arXiv subjects

Tahir Choulli

Publications and source records attributed to Tahir Choulli.

At least 19 recordsLinked to original sources

arXiv2024

Second-Order Esscher Pricing for L\'evy Models with Applications: Risk Management and Fear Quantification

This paper proposes the second-order Esscher transform as a tractable extension of the classical Esscher framework for option pricing and risk management in L\'evy-driven markets. For a general L\'evy process, we derive the associated densities and equivalent pricing measures, characterize the martingale condition in closed form, and obtain FFT-based valuation formulas for European call options. For jump-diffusion models, we establish explicit pricing formulas under the second-order Esscher measure and show that the resulting option prices lie in an interval bounded below by the Black--Scholes price and above by the underlying asset value. For the constant jump-diffusion model, we further prove monotonicity of option prices with respect to the second-order Esscher parameter. An empirical analysis based on market data shows that this additional parameter provides a tractable tool for stress testing, delta-hedging evaluation, and the construction of interval-valued risk measures in incomplete markets. We further document a strong association between the estimated second-order Esscher parameter and standard indicators of market stress, including the VIX, news sentiment, and crisis regimes. The proposed framework preserves analytical tractability while enlarging the class of admissible pricing measures, thereby supporting pricing, hedging, and stress-based risk assessment in incomplete markets with jump and general L\'evy dynamics.

q-fin.MF↗

arXiv2024

The second-order Esscher martingale densities for continuous-time market models

In this paper, we introduce the second-order Esscher pricing notion for continuous-time models. Depending whether the stock price $S$ or its logarithm is the main driving noise/shock in the Esscher definition, we obtained two classes of second-order Esscher densities called linear class and exponential class respectively. Using the semimartingale characteristics to parametrize $S$, we characterize the second-order Esscher densities (exponential and linear) using pointwise equations. The role of the second order concept is highlighted in many manners and the relationship between the two classes is singled out for the one-dimensional case. Furthermore, when $S$ is a compound Poisson model, we show how both classes are related to the Delbaen-Haenzendonck's risk-neutral measure. Afterwards, we restrict our model $S$ to follow the jump-diffusion model, for simplicity only, and address the bounds of the stochastic Esscher pricing intervals. In particular, no matter what is the Esscher class, we prove that both bounds (upper and lower) are solutions to the same linear backward stochastic differential equation (BSDE hereafter for short) but with two different constraints. This shows that BSDEs with constraints appear also in a setting beyond the classical cases of constraints on gain-processes or constraints on portfolios. We prove that our resulting constrained BSDEs have solutions in our framework for a large class of claims' payoffs including any bounded claim, in contrast to the literature, and we single out the monotonic sequence of BSDEs that ``naturally" approximate it as well.

q-fin.MF↗

arXiv2024

Optimal stopping problem under random horizon

This paper considers a pair $(\mathbb{F},τ)$, where $\mathbb{F}$ is a filtration representing the "public" flow of information which is available to all agents overtime, and $τ$ is a random time which might not be an $\mathbb{F}$-stopping time. This setting covers the case of credit risk framework where $τ$ models the default time of a firm or client, and the setting of life insurance where $τ$ is the death time of an agent. It is clear that random times can not be observed before their occurrence. Thus the larger filtration $\mathbb{G}$, which incorporates $\mathbb{F}$ and makes $τ$ observable, results from the progressive enlargement of $\mathbb{F}$ with $τ$. For this informational setting, governed by $\mathbb{G}$, we analyze the optimal stopping problem in three main directions. The first direction consists of characterizing the existence of the solution to this problem in terms of $\mathbb{F}$-observable processes. The second direction lies in deriving the {\it mathematical structures} of the value process of this control problem, while the third direction singles out the associated optimal stopping problem under $\mathbb{F}$. These three aspects allow us to quantify deeply how $τ$ impact the optimal stopping problem, while they are also vital for studying reflected backward stochastic differential equations which arise {\it naturally} from pricing and hedging of vulnerable claims.

math.PR↗

arXiv2024

New Stochastic Fubini Theorems

The classic stochastic Fubini theorem says that if one stochastically integrates with respect to a semimartingale $S$ an $η(dz)$-mixture of $z$-parametrized integrands $ψ^z$, the result is just the $η(dz)$-mixture of the individual $z$-parametrized stochastic integrals $\intψ^z{d}S.$ But if one wants to use such a result for the study of Volterra semimartingales of the form $ X_t =\int_0^t Ψ_{t,s}dS_s, t \geq0,$ the classic assumption that one has a fixed measure $η$ is too restrictive; the mixture over the integrands needs to be taken instead with respect to a stochastic kernel on the parameter space. To handle that situation and prove a corresponding new stochastic Fubini theorem, we introduce a new notion of measure-valued stochastic integration with respect to a general multidimensional semimartingale. As an application, we show how this allows to handle a class of quite general stochastic Volterra semimartingales.

math.PR↗

arXiv2024

Super-hedging-pricing formulas and Immediate-Profit arbitrage for market models under random horizon

In this paper, we consider the discrete-time setting, and the market model described by (S,F,T)$. Herein F is the ``public" flow of information which is available to all agents overtime, S is the discounted price process of d-tradable assets, and T is an arbitrary random time whose occurrence might not be observable via F. Thus, we consider the larger flow G which incorporates F and makes T an observable random time. This framework covers the credit risk theory setting, the life insurance setting and the setting of employee stock option valuation. For the stopped model (S^T,G) and for various vulnerable claims, based on this model, we address the super-hedging pricing valuation problem and its intrinsic Immediate-Profit arbitrage (IP hereafter for short). Our first main contribution lies in singling out the impact of change of prior and/or information on conditional essential supremum, which is a vital tool in super-hedging pricing. The second main contribution consists of describing as explicit as possible how the set of super-hedging prices expands under the stochasticity of T and its risks, and we address the IP arbitrage for (S^T,G) as well. The third main contribution resides in elaborating as explicit as possible pricing formulas for vulnerable claims, and singling out the various informational risks in the prices' dynamics.

q-fin.MF↗

arXiv2022

Representation for martingales living after a random time with applications

Our financial setting consists of a market model with two flows of information. The smallest flow F is the "public" flow of information which is available to all agents, while the larger flow G has additional information about the occurrence of a random time T. This random time can model the default time in credit risk or death time in life insurance. Hence the filtration G is the progressive enlargement of F with T. In this framework, under some mild assumptions on the pair (F, T), we describe explicitly how G-local martingales can be represented in terms of F-local martingale and parameters of T. This representation complements Choulli, Daveloose and Vanmaele \cite{ChoulliDavelooseVanmaele} to the case when martingales live "after T". The application of these results to the explicit parametrization of all deflators under G is fully elaborated. The results are illustrated on the case of jump-diffusion model and the discrete-time market model.

q-fin.MF↗

arXiv2022

Log-optimal portfolio after a random time: Existence, description and sensitivity analysis

In this paper, we consider an informational market model with two flows of informations. The smallest flow F, which is available to all agents, is the filtration of the initial market model(S,F,P), where S is the assets' prices and P is a probability measure. The largest flow G contains additional information about the occurrence of a random time T. This setting covers credit risk theory where T models the default time of a firm, and life insurance where T represents the death time of an insured. For the model (S-S^T,G,P), we address the log-optimal portfolio problem in many aspects. In particular, we answer the following questions and beyond: 1) What are the necessary and sufficient conditions for the existence of log-optimal portfolio of the model under consideration? 2) what are the various type of risks induced by T that affect this portfolio and how? 3) What are the factors that completely describe the sensitivity of the log-portfolio to the parameters of T? The answers to these questions and other related discussions definitely complement the work of Choulli and Yansori [12] which deals with the stopped model (S^T,G).

q-fin.PM↗

arXiv2021

Reflected backward stochastic differential equations under stopping with an arbitrary random time

This paper addresses reflected backward stochastic differential equations (RBSDE hereafter) that take the form of \begin{eqnarray*} \begin{cases} dY_t=f(t,Y_t, Z_t)d(t\wedgeτ)+Z_tdW_t^τ+dM_t-dK_t,\quad Y_τ=ξ, Y\geq S\quad\mbox{on}\quad \Lbrack0,τ\Lbrack,\quad \displaystyle\int_0^τ(Y_{s-}-S_{s-})dK_s=0\quad P\mbox{-a.s..}\end{cases} \end{eqnarray*} Here $τ$ is an arbitrary random time that might not be a stopping time for the filtration $\mathbb F$ generated by the Brownian motion $W$. We consider the filtration $\mathbb G$ resulting from the progressive enlargement of $\mathbb F$ with $τ$ where this becomes a stopping time, and study the RBSDE under $\mathbb G$. Precisely, we focus on answering the following problems: a) What are the sufficient minimal conditions on the data $(f, ξ, S, τ)$ that guarantee the existence of the solution of the $\mathbb G$-RBSDE in $L^p$ ($p>1$)? b) How can we estimate the solution in norm using the triplet-data $(f, ξ, S)$? c) Is there an RBSDE under $\mathbb F$ that is intimately related to the current one and how their solutions are related to each other? We prove that for any random time, having a positive Azéma supermartingale, there exists a positive discount factor ${\widetilde{\cal E}}$ that is vital in answering our questions without assuming any further assumption on $τ$, and determining the space for the triplet-data $(f,ξ, S)$ and the space for the solution of the RBSDE as well.

math.PR↗

arXiv2021

Explicit description of all deflators for market models under random horizon with applications to NFLVR

This paper considers an initial market model, specified by its underlying assets $S$ and its flow of information $\mathbb F$, and an arbitrary random time $τ$ which might not be an $\mathbb F$-stopping time. As the death time and the default time (that $τ$ might represent) can be seen when they occur only, the progressive enlargement of $\mathbb F$ with $τ$ sounds tailor-fit for modelling the new flow of information $\mathbb G$ that incorporates both $\mathbb F$ and $τ$. In this setting of informational market, the first principal goal resides in describing as explicitly as possible the set of all deflators for $(S^τ, \mathbb G)$, while the second principal goal lies in addressing the No-Free-Lunch-with-Vanishing-Risk concept (NFLVR hereafter) for $(S^τ, \mathbb G)$. Besides this direct application to NFLVR, the set of all deflators constitutes the dual set of all "admissible" wealth processes for the stopped model $(S^τ,\mathbb G)$, and hence it is vital in many hedging and pricing related optimization problems. Thanks to the results of Choulli et al. [7], on martingales classification and representation for progressive enlarged filtration, our two main goals are fully achieved in different versions, when the survival probability never vanishes. The results are illustrated on the two particular cases when $(S,\mathbb F)$ follows the jump-diffusion model and the discrete-time model.

q-fin.MF↗

arXiv2020

Log-optimal portfolio and numéraire portfolio for market models stopped at a random time

This paper focuses on numéraire portfolio and log-optimal portfolio (portfolio with finite expected utility that maximizes the expected logarithm utility from terminal wealth), when a market model $(S,\mathbb F)$ -specified by its assets' price $S$ and its flow of information $\mathbb F$- is stopped at a random time $τ$. This setting covers the areas of credit risk and life insurance, where $τ$ represents the default time and the death time respectively. Thus, the progressive enlargement of $\mathbb F$ with $τ$, denoted by $\mathbb G$, sounds tailor-fit for modelling the new flow of information that incorporates both $\mathbb F$ and $τ$. For the resulting stopped model $(S^τ,\mathbb G)$, we study the two portfolios in different manners, and describe their computations in terms of the $\mathbb F$-observable parameters of the pair $(S, τ)$.

q-fin.MF↗

arXiv2018

Structure conditions under progressively added information

It has been understood that the "local" existence of the Markowitz' optimal portfolio or the solution to the local-risk minimization problem is guaranteed by some specific mathematical structures on the underlying assets price processes known in the literature as "{\it Structure Conditions}". In this paper, we consider a semi-martingale market model, and an arbitrary random time that is not adapted to the information flow of the market model. This random time may model the default time of a firm, the death time of an insured, or any the occurrence time of an event that might impact the market model somehow. By adding additional uncertainty to the market model, via this random time, the {\it structures conditions} may fail and hence the Markowitz's optimal portfolio and other quadratic-optimal portfolios might fail to exist. Our aim is to investigate the impact of this random time on the structures conditions from different perspectives. Our analysis allows us to conclude that under some mild assumptions on the market model and the random time, these structures conditions will remain valid on the one hand. Furthermore, we provide two examples illustrating the importance of these assumptions. On the other hand, we describe the random time models for which these structure conditions are preserved for any market model. These results are elaborated separately for the two contexts of stopping with the random time and incorporating totally a specific class of random times respectively.

q-fin.RM↗

arXiv2018

Log-optimal portfolio without NFLVR: existence, complete characterization, and duality

This paper addresses the log-optimal portfolio for a general semimartingale model. The most advanced literature on the topic elaborates existence and characterization of this portfolio under no-free-lunch-with-vanishing-risk assumption (NFLVR). There are many financial models violating NFLVR, while admitting the log-optimal portfolio on the one hand. On the other hand, for financial markets under progressively enlargement of filtration, NFLVR remains completely an open issue, and hence the literature can be applied to these models. Herein, we provide a complete characterization of log-optimal portfolio and its associated optimal deflator, necessary and sufficient conditions for their existence, and we elaborate their duality as well without NFLVR.

q-fin.MF↗

arXiv2018

Mortality/longevity Risk-Minimization with or without securitization

This paper addresses the risk-minimization problem, with and without mortality securitization, à la Föllmer-Sondermann for a large class of equity-linked mortality contracts when no model for the death time is specified. This framework includes the situation where the correlation between the market model and the time of death is arbitrary general, and hence leads to the case of a market model where there are two levels of information. The public information which is generated by the financial assets, and a larger flow of information that contains additional knowledge about a death time of an insured. By enlarging the filtration, the death uncertainty and its entailed risk are fully considered without any mathematical restriction. Our key tool lies in our optional martingale representation that states that any martingale in the large filtration stopped at the death time can be decomposed into precise orthogonal local martingales. This allows us to derive the dynamics of the value processes of the mortality/longevity securities used for the securitization, and to decompose any mortality/longevity liability into the sum of orthogonal risks by means of a risk basis. The first main contribution of this paper resides in quantifying, as explicit as possible, the effect of mortality uncertainty on the risk-minimizing strategy by determining the optimal strategy in the enlarged filtration in terms of strategies in the smaller filtration. Our second main contribution consists of finding risk-minimizing strategies with insurance securitization by investing in stocks and one (or more) mortality/longevity derivatives such as longevity bonds. This generalizes the existing literature on risk-minimization using mortality securitization in many directions.

q-fin.MF↗

arXiv2018

A martingale representation theorem and valuation of defaultable securities

We consider a market model where there are two levels of information. The public information generated by the financial assets, and a larger flow of information that contains additional knowledge about a random time. This random time can represent many economic and financial settings, such as the default time of a firm for credit risk, and the death time of an insured for life insurance. By using the expansion of filtration, the random time uncertainty and its entailed risk are fully considered without any mathematical restriction. In this context with no model's specification for the random time, the main challenge lies in finding the dynamics and the structures for the value processes of defaultable or mortality and/or longevity securities which are vital for the insurance securitization. To overcome this obstacle, we elaborate our optional martingale representation results, which state that any martingale in the large filtration stopped at the random time can be decomposed into precise and unique orthogonal local martingales (i.e. local martingales whose product remains a local martingale). This constitutes our first and probably the principal contribution. Even though the driving motivation for this representation resides in credit risk theory, our results are applicable to several other financial and economics contexts, such as life insurance and financial markets with random horizon. Thanks to this optional representation, we decompose any defaultable or mortality and/or longevity liability into the sum of "non-correlated" risks using a risk basis. This constitutes our second contribution.

q-fin.MF↗

arXiv2018

Thin times and random times' decomposition

The paper studies thin times which are random times whose graph is contained in a countable union of the graphs of stopping times with respect to a reference filtration $\mathbb F$. We show that a generic random time can be decomposed into thin and thick parts, where the second is a random time avoiding all $\mathbb F$-stopping times. Then, for a given random time $τ$, we introduce ${\mathbb F}^τ$, the smallest right-continuous filtration containing $\mathbb F$ and making $τ$ a stopping time, and we show that, for a thin time $τ$, each $\mathbb F$-martingale is an ${\mathbb F}^τ$-semimartingale, i.e., the hypothesis $({\mathcal H}^\prime)$ for $(\mathbb F, {\mathbb F}^τ)$ holds. We present applications to honest times, which can be seen as last passage times, showing classes of filtrations which can only support thin honest times, or can accommodate thick honest times as well.

math.PR↗

arXiv2016

Non-Arbitrage under a Class of Honest Times

This paper quantifies the interplay between the non-arbitrage notion of No-Unbounded-Profit-with-Bounded-Risk (NUPBR hereafter) and additional information generated by a random time. This study complements the one of Aksamit/Choulli/Deng/Jeanblanc [1] in which the authors studied similar topics for the case of stopping at the random time instead, while herein we are concerned with the part after the occurrence of the random time. Given that all the literature -up to our knowledge- proves that the NUPBR notion is always violated after honest times that avoid stopping times in a continuous filtration, herein we propose a new class of honest times for which the NUPBR notion can be preserved for some models. For this family of honest times, we elaborate two principal results. The first main result characterizes the pairs of initial market and honest time for which the resulting model preserves the NUPBR property, while the second main result characterizes the honest times that preserve the NUPBR property for any quasi-left continuous model. Furthermore, we construct explicitly "the-after-tau" local martingale deflators for a large class of initial models (i.e. models in the small filtration) that are already risk-neutralized.

q-fin.PR↗

arXiv2015

Non-Arbitrage Under Additional Information for Thin Semimartingale Models

This paper completes the two studies undertaken in \cite{aksamit/choulli/deng/jeanblanc2} and \cite{aksamit/choulli/deng/jeanblanc3}, where the authors quantify the impact of a random time on the No-Unbounded-Risk-with-Bounded-Profit concept (called NUPBR hereafter) when the stock price processes are quasi-left-continuous (do not jump on predictable stopping times). Herein, we focus on the NUPBR for semimartingales models that live on thin predictable sets only and the progressive enlargement with a random time. For this flow of information, we explain how far the NUPBR property is affected when one stops the model by an arbitrary random time or when one incorporates fully an honest time into the model. This also generalizes \cite{choulli/deng} to the case when the jump times are not ordered in anyway. Furthermore, for the current context, we show how to construct explicitly local martingale deflator under the bigger filtration from those of the smaller filtration.

q-fin.MF↗

arXiv2014

Non-arbitrage for Informational Discrete Time Market Models

This paper focuses on the stability of the non-arbitrage condition in discrete time market models when some unknown information $τ$ is partially/fully incorporated into the market. Our main conclusions are twofold. On the one hand, for a fixed market $S$, we prove that the non-arbitrage condition is preserved under a mild condition. On the other hand, we give the necessary and sufficient equivalent conditions on the unknown information $τ$ to ensure the validity of the non-arbitrage condition for any market. Two concrete examples are presented to illustrate the importance of these conditions, where we calculate explicitly the arbitrage opportunities when they exist.

q-fin.MF↗