Umut Çetin

Modeling Credit Risk with Partial Information

This paper provides an alternative approach to Duffie and Lando [Econometrica 69 (2001) 633-664] for obtaining a reduced form credit risk model from a structural model. Duffie and Lando obtain a reduced form model by constructing an economy where the market sees the managers information set plus noise. The noise makes default a surprise to the market. In contrast, we obtain a reduced form model by constructing an economy where the market sees a reduction of the managers information set. The reduced information makes default a surprise to the market. We provide an explicit formula for the default intensity based on an Azema martingale, and we use excursion theory of Brownian motions to price risky debt.

MODELING LIQUIDITY EFFECTS IN DISCRETE TIME

We study optimal portfolio choices for an agent with the aim of maximising utility from terminal wealth within a market with liquidity costs. Under some mild conditions, we show the existence of optimal portfolios and that the marginal utility of the optimal terminal wealth serves as a change of measure to turn the marginal price process of the optimal strategy into a martingale. Finally, we illustrate our results numerically in a Cox-Ross-Rubinstein binomial model with liquidity costs and find the reservation ask prices for simple European put options.

PRICING AND HEDGING IN CARBON EMISSIONS MARKETS

We propose a model for trading in emission allowances in the EU Emission Trading Scheme (ETS). Exploiting an arbitrage relationship we derive the spot prices of carbon allowances given a forward contract whose price is exogenous to the model. The modeling is done under the assumption of no banking of carbon allowances (which is valid during the Phase I of Kyoto protocol), however, we also discuss how the model can be extended when banking of permits is available. We employ results from filtering theory to derive the spot prices of permits and suggest hedging formulas using a local risk minimisation approach. We also consider the effect of intermediate announcements regarding the net position of the ETS zone on the prices and show that the jumps in the prices can be attributed to information release on the net position of the zone. We also provide a brief numerical simulation for the price processes of carbon allowances using our model to show the resemblance to the actual data.

Insider trading in an equilibrium model with default: a passage from reduced-form to structural modelling

Abstract We study, in the framework of Back [Rev. Financial Stud. 5(3), 387–409 (1992)], an equilibrium model for the pricing of a defaultable zero coupon bond issued by a firm. The market consists of a risk-neutral informed agent, noise traders, and a market maker who sets the price using the total order. When the insider does not trade, the default time possesses a default intensity in the market’s view as in reduced-form credit risk models. However, we show that, in equilibrium, the modelling becomes structural in the sense that the default time becomes the first time that some continuous observation process falls below a certain barrier. Interestingly, the firm value is still not observable. We also establish the no expected trade theorem that the insider’s trades are inconspicuous.

Bayesian Sequential Estimation of a Drift of Fractional Brownian Motion

Abstract We solve explicitly a Bayesian sequential estimation problem for the drift parameter μ of a fractional Brownian motion under the assumptions that a prior density of μ is Gaussian and that a penalty function is quadratic or Dirac-delta. The optimal stopping time for this case is deterministic.

Equilibrium model with default and dynamic insider information

We consider an equilibrium model à la Kyle–Back for a defaultable claim issued by a given firm. In such a market the insider observes continuously in time the value of the firm, which is unobservable by the market makers. Using the construction in Campi et al. (http://hal.archives-ouvertes.fr/hal-00534273/en/, 2011) of a dynamic three-dimensional Bessel bridge, we provide the equilibrium price and the insider’s optimal strategy. As in Campi and Çetin (Finance Stoch. 11:591–602, 2007), the information released by the insider while trading optimally makes the default time predictable in the market’s view at the equilibrium. We conclude the paper by comparing the insider’s expected profits in the static and dynamic private information case. We also compute explicitly the value of the insider’s information in the special cases of a defaultable stock and a bond.

On certain integral functionals of squared Bessel processes

For a squared Bessel process, , the Laplace transforms of joint laws of are studied where is the first hitting time of by and is a random variable measurable with respect to the history of X until . A subset of these results are then used to solve the associated small ball problems for and to determine a Chungs law of the iterated logarithm. is also considered as a purely discontinuous increasing Markov process and its infinitesimal generator is found. The findings are then used to price a class of exotic derivatives on interest rates and to determine the asymptotics for the prices of some put options that are only slightly in-the-money.

Financial equilibrium with asymmetric information and random horizon

We study in detail and explicitly solve the version of Kyle’s model introduced in a specific case in Back and Baruch (Econometrica 72:433–465, 2004), where the trading horizon is given by an exponentially distributed random time. The first part of the paper is devoted to the analysis of time-homogeneous equilibria using tools from the theory of one-dimensional diffusions. It turns out that such an equilibrium is only possible if the final payoff is Bernoulli distributed as in Back and Baruch (Econometrica 72:433–465, 2004). We show in the second part that the signal the market makers use in the general case is a time-changed version of the one they would have used had the final payoff had a Bernoulli distribution. In both cases, we characterise explicitly the equilibrium price process and the optimal strategy of the informed trader. In contrast to the original Kyle model, it is found that the reciprocal of the market’s depth, i.e., Kyle’s lambda, is a uniformly integrable supermartingale. While Kyle’s lambda is a potential, i.e., converges to 0, for the Bernoulli-distributed final payoff, its limit in general is different from 0.

Financial Markets with Informational Asymmetries and Equilibrium

This chapter introduces the setup for the equilibrium models that extends, among others, the works of Kyle and Back. It also contains some key results that will be relevant for the characterisation of the equilibrium. Finally the equilibrium will be derived and discussed in Chaps. 7 and 8.

Kyle—Back Model with Default and Dynamic Information

In this chapter we will continue applying the dynamic bridge construction from Chap. 5 to solve the Kyle–Back model in the case of dynamic information and default risk.