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

Long, Mingsi; Zhang, Hongzhong

doi:10.1016/j.spa.2018.08.005

Cited by 10 publications

(10 citation statements)

References 22 publications

(57 reference statements)

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“…In this case, we always discount at rate

r > 0

. Given that

z \mapsto U false(e^{z} false) - \frac{1}{Φ (r)} \frac{normald}{d z} U false(e^{z} false) = {normale}^{(1 - ρ) z} (\frac{1}{1 - ρ} - \frac{1}{normalΦ false(r false)}) - \frac{1}{1 - ρ}

is strictly increasing over

double-struckR

, we know from Long and Zhang (2019, Theorem 2.2) (see also footnote 4) that problem (5) is solved by a take‐profit (up‐crossing) selling strategy when the log price reaches the target

\frac{1}{1 - ρ} \log false(\frac{normalΦ false(r false)}{normalΦ false(r false) - 1 + ρ} false) > 0

.…”

Section: Model and Main Resultsmentioning

confidence: 99%

“…Our preliminary approach is inspired by the optimality of threshold‐type strategies (Long & Zhang, 2019). We first consider the benchmark case of no anxiety, that is,

q = 0

.…”

Section: Model and Main Resultsmentioning

confidence: 99%

“…In case

\bar{u} = - \infty

, the function

g (\cdot)

is nondecreasing over

double-struckR

, so for any fixed

y \in R

, the mapping

z \mapsto U false(e^{z} false) - \frac{1}{Λ (z - y)} \frac{normald}{d z} U false(e^{z} false) = {normale}^{(1 - ρ) y} g false(z - y false) - \frac{1}{1 - ρ}, \forall z \in R

is also nondecreasing. By Long and Zhang (2019, Theorem 2.2) and Lemma A.2 (see also footnote 4), we immediately know that (5) is also solved by a take‐profit selling strategy, regardless of the risk tolerance level

y

. In this case, the selling target log price is given by5

z^{★} false(y false) : = y + inf \{u > \bar{u} : g (u) > e^{- false(1 - ρ false) y} / (1 - ρ)\} .

Equivalently,

z^{★} false(y false)

is the largest root over

(y + \bar{u}, \infty)

, which is simply

double-struckR

when

\bar{u} = - \infty

, to equation

g false(z - y

…”

Section: Model and Main Resultsmentioning

confidence: 99%

“…To begin, we note that the optimal stopping of reward

U ({normale}^{X})

with a constant discounting rate

r + q

, is solved by a take‐profit selling strategy with target log price

\underset{̲}{b} = \frac{1}{1 - ρ} \log (\frac{normalΦ false(r + q false)}{normalΦ false(r + q false) - 1 + ρ}) > 0,

and the value function is given by

\begin{matrix} true \\ right \underset{̲}{v} (x) & left : = sup_{τ \in scriptT} E_{x} ([], {normale}^{- (r + q) τ}, U, (e^{X_{τ}}), {bold1}_{{τ < \infty}}) = 1_{false{x \leq \underset{̲}{b} false}} e^{normalΦ (r + q) (x - \underset{̲}{b})} U ({normale}^{\underset{̲}{b}}) + 1_{false{x > \underset{̲}{b} false}} U ({normale}^{x}) . \end{matrix}

Function

\underset{̲}{v} false(\cdot false)

is smooth everywhere off the set

{\underset{̲}{b}}

, and is continuously differentiable over

double-struckR

. These results follow from Long and Zhang (2019, Theorem 2.2) together with the increasing property of mapping (7) as we replace discount rate

r

r + q

.…”

Section: Model and Main Resultsmentioning

confidence: 99%

See 3 more Smart Citations

When to sell an asset amid anxiety about drawdowns

Rodosthenous

Zhang

2020

Mathematical Finance

Self Cite

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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.

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“…In this case, we always discount at rate

r > 0

. Given that

z \mapsto U false(e^{z} false) - \frac{1}{Φ (r)} \frac{normald}{d z} U false(e^{z} false) = {normale}^{(1 - ρ) z} (\frac{1}{1 - ρ} - \frac{1}{normalΦ false(r false)}) - \frac{1}{1 - ρ}

is strictly increasing over

double-struckR

, we know from Long and Zhang (2019, Theorem 2.2) (see also footnote 4) that problem (5) is solved by a take‐profit (up‐crossing) selling strategy when the log price reaches the target

\frac{1}{1 - ρ} \log false(\frac{normalΦ false(r false)}{normalΦ false(r false) - 1 + ρ} false) > 0

.…”

Section: Model and Main Resultsmentioning

confidence: 99%

“…Our preliminary approach is inspired by the optimality of threshold‐type strategies (Long & Zhang, 2019). We first consider the benchmark case of no anxiety, that is,

q = 0

.…”

Section: Model and Main Resultsmentioning

confidence: 99%

“…In case

\bar{u} = - \infty

, the function

g (\cdot)

is nondecreasing over

double-struckR

, so for any fixed

y \in R

, the mapping

z \mapsto U false(e^{z} false) - \frac{1}{Λ (z - y)} \frac{normald}{d z} U false(e^{z} false) = {normale}^{(1 - ρ) y} g false(z - y false) - \frac{1}{1 - ρ}, \forall z \in R

y

. In this case, the selling target log price is given by5

z^{★} false(y false) : = y + inf \{u > \bar{u} : g (u) > e^{- false(1 - ρ false) y} / (1 - ρ)\} .

Equivalently,

z^{★} false(y false)

is the largest root over

(y + \bar{u}, \infty)

, which is simply

double-struckR

when

\bar{u} = - \infty

, to equation

g false(z - y

…”

Section: Model and Main Resultsmentioning

confidence: 99%

“…To begin, we note that the optimal stopping of reward

U ({normale}^{X})

with a constant discounting rate

r + q

, is solved by a take‐profit selling strategy with target log price

\underset{̲}{b} = \frac{1}{1 - ρ} \log (\frac{normalΦ false(r + q false)}{normalΦ false(r + q false) - 1 + ρ}) > 0,

and the value function is given by

\begin{matrix} true \\ right \underset{̲}{v} (x) & left : = sup_{τ \in scriptT} E_{x} ([], {normale}^{- (r + q) τ}, U, (e^{X_{τ}}), {bold1}_{{τ < \infty}}) = 1_{false{x \leq \underset{̲}{b} false}} e^{normalΦ (r + q) (x - \underset{̲}{b})} U ({normale}^{\underset{̲}{b}}) + 1_{false{x > \underset{̲}{b} false}} U ({normale}^{x}) . \end{matrix}

Function

\underset{̲}{v} false(\cdot false)

is smooth everywhere off the set

{\underset{̲}{b}}

, and is continuously differentiable over

double-struckR

. These results follow from Long and Zhang (2019, Theorem 2.2) together with the increasing property of mapping (7) as we replace discount rate

r

r + q

.…”

Section: Model and Main Resultsmentioning

confidence: 99%

See 2 more Smart Citations

When to sell an asset amid anxiety about drawdowns

Rodosthenous

Zhang

2020

Mathematical Finance

Self Cite

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show abstract

“…Several key studies have been performed on the application of scale functions in optimal stopping in a continuous observation setting (see e.g. Alili and Kyprianou [3], Avram et al [7], Long and Zhang [43], Rodosthenous and Zhang [51] and Surya [53,Chap. 6]).…”

Section: Related Literaturementioning

confidence: 99%

The Leland–Toft optimal capital structure model under Poisson observations

et al. 2020

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This paper revisits the optimal capital structure model with endogenous bankruptcy, first studied by Leland (J. Finance 49:1213–1252, 1994) and Leland and Toft (J. Finance 51:987–1019, 1996). Unlike in the standard case where shareholders continuously observe the asset value and bankruptcy is executed instantaneously without delay, the information of the asset value is assumed to be updated periodically at the jump times of an independent Poisson process. Under a spectrally negative Lévy model, we obtain the optimal bankruptcy strategy and the corresponding capital structure. A series of numerical studies provide an analysis of the sensitivity, with respect to the observation frequency, of the optimal strategies, optimal leverage and credit spreads.

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Perpetual American Options with Asset-Dependent Discounting

Al-Hadad,

Palmowski

2023

Appl Math Optim

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In this paper we consider the following optimal stopping problem $$\begin{aligned} V^{\omega }_{\text {A}}(s) = \sup _{\tau \in {\mathcal {T}}} {\mathbb {E}}_{s}[e^{-\int _0^\tau \omega (S_w) dw} g(S_\tau )], \end{aligned}$$ V A ω ( s ) = sup τ ∈ T E s [ e - ∫ 0 τ ω ( S w ) d w g ( S τ ) ] , where the process $$S_t$$ S t is a jump-diffusion process, $${\mathcal {T}}$$ T is a family of stopping times while g and $$\omega $$ ω are fixed payoff function and discount function, respectively. In a financial market context, if $$g(s)=(K-s)^+$$ g ( s ) = ( K - s ) + or $$g(s)=(s-K)^+$$ g ( s ) = ( s - K ) + and $${\mathbb {E}}$$ E is the expectation taken with respect to a martingale measure, $$V^{\omega }_{\text {A}}(s)$$ V A ω ( s ) describes the price of a perpetual American option with a discount rate depending on the value of the asset process $$S_t$$ S t . If $$\omega $$ 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 $$V^{\omega }_{\text {A}}(s)$$ V A ω ( s ) . This allows us to determine the stopping region as a certain interval and hence we are able to identify the form of $$V^{\omega }_{\text {A}}(s)$$ V A ω ( s ) . We also present a put-call symmetry for American options with asset-dependent discounting. In the case when $$S_t$$ S t is a spectrally negative geometric Lévy process we give exact expressions using the so-called omega scale functions introduced in [30]. We show 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 $${V}^{\omega }_{\text {A}}(s)$$ V A ω ( s ) .

show abstract

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

Cited by 10 publications

References 22 publications

When to sell an asset amid anxiety about drawdowns

When to sell an asset amid anxiety about drawdowns

The Leland–Toft optimal capital structure model under Poisson observations

Perpetual American Options with Asset-Dependent Discounting

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