A martingale representation theorem and valuation of defaultable securities

Considerably much work has been done on Backward Stochastic Differential Equations (BSDEs) in continuous-time with deterministic terminal horizon or stopping times. Various new models have been introduced in these years in order to generalize BSDEs to solve new practical financial problems.
\nOne strand is focused on discrete-time models. Backward Stochastic Difference Equations (also called BSDEs if no ambiguity) on discrete-time finite-state space were introduced by Cohen and Elliott (2010a). The associated theory required only weak assumptions. In the first topic of this thesis, properties of non-linear expectations defined using the discrete-time finite-state BSDEs were studied. A converse comparison theorem was established. Properties of risk measures defined by non-linear expectations, especially the representation theorems, were given. Then the theory of BSDEs was applied to optimal design of dynamic risk measures. Another strand is about a general random terminal time, which is not necessarily a stopping time. The motivation of this model is a financial problem of hedging of defaultable contingent claims, where BSDEs with stopping times are not applicable. In the second topic of this thesis, discrete-time finite-state BSDEs under progressively enlarged filtration were considered. Martingale representation
\ntheorem, existence and uniqueness theorem and comparison theorem were established. Application to nonlinear expectations was also explored. Using the theory of BSDEs, the explicit solution for optimal design of dynamic default risk measures was obtained.
\nIn recent work on continuous-time BSDEs under progressively enlarged filtration, the reference filtration is generated by Brownian motions. In order to deal with cases with jumps, in the third topic of this thesis, a general reference filtration with predictable representation property and an initial time with immersion property were considered. The martingale representation theorem for square-integrable martingales under progressively enlarged filtration was established. Then the existence and uniqueness theorem of BSDEs under enlarged filtration using Lipschitz continuity of the driver was proved. Conditions for a comparison theorem were also presented. Finally applications to nonlinear expectations and hedging of defaultable contingent claims on Brownian-Poisson setting were explored.

ABSTRACTIn this article, we consider an optimization problem of expected utility maximization of continuous-time trading in a financial market. This trading is constrained by a benchmark for a utility-based shortfall risk measure. The market consists of one asset whose price process is modelled by a Geometric Brownian motion where the market parameters change at a random time. The information flow is modelled by initially and progressively enlarged filtrations which represent the knowledge about the price process, the Brownian motion and the random time. We solve the maximization problem and give the optimal terminal wealth depending on these different filtrations for general utility functions by using martingale representation results for the corresponding filtration.

This paper considers the setting governed by , where is the "public" flow of information, and τ is a random time which might not be -observable. This framework covers credit risk and life insurance. In this setting, is assumed to be generated by a d-dimensional Brownian motion W and ξ is a vulnerable claim, whose payment's policy depends essentially on the occurrence of τ. The hedging problems, in many directions, for this claim led to the question of studying the linear reflected-backward-stochastic differential equations (RBSDE hereafter), This is the objective of this paper. For this RBSDE and without any further assumption on τ that might neglect any risk intrinsic to its stochasticity, we answer the following: a) What are the sufficient minimal conditions on the data that guarantee the existence of the solution to this RBSDE? b) How can we estimate the solution in norm using ? c) Is there an -RBSDE that is intimately related to the current one and how their solutions are related to each other? This latter question has practical and theoretical leitmotivs.

In this chapter, we will briefly introduce the martingale theory and Ito’s stochastic analysis. First we introduce some basic concepts of continuous time stochastic processes and the definitions of four basic types of process: Markov process, martingale, Poisson process, and Brownian motion, as well as their basic properties, including Doob-Meyer’s decompositions of continuous local submartingales and quadratic variation processes of continuous local martingales and continuous semimartingales. Then we introduce the stochastic integrals of measurable adapted processes w.r.t. the Brownian motion. Finally, we introduce some useful tools for Ito calculus, such as Ito’s formula, Girsanov’s theorem, and the martingale representation theorem. Ito stochastic differential equations and Feynman-Kac formula are also presented. This chapter is self-contained. For a small number of results, we omit their proofs and refer to Karatzas and Shreve (1991) or Revuz and Yor (1999).

Recently, van Neerven, Weis and the author, constructed a theory for stochastic integration of UMD Banach space valued processes. Here the authors use a (cylindrical) Brownian motion as an integrator. In this note we show how one can extend these results to the case where the integrator is an arbitrary real-valued continuous local martingale. We give several characterizations of integrability and prove a version of the Itô isometry, the Burkholder–Davis–Gundy inequality, the Itô formula and the martingale representation theorem.

It is proved that the martingale representation theorem for set-valued martingales proposed by Kisielewicz (2014b) holds only in the degenerate case. A revised representation theorem for a special kind of non-degenerate set-valued martingales is presented.

Example 3.1.9 illustrates that the basic definition of Ito integrals is not very useful when we try to evaluate a given integral. This is similar to the situation for ordinary Riemann integrals, where we do not use the basic definition but rather the fundamental theorem of calculus plus the chain rule in the explicit calculations.

In this first chapter, we show that a CDO tranche payoff can be perfectly replicated with a self-financed strategy based on the underlying credit default swaps. This extends to any payoff which depends only upon default arrivals, such as basket default swaps. Clearly, the replication result is model dependent and relies on two critical assumptions. First, we preclude the possibility of simultaneous defaults. The other assumption is that credit default swap premiums are adapted to the filtration of default times which therefore can be seen as the relevant information set on economic grounds. Our framework corresponds to a pure contagion model, where the arrivals of defaults lead to jumps in the credit spreads of survived names, the magnitude of which depending upon the names in question, and the whole history of defaults up to the current time. These jumps can be related to the derivatives of the joint survival function of default times. The dynamics of replicating prices of CDO tranches follows the same way. In other words, we only deal with default risks and not with spread risks. Unsurprisingly, the possibility of perfect hedging is associated with a martingale representation theorem under the filtration of default times. Subsequently, we exhibit a new probability measure under which the short term credit spreads (up to some scaling factor due to positive recovery rates) are the intensities associated with the corresponding default times. For ease of presentation, we introduced first some instantaneous default swaps as a convenient basis of hedging instruments. Eventually, we can exhibit a replicating strategy of a CDO tranche payoff with respect to actually traded credit default swaps, for instance, with the same maturity as the CDOtranche. Letus note that no Markovian assumption is required for the existence of such a replicating strategy. However, the practical implementation of actual hedging strategies requires some extra assumptions. We assume that all pre-default intensities are equal and only depend upon the current number of defaults. We also assume that all recovery rates are constant across names and time. In that framework, it can be shown that the aggregate loss process is a homogeneous Markov chain, more precisely a pure death process. Thanks to these restrictions, the model involves as many unknown parameters as the number of underlying names. Such Markovian model is also known as a local intensity model, the simplest form of aggregate loss models. As in local volatility models in the equity derivatives world, there is a perfect match of unknown parameters from a complete set of CDO tranches quotes. Numerical implementation can be achieved through a binomial tree, well-known to finance people, or by means of Markov chain techniques. We provide some examples and show that the market quotes of CDOs are associated with pronounced contagion effects. We can therefore explain the dynamics of the amount of hedging CDS and relate them to deltas computed by market practitioners. The figures are hopefully roughly the same, the discrepancies being mainly explained by contagion effects leading to an increase of dependence between default times after some defaults.KeywordsCredit Default SwapCredit SpreadGaussian CopulaCredit DerivativeDefault TimeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

After introducing cylindrical Gaussian random variables and Hilbert space valued Gaussian random variables, we define cylindrical Wiener process and Hilbert space valued Wiener process in a natural way. We develop the Itô stochastic integral with respect to cylindrical and Hilbert space valued Wiener processes simultaneously as they share many common features. Our construction of the Itô integral is consistent with the construction of a stochastic integral with respect to square integrable martingales, however, regarding measurability, we assume only that the integrand is adapted, following the classical ideas as presented by Liptser and Shiryayev (Statistics of Stochastic Processes. Nauka, ) and by Øksendal (Stochastic Differential Equations. Springer, New York, ). We provide detailed proofs of properties of the Itô integral and follow with the Martingale Representation Theorem. Our unified approach to stochastic integration with respect to cylindrical and Hilbert space valued Wiener processes allowed us to present the Stochastic Fubini Theorem and the Itô Formula in both cases in an almost identical way.

This paper firstly discuss some related properties for a type of G-function by introducing a G-Brownian motion martingale. And a martingale representation theorem is then proved by using Multiple G-Itô integral appeared in [1].

Research background: The contagious impact of the COVID-19 pandemic has heightened financial market's volatility, nonlinearity, asymmetric and nonstationary dynamics. Hence, the existing relationship among financial assets may have been altered. Moreover, the level of investor risk aversion and market opportunities could also alter in the pandemic. Predictably, investors in the heat of the moment are concerned about minimizing losses. In order to determine the level of hedge risks between implied volatilities in the COVID-19 pandemic through information flow, it is required to take into account the increased vagueness of economic projections as well as the increased uncertainty in asset values as a result of the pandemic. Purpose of the article: The study aims to examine the transmission of information between the VIX-implied volatility index for S&P 500 and fifteen other implied volatility indices in the COVID-19 pandemic. Methods: We relied on daily changes in the VIX and fifteen other implied volatility indices from commodities, currencies, and stocks. The study employed the improved complete ensemble empirical mode decomposition with adaptive noise which is in line with the heterogeneous expectations of market participants to denoise the data and extract intrinsic mode functions (IMFs). Subsequently, we clustered the IMFs based on common features into high, low, and medium frequencies. The analysis was carried out using Rényi transfer entropy (RTE), which allowed for the evaluation of both linear and non-linear, as well as varied distributions of the market dynamics. Findings & value added: Findings from the RTE revealed a bi-directional flow of negative information amid the VIX and each of the volatility indices, particularly in the long term. We found this behavior of the markets to be consistent at varying levels of investors' risk aversion. The findings help investors with their portfolio strategies in the time of the pandemic, which has resulted in fluctuating levels of risk aversion. Our findings characterize global financial markets to be ?non-linear heterogeneous evolutionary systems?. The results also lend support to the emerging delayed volatility of market competitiveness and external shocks hypothesis.

In this chapter, the problem of a quasi-explicit evaluation of various conditional expectations is studied in the case when the only filtration available in calculations is the natural filtration of a random time. At the intuitive level, we consider here an individual who is able to observe a certain random time τ, but has no access to any other information. A detailed analysis of a more interesting and practically more relevant case — when an additional flow of information is also available — is postponed to the next chapter.

In this paper, we present two related results. First, we shall obtain a sufficient condition under which a second order sample-continuous martingale can be represented as a stochastic integral in terms of a Brownian motion. Secondly, we shall show that if X and Y are sample-continuous local martingales (not necessarily with respect to the same family of ($\sigma $-algebras) and if either $X + Y$ or $X - Y$ is almost surely of bounded variation, then the quadratic variations of the two martingales are equal. This rather simple result has some surprising consequences.

Information and Capital Flows Revisited: The Internet as a Determinant of Transactions in Financial Assets

The 2030 Agenda for Sustainable Development Goals (SDGs) contribute to the realization of economic, social, and cultural rights. However, the progress of these goals now faces a significant setback due to the impact of the COVID-19 pandemic, especially in Indonesia. This article discusses the impact of the social restriction policy, which results in setbacks in several goals that have further implications for fulfilling the rights to employment and social security. Also, this article elaborates on the role of the human rights approach in recovery efforts that the state must undertake due to the pandemic’s impact. By using the Maastricht Guidelines and the International Covenant on Economic, Social, and Cultural Rights as the theoretical framework, it can be concluded that the impact of the COVID-19 pandemic has caused setbacks in some of the realizations of the SDGs, particularly in economic, social, and cultural rights. These setbacks must be anticipated with recovery policies such as reform of the social security system and improving data collection system. Recovery efforts must be implemented promptly in a measured strategy to prevent the more fatal setbacks that lead to human rights violations.
 Keywords: COVID-19, Human rights, SDGs