arXiv subjects

Zbigniew Palmowski

Publications and source records attributed to Zbigniew Palmowski.

At least 55 records · Page 3Linked to original sources

arXiv2021

Perpetual American options with asset-dependent discounting

In this paper we consider the following optimal stopping problem $$V^ω_{\rm A}(s) = \sup_{τ\in\mathcal{T}} \mathbb{E}_{s}[e^{-\int_0^τω(S_w) dw} g(S_τ)],$$ where the process $S_t$ is a jump-diffusion process, $\mathcal{T}$ is a family of stopping times while $g$ and $ω$ are fixed payoff function and discount function, respectively. In a financial market context, if $g(s)=(K-s)^+$ or $g(s)=(s-K)^+$ and $\mathbb{E}$ is the expectation taken with respect to a martingale measure, $V^ω_{\rm A}(s)$ describes the price of a perpetual American option with a discount rate depending on the value of the asset process $S_t$. If $ω$ is a constant, the above problem produces the standard case of pricing perpetual American options. In the first part of this paper we find sufficient conditions for the convexity of the value function $V^ω_{\rm A}(s)$. This allows us to determine the stopping region as a certain interval and hence we are able to identify the form of $V^ω_{\rm A}(s)$. We also prove a put-call symmetry for American options with asset-dependent discounting. In the case when $S_t$ is a geometric Lévy process we give exact expressions using the so-called omega scale functions introduced in Li and Palmowski (2018). We prove that the analysed value function satisfies the HJB equation and we give sufficient conditions for the smooth fit property as well. Finally, we present a few examples for which we obtain the analytical form of the value function $V^ω_{\rm A}(s)$.

q-fin.MF↗

arXiv2020

A dual risk model with additive and proportional gains: ruin probability and dividends

We consider a dual risk model with constant expense rate and i.i.d. exponentially distributed gains $C_i$ ($i=1,2,\dots$) that arrive according to a renewal process with general interarrival times. We add to this classical dual risk model the proportional gain feature, that is, if the surplus process just before the $i$th arrival is at level $u$, then for $a>0$ the capital jumps up to the level $(1+a)u+C_i$. The ruin probability and the distribution of the time to ruin are determined. We furthermore identify the value of discounted cumulative dividend payments, for the case of a Poisson arrival process of proportional gains. In the dividend calculations, we also consider a random perturbation of our basic risk process modeled by an independent Brownian motion with drift.

math.PR↗

arXiv2020

Distributional properties of fluid queues busy period and first passage times

In this paper we analyze the distributional properties of a busy period in an on-off fluid queue and the a first passage time in a fluid queue driven by a finite state Markov process. In particular, we show that in Anick-Mitra-Sondhi model the first passage time has a IFR distribution and the busy period has a DFR distribution.

math.PR↗

arXiv2020

Modeling social media contagion using Hawkes processes

The contagion dynamics can emerge in social networks when repeated activation is allowed. An interesting example of this phenomenon is retweet cascades where users allow to re-share content posted by other people with public accounts. To model this type of behaviour we use a Hawkes self-exciting process. To do it properly though one needs to calibrate model under consideration. The main goal of this paper is to construct moments method of estimation of this model. The key step is based on identifying of a generator of a Hawkes process. We perform numerical analysis on real data as well.

cs.SI↗

arXiv2020

Subexponential potential asymptotics with applications

Let $X_t^\sharp$ be a multivariate process of the form $X_t =Y_t - Z_t$, $X_0=x$, killed at some terminal time $T$, where $Y_t$ is a Markov process having only jumps of the length smaller than $δ$, and $Z_t$ is a compound Poisson process with jumps of the length bigger than $δ$ for some fixed $δ>0$. Under the assumptions that the summands in $Z_t$ are sub-exponential, we investigate the asymptotic behaviour of the potential function $u(x)= E^x \int_0^\infty \ell(X_s^\sharp)ds$. The case of heavy-tailed entries in $Z_t$ corresponds to the case of "big claims" in insurance models and is of practical interest. The main approach is based on fact that $u(x)$ satisfies a certain renewal equation.

math.PR↗

arXiv2020

Branching processes with immigration in atypical random environment

Motivated by a seminal paper of Kesten et al. (1975) we consider a branching process with a geometric offspring distribution with i.i.d. random environmental parameters $A_n$, $n\ge 1$ and size -1 immigration in each generation. In contrast to above mentioned paper we assume that the environment is long-tailed, that is that the distribution $F$ of $ξ_n := \log ((1-A_n)/A_n)$ is long-tailed. We prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of the distribution of the population size in the n-th generation which becomes even heavier with increase of n. More precisely, we prove that, for any n, the distribution tail $\mathbb{P}(Z_n > m)$ of the $n$-th population size $Z_n$ is asymptotically equivalent to $n\overline{F}(\log m)$ as $m$ grows. In this way we generalize Bhattacharya and Palmowski (2019) who proved this result in the case $n=1$ for regularly varying environment $F$ with parameter $α>1$. Further, for a subcritical branching process with subexponentially distributed $ξ_n$, we provide the asymptotics for the distribution tail $\mathbb{P}(Z_n>m)$ which are valid uniformly for all $n$, and also for the stationary tail distribution. Then we establish the "principle of a single atypical environment" which says that the main cause for the number of particles to be large is a presence of a single very small environmental parameter $A_k$.

math.PR↗

arXiv2020

Importance sampling for maxima on trees

We consider the distributional fixed-point equation: $$R \stackrel{\mathcal{D}}{=} Q \vee \left( \bigvee_{i=1}^N C_i R_i \right),$$ where the $\{R_i\}$ are i.i.d.~copies of $R$, independent of the vector $(Q, N, \{C_i\})$, where $N \in \mathbb{N}$, $Q, \{C_i\} \geq 0$ and $P(Q > 0) > 0$. By setting $W = \log R$, $X_i = \log C_i$, $Y = \log Q$ it is equivalent to the high-order Lindley equation $$W \stackrel{\mathcal{D}}{=} \max\left\{ Y, \, \max_{1 \leq i \leq N} (X_i + W_i) \right\}.$$ It is known that under Kesten assumptions, $$P(W > t) \sim H e^{-αt}, \qquad t \to \infty,$$ where $α>0$ solves the Cramér-Lundberg equation $E \left[ \sum_{j=1}^N C_i ^α\right] = E\left[ \sum_{i=1}^N e^{αX_i} \right] = 1$. The main goal of this paper is to provide an explicit representation for $P(W > t)$, which can be directly connected to the underlying weighted branching process where $W$ is constructed and that can be used to construct unbiased and strongly efficient estimators for all $t$. Furthermore, we show how this new representation can be directly analyzed using Alsmeyer's Markov renewal theorem, yielding an alternative representation for the constant $H$. We provide numerical examples illustrating the use of this new algorithm.

math.PR↗

arXiv2020

Note on simulation pricing of $π$-options

In this work, we adapt a Monte Carlo algorithm introduced by Broadie and Glasserman (1997) to price a $π$-option. This method is based on the simulated price tree that comes from discretization and replication of possible trajectories of the underlying asset's price. As a result this algorithm produces the lower and the upper bounds that converge to the true price with the increasing depth of the tree. Under specific parametrization, this $π$-option is related to relative maximum drawdown and can be used in the real-market environment to protect a portfolio against volatile and unexpected price drops. We also provide some numerical analysis.

q-fin.CP↗

arXiv2020

Yaglom limit for Stochastic Fluid Models

In this paper we provide the analysis of the limiting conditional distribution (Yaglom limit) for stochastic fluid models (SFMs), a key class of models in the theory of matrix-analytic methods. So far, transient and stationary analyses of the SFMs have been only considered in the literature. The limiting conditional distribution gives useful insights into what happens when the process has been evolving for a long time, given its busy period has not ended yet. We derive expressions for the Yaglom limit in terms of the singularity $s^*$ such that the key matrix of the SFM, ${\bfΨ}(s)$, is finite (exists) for all $s\geq s^*$ and infinite for $s<s^*$. We show the uniqueness of the Yaglom limit and illustrate the application of the theory with simple examples.

math.PR↗

arXiv2020

Phase-type approximations perturbed by a heavy-tailed component for the Gerber-Shiu function of risk processes with two-sided jumps

We consider in this paper a risk reserve process where the claims and gains arrive according to two independent Poisson processes. While the gain sizes are phase-type distributed, we assume instead that the claim sizes are phase-type perturbed by a heavy-tailed component; that is, the claim size distribution is formally chosen to be phase-type with large probability $1-ε$ and heavy-tailed with small probability $ε$. We analyze the seminal Gerber-Shiu function coding the joint distribution of the time to ruin, the surplus immediately before ruin, and the deficit at ruin. We derive its value as an expansion with respect to powers of $ε$ with known coefficients and we construct approximations from the first two terms of the aforementioned series. The main idea is based on the so-called fluid embedding that allows to put the considered risk process into the framework of spectrally negative Markov-additive processes and use its fluctuation theory developed in Ivanovs and Palmowski (2012).

math.PR↗

arXiv2020

Optimal valuation of American callable credit default swaps under drawdown of Lévy insurance risk process

This paper discusses the valuation of credit default swaps, where default is announced when the reference asset price has gone below certain level from the last record maximum, also known as the high-water mark or drawdown. We assume that the protection buyer pays premium at fixed rate when the asset price is above a pre-specified level and continuously pays whenever the price increases. This payment scheme is in favour of the buyer as she only pays the premium when the market is in good condition for the protection against financial downturn. Under this framework, we look at an embedded option which gives the issuer an opportunity to call back the contract to a new one with reduced premium payment rate and slightly lower default coverage subject to paying a certain cost. We assume that the buyer is risk neutral investor trying to maximize the expected monetary value of the option over a class of stopping time. We discuss optimal solution to the stopping problem when the source of uncertainty of the asset price is modelled by Lévy process with only downward jumps. Using recent development in excursion theory of Lévy process, the results are given explicitly in terms of scale function of the Lévy process. Furthermore, the value function of the stopping problem is shown to satisfy continuous and smooth pasting conditions regardless of regularity of the sample paths of the Lévy process. Optimality and uniqueness of the solution are established using martingale approach for drawdown process and convexity of the scale function under Esscher transform of measure. Some numerical examples are discussed to illustrate the main results.

q-fin.MF↗

arXiv2020

An application of dynamic programming to assign pressing tanks at wineries

This paper describes an application of dynamic programming to determine the optimal strategy for assigning grapes to pressing tanks in one of the largest Portuguese wineries. To date, linear programming has been employed to generate proposed solutions to analogous problems, but this approach lacks robustness and may, in fact, result in severe losses in cases of sudden changes, which frequently occur in weather-dependent wine factories. Hence, we endowed our model with stochasticity, thereby rendering it less vulnerable to such changes. Our analysis, which is based on real-world data, demonstrates that the proposed algorithm is highly efficient and, after calibration, can be used to support winery's decision-making. The solution proposed herein could also be applied in numerous other contexts where production processes rely on outside supplies.

math.OC↗

arXiv2020

Double continuation regions for American options under Poisson exercise opportunities

We consider the Lévy model of the perpetual American call and put options with a negative discount rate under Poisson observations. Similar to the continuous observation case as in De Donno et al. [24], the stopping region that characterizes the optimal stopping time is either a half-line or an interval. The objective of this paper is to obtain explicit expressions of the stopping and continuation regions and the value function, focusing on spectrally positive and negative cases. To this end, we compute the identities related to the first Poisson arrival time to an interval via the scale function and then apply those identities to the computation of the optimal strategies. We also discuss the convergence of the optimal solutions to those in the continuous observation case as the rate of observation increases to infinity. Numerical experiments are also provided.

math.OC↗

arXiv2020

The Leland-Toft optimal capital structure model under Poisson observations

We revisit the optimal capital structure model with endogenous bankruptcy first studied by Leland \cite{Leland94} and Leland and Toft \cite{Leland96}. Differently from the standard case, where shareholders observe continuously the asset value and bankruptcy is executed instantaneously without delay, we assume that the information of the asset value is updated only at intervals, modeled by the jump times of an independent Poisson process. Under the spectrally negative Lévy model, we obtain the optimal bankruptcy strategy and the corresponding capital structure. A series of numerical studies are given to analyze the sensitivity of observation frequency on the optimal solutions, the optimal leverage and the credit spreads.

q-fin.PR↗

arXiv2020

A Multiplicative Version of the Lindley Recursion

This paper presents an analysis of the stochastic recursion $W_{i+1} = [V_iW_i+Y_i]^+$ that can be interpreted as an autoregressive process of order 1, reflected at 0. We start our exposition by a discussion of the model's stability condition. Writing $Y_i=B_i-A_i$, for independent sequences of non-negative i.i.d.\ random variables $\{A_i\}_{i\in N_0}$ and $\{B_i\}_{i\in N_0}$, and assuming $\{V_i\}_{i\in N_0}$ is an i.i.d. sequence as well (independent of $\{A_i\}_{i\in N_0}$ and $\{B_i\}_{i\in N_0}$), we then consider three special cases: (i) $V_i$ attains negative values only and $B_i$ has a rational LST, (ii) $V_i$ equals a positive value $a$ with certain probability $p\in (0,1)$ and is negative otherwise, and both $A_i$ and $B_i$ have a rational LST, (iii) $V_i$ is uniformly distributed on $[0,1]$, and $A_i$ is exponentially distributed. In all three cases we derive transient and stationary results, where the transient results are in terms of the transform at a geometrically distributed epoch.

math.PR↗

arXiv2020

Optimal Dividends Paid in a Foreign Currency for a Lévy Insurance Risk Model

This paper considers an optimal dividend distribution problem for an insurance company where the dividends are paid in a foreign currency. In the absence of dividend payments, our risk process follows a spectrally negative Lévy process. We assume that the exchange rate is described by a an exponentially Lévy process, possibly containing the same risk sources like the surplus of the insurance company under consideration. The control mechanism chooses the amount of dividend payments. The objective is to maximise the expected dividend payments received until the time of ruin and a penalty payment at the time of ruin, which is an increasing function of the size of the shortfall at ruin. A complete solution is presented to the corresponding stochastic control problem. Via the corresponding Hamilton--Jacobi--Bellman equation we find the necessary and sufficient conditions for optimality of a single dividend barrier strategy. A number of numerical examples illustrate the theoretical analysis.

q-fin.MF↗

arXiv2019

Slower variation of the generation sizes induced by heavy-tailed environment for geometric branching

Motivated by seminal paper of Kozlov et al.(1975) we consider in this paper a branching process with a geometric offspring distribution parametrized by random success probability $A$ and immigration equals $1$ in each generation. In contrast to above mentioned article, we assume that environment is heavy-tailed, that is $\log A^{-1} (1-A)$ is regularly varying with a parameter $α>1$, that is that ${\bf P} \Big( \log A^{-1} (1-A) > x \Big) = x^{-α} L(x)$ for a slowly varying function $L$. We will prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of distribution of the population at $n$-th generation which gets even heavier with $n$ increasing. Precisely, in this work, we prove that asymptotic tail ${\bf P}(Z_l \ge m)$ of $l$-th population $Z_l$ is of order $ \Big(\log^{(l)} m \Big)^{-α} L \Big(\log^{(l)} m \Big)$ for large $m$, where $\log^{(l)} m = \log \ldots \log m$. The proof is mainly based on Tauberian theorem. Using this result we also analyze the asymptotic behaviour of the first passage time $T_n$ of the state $n \in \mathbb{Z}$ by the walker in a neighborhood random walk in random environment created by independent copies $(A_i : i \in \mathbb{Z})$ of $(0,1)$-valued random variable $A$. This version differs from the final version as it contains an alternative proof for the tail behavior for generation sizes which is not very sharp (lacks constant) but completely avoids arguments based on Tauberian theorem. This proof may be of an independent interest.

math.PR↗

arXiv2019

Double continuation regions for American and Swing options with negative discount rate in Lévy models

In this paper we study perpetual American call and put options in an exponential Lévy model. We consider a negative effective discount rate which arises in a number of financial applications including stock loans and real options, where the strike price can potentially grow at a higher rate than the original discount factor. We show that in this case a double continuation region arises and we identify the two critical prices. We also generalize this result to multiple stopping problems of Swing type, that is, when successive exercise opportunities are separated by i.i.d. random refraction times. We conduct an extensive numerical analysis for the Black-Scholes model and the jump-diffusion model with exponentially distributed jumps.

q-fin.MF↗