1,721,031 research outputs found
Credit risk models under partial information
Full text linkThis Ph.D. thesis consists of five independent parts (Introduction included) devoted to the modeling and to studying problems related to default risk, under partial information.
The first part constitutes the Introduction.
The second part is devoted to the computation of survival probabilities of a firm, conditionally to the information available to the investor, in a structural model, under partial information. We exploit a numerical hybrid technique based on the application of the Monte Carlo method and of optimal quantization. As an application, we trace the credit spreads curve for zero coupon bonds for different maturities, showing that (as in practice on the market) the spreads in the neighborhood of the maturity are not null, i.e., under partial information there is some residual risk on the market, even if we are close to maturity.
Calibration to real data completes this second part.
In the third part we deal, by means of the Dynamic Programming, with a discrete time maximization of the expected utility from terminal wealth problem, in a market where defaultable assets are traded. Contagion risk between the default times is modeled, as well as model uncertainty, by working under partial information. In the part devoted to numerics we study the robustness of the solution found under partial information.
In the fourth part we are interested in studying the problem linked to the uncertainty of the investment horizon. In particular, in a complete market model subject to default risk, we solve, both with a direct martingale approach and with the Dynamic Programming, three different consumption maximization problems. More specifically, denoting by the default time, where is an exogenous positive random variable, we consider three problems of maximization of expected utility from consumption: when the investment horizon is fixed and equal to T, when it is finite, but possibly uncertain, equal to T ^, and when it is
infinite. First we consider the general stochastic coefficients case, then, in order to obtain explicit results in the logarithmic and power utility cases, we pass to the constant coefficients case.
Finally, in the fifth part we deal with a totally different problem, given that it is purely theoretical. In the context of enlargement of filtrations our aim is to retrieve, in a specific setting, the already known results on martingales’ characterization, on the decomposition of
martingales with respect to the reference filtration as semi-martingales in the progressively and in the initially enlarged filtrations and the Predictable Representation Theorem. Some of these results were used in the fourth part of this thesis. The interest in this study is pedagogical: in our specific context most of the results are found more easily, by exploiting "basic" tools, such as Girsanov’s Theorem and by computing conditional expectations
An application to credit risk of a hybrid Monte Carlo-optimal quantization method
No full textIn this paper, we use a hybrid Monte Carlo-optimal quantization method to approximate the conditional survival probabilities of a firm, given a structural model for its credit default, under partial information. We consider the case when the firm’s value is a nonobservable stochastic process (Vt)t≽0 and investors in the market have access to a process (St)t≽0, whose value at each time t is related to (Vs,0 ≼ s ≼ t). We are interested in the computation of the conditional survival probabilities of the firm given the “investor’s information”. As an application, we analyze the shape of the credit spread curve for zero-coupon bonds in two examples. Calibration to available market data is also analyzed
Optimal portfolio for HARA utility functions in a pure jump multidimensional incomplete market
No full textIn this paper we analyse a pure jump incomplete market where the
risky assets can jump upwards or downwards. In this market we show that, when an investor wants to maximise a HARA utility function of his/her terminal wealth, his/her optimal strategy consists of keeping constant proportions of wealth in the risky assets, thus extending the classical Merton result to this market. Finally, we compare our results with the classical ones in the diffusion case in terms of scalar dependence of portfolio proportions on the risk-aversion coefficient
Quantized calibration in local volatility
No full textPricing of a derivative should be fast and accurate, otherwise it cannot be calibrated efficiently. Here, Giorgia Callegaro,
Lucio Fiorin and Martino Grasselli apply a fast quantization methodology, in a local volatility context, to the pricing of vanilla
and barrier options that overcomes the numerical problems in existing method
Pricing via recursive Quantization in Stochastic Volatility Models
No full textWe provide the first recursive quantization-based approach for pricing options in the presence of
stochastic volatility. This method can be applied to any model for which an Euler scheme is available
for the underlying price process and it allows to price vanillas, as well as exotics, thanks to the
knowledge of the transition probabilities for the discretized stock process.We apply the methodology
to some celebrated stochastic volatility models, including the Stein and Stein [Rev. Financ. Stud. 1991,
(4), 727–752] model and the SABR model introduced in Hagan et al. [Wilmott Mag., 2002, 84–108].
A numerical exercise shows that the pricing of vanillas turns out to be accurate; in addition, when
applied to some exotics like equity-volatility options, the quantization-based method overperforms
by far the Monte Carlo simulation
Portfolio Optimization in Discontinuous Markets under Incomplete Information
No full textWe consider the problem of maximization of expected utility from terminal wealth for log and power utility functions in a market model that leads to purely discontinuous processes. We study this problem as a stochastic control problem both under complete as well as incomplete information. Our contribution consists in showing that the optimal strategy can be obtained by solving a system of equations that in some cases is linear and that a certainty equivalence property holds not only for log-utility but also for a power utility function. For the case of a power utility under incomplete information we also present an independent direct approach based on a Zakai-type equation
Optimal Reduction of Public Debt under Partial Observation of the Economic Growth
Full text linkCallegaro G, Ceci C, Ferrari G. Optimal Reduction of Public Debt under Partial Observation of the Economic Growth. Center for Mathematical Economics Working Papers. Vol 608. Bielefeld: Center for Mathematical Economics; 2019.We consider a government that aims at reducing the debt-to-gross domestic product
(GDP) ratio of a country. The government observes the level of the debt-to-GDP ratio and an
indicator of the state of the economy, but does not directly observe the development of the underlying
macroeconomic conditions. The government's criterion is to minimize the sum of the total expected
costs of holding debt and of debt's reduction policies. We model this problem as a singular stochastic
control problem under partial observation. The contribution of the paper is twofold. Firstly, we
provide a general formulation of the model in which the level of debt-to-GDP ratio and the value of
the macroeconomic indicator evolve as a diffusion and a jump-diffusion, respectively, with coefficients
depending on the regimes of the economy. These are described through a finite-state continuous-time
Markov chain. We reduce via filtering techniques the original problem to an equivalent one with full
information (the so-called separated problem), and we provide a general verification result in terms of
a related optimal stopping problem under full information. Secondly, we specialize to a case study in
which the economy faces only two regimes, and the macroeconomic indicator has a suitable diffusive
dynamics. In this setting we provide the optimal debt reduction policy. This is given in terms of the
continuous free boundary arising in an auxiliary fully two-dimensional optimal stopping problem
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