arXiv subjects

Kristoffer Glover

Publications and source records attributed to Kristoffer Glover.

6 recordsLinked to original sources

arXiv2022

With or without replacement? Sampling uncertainty in Shepp's urn scheme

We introduce a variant of Shepp's classical urn problem in which the optimal stopper does not know whether sampling from the urn is done with or without replacement. By considering the problem's continuous-time analog, we provide bounds on the value function and in the case of a balanced urn (with an equal number of each ball type) an explicit solution is found. Surprisingly, the optimal strategy for the balanced urn is the same as in the classical urn problem.

math.PR↗

arXiv2020

Dynkin games with incomplete and asymmetric information

We study the value and the optimal strategies for a two-player zero-sum optimal stopping game with incomplete and asymmetric information. In our Bayesian set-up, the drift of the underlying diffusion process is unknown to one player (incomplete information feature), but known to the other one (asymmetric information feature). We formulate the problem and reduce it to a fully Markovian setup where the uninformed player optimises over stopping times and the informed one uses randomised stopping times in order to hide their informational advantage. Then we provide a general verification result which allows us to find the value of the game and players' optimal strategies by solving suitable quasi-variational inequalities with some non-standard constraints. Finally, we study an example with linear payoffs, in which an explicit solution of the corresponding quasi-variational inequalities can be obtained.

math.PR↗

arXiv2020

Optimally Stopping a Brownian Bridge with an Unknown Pinning Time: A Bayesian Approach

We consider the problem of optimally stopping a Brownian bridge with an unknown pinning time so as to maximise the value of the process upon stopping. Adopting a Bayesian approach, we assume the stopper has a general continuous prior and is allowed to update their belief about the value of the pinning time through sequential observations of the process. Uncertainty in the pinning time influences both the conditional dynamics of the process and the expected (random) horizon of the optimal stopping problem. We analyse certain gamma and beta distributed priors in detail. Remarkably, the optimal stopping problem in the gamma case becomes time homogeneous and is completely solvable in closed form. Moreover, in the beta case we find that the optimal stopping boundary takes on a square-root form, similar to the classical solution with a known pinning time.

math.PR↗

arXiv2019

Short Selling with Margin Risk and Recall Risk

Short sales are regarded as negative purchases in textbook asset pricing theory. In reality, however, the symmetry between purchases and short sales is broken by a variety of costs and risks peculiar to the latter. We formulate an optimal stopping model in which the decision to cover a short position is affected by two short sale-specific frictions---margin risk and recall risk. Margin risk refers to the fact that short sales are collateralised transactions, which means that short sellers may be forced to close out their positions involuntarily if they cannot fund margin calls. Recall risk refers to a peculiarity of the stock lending market, which permits lenders to recall borrowed stock at any time, once again triggering involuntary close-outs. We examine the effect of these frictions on the optimal close-out strategy and quantify the loss of value resulting from each. Our results show that realistic short selling constraints have a dramatic impact on the optimal behaviour of a short seller, and are responsible for a substantial loss of value relative to the first-best situation without them. This has implications for many familiar no-arbitrage identities, which are predicated on the assumption of unfettered short selling.

q-fin.MF↗

arXiv2013

Optimal Prediction of the Last-Passage Time of a Transient Diffusion

We identify the integrable stopping time $τ_*$ with minimal $L^1$-distance to the last-passage time $γ_z$ to a given level $z>0$, for an arbitrary non-negative time-homogeneous transient diffusion $X$. We demonstrate that $τ_*$ is in fact the first time that $X$ assumes a value outside a half-open interval $[0,r_*)$. The upper boundary $r_*>z$ of this interval is characterised either as the solution for a one-dimensional optimisation problem, or as part of the solution for a free-boundary problem. A number of concrete examples illustrate the result.

math.PR↗

arXiv2013

Three-dimensional Brownian motion and the golden ratio rule

Let $X=(X_t)_{t\ge0}$ be a transient diffusion process in $(0,\infty)$ with the diffusion coefficient $σ>0$ and the scale function $L$ such that $X_t\rightarrow\infty$ as $t\rightarrow \infty$, let $I_t$ denote its running minimum for $t\ge0$, and let $θ$ denote the time of its ultimate minimum $I_{\infty}$. Setting $c(i,x)=1-2L(x)/L(i)$ we show that the stopping time \[τ_*=\inf\{t\ge0\vert X_t\ge f_*(I_t)\}\] minimizes $\mathsf{E}(\vertθ-τ\vert-θ)$ over all stopping times $τ$ of $X$ (with finite mean) where the optimal boundary $f_*$ can be characterized as the minimal solution to \[f'(i)=-\frac{σ^2(f(i))L'(f(i))}{c(i,f(i))[L(f(i))-L(i)]}\int_i^{f(i)}\frac{c_i'(i,y)[L(y) -L(i)]}{σ^2(y)L'(y)}\,dy\] staying strictly above the curve $h(i)=L^{-1}(L(i)/2)$ for $i>0$. In particular, when $X$ is the radial part of three-dimensional Brownian motion, we find that \[τ_ *=\inf\biggl\{t\ge0\Big\vert\frac{X_t-I_t}{I_t}\geφ\biggr\},\] where $φ=(1+\sqrt{5})/2=1.61\ldots$ is the golden ratio. The derived results are applied to problems of optimal trading in the presence of bubbles where we show that the golden ratio rule offers a rigorous optimality argument for the choice of the well-known golden retracement in technical analysis of asset prices.

math.PR↗