arXiv subjects

Traian A. Pirvu

Publications and source records attributed to Traian A. Pirvu.

At least 19 recordsLinked to original sources

arXiv2026

Sharpe Ratio and Return-VaR Ratio Maximization for Option Portfolios with Skew-Elliptical $t$ Underlying Returns

We provide a formulation for optimal option portfolios under Sharpe Ratio maximization when the underlying returns follow a skew-elliptical t-distribution. This departs from the traditional normal returns setting in the context of Sharpe ratio maximization by allowing the modelling of heavy-tailed and skewed dynamics. The novelty of this paper and our main result is to provide explicit formulas for the portfolio weights when maximizing the Sharpe ratio and return-to-Value-at-Risk (VaR) ratio in the skew-elliptical setting. Numerical experiments reveal that the optimal portfolios for the two ratios are different.

q-fin.PM↗

arXiv2026

Optimal Option Portfolios for Skew-Elliptical t Returns

This paper explores option portfolio optimization when the underlying returns are skew-elliptical t-distributed. We use the variance and value at risk (VaR) to measure portfolio risk. The novelty of our work is the departure from the traditional normal returns setting, allowing investors to capture both heavy-tailed and skewed market dynamics. We provide explicit portfolio weights for the variance and VaR approximation. Our second contribution is the numerical representation of portfolio weights, obtained from numerical optimization for better VaR approximations. The effect of skewness on the portfolio weights is quantified by comparing our optimal skew t weights with those generated in the Student t setting. We also find that, as expected, a better VaR approximation risk measure yields optimal portfolio weights which are more different than the variance optimal weights.

q-fin.PM↗

arXiv2026

Optimal Investment with Switching Preferences

Major life events can significantly increase individuals' risk aversion over a sustained period of time, as empirical studies reveal. How such an event-triggered shift of risk preferences impacts optimal investment is the focus of this paper. On a finite time horizon where a major life event may occur independently of the financial market, an investor aims to maximize her expected utility from terminal wealth while foreseeing a potential change in her risk aversion. We find that the associated Hamilton--Jacobi--Bellman (HJB) equation involves the post-event optimal value function (under elevated but fixed risk aversion after the event's occurrence), and the Fenchel--Legendre transform fails to linearize this HJB equation: it yields a parabolic equation with a fully nonlinear term, induced precisely by the post-event optimal value function. Through a combination of fixed-point, compactness, and verification arguments, we establish the existence of a positive convex classical solution with suitable growth to the fully-nonlinear parabolic equation. The convex conjugate of this solution is shown to satisfy the HJB equation and coincides with the pre-event optimal value function. The optimal trading strategy is obtained by concatenating the optimal pre-event and post-event strategies -- the former is expressed in terms of the solution to the HJB equation and the latter is traditional Merton's ratio.

math.OC↗

arXiv2023

Portfolio Time Consistency and Utility Weighted Discount Rates

Merton portfolio management problem is studied in this paper within a stochastic volatility, non constant time discount rate, and power utility framework. This problem is time inconsistent and the way out of this predicament is to consider the subgame perfect strategies. The later are characterized through an extended Hamilton Jacobi Bellman (HJB) equation. A fixed point iteration is employed to solve the extended HJB equation. This is done in a two stage approach: in a first step the utility weighted discount rate is introduced and characterized as the fixed point of a certain operator; in the second step the value function is determined through a linear parabolic partial differential equation. Numerical experiments explore the effect of the time discount rate on the subgame perfect and precommitment strategies.

q-fin.PM↗

arXiv2023

A nonlinear population model

This paper considers a nonlinear model for population dynamics with age structure. The fertility rate with respect to age is non constant and has the form proposed by [17]. Moreover, its multiplicative structure and the multiplicative structure of mortality makes the model separable. In this setting it is shown that the number of births in unit time is given by a system of nonlinear ordinary differential equations. The steady state solution together with the equilibrium solution is found explicitly.

math.GM↗

arXiv2022

Optimal annuitization post-retirement with labor income

Evidence shows that the labor participation rate of retirement age cohorts is non-negligible, and it is a widespread phenomenon globally. In the United States, the labor force participation rate for workers age 75 and older is projected to be over 10 percent by 2026 as reported by the Bureau of Labor Statistics. The prevalence of post-retirement work changes existing considerations of optimal annuitization, a research question further complicated by novel factors such as post-retirement labor rates, wage rates, and capacity or willingness to work. To our knowledge, this poses a practical and theoretical problem not previously investigated in actuarial literature. In this paper, we study the problem of post-retirement annuitization with extra labor income in the framework of stochastic control, optimal stopping, and expected utility maximization. The utility functions are of the Cobb-Douglas type. The martingale methodology and duality techniques are employed to obtain closed-form solutions for the dual and primal problems. The effect of labor income is investigated by exploiting the explicit solutions and Monte-Carlo simulation. The latter reveals that the optimal annuitization time is strongly linear with respect to the initial wealth, with or without labor income. When it comes to optimal annuitization, we find that the wage and labor rates may play opposite roles. However, their impact is mediated by the leverage ratio.

q-fin.PM↗

arXiv2021

Stochastic production planning with regime switching

This paper considers a stochastic production planning problem with regime switching. There are two regimes corresponding to different economic cycles. A factory is planning its production so as to minimize production costs. We analyze this problem through the value function approach. The optimal production is characterized through the solution of an elliptic system of partial differential equations which is shown to have a solution.

math.OC↗

arXiv2020

Numerical Simulation of Exchange Option with Finite Liquidity: Controlled Variate Model

In this paper we develop numerical pricing methodologies for European style Exchange Options written on a pair of correlated assets, in a market with finite liquidity. In contrast to the standard multi-asset Black-Scholes framework, trading in our market model has a direct impact on the asset's price. The price impact is incorporated into the dynamics of the first asset through a specific trading strategy, as in large trader liquidity model. Two-dimensional Milstein scheme is implemented to simulate the pair of assets prices. The option value is numerically estimated by Monte Carlo with the Margrabe option as controlled variate. Time complexity of these numerical schemes are included. Finally, we provide a deep learning framework to implement this model effectively in a production environment.

q-fin.PR↗

arXiv2019

An elliptic partial differential equations system and its application

This paper deals with the existence of solutions for an elliptic system of partial differential equations. The solution method is based on the sub- and super-solutions approach. An application to a stochastic control problem is presented. This system seemed not considered before.

math.AP↗

arXiv2019

A Stochastic production planning problem

Stochastic production planning problems were studied in several works; the model with one production good was discussed in [3]. The extension to several economic goods is not a trivial issue as one can see from the recent works [4], [5] and [6]. The following qualitative aspects of the problem are analyzed in [5]; the existence of a solution and its characterization through dynamic programming/HJB equation, as well as the verification (i.e., the solution of the HJB equation yields the optimal production of the goods). In this paper, we stylize the model of [4] and [5] in order to provide some quantitative answers to the problem. This is possible especially because we manage to solve the HJB equation in closed form. Among other results, we find that the optimal production rates are the same across all the goods and they also turn to be independent of some model parameters. Moreover we show that production rates are increasing in the aggregate number of goods produced, and they are also uniformly bounded. Numerical experiments show some patterns of the output.

math.OC↗

arXiv2019

An elliptic partial differential equation and its application

This paper deals with the following elliptic equation \begin{equation*} -2σ^{2}Δz+\left\| \nabla z\right\| ^{2}+4αz=4\left\| x\right\| ^{2}\text{ for }x\in \mathbb{R}^{N}\text{, (}% N\geq 1\text{),} \end{equation*}% where $α>0,$ $σ>0$ are some real parameters. The solution method is based on the sub- and super-solutions approach. The case $N>1$ seemed not considered before. This equation models a stochastic production planning problem.

math.AP↗

arXiv2019

Multi-period investment strategies under Cumulative Prospect Theory

In this article, inspired by Shi, et al. we investigate the optimal portfolio selection with one risk-free asset and one risky asset in a multiple period setting under cumulative prospect theory (CPT). Compared with their study, our novelty is that we consider a stochastic benchmark, and portfolio constraints. We test the sensitivity of the optimal CPT-investment strategies to different model parameters by performing a numerical analysis.

q-fin.PM↗

arXiv2019

Optimal Sharing Rule for a Household with a Portfolio Management Problem

We study the Merton problem of optimal consumption-investment for the case of two investors sharing a final wealth. The typical example would be a husband and wife sharing a portfolio looking to optimize the expected utility of consumption and final wealth. Each agent has different utility function and discount factor. An explicit formulation for the optimal consumptions and portfolio can be obtained in the case of a complete market. The problem is shown to be equivalent to maximizing three different utilities separately with separate initial wealths. We study a numerical example where the market price of risk is assumed to be mean reverting, and provide insights on the influence of risk aversion or discount rates on the initial optimal allocation.

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arXiv2017

An Extension of Clark-Haussman Formula and Applications

This work considers a stochastic model in which the uncertainty is driven by a multidimensional Brownian motion. The market price of risk process makes the transition between real world probability measure and risk neutral probability measure. Traditionally, the martingale representation formulas under the risk neutral probability measure requires the market price of risk process to be bounded. However, in several financial models the boundedness assumption of the market price of risk fails. One example is a stock price model with the market price of risk following an Ornstein-Uhlenbeck process. This work extends Clark-Haussmann formula to underlying stochastic processes which fail to satisfy the standard requirements. Our result can be applied to hedging and optimal investment in stock markets with unbounded market price of risk.

math.PR↗

arXiv2017

Risk management under Omega measure

We prove that the Omega measure, which considers all moments when assessing portfolio performance, is equivalent to the widely used Sharpe ratio under jointly elliptic distributions of returns. Portfolio optimization of the Sharpe ratio is then explored, with an active-set algorithm presented for markets prohibiting short sales. When asymmetric returns are considered we show that the Omega measure and Sharpe ratio lead to different optimal portfolios.

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arXiv2016

One bank problem in the federal funds market

The model of this paper gives a convenient strategy that a bank in the federal funds market can use in order to maximize its profit in a contemporaneous reserve requirement (CRR) regime. The reserve requirements are determined by the demand deposit process, modelled as a Brownian motion with drift. We propose a new model in which the cumulative funds purchases and sales are discounted at possible different rates. We formulate and solve the problem of finding the bank's optimal strategy. The model can be extended to involve the bank's asset size and we obtain that, under some conditions, the optimal upper barrier for fund sales is a linear function of the asset size. As a consequence, the bank net purchase amount is linear in the asset size.

q-fin.PR↗

arXiv2014

Risk minimization and portfolio diversification

We consider the problem of minimizing capital at risk in the Black-Scholes setting. The portfolio problem is studied given the possibility that a correlation constraint between the portfolio and a financial index is imposed. The optimal portfolio is obtained in closed form. The effects of the correlation constraint are explored; it turns out that this portfolio constraint leads to a more diversified portfolio.

q-fin.PM↗

arXiv2012

A Multi Period Equilibrium Pricing Model

In this paper, we propose an equilibrium pricing model in a dynamic multi-period stochastic framework with uncertain income streams. In an incomplete market, there exist two traded risky assets (e.g. stock/commodity and weather derivative) and a non-traded underlying (e.g. temperature). The risk preferences are of exponential (CARA) type with a stochastic coefficient of risk aversion. Both time consistent and time inconsistent trading strategies are considered. We obtain the equilibriums prices of a contingent claim written on the risky asset and non-traded underlying. By running numerical experiments we examine how the equilibriums prices vary in response to changes in model parameters.

math.OC↗