Figure 4 - uploaded by Zbigniew Palmowski
Content may be subject to copyright.
The value g(y, p, θ) for Cramér-Lundberg model with respect to the θ value for different levels of y (first) and p (second). Parameters: r = 0.01, µ = 0.05, β = 0.1, ρ = 2.5, a = 10, α = 100, c = 50, p = 0.51, y = 8.
Source publication
In this paper we consider some insurance policies related with drawdown and drawup events of log-returns for an underlying asset modeled by a spectrally negative geometric L\'evy process. We consider four contracts among which three were introduced in Zhang et al. (2013) for a geometric Brownian motion. The first one is an insurance contract where...
Context in source publication
Similar publications
In this paper, we analyse some equity-linked contracts that are related to drawdown and drawup events based on assets governed by a geometric spectrally negative L\'evy process. Drawdown and drawup refer to the differences between the historical maximum and minimum of the asset price and its current value, respectively. We consider four contracts....
Citations
... Therefore, the list of papers pricing contracts with drawdown or drawup feature is quite long; see, e.g. [11,21,26,32,33,34,35,37,40,41,44,45] and references therein. This paper is organised as follows. ...
... Such stopping times allow us to define an approximation of the value by (32) U M 0 = max(Z 0 , EZ τ M 1 ). As Z 0 is deterministic, it is sufficient to find EZ τ M 1 to be able to obtain the approximation of the option price. ...
In this paper, we adopt the least squares Monte Carlo (LSMC) method to price time-capped American options. The aforementioned cap can be an independent random variable or dependent on asset price at random time. We allow various time caps. In particular, we give an algorithm for pricing the American options capped by the first drawdown epoch. We focus on the geometric L\'evy market. We prove that our estimator converges to the true price as one takes the discretisation step tending to zero and the number of trajectories going to infinity.
... Thirdly, we study the random time η r and derive the joint Laplace transform of (θ r , θ r − η r ) through a probabilistic decomposition, generalizing the result for a Brownian motion in [6]. Lastly, using the our theoretical results, we consider a few insurance contracts with different features designed to hedge against the adverse events associated with the range process as in [22] for pricing of drawdown-insurance contract for the geometric Brownian motion and [17] for exponential SNLPs. ...
... From this perspective, one would expect that the fair premium for this contract should be lower than that of the traditional drawdown insurance. This can be verified by comparing the fair premium in Eq. (58) with the fair premium for the traditional drawdown insurance given in Theorem 2 of Palmowski and Tumilewicz [17], that is ...
... This coincides with the drawdown insurance with a drawup contingency studied by Palmowski and Tumilewicz [17]. From Figure 4(B), we observe that the above fair premium p * decreases with respect to the range level r, which makes sense asθ r occurs later as r increases. ...
... On the application side, Angoshtari et al. (2016) examined the optimal investment strategy that minimizes the probability of lifetime drawdown. Palmowski and Tumilewicz (2018) studied the drawdown insurance where the underlying asset is modeled by a geometric SNLP. Interested readers are referred to Landriault et al. (2015), Landriault et al. (2017), Palmowski and Surya (2020) and references therein for more discussions on this topic. ...
In recent years, there has been a significant amount of work dedicated to the study of the generalized drawdown process with its extensive applications in insurance and finance. While existing studies have primarily focused on analyzing the associated first passage times, which signal early warnings, the investigation of last passage times should not be overlooked. Last passage times involve knowledge of the future and can thus offer additional insights. This paper aims to fill this gap in the literature by studying the last passage times for the generalized drawdown process with an independent exponential killing and discussing their applications to insurance risk. Our analysis focuses on the Lévy insurance risk processes, for which we derive the Laplace transforms for these random times. Additionally, we obtain new results on the joint distribution of the duration of the drawdown and the surplus level at killing. As applications, we implement our results in the loss-carry-forward tax and dividend models and investigate the valuation of an European digital drawdown option. Detailed numerical examples are presented for illustrative purposes.
... A more general treatment of drawdown quantities the Lévy insurance risk model has been studied by Mijatovic et al. [22] and Baurdoux et al. [6]. Building on the aforementioned papers, Palmowski and Tumilewicz [23] studied fair valuation of insurance contracts against drawdown (and drawup) of log-returns of stock price modelled by a spectrally negative geometric Lévy process. Also, Avram et al. [3] studied the pricing of Russian options for the same model. ...
p>In this paper, we study the magnitude and the duration of deep drawdowns for the Lévy insurance risk model through the characterization of the Laplace transform of a related stopping time. Relying on a temporal approximation approach (e.g., Li et al. (2018)), the proposed methodology allows for a unified treatment of processes with bounded and unbounded variation paths whereas these two cases used to be treated separately. In particular, we extend the results of Landriault et al. (2017) and Surya (2019). We later analyze certain limiting cases of our main results where consistency with some known drawdown results in the literature will be shown.</p
... In this paper we follow Zhang et al. [21] and Palmowski and Tumilewicz [11]. Zhang et al. [21] considered the Black-Scholes model, in contrast to our more general, Lévy-type market. ...
... In Zhang et al. [21], and Palmowski and Tumilewicz [11] the insured amount and penalty fee are fixed and constant. In this paper, we allow these quantities to depend on level of drawdown at the maturity of the contract or at the stopping epoch. ...
... Example 1 (continued) The linear Brownian motion given in (2) is a continuous process and, therefore, D τ + D (a) = a. The paid reward is always equal to α(a) := α, which corresponds to the results for constant reward function in [11]. The value function equals: ...
In this paper, we analyse some equity-linked contracts that are related to drawdown and drawup events based on assets governed by a geometric spectrally negative L\'evy process. Drawdown and drawup refer to the differences between the historical maximum and minimum of the asset price and its current value, respectively. We consider four contracts. In the first contract, a protection buyer pays a premium with a constant intensity p until the drawdown of fixed size occurs. In return, he/she receives a certain insured amount at the drawdown epoch, which depends on the drawdown level at that moment. Next, the insurance contract may expire earlier if a certain fixed drawup event occurs prior to the fixed drawdown. The last two contracts are extensions of the previous ones but with an additional cancellable feature that allows the investor to terminate the contracts earlier. In these cases, a fee for early stopping depends on the drawdown level at the stopping epoch. In this work, we focus on two problems: calculating the fair premium p for basic contracts and finding the optimal stopping rule for the polices with a cancellable feature. To do this, we use a fluctuation theory of L\'evy processes and rely on a theory of optimal stopping.
... Such risk may be protected against using an insurance contract. In their recent works, Zhang et al. [21], Palmowski and Tumilewicz [19] discussed fair valuation and design of such insurance contract. ...
This paper presents some new results on Parisian ruin under Lévy insurance risk process, where ruin occurs when the process has gone below a fixed level from the last record maximum, also known as the high-water mark or drawdown, for a fixed consecutive periods of time. The law of ruin-time and the position at ruin is given in terms of their joint Laplace transforms. Identities are presented semi-explicitly in terms of the scale function and the law of the Lévy process. They are established using recent developments on fluctuation theory of drawdown of spectrally negative Lévy process. In contrast to the Parisian ruin of Lévy process below a fixed level, ruin under drawdown occurs in finite time with probability one.
... Such risk may be protected against using an insurance contract. In their recent works, Zhang et al. [21], Palmowski and Tumilewicz [19] discussed fair valuation and design of such insurance contract. ...
This paper presents some new results on Parisian ruin under Levy insurance risk process, where ruin occurs when the process has gone below a fixed level from the last record maximum, also known as the high-water mark or drawdown, for a fixed consecutive periods of time. The law of ruin-time and the position at ruin is given in terms of their joint Laplace transforms. Identities are presented semi-explicitly in terms of the scale function and the law of the Levy process. They are established using recent developments on fluctuation theory of drawdown of spectrally negative Levy process. In contrast to the Parisian ruin of Levy process below a fixed level, ruin under drawdown occurs in finite time with probability one.
... In this paper we follow Zhang et al. [24] and Palmowski and Tumilewicz [13]. Zhang et al. [24] considered the Black-Scholes model, in contrast to our more general, Lévy-type market. ...
... In Zhang et al. [24], and Palmowski and Tumilewicz [13] the insured amount and penalty fee are fixed and constant. In this paper, we allow these quantities to depend on level of drawdown at the maturity of the contract or at the stopping epoch. ...
... The linear Brownian motion given in (2) is a continuous process and, therefore, D τ + D (a) = a. The paid reward is always equal to α(a) := α, which corresponds to the results for constant reward function in [13]. The value function equals: ...
In this paper, we analyse some equity-linked contracts that are related to drawdown and drawup events based on assets governed by a geometric spectrally negative L\'evy process. Drawdown and drawup refer to the differences between the historical maximum and minimum of the asset price and its current value, respectively. We consider four contracts. In the first contract, a protection buyer pays a premium with a constant intensity p until the drawdown of fixed size occurs. In return, he/she receives a certain insured amount at the drawdown epoch, which depends on the drawdown level at that moment. Next, the insurance contract may expire earlier if a certain fixed drawup event occurs prior to the fixed drawdown. The last two contracts are extensions of the previous ones but with an additional cancellable feature that allows the investor to terminate the contracts earlier. In these cases, a fee for early stopping depends on the drawdown level at the stopping epoch. In this work, we focus on two problems: calculating the fair premium p for basic contracts and finding the optimal stopping rule for the polices with a cancellable feature. To do this, we use a fluctuation theory of L\'evy processes and rely on a theory of optimal stopping.
In this paper, inspired by the ideas of Parisian ruin and ultimate bankruptcy, we introduce two new stopping times for the (general) drawdown process, namely, the Parisian drawdown and ultimate drawdown under the exponential implementation delays. We provide quantitative analysis of their distributional properties of interest through the generalized scale functions, whose properties are examined as well. We then discuss their relationships with the existing results on exit times and occupation times as the application of our main results. Another application in the fair market valuation of drawdown insurance is also presented and illustrated in a numerical example.
