Beating the omega clock: An optimal stopping problem with random time-horizon under spectrally negative Lévy models

Rodosthenous, Neofytos; Zhang, Hongzhong

doi:10.1214/17-aap1322

Cited by 22 publications

(50 citation statements)

References 24 publications

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“…With anxiety, that is,

q > 0

, we consider the function

g (\cdot)

given by

\begin{matrix} true \\ right g (x) = e^{false(1 - ρ false) x} (() \frac{1}{1 - ρ} - \frac{1}{Λ (x)}) with normalΛ (x) : = \frac{d}{normald x} \log I^{false(r, q false)} (x), \end{matrix}

where

{scriptI}^{(r, q)} false(\cdot false)

is defined by

{scriptI}^{(r, q)} false(x false) : = \int_{0}^{\infty} {normale}^{- Φ (r + q) u} W^{(r)} false(u + x false) d u, \forall x \in R .

The function

g (\cdot)

is continuous everywhere with one possible discontinuity at 0. It is known from Rodosthenous and Zhang (2018, Lemma 4.2) that

Λ (\cdot)

is strictly decreasing over

R_{+}

, with limits

Λ (0 +) \leq Φ (r + q)

and

Λ (\infty) = Φ (r

…”

Section: Model and Main Resultsmentioning

confidence: 99%

“…Part of our analysis builds on the results obtained in Theorems 4.2 and 4.3 for the problem (5), which generalizes Rodosthenous and Zhang (2018, Theorems 2.4 and 2.5) to the class of CRRA utility functions. To be more precise, we show that the original optimal stopping problem (2) reduces to its simplified version (5) for specific ranges of parameter values and certain levels of asset price best performance.…”

Section: Introductionmentioning

confidence: 98%

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When to sell an asset amid anxiety about drawdowns

Rodosthenous

Zhang

2020

Mathematical Finance

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We consider risk‐averse investors with different levels of anxiety about asset price drawdowns. The latter is defined as the distance of the current price away from its best performance since inception. These drawdowns can increase either continuously or by jumps, and will contribute toward the investor's overall impatience when breaching the investor's private tolerance level. We investigate the unusual reactions of investors when aiming to sell an asset under such adverse market conditions. Mathematically, we study the optimal stopping of the utility of an asset sale with a random discounting that captures the investor's overall impatience. The random discounting is given by the cumulative amount of time spent by the drawdowns in an undesirable high region, fine‐tuned by the investor's personal tolerance and anxiety about drawdowns. We prove that in addition to the traditional take‐profit sales, the real‐life employed stop‐loss orders and trailing stops may become part of the optimal selling strategy, depending on different personal characteristics. This paper thus provides insights on the effect of anxiety and its distinction with traditional risk aversion on decision making.

show abstract

“…With anxiety, that is,

q > 0

, we consider the function

g (\cdot)

given by

\begin{matrix} true \\ right g (x) = e^{false(1 - ρ false) x} (() \frac{1}{1 - ρ} - \frac{1}{Λ (x)}) with normalΛ (x) : = \frac{d}{normald x} \log I^{false(r, q false)} (x), \end{matrix}

where

{scriptI}^{(r, q)} false(\cdot false)

is defined by

{scriptI}^{(r, q)} false(x false) : = \int_{0}^{\infty} {normale}^{- Φ (r + q) u} W^{(r)} false(u + x false) d u, \forall x \in R .

The function

g (\cdot)

is continuous everywhere with one possible discontinuity at 0. It is known from Rodosthenous and Zhang (2018, Lemma 4.2) that

Λ (\cdot)

is strictly decreasing over

R_{+}

, with limits

Λ (0 +) \leq Φ (r + q)

and

Λ (\infty) = Φ (r

…”

Section: Model and Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

When to sell an asset amid anxiety about drawdowns

Rodosthenous

Zhang

2020

Mathematical Finance

Self Cite

View full text Add to dashboard Cite

show abstract

“…Optimal stopping problems with a random discounting find their applications in many areas such as finance and applied probability (for instance, in problems driven by continuously time-changed Markov processes [4], or problems with random maturity [20]). Perpetual optimal stopping of time-homogeneous diffusions with a random discounting rate were studied in Dayanik [5], by exploiting Dynkin's concave characterizations of the excessive functions.…”

Section: Introductionmentioning

confidence: 99%

“…While this is a very promising approach within diffusion framework, it shows limitations when jumps present. For example, for perpetual American call options on exponential Lévy models under an occupation time type discounting [20], it is shown that there can be two disjoint components for both the continuation and the stopping regions, which appear alternately. Possible overshoots thus make it difficult to apply [5]'s approach directly.…”

Section: Introductionmentioning

confidence: 99%

On the optimality of threshold type strategies in single and recursive optimal stopping under Lévy models

Long

Zhang

2019

Stochastic Processes and their Applications

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In the spirit of Surya [22], we develop an average problem approach to prove the optimality of threshold type strategies for optimal stopping of Lévy models with a continuous additive functional (CAF) discounting. Under spectrally negative models, we specialize this in terms of conditions on the reward function and random discounting, where we present two examples of local time and occupation time discounting. We then apply this approach to recursive optimal stopping problems, and present simpler and neater proofs for a number of important results on qualitative properties of the optimal thresholds, which are only known under a few special cases [3,15,23]. . 1 By considering the Lévy process −X·, one can easily adjust the argument to incorporate the running minimum of X·.where M is a random variable whose law under P Xt is identical to the conditional law of sup [t,ζ] X s under P x given F t and {t < ζ}.Second, we identify v(·) as the expected payoff of the up-crossing strategyIn fact, for any z ∈ R, by conditioning,On the event {T + z < ζ}, the identitywhere M is a random variable whose law under P X T + z is identical to the conditional law of sup t∈[T + z ,ζ] X t under P x given F T + z and {T + z < ζ}, from which the last equality results. It follows from the tower property of conditional expectations that

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“…It turned out that the resulting value functions and optimal stopping boundaries in models with exponentially distributed time horizons independent of the underlying processes are analytically more tractable than those obtained in models with fixed time horizons. Other optimal stopping problems for exponentially distributed time horizons which are dependent of the underlying Lévy process were recently considered in Rodosthenous and Zhang (2018). Glattfelder et al (2011) suggested a new paradigm, the directional changes, that summarises the price dynamics in the financial market.…”

mentioning

confidence: 99%

On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes

Gapeev

Rodosthenous

Chinthalapati

2019

Risks

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We obtain closed-form expressions for the value of the joint Laplace transform of the running maximum and minimum of a diffusion-type process stopped at the first time at which the associated drawdown or drawup process hits a constant level before an independent exponential random time. It is assumed that the coefficients of the diffusion-type process are regular functions of the current values of its running maximum and minimum. The proof is based on the solution to the equivalent inhomogeneous ordinary differential boundary-value problem and the application of the normal-reflection conditions for the value function at the edges of the state space of the resulting three-dimensional Markov process. The result is related to the computation of probability characteristics of the take-profit and stop-loss values of a market trader during a given time period. Risks 2019, 7, 87 2 of 15The joint Laplace transform of the first time at which a Brownian motion with linear drift hits a given drawdown value and the running maximum stopped at the same time was computed by Taylor (1975). The joint distribution of the same random variables was obtained by Lehoczky (1977). The mean value and the density of the maximum drawdown of a Brownian motion with linear drift were explicitly derived by (Douady et al. 2000;Magdon-Ismail et al. 2004), respectively. More recently, Pospisil et al. (2009) computed the probability of the event that the drawdown of a one-dimensional diffusion reaches a fixed value occurs before the drawup of the same process reaches another fixed value. Mijatović and Pistorius (2012) obtained the distribution laws of the first-passage times of spectrally positive and negative Lévy processes over constant levels and derived explicit expressions for several related characteristics for the drawdowns and drawups in those models. An extensive overview of various probabilistic and practically applied aspects of drawdowns such as the speed of market crashes and others was recently provided in the monograph of Zhang (2018).The diffusion-type processes can be considered as immediate generalisations of the diffusion processes particularly arising in the so-called local volatility models introduced by Dupire (1997), where the local drift and diffusion coefficients depend only on the running value of the original process. Other generalisations of the original processes with diffusion coefficients depending on the running values of the initial processes and their running minima were constructed by Forde (2011) for given joint laws of the terminal level and supremum at an independent exponential time (see also Forde et al. 2013;Zhang 2014) for other important probability characteristics of processes of such type). The valuation functional equations for general functional path-dependent volatility models were derived in (Cont and Fournié 2013; Fournié 2010), who also considered the sensitivity analysis of path-dependent financial derivative securities. Henry-Labordère (2009) and Ren et al. (2007), among others, considered the o...

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Beating the omega clock: An optimal stopping problem with random time-horizon under spectrally negative Lévy models

Cited by 22 publications

References 24 publications

When to sell an asset amid anxiety about drawdowns

When to sell an asset amid anxiety about drawdowns

On the optimality of threshold type strategies in single and recursive optimal stopping under Lévy models

On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes

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