Global $C^{1}$ regularity of the value function in optimal stopping problems

Angelis, Tiziano De; Peškir, Goran

doi:10.1214/19-aap1517

Cited by 47 publications

(52 citation statements)

References 36 publications

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“…Remark 3.11. We could not infer smooth fit across the stopping boundary diectly from [10] because our underlying process is killed at points a, b. Instead we adapted the line of arguments in the aforementioned paper and used the particular characteristics of our optimal stopping problem.…”

Section: We Can Immediately Claim Thatmentioning

confidence: 99%

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Optimal Hedging of a Perpetual American Put with a Single Trade

Cai¹,

Angelis²,

Palczewski³

2021

SIAM J. Finan. Math.

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It is well-known that using delta hedging to hedge financial options is not feasible in practice. Traders often rely on discrete-time hedging strategies based on fixed trading times or fixed trading prices (i.e., trades only occur if the underlying asset's price reaches some predetermined values). Motivated by this insight and with the aim of obtaining explicit solutions, we consider the seller of a perpetual American put option who can hedge her portfolio once until the underlying stock price leaves a certain range of values (a, b). We determine optimal trading boundaries as functions of the initial stock holding, and an optimal hedging strategy for a bond/stock portfolio. Optimality here refers to the variance of the hedging error at the (random) time when the stock leaves the interval (a, b). Our study leads to analytical expressions for both the optimal boundaries and the optimal stock holding, which can be evaluated numerically with no effort.

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Section: We Can Immediately Claim Thatmentioning

confidence: 99%

“…(Continuity of optimal stopping times). This part of the proof is based on ideas from [10] (see also, e.g., [21]). Letτ *…”

Section: This Is Equivalent Tomentioning

confidence: 99%

Optimal Hedging of a Perpetual American Put with a Single Trade

Cai¹,

Angelis²,

Palczewski³

2021

SIAM J. Finan. Math.

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“…In particular, since -a.s. for any , by its definition (10), this implies that -a.s. as well for . This means that the boundary is regular for the interior of the stopping set in the sense of diffusions (see, e.g., [7]).…”

Section: Regularity Of the Value Function And Integral Equationsmentioning

confidence: 99%

“…It is therefore possible to prove (see, e.g., Corollary 6 in [7] and Proposition 5.2 in [5]) that for any (i.e. ) and any sequence converging to as , we have Now we can use this property of the optimal stopping time and some related ideas from [7] to establish regularity of the value function.…”

Section: Regularity Of the Value Function And Integral Equationsmentioning

confidence: 99%

Optimal stopping for the exponential of a Brownian bridge

Angelis

Milazzo

2020

J. Appl. Probab.

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We study the problem of stopping a Brownian bridge X in order to maximise the expected value of an exponential gain function. The problem was posed by Ernst and Shepp (2015), and was motivated by bond selling with non-negative prices.Due to the non-linear structure of the exponential gain, we cannot rely on methods used in the literature to find closed-form solutions to other problems involving the Brownian bridge. Instead, we must deal directly with a stopping problem for a time-inhomogeneous diffusion. We develop techniques based on pathwise properties of the Brownian bridge and martingale methods of optimal stopping theory, which allow us to find the optimal stopping rule and to show the regularity of the value function.

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“…More recently, Ernst et al [16] solved the optimal stopping problem for a two-dimensional diffusion process related to the optimal real-time detection of a drifting Brownian coordinate which is is equivalent to a free-boundary problem the associated partial differential operator of elliptic type. The important recent results in the area comprise the continuity of the optimal stopping boundaries in optimal stopping problems for two-dimensional diffusions proved by Peskir [44] and the global C 1 -regularity of the value function in two-dimensional optimal stopping problems studied by De Angelis and Peskir [9].…”

Section: Introductionmentioning

confidence: 99%

Discounted optimal stopping problems in continuous hidden Markov models

Gapeev

2021

Stochastics

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We study a two-dimensional discounted optimal stopping problem related to the pricing of perpetual commodity equities in a model of financial markets in which the behaviour of the underlying asset price follows a generalized geometric Brownian motion and the dynamics of the convenience yield are described by an unobservable continuous-time Markov chain with two states. It is shown that the optimal time of exercise is the first time at which the commodity spot price paid in return to the fixed coupon rate hits a lower stochastic boundary being a monotone function of the running value of the filtering estimate of the state of the chain. We rigorously prove that the optimal stopping boundary is regular for the stopping region relative to the resulting two-dimensional diffusion process and the value function is continuously differentiable with respect to the both variables. It is verified by means of a change-of-variable formula with local time on surfaces that the value function and the boundary are determined as a unique solution of the associated parabolic-type free-boundary problem. We also give a closed-form solution to the optimal stopping problem for the case of an observable Markov chain.

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Global $C^{1}$ regularity of the value function in optimal stopping problems

Cited by 47 publications

References 36 publications

Optimal Hedging of a Perpetual American Put with a Single Trade

Optimal Hedging of a Perpetual American Put with a Single Trade

Optimal stopping for the exponential of a Brownian bridge

Discounted optimal stopping problems in continuous hidden Markov models

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