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The p * values for drawdown contract for the linear Brownian motion and various reward values α. Parameters: r = 0.01, µ = 0.03, σ = 0.4, a = 10.
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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....
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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. ...
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.
... Following [14,17,21], default is announced as soon as the underlying Lévy process has gone below a fixed level b > 0 from its last record maximum, ...
... , t ≥ 0} is supermartingale. Hence, by Lemma 7 of Palmowski and Tumilewicz [14], positivity of V b (y), and (iii) of Proposition 4.3 we have for any stopping time θ, ...
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.
... Such risk may be protected against using an insurance. In their recent works, Zhang et al. [18] and Palmowski and Tumilewicz [11] discussed fair valuation and design of such insurance. Analysis of the Parisian type of default with reference to the last record maximum is given in Surya [14]. ...
... Following [11,14,18], default is announced as soon as the underlying Lévy process has gone below a fixed level b > 0 from its last record maximum, ...
... , t ≥ 0} is supermartingale. Hence, by Lemma 7 of Palmowski and Tumilewicz [11], positivity of V b (y), and (iii) of Proposition 4.3 we have for any stopping time θ, ...
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 L\'evy process with only downward jumps. Using recent development in excursion theory of L\'evy process, the results are given explicitly in terms of scale function of the L\'evy 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 L\'evy 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.
As highlighted by the recent market turmoil following COVID‐19, markets can experience significant retracements or drawdowns. While these recent market moves have definitely been large, significant drawdowns have been around since the start of financial markets. Various risk metrics such as Value at Risk and volatility are used to describe risk. The intuitive drawdown risk measure, which is often used in practice alongside the above metrics, is receiving more and more academic attention. In this article we provide a systematic review of the literature on the drawdown risk measures. We describe two different methodologies for calculating drawdowns and analyze drawdown based risk measures used in risk management, portfolio construction and optimization. Finally we discuss the statistical properties related to drawdowns. Based on the research done so far, we identify several areas for further research.
W artykule rozważamy ubezpieczenie zabezpieczające inwestora przed dużym spadkiem wartości aktywa poniżej jego maksymalnej wartości o pewną określoną wartość. Opisane zachowanie wartości bazowego instrumentu może powodować olbrzymie straty udziałowcom, a w związku z tym, jest równocześnie sytuacją, przed którą należy się zabezpieczyć.
W pracy modelujemy wartości aktywa bazowego poprzez geometryczny ruch Browna ze skokami typu fazowego. Definiujemy proces spadku jako różnicę pomiędzy supremum tego procesu a jego obecną wartością. Dla powyższego modelu konstruujemy ubezpieczenie zabezpieczające inwestora przed dużym spadkiem. W kontrakcie tym, firma ubezpieczeniowa zobowiązuje się wypłacić wynagrodzenie, określone jako funkcja od wielkości spadku, w momencie, gdy zaobserwujemy spadek o co najmniej ustaloną z góry wartość. W zamian, ubezpieczyciel otrzymuje składkę płatną w sposób ciągły o stałej intensywności aż do momentu wypłaty uzgodnionego wynagrodzenia.
Praca opiera się na artykule Palmowskiego i Tumilewicz (2018), w którym autorzy przedstawili ogólny model matematyczny ubezpieczeń zabezpieczających przeciwko spadkowi cen aktywów w modelu typu Lévy’ego. Nasze rozważania dotyczą aplikacyjnego zastosowania tych wyników dla procesów cen akcji ze skokami modelowanymi przez pewne rozkłady fazowe.
