and
T. Tsuchiya)
**

Speaker

Jiro Akahori

Abstract

We construct interest rate models in the spirit of the well-known Markov funcional models: our focus is analytic tractability of the models and generality of the approach. We work in the setting of state price densities and construct models by means of the so called propagation property. The propagation property can be found implicitly in all of the popular state price density approaches, in particular heat kernels share the propagation property (wherefrom we deduced the name of the approach). As a related matter, an interesting property of heat kernels is presented, too.

Speaker

Takuji Arai

Speaker

Freddy Delbaen

Abstract

Using the structure of time consistent utility functions we will show that there is a relation with non-linear PDE and BSDE. If the driver is not sub-quadratic, existence and uniqueness of solutions is not guaranteed. In the special case of Markov processes we will give an example where the PDE has non-unique solution.

Speaker

Hans Föllmer

Abstract

Convex risk measures quantify the risk of a financial position as the worst expected loss with respect to a class of probabilistic models which are taken more or less seriously, and this is made precise by a penalty function. In a dynamic setting, where the risk assessment is updated as new information comes in, and where positions are replaced by uncertain cash flows, different notions of time-consistency correspond to different supermartingale properties of the process of penalty functions. In particular we discuss the appearance of "bubbles" in the penalty process which may lead to an excessive penalization of relevant models and thus to an underestimation of the model risk. The talk will be based on joint work with Irina Penner and Beatrice Accaio.

Speaker

David Hobson

Abstract

It is well-known how to determine the price of perpetual American options
if the underlying stock price is a time-homogeneous diffusion. In the
present paper we consider the inverse problem, i.e. given prices of
perpetual American options for different strikes we ask, when is it
possible to construct a time-homogeneous model for the stock price which
reproduces the given option prices?

This is joint work with Erik Ekstrom (Uppsala).

Speaker

Lane P Hughston

Abstract

The major determiner of price changes is new information. When a new
piece of information circulates in a financial market (whether true,
partly true, misleading, or bogus), the prices of related assets are
adjusted in response. In this talk I discuss some of the issues involved in modelling the flow of information in financial markets, and I present some elementary models for information in various situations.
Some applications to the pricing of various types of financial products will be indicated.
In particular I shall look at dynamic models for stochastic volatility and correlation. Finally, I shall make a few remarks about statistical arbitrage strategies, and about price formation in inhomogeneous markets. The approach taken as regards the role of filtering in finance can be summarized in more detail as follows. We take the view that financial assets are defined by the cash flows that they generate. Each cash flow is modelled by a random variable that can be expressed as a function of a collection of independent random variables called market factors. With each such X-factor we associate a so-called market information process, the values of which we assume are accessible to market participants. In the simplest version of the model, each market information process consists of a sum of two terms; one contains true information about the value of the associated market factor, and the other represents ``noise''. The noise term is modelled by an independent Brownian bridge process that spans the time interval from the present to the time at which the value of the given market factor is revealed. Other types of information processes can also be considered. The market filtration is assumed to be that generated collectively by the market information processes. The price of an asset is given by the expectation of the discounted cash flows in the risk neutral measure, conditional on the information provided by the market filtration thus constructed. This talk is based on work carried out variously in collaboration with D. Brody, A. Macrina, E. Hoyle, M. Davis, R. Friedman. References:

D.C. Brody, L.P. Hughston & A. Macrina (2007) "Beyond hazard
rates: a new framework for credit-risk modelling''. In Advances
in Mathematical Finance, Festschrift volume in honour of Dilip
Mada}. R. Elliott, M. Fu, R. Jarrow and Ju-Yi Yen, eds. (Basel:
Birkhauser).

D.C. Brody, L.P. Hughston & A. Macrina (2008)
"Information-based asset pricing'', Int. J. Theor. Appl.
Fin. 11, 107-142.

D.C. Brody, L.P. Hughston & A. Macrina (2008) "Dam rain and
cumulative gain'', Proc. Roy. Soc. Lond. A 464
1801-1822.

L. P. Hughston & A. Macrina (2008) "Information, Inflation,
and Interest". In Advances in Mathematics of Finance, Banach
Center Publications 83 (Institute of Mathematics, Polish
Academy of Sciences).

D.C. Brody, M.H.A. Davis, R.L. Friedman & L.P. Hughston
(2009) "Informed traders''. Proc. Roy. Soc. Lond. A
465, 1103-1122.

Title

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Max-plus stochastic control and risk-sensitivity: general framework
related with risk-averse limit of optimal consumption problem
**

Speaker

Hidehiro KAISE

Abstract

This is a joint work with W.H. Fleming and S.-J. Sheu. We consider max-plus stochastic optimal control problems with max-plus additive functionals. This type of problems can be derived from an optimal consumption problem with power utility function via risk-averse limit. In this talk, we discuss general problems by using the notion of max-plus probability. We show that the value function of the max-plus control problem is the unique viscosity solution of a quasi-variational inequality with complex nonlinearity. We also obtain our max-plus control problem from a risk-sensitive type stochastic control by taking risk-averse limit.

Speaker

Arturo KOHATSU HIGA

Abstract

Weak approximations have been developed to calculate the expectation value of functionals of stochastic differential equations, and various numerical discretization schemes (Euler, Milshtein) have been studied by many authors. We present first an error study of a scheme with random time partition for SDE's driven by pure jump Lévy processes which shows that due to the concentration of jumps around zero one can define schemes with fast convergence rate. On the other hand, we define other schemes that consider few jumps which combined with Euler-like schemes lead to methods where the error due to each approximation (Brownian and jump part) contribute the same to the error. In order to do this, we study an operator decomposition method applicable to jump driven SDEs. This leads to alternative schemes and a clear decomposition of the error analysis.

Speaker

Yoshio Miyahara

Abstract

The geometric stable process (=exponential stable process) is one of the remarkable model for the asset price
processes with the strong fat tail property. This process is an incomplete market model, and so there many equivalent
martingale measures.

The importance of the stable process models was pointed by Fama('63), and the option pricing models based on the
stable processes have been studied by many researchers (Edelmann('95),
Rachev and Mittnik('00), Carr and Wu('03), etc.)
But in their pricing models the martingale measure were nor clear.

We see that, by adopting the minimal entropy martingale measure (MEMM) as the suitable martingale measure, we can
construct the option pricing model based on the stable processes in general setting. And we also see that this model is
fitting very well to the market prices of currency options.

Speaker

Huyên Pham

Abstract

We consider a financial market with a stock exposed to a counterparty risk inducing a jump in the price, and which can still be traded after this default time. This jump represents a loss or a gain of the asset value at the default of the counterparty. We use a default-density modeling approach, and address in this incomplete market context the expected utility maximization from terminal wealth. We show how this problem can be suitably decomposed in two optimization problems in a default-free framework: an after-default utility maximization and a global before-default optimization problem involving the former one. These two optimization problems are solved explicitly, respectively by duality and dynamic programming approaches, and provide a fine description of the optimal strategy. We give some numerical results illustrating the impact of counterparty risk and the loss or gain given default on optimal trading strategies, in particular with respect to the Merton portfolio selection problem. For example, this explains how an investor can take advantage of a large loss of the asset value at default in extreme situations observed during the financial crisis. Based on jont work with Y. Jiao (Paris 7).

Speaker

Alexander Schied

Abstract

The talk focusses on market impact models that are based on the dynamics of electronic limit order books. A first question is whether these models are viable or whether they give incentives for applying price manipulation strategies. We approach this question by analyzing liquidation or acquisition strategies that minimize the expected liquidity costs. It follows from this analysis that requiring the absence of price manipulation strategies in the usual sense is not sufficient to guarantee the viability of market impact models. We therefore propose additional requirements and single out a class of models that satisfies them. The talk is based on joint work with Aurélien Alfonsi and Alla Slynko.

Speaker

Martin Schweizer

Abstract

We consider the standard problem of maximising expected utility from terminal wealth over self-financing strategies, with a utility function which is also allowed to depend on time and omega. For each fixed time T, denote by X_T the optimal terminal wealth for the problem with time horizon T; so each X_T is a stochastic integral of the underlying price process S, but of course the integrand depends on the chosen time horizon T. Now consider the collection X_T (for T>0) of these random variables as a stochastic process. What kind of process is this, and what is its structure? Understanding this question is crucial for understanding how optimal portfolio choice depends on the time horizon. Answering this question also brings up other challenging mathematical problems in a natural way. The results given in this talk are joint work with Tahir Choulli (University of Alberta, Edmonton, Canada).

Speaker

Jun Sekine

Abstract

In a complete market situation, large deviations probability maximizing/minimizing portfolios with long horizon are re-derived, starting with quantile hedging results and utilizing Gartner-Ellis's theorem for the family of random variables (L_T ;T>0) with respect to the physical probability P and the risk-neutral probability Q, where L_T:=log(dQ/dP) / T is the (time-averaged) log-likelihood restricted to the horizon T. This provides a different solution method to the above large deviations control problems, which have been studied by Pham (2003), Hata-Nagai-Sheu (2007) and Nagai (2008), etc. Also, exact asymptotics of the optimized probabilities are computed.

Speaker

Akihiko Takahashi

Abstract

This presentation reviews recent results of an asymptotic expansion approach to numerical problems for pricing and hedging derivatives. In particular, expansions around a normal and a log-normal distributions are considered. As examples, the approximations for the Lambda-SABR model and a cross-currency Libor market model with a general stochastic volatility model of the spot foreign exchange rate are presented. Also applications to an implied volatility expansion and to pricing average and barrier options are shown.

Speaker

Nizar Touzi

Abstract

We provide a new point of view for the formulation of second order stochastic target problems. The main new ingredient is to modify the reference probability so as to allow for different scales. Our main result is a dual formulation of this control problem as a supremum of the solutions of standard backward stochastic differential equations. In particular, in the Markov case the dual problem is immediately seen to correspond to a fully nonlinear PDE, thus avoiding the heavy technicalities in our previous work.

Speaker

Marc Yor

Abstract

It is known (Kellerer ) that aprocess increasing in the convex order admits the 1-dimensional marginals of amartingale . but ,there is no general algorithm to produce such a martingale . 4 methods shall be presented to remedy y this lack of explicit construction . I call them :the Brownian sheet ,the time reversal ,the time inversion ,the self -decomposability methods .They allow to construct martingales for many interesting classes of processes increasing in the convex order .

Speaker

Dorje Brody

Abstract

An asymmetric information model is introduced for the situation in which there is a small agent who is more susceptible to the flow of information in the market than the general market participant, and who tries to implement strategies based on the additional information. The informed trader uses the extraneous information source to seek statistical arbitrage opportunities, while at the same time accommodating the additional risk. Explicit trading strategies leading to statistical arbitrage opportunities, taking advantage of the additional information, are constructed, illustrating how excess information can be translated into profit. The work is carried out in collaboration with Mark Davis, Robyn Friedman, and Lane Hughston.

Speaker

Stefan Geiss

Speaker

Takashi Kato

Speaker

Toshiyuki Nakayama

Abstract

Let us introduce support theorem and viability theorem for SPDE's including HJM framework. This is closely related to the consistency problem between methods of making yield curves and volatility structures in HJM models. The consistency is very important to stabilize daily volatility calibration.

Speaker

Ashkan Nikeghbali

Abstract

Madan, Roynette and Yor have recently been able to express the price of a European put option in terms of the probability distribution of some last passage times. This situation corresponds to a special case of a more general framework, including some examples from penalization of the Wiener measure. In all cases, some sigma-finite measure and some last passage time play a crucial role. In this talk, we will solve the problem stated by Madan-Roynette-Yor of characterizing situations in which a nonnegative submartingale can be characterized by a sigma-finite measure and a last passage time. Our construction will provide a unified framework for the known examples and an extension of the stopping theorem.

Speaker

Nicolas Privault

Abstract

The Dothan model is a short term interest rate model based on geometric Brownian motion. In this talk we will present several computations of zero coupon bond prices in the Dothan model using both the PDE approach of heat kernels and Yor's representation formula for the law of the time integral of geometric Brownian motion. The formulas obtained for the price P(t,T) at time t > 0 of a bond with maturity T > 0 complete those of the original paper by Dothan, which are shown not to always satisfy the boundary condition P(T,T) = 1.