Selling a Stock at the Ultimate Maximum

  title={Selling a Stock at the Ultimate Maximum},
  author={Jacques Du Toit and Goran Peskir},
  journal={Annals of Applied Probability},
Assuming that the stock price $Z=(Z_t)_{0\leq t\leq T}$ follows a geometric Brownian motion with drift $\mu\in\mathbb{R}$ and volatility $\sigma>0$, and letting $M_t=\max_{0\leq s\leq t}Z_s$ for $t\in[0,T]$, we consider the optimal prediction problems \[V_1=\inf_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{M_T}{Z_{\tau}}\biggr)\quadand\quad V_2=\sup_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{Z_{\tau}}{M_T}\biggr),\] where the infimum and supremum are taken over all stopping times $\tau$ of $Z$. We show… 

Figures from this paper

Optimal mean–variance selling strategies
Assuming that the stock price X follows a geometric Brownian motion with drift $$\mu \in \mathbb {R}$$μ∈R and volatility $$\sigma >0$$σ>0, and letting $$\mathsf {P}_{\!x}$$Px denote a probability
Quickest detection of a hidden target and extremal surfaces
Let $Z=(Z_t)_{t\ge0}$ be a regular diffusion process started at $0$, let $\ell$ be an independent random variable with a strictly increasing and continuous distribution function $F$, and let
Optimal selling time in stock market over a finite time horizon
AbstractIn this paper, we examine the best time to sell a stock at a price being as close as possible to its highest price over a finite time horizon [0, T], where the stock price is modelled by a
Detecting the Maximum of a Scalar Diffusion with Negative Drift
For the quadratic loss function and under mild conditions, it is proved that an optimal stopping time exists and is defined by: $\theta^*=T_0\wedge\inf\{t\geq 0;~X^*_t \geq \gamma(X_t)\},$ where the boundary $\gamma$ is explicitly characterized as the concatenation of the solutions of two equations.
A Recursive Algorithm for Selling at the Ultimate Maximum in Regime-Switching Models
We propose a recursive algorithm for the numerical computation of the optimal value function inft≤τ≤T𝔼sup0≤s≤TYs/YτFt$\inf _{t\le \tau \le T} \mathbb {E} \left [\sup _{0\le s\le T } Y_{s} / Y_{\tau
Time-Randomized Stopping Problems for a Family of Utility Functions
Some properties of V are obtained and solutions for the optimal strategies to follow are offered and a technique to build numerical approximations of stopping boundaries for the fixed terminal time optimal stopping problem is built.
A Unified “Bang-Bang” Principle with Respect to ${\ccR}$-Invariant Performance Benchmarks
This work provides a unified proof of all similar problems for Brownian motion considered in the existing literature an “bang-bang''-type optimal solution to the optimal selling time of a stock.
A General ‘Bang-Bang’ Principle for Predicting the Maximum of a Random Walk
  • P. Allaart
  • Mathematics
    Journal of Applied Probability
  • 2010
Let (B t )0≤t≤T be either a Bernoulli random walk or a Brownian motion with drift, and let M t := max{B s: 0 ≤ s ≤ t}, 0 ≤ t ≤ T. In this paper we solve the general optimal prediction problem
Optimal Prediction of the Last-Passage Time of a Transient Diffusion
We identify the integrable stopping time $\tau_*$ with minimal $L^1$-distance to the last-passage time $\gamma_z$ to a given level $z>0$, for an arbitrary non-negative time-homogeneous transient
Predicting the Supremum: Optimality of 'Stop at Once or Not at All'
Under fairly general conditions on the process X_t, it is shown that the optimal stopping time tau is of "bang-bang" form: it is either optimal to stop at time 0 or at time T.


Stopping Brownian Motion Without Anticipation as Close as Possible to Its Ultimate Maximum
Let $B=(B_t)_{0 \le t \le 1}$ be the standard Brownian motion started at 0, and let $S_t=$ $\max_{ 0 \le r \le t} B_r$ for $0 \le t \le 1$. Consider the optimal stopping problem $\displaystyle V_*=
On Conditional-Extremal Problems of the Quickest Detection of Nonpredictable Times of the Observable Brownian Motion
We consider nonpredictable stopping times $\theta=\inf\{t\le1 : B_t=\max_{0\le s\le1}B_s\}$, $g=\sup\{t\le1 : B_t=0\}$ for the Brownian motion $B=(B_t)_{0\le t\le1}$. The main results of the paper
On a Property of the Moment at Which Brownian Motion Attains Its Maximum and Some Optimal Stopping Problems
Let $B=(B_t)_{0\le t\le 1}$ be a standard Brownian motion and $\theta$ be the moment at which B attains its maximal value, i.e., $B_\theta=\max_{0\le t\le 1}B_t$. Denote by $({\cal F}^B_t)_{0\le t\le
A Change-of-Variable Formula with Local Time on Curves
AbstractLet $$X = (X_t)_{t \geq 0}$$ be a continuous semimartingale and let $$b: \mathbb{R}_+ \rightarrow \mathbb{R}$$ be a continuous function of bounded variation. Setting $$C = \{(t, x) \in
Predicting the Time of the Ultimate Maximum for Brownian Motion with Drift
where the infimum is taken over all stopping times τ of B . Reducing the optimal prediction problem to a parabolic free-boundary problem and making use of local time-space calculus techniques, we
Predicting the last zero of Brownian motion with drift
Given a standard Brownian motion with drift μ ∈ IR and letting g denote the last zero of before T, we consider the optimal prediction problem where the infimum is taken over all stopping times τ of B
Thou shalt buy and hold
An investor holding a stock needs to decide when to sell it over a given investment horizon. It is tempting to think that she should sell at the maximum price over the entire horizon, which is
On Reflecting Brownian Motion with Drift
Let B = (Bt)t≥0 be a standard Brownian motion started at zero, and let μ ∈ R be a given and fixed constant. Set B t = Bt+μt and S t = max 0≤s≤t B s for t ≥ 0 . Then the process: (x ∨ Sμ)−Bμ = ((x ∨ S
The trap of complacency in predicting the maximum.
„where t⁄ 2 [0;T) and the functions t 7! b1(t) and t 7! b2(t) are continuous on [t⁄;T] with b1(T)=0 and b2(T)=1=2„ . If „ > 0 then b1 is decreasing and b2 is increasing on [t⁄;T] with b1(t⁄) = b2(t⁄)