arXiv subjects

Abass Sagna

Publications and source records attributed to Abass Sagna.

11 recordsLinked to original sources

arXiv2025

Fair profit sharing ratios of Islamic investment contracts

The aim of this work is to calculate the fair profit-sharing ratios and the expected payoffs at maturity for each partner in islamic investment contrats (or instruments), based on profit and loss-sharing (PL-sharing). These investment contracts, known as {\em mudarabah} and {\em musharakah}, can be compared to {\em limited partnerships} and {\em joint ventures} (including all types of venture, such as joint-stock companies, partnerships, etc) in conventional finance. To compute these quantities, we introduce the notion of c-fair profit-sharing ratios, where $c = (c_1, \ldots,c_d) \in (\mathbb R^{\star})^d$ and $d$ is the number of partners. This constitutes an equilibrium approach that accounts for the contributions of the contracting parties in terms of both capital and labour. We show that the $c$-fair profit-sharing ratio of each partner is the sum of their contributions to capital and labour, weighted by some economic factors that we identify as {\em investment risk and opportunity} respectively. We deduce that, in the $c$-fair model, the expected investment profit is distributed among the contracts partners according to the shares $\varpi_{\ell} = c_{\ell} \, / \, (c_1 + \ldots + c_d)$, that correspond to their respective contribution weights to the venture's overall success. We extend these results to mixed contract that combine one or both previous contracts with an agency (known as {\em wakalah}) contract.

q-fin.PR↗

arXiv2019

Conditional survival probabilities under partial information: a recursive quantization approach with applications

We consider a structural model where the survival/default state is observed together with a noisy version of the firm value process. This assumption makes the model more realistic than most of the existing alternatives, but triggers important challenges related to the computation of conditional default probabilities. In order to deal with general diffusions as firm value process, we derive a numerical procedure based on the recursive quantization method to approximate it. Then, we investigate the error approximation induced by our procedure. Eventually, numerical tests are performed to evaluate the performance of the method, and an application is proposed to the pricing of CDS options.

q-fin.MF↗

arXiv2018

A general weak and strong error analysis of the recursive quantization with an application to jump diffusions

Observing that the recent developments of the recursive (product) quantization method induces a family of Markov chains which includes all standard discretization schemes of diffusions processes , we propose to compute a general error bound induced by the recursive quantization schemes using this generic markovian structure. Furthermore, we compute a marginal weak error for the recursive quantization. We also extend the recursive quantization method to the Euler scheme associated to diffusion processes with jumps, which still have this markovian structure, and we say how to compute the recursive quantization and the associated weights and transition weights.

math.PR↗

arXiv2017

Product Markovian quantization of an R^d -valued Euler scheme of a diffusion process with applications to finance

We introduce a new approach to quantize the Euler scheme of an $\mathbb{R}^d$-valued diffusion process. This method is based on a Markovian and componentwise product quantization and allows us, from a numerical point of view, to speak of {\em fast online quantization} in dimension greater than one since the product quantization of the Euler scheme of the diffusion process and its companion weights and transition probabilities may be computed quite instantaneously. We show that the resulting quantization process is a Markov chain, then, we compute the associated companion weights and transition probabilities from (semi-) closed formulas. From the analytical point of view, we show that the induced quantization errors at the $k$-th discretization step $t_k$ is a cumulative of the marginal quantization error up to time $t_k$. Numerical experiments are performed for the pricing of a Basket call option, for the pricing of a European call option in a Heston model and for the approximation of the solution of backward stochastic differential equations to show the performances of the method.

math.PR↗

arXiv2015

Recursive marginal quantization of the Euler scheme of a diffusion process

We propose a new approach to quantize the marginals of the discrete Euler diffusion process. The method is built recursively and involves the conditional distribution of the marginals of the discrete Euler process. Analytically, the method raises several questions like the analysis of the induced quadratic quantization error between the marginals of the Euler process and the proposed quantizations. We show in particular that at every discretization step $t\_k$ of the Euler scheme, this error is bounded by the cumulative quantization errors induced by the Euler operator, from times $t\_0=0$ to time $t\_k$. For numerics, we restrict our analysis to the one dimensional setting and show how to compute the optimal grids using a Newton-Raphson algorithm. We then propose a closed formula for the companion weights and the transition probabilities associated to the proposed quantizations. This allows us to quantize in particular diffusion processes in local volatility models by reducing dramatically the computational complexity of the search of optimal quantizers while increasing their computational precision with respect to the algorithms commonly proposed in this framework. Numerical tests are carried out for the Brownian motion and for the pricing of European options in a local volatility model. A comparison with the Monte Carlo simulations shows that the proposed method may sometimes be more efficient (w.r.t. both computational precision and time complexity) than the Monte Carlo method.

math.PR↗

arXiv2012

Conditional hitting time estimation in a nonlinear filtering model by the Brownian bridge method

The model consists of a signal process $X$ which is a general Brownian diffusion process and an observation process $Y$, also a diffusion process, which is supposed to be correlated to the signal process. We suppose that the process $Y$ is observed from time 0 to $s>0$ at discrete times and aim to estimate, conditionally on these observations, the probability that the non-observed process $X$ crosses a fixed barrier after a given time $t>s$. We formulate this problem as a usual nonlinear filtering problem and use optimal quantization and Monte Carlo simulations techniques to estimate the involved quantities.

math.PR↗

arXiv2012

Asymptotics of the maximal radius of an $L^r$-optimal sequence of quantizers

Let $P$ be a probability distribution on $\mathbb{R}^d$ (equipped with an Euclidean norm $|\cdot|$). Let $ r> 0 $ and let $(α_n)_{n \geq1}$ be an (asymptotically) $L^r(P)$-optimal sequence of $n$-quantizers. We investigate the asymptotic behavior of the maximal radius sequence induced by the sequence $(α_n)_{n \geq1}$ defined for every $n \geq1$ by $ρ(α_n) = \max{|a|, a \inα_n}$. When $\card(\supp(P))$ is infinite, the maximal radius sequence goes to $\sup{|x|, x \in\operatorname{supp}(P)}$ as $n$ goes to infinity. We then give the exact rate of convergence for two classes of distributions with unbounded support: distributions with hyper-exponential tails and distributions with polynomial tails. In the one-dimensional setting, a sharp rate and constant are provided for distributions with hyper-exponential tails.

math.PR↗

arXiv2011

Quantization based recursive Importance Sampling

We investigate in this paper an alternative method to simulation based recursive importance sampling procedure to estimate the optimal change of measure for Monte Carlo simulations. We propose an algorithm which combines (vector and functional) optimal quantization with Newton-Raphson zero search procedure. Our approach can be seen as a robust and automatic deterministic counterpart of recursive importance sampling by means of stochastic approximation algorithm which, in practice, may require tuning and a good knowledge of the payoff function in practice. Moreover, unlike recursive importance sampling procedures, the proposed methodology does not rely on simulations so it is quite generic and can come along on the top of Monte Carlo simulations. We first emphasize on the consistency of quantization for designing an importance sampling algorithm for both multi-dimensional distributions and diffusion processes. We show that the induced error on the optimal change of measure is controlled by the mean quantization error. We illustrate the effectiveness of our algorithm by pricing several options in a multi-dimensional and infinite dimensional framework.

math.PR↗

arXiv2010

Pricing of barrier options by marginal functional quantization

This paper is devoted to the pricing of Barrier options by optimal quadratic quantization method. From a known useful representation of the premium of barrier options one deduces an algorithm similar to one used to estimate nonlinear filter using quadratic optimal functional quantization. Some numerical tests are fulfilled in the Black-Scholes model and in a local volatility model and a comparison to the so called Brownian Bridge method is also done.

q-fin.PR↗

arXiv2009

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.

q-fin.CP↗

arXiv2007

Universal L^s -rate-optimality of L^r-optimal quantizers by dilatation and contraction

Let $ r, s>0 $. For a given probability measure $P$ on $\mathbb{R}^d$, let $(α_n)_{n \geq 1}$ be a sequence of (asymptotically) $L^r(P)$- optimal quantizers. For all $μ\in \mathbb{R}^d $ and for every $θ>0$, one defines the sequence $(α_n^{θ, μ})_{n \geq 1}$ by : $\forall n \geq 1, α_n^{θ, μ} = μ+ θ(α_n - μ) = \{μ+ θ(a- μ), a \in α_n \} $. In this paper, we are interested in the asymptotics of the $L^s$-quantization error induced by the sequence $(α_n^{θ, μ})_{n \geq 1}$. We show that for a wide family of distributions, the sequence $(α_n^{θ, μ})_{n \geq 1}$ is $L^s$-rate-optimal. For the Gaussian and the exponential distributions, one shows how to choose the parameter $θ$ such that $(α_n^{θ, μ})_{n \geq 1}$ satisfies the empirical measure theorem and probably be asymptotically $L^s$-optimal.

math.PR↗