An application to credit risk of a hybrid Monte Carlo-Optimal quantization method

In 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 defaul, under partial information. We consider the case when the firm's value is a non-observable stochastic process $(V_t)_{t \geq 0}$ and inverstors in the market have access to a process $(S_t)_{t \geq 0}$, whose value at each time t is related to $(V_s, s \leq t)$. We are interested in the computation of the conditional survival probabilities of the firm given the "investor information". As a application, we analyse the shape of the credit spread curve for zero coupon bonds in two examples.credit risk, structural approach, survival probabilities, partial information, filtering, optimal quantization, Monte Carlo method.

Fast Hybrid Schemes for Fractional Riccati Equations (Rough is not so Tough)

We solve a family of fractional Riccati differential equations with constant (possibly complex) coefficients. These equations arise, e.g., in fractional Heston stochastic volatility models, that have received great attention in the recent financial literature thanks to their ability to reproduce a rough volatility behavior. We first consider the case of a zero initial value corresponding to the characteristic function of the log-price. Then we investigate the case of a general starting value associated to a transform also involving the volatility process. The solution to the fractional Riccati equation takes the form of power series, whose convergence domain is typically finite. This naturally suggests a hybrid numerical algorithm to explicitly obtain the solution also beyond the convergence domain of the power series representation. Our numerical tests show that the hybrid algorithm turns out to be extremely fast and stable. When applied to option pricing, our method largely outperforms the only available alternative in the literature, based on the Adams method.Comment: 48 pages, 4 figure

Utility indifference pricing and hedging for structured contracts in energy markets

In this paper we study the pricing and hedging of structured products in energy markets, such as swing and virtual gas storage, using the exponential utility indifference pricing approach in a general incomplete multivariate market model driven by finitely many stochastic factors. The buyer of such contracts is allowed to trade in the forward market in order to hedge the risk of his position. We fully characterize the buyer's utility indifference price of a given product in terms of continuous viscosity solutions of suitable nonlinear PDEs. This gives a way to identify reasonable candidates for the optimal exercise strategy for the structured product as well as for the corresponding hedging strategy. Moreover, in a model with two correlated assets, one traded and one nontraded, we obtain a representation of the price as the value function of an auxiliary simpler optimization problem under a risk neutral probability, that can be viewed as a perturbation of the minimal entropy martingale measure. Finally, numerical results are provided.Comment: 32 pages, 5 figure

Credit risk models under partial information

This 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\u2019 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\u2019s Theorem and by computing conditional expectations