arXiv subjects

Hasanjan Sayit

Publications and source records attributed to Hasanjan Sayit.

18 recordsLinked to original sources

arXiv2026

Ruin theory incorporating MIPP-type jumps

The paper investigates the ruin probability of an insurer's surplus process when claims follow a Multiply Iterated Poisson Process (MIPP). This setting extends the classical Cramer-Lundberg model by allowing for clustered claim arrivals, making it particularly appropriate for modeling catastrophic insurance losses. The primary contribution is the derivation of explicit criteria for ultimate ruin, showing that the ruin probability is largely determined by the jump intensity parameter and the iteration count. Furthermore, we establish integro-differential equations for the survival and ruin probabilities and compute their Laplace transforms. These transforms are then linked via recursive formulas that relate ruin probabilities at consecutive iteration levels. We also introduce a Cramer-Lundberg-style approximation, which provides asymptotic expressions for ruin probabilities along with a Lundberg-type inequality. Numerical examples, including comparisons with the classical model, are provided to validate the theoretical results and to highlight the impact of claim clustering on the insurer's risk of ruin.

math.PR↗

arXiv2026

Beyond Lognormal Sums: A Four-Moment Probability Framework for Basket and Spread Option Pricing

Basket options are difficult to value under correlated lognormal dynamics because weighted sums and differences of lognormal variables have no tractable distribution. This paper develops a probability-based four-moment framework that separates the exact pricing representation from the distributional approximation. A change of measure first writes a basket price as a linear combination of probabilities. For a standard basket with one positive weight, these probabilities become CDF values of positive correlated lognormal sums. Each sum is approximated by a shifted lognormal variance mixture matched to its first four moments. For an unrestricted mixed-sign basket, a signed shifted lognormal proxy gives an analytical call-price formula. We state admissibility conditions, provide a practical root-selection rule, establish the main strike-based financial properties of the direct proxy, and derive exact pricing-error identities in terms of cumulative distribution function (CDF) discrepancies. The numerical analysis combines standard-basket benchmarks with an empirical application to a normalized $3{:}2{:}1$ crack spread constructed from RBOB gasoline, ULSD or heating oil, and WTI futures. The results show that the probability reformulation and the fourth-moment condition improve the distributional fit and pricing accuracy, particularly when maturity and tail asymmetry increase. The framework remains analytical, transparent, and suitable for repeated valuation across strikes and maturities.

q-fin.PR↗

arXiv2025

On certain integral functionals of integer-valued subordinators

It is known that the exponential functional of a Poisson process admits a probability density function in the form of an infinite series. In this paper, we obtain an explicit expression for the density function of the exponential functional of any integer-valued subordinator, and by extension, limit representations for that of an arbitrary pure-jump subordinator. With an added positive drift, the density function is expressed via piecewise basis functions governed by a functional relation. Closed-form density functions for these cases have been established only for a few special instances of L\'{e}vy processes in the past literature. Our work substantially advances this line of research by providing an analytical perspective on the distribution of a broad class of exponential L\'{e}vy functionals, also suggesting potential methodological extensions to general purely discontinuous L\'{e}vy processes. Moreover, we consider arbitrary decreasing functionals of integer-valued subordinators by deriving sufficient and necessary conditions for their convergence, which are then applied to obtain limit-series representations for the density functions of inverse-power functionals. The numerical performance of the proposed formulae is demonstrated through various examples of well-known distributions.

math.PR↗

arXiv2025

Iterated Poisson Processes for Catastrophic Risk Modeling in Ruin Theory

This paper studies the properties of the Multiply Iterated Poisson Process (MIPP), a stochastic process constructed by repeatedly time-changing a Poisson process, and its applications in ruin theory. Like standard Poisson processes, MIPPs have exponentially distributed sojourn times (waiting times between jumps). We explicitly derive the probabilities of all possible jump sizes at the first jump and obtain the Laplace transform of the joint distribution of the first jump time and its corresponding jump size. In ruin theory, the classical Cram\'er-Lundberg model assumes that claims arrive independently according to a Poisson process. In contrast, our model employs an MIPP to allow for clustered arrivals, reflecting real-world scenarios, such as catastrophic events. Under this new framework, we derive the corresponding scale function in closed form, facilitating accurate calculations of the probability of ruin in the presence of clustered claims. These results improve the modeling of extreme risks and have practical implications for insurance solvency assessments, reinsurance pricing, and capital reserve estimation.

math.PR↗

arXiv2024

Two-fund separation under hyperbolically distributed returns and concave utility functions

Portfolio selection problems that optimize expected utility are usually difficult to solve. If the number of assets in the portfolio is large, such expected utility maximization problems become even harder to solve numerically. Therefore, analytical expressions for optimal portfolios are always preferred. In our work, we study portfolio optimization problems under the expected utility criterion for a wide range of utility functions, assuming return vectors follow hyperbolic distributions. Our main result demonstrates that under this setup, the two-fund monetary separation holds. Specifically, an individual with any utility function from this broad class will always choose to hold the same portfolio of risky assets, only adjusting the mix between this portfolio and a riskless asset based on their initial wealth and the specific utility function used for decision making. We provide explicit expressions for this mutual fund of risky assets. As a result, in our economic model, an individual's optimal portfolio is expressed in closed form as a linear combination of the riskless asset and the mutual fund of risky assets. Additionally, we discuss expected utility maximization problems under exponential utility functions over any domain of the portfolio set. In this part of our work, we show that the optimal portfolio in any given convex domain of the portfolio set either lies on the boundary of the domain or is the unique globally optimal portfolio within the entire domain.

q-fin.PM↗

arXiv2024

Weak convergence implies convergence in mean within GGC

We prove that weak convergence within generalized gamma convolution (GGC) distributions implies convergence in the mean value. We use this fact to show the robustness of the expected utility maximizing optimal portfolio under exponential utility function when return vectors are modelled by hyperbolic distributions.

q-fin.MF↗

arXiv2023

Pricing basket options with the first three moments of the basket: log-normal models and beyond

Options on baskets (linear combinations) of assets are notoriously challenging to price using even the simplest log-normal continuous-time stochastic models for the individual assets. The paper [5] gives a closed form approximation formula for pricing basket options with potentially negative portfolio weights under log-normal models by moment matching. This approximation formula is conceptually simple, methodologically sound, and turns out to be highly accurate. However it involves solving a system of nonlinear equations which usually produces multiple solutions and which is sensitive to the selection of initial values in the numerical procedures, making the method computationally challenging. In the current paper, we take the moment-matching methodology in [5] a step further by obtaining a closed form solution for this non-linear system of equations, by identifying a unary cubic equation based solely on the basket's skewness, which parametrizes all model parameters, and we use it to express the approximation formula as an explicit function of the mean, variance, and skewness of the basket. Numerical comparisons with the baskets considered in [5] show a very high level of agreement, and thus of accuracy relative to the true basket option price.

q-fin.PR↗

arXiv2022

Exponential utility maximization in small/large financial markets

Obtaining utility maximizing optimal portfolios in closed form is a challenging issue when the return vector follows a more general distribution than the normal one. In this note, we give closed form expressions, in markets based on finitely many assets, for optimal portfolios that maximize the expected exponential utility when the return vector follows normal mean-variance mixture models. We then consider large financial markets based on normal mean-variance mixture models also and show that, under exponential utility, the optimal utilities based on small markets converge to the optimal utility in the large financial market. This result shows, in particular, that to reach optimal utility level investors need to diversify their portfolios to include infinitely many assets into their portfolio and with portfolios based on any set of only finitely many assets, they never be able to reach optimum level of utility. In this paper, we also consider portfolio optimization problems with more general class of utility functions and provide an easy-to-implement numerical procedure for locating optimal portfolios. Especially, our approach in this part of the paper reduces a high dimensional problem in locating optimal portfolio into a three dimensional problem for a general class of utility functions.

q-fin.PM↗

arXiv2022

Efficient frontiers for portfolios under SSD and law-invariant risk measures with hyperbolic return distributions

In the classical Markowitz mean variance framework, risk is measured by variance, and the portfolios on the efficient frontier can be derived in closed form using standard optimization methods. For broader mean risk formulations, however, obtaining closed form optimal portfolios is typically difficult. In this work, we derive explicit expressions for frontier portfolios corresponding to arbitrary law-invariant convex risk measures, assuming that the return vector follows a normal mean variance mixture distribution. Our approach first establishes stochastic dominance relations within the family of normal mean variance mixture models, and then leverages these relations to derive closed-form representations of frontier portfolios. The central finding demonstrates that when asset returns are described by a normal mean variance mixture models the associated mean risk efficient frontier can be obtained by solving a Markowitz mean variance problem for a suitably transformed return vector. The paper also extends the CAPM framework to the context of a normal mean variance mixture distribution.

q-fin.MF↗

arXiv2021

Portfolio analysis with mean-CVaR and mean-CVaR-skewness criteria based on mean-variance mixture models

The paper Zhao et al. (2015) shows that mean-CVaR-skewness portfolio optimization problems based on asymetric Laplace (AL) distributions can be transformed into quadratic optimization problems under which closed form solutions can be found. In this note, we show that such result also holds for mean-risk-skewness portfolio optimization problems when the underlying distribution is a larger class of normal mean-variance mixture (NMVM) models than the class of AL distributions. We then study the value at risk (VaR) and conditional value at risk (CVaR) risk measures on portfolios of returns with NMVM distributions. They have closed form expressions for portfolios of normal and more generally elliptically distributed returns as discussed in Rockafellar & Uryasev (2000) and in Landsman & Valdez (2003). When the returns have general NMVM distributions, these risk measures do not give closed form expressions. In this note, we give approximate closed form expressions for VaR and CVaR of portfolios of returns with NMVM distributions. Numerical tests show that our closed form formulas give accurate values for VaR and CVaR and shortens the computational time for portfolio optimization problems associated with VaR and CVaR considerably.

q-fin.PM↗

arXiv2021

A note on closed-form spread option valuation under log-normal models

In the papers Carmona and Durrleman [7] and Bjerksund and Stensland [1], closed form approximations for spread call option prices were studied under the log normal models. In this paper, we give an alternative closed form formula for the price of spread call options under the log-normal models also. Our formula can be seen as a generalization of the closed-form formula presented in Bjerksund and Stensland [1] as their formula can be obtained by selecting special parameter values to our formula. Numerical tests show that our formula performs better for certain range of model parameters than the closed-form formula presented in Bjerksund and Stensland [1].

q-fin.MF↗

arXiv2021

Moment Matching Method for Pricing Spread Options with Mean-Variance Mixture L\'evy Motions

The paper Borovkova et al. [4] uses moment matching method to obtain closed form formulas for spread and basket call option prices under log normal models. In this note, we also use moment matching method to obtain semi-closed form formulas for the price of spread options under exponential L\'evy models with mean-variance mixture. Unlike the semi-closed form formulas in Caldana and Fusai [5], where spread prices were expressed by using Fourier inversion formula for general price dynamics, our formula expresses spread prices in terms of the mixing distribution. Numerical tests show that our formulas give accurate spread prices also

q-fin.PR↗

arXiv2017

Sticky processes, local and true martingales

We prove that for a so-called sticky process $S$ there exists an equivalent probability $Q$ and a $Q$-martingale $\tilde{S}$ that is arbitrarily close to $S$ in $L^p(Q)$ norm. For continuous $S$, $\tilde{S}$ can be chosen arbitrarily close to $S$ in supremum norm. In the case where $S$ is a local martingale we may choose $Q$ arbitrarily close to the original probability in the total variation norm. We provide examples to illustrate the power of our results and present applications in mathematical finance.

q-fin.MF↗

arXiv2014

Sticky continuous processes have consistent price systems

Under proportional transaction costs, a price process is said to have a consistent price system, if there is a semimartingale with an equivalent martingale measure that evolves within the bid-ask spread. We show that a continuous, multi-asset price process has a consistent price system, under arbitrarily small proportional transaction costs, if it satisfies a natural multi-dimensional generalization of the stickiness condition introduced by Guasoni [Math. Finance 16(3), 569-582 (2006)].

q-fin.PR↗

arXiv2013

On the Existence of Consistent Price Systems

We formulate a sufficient condition for the existence of a consistent price system (CPS), which is weaker than the conditional full support condition (CFS) introduced by Guasoni, Rasonyi, and Schachermayer [Ann. Appl. Probab., 18(2008), pp. 491-520] . We use the new condition to show the existence of CPSs for certain processes that fail to have the CFS property. In particular this condition gives sufficient conditions, under which a continuous function of a process with CFS admits a CPS, while the CFS property might be lost.

q-fin.GN↗

arXiv2009

On the Stickiness Property

In [2] the notion of stickiness for stochastic processes was introduced. It was also shown that stickiness implies absense of arbitrage in a market with proportional transaction costs. In this paper, we investigate the notion of stickiness further. In particular, we give examples of processes that are not semimartingales but are sticky.

q-fin.PR↗

arXiv2009

No arbitrage without semimartingales

We show that with suitable restrictions on allowable trading strategies, one has no arbitrage in settings where the traditional theory would admit arbitrage possibilities. In particular, price processes that are not semimartingales are possible in our setting, for example, fractional Brownian motion.

math.PR↗

arXiv2009

No Arbitrage Conditions For Simple Trading Strategies

Strict local martingales may admit arbitrage opportunities with respect to the class of simple trading strategies. (Since there is no possibility of using doubling strategies in this framework, the losses are not assumed to be bounded from below.) We show that for a class of non-negative strict local martingales, the strong Markov property implies the no arbitrage property with respect to the class of simple trading strategies. This result can be seen as a generalization of a similar result on three dimensional Bessel process in [3]. We also pro- vide no arbitrage conditions for stochastic processes within the class of simple trading strategies with shortsale restriction.

q-fin.PR↗