# Perpetual American Cancellable Standard Options in Models with Last Passage Times
## 1 Introduction  
The main aim of this paper is to compute closed-form expressions for the value functions of the discounted optimal stopping problems:  
\[\overline{V}(x)=\sup_{\tau}E\big{[}e^{-\tau\tau}\,(L-X_{\tau})\,I(\tau\ <\ \theta)\big{]}\quad\text{and}\quad\overline{U}(x)\ =\ \sup_{\zeta}E\big{[}e^{-\tau\zeta}\,(X_{\zeta}-K)\,I(\zeta\ <\ \eta)\big{]} \tag{1}\]  
with some \(L,\ K\ >\ 0\) fixed, where \(I(\cdot)\) denotes the indicator function. Here, for precise formulation of the problems, we consider a probability space \((\Omega,\mathcal{F},P)\) with a standard Brownian motion \(B=(B_{t})_{t\geq 0}\). Here, the process \(X=(X_{t})_{t\geq 0}\) is given by:  
\[X_{t}\ =\ x\,\exp\Big{(}\big{(}r-\delta-\sigma^{2}/2\big{)}\,t+\sigma\,B_{t} \Big{)} \tag{2}\]  
which solves the stochastic differential equation:  
\[dX_{t}\ =\ (r-\delta)\,X_{t}\,dt+\sigma\,X_{t}\,dB_{t}\quad(X_{0}=x) \tag{3}\]  
where \(x\ >\ 0\) is fixed, and \(r\ >\ 0\), \(\delta\ >\ 0\), and \(\sigma\ >\ 0\) are some given constants. Assume that the process \(X\) describes the price of risky assets on a financial market, where \(r\) is the riskless interest rate, \(\delta\) is the dividend rate paid to the asset holders, and \(\sigma\) is the volatility rate. We also introduce the random times \(\theta\) and \(\eta\) by:  
\[\theta\ =\ \sup\{t\geq 0\,|\ X_{t}\geq h\}\quad\text{and}\quad\eta\ =\ \sup\{t\geq 0\,|\ X_{t}\leq g\} \tag{4}\]  
for some \(h\ >\ L\ >\ 0\) and \(0\ <\ g\ <\ K\) fixed, which are not stopping times with respect to the natural filtration \((\mathcal{F}_{t})_{t\geq 0}\) of the process \(X\), but they are honest times in the sense of Nikeghbali and Yor [1]. Assume that the process \(X\) describes the price of a risky asset ina financial market, where \(r\) is the riskless interest rate, \(\delta\) is the dividend rate paid to the asset holders, and \(\sigma\) is the volatility rate. Suppose that the suprema in (1) are taken over all stopping times \(\tau\) and \(\zeta\) with respect to the filtration \((\mathcal{F}_{t})_{t\geq 0}\), and the expectations there are taken with respect to the risk-neutral probability measure \(P\). In this view, the values \(\overline{V}\) and \(\overline{U}\) in (1) are the no-arbitrage prices of the perpetual American cancellable options in the Black-Merton-Scholes model (see, e.g., [2] ([Chapter VII, Section 3g])). Note that other perpetual American cancellable options with another game-type payoff structure were recently studied by [PERSON] [3] (see also references therein). Some extensive overviews of the perpetual American options in diffusion models of financial markets and other related results in the area are provided in [PERSON] [2] ([Chapter VIII; Section 2a]), [PERSON] and [PERSON] [4] ([Chapter VII; Section 25]), and [PERSON] [5] among others. The model studied here differs from models studied in existing works such as [PERSON] [6], [PERSON] and [PERSON] [7], [PERSON] and [PERSON] [8], [PERSON] et al. [9], and [PERSON] et al. [10], as neither the immersion hypothesis nor the density hypothesis is satisfied by the random times (or default times) \(\theta\) and \(\eta\), and the default intensity process simply does not exists in our setting (see, e.g., [PERSON] and [PERSON] [11]). We see clearly in (6) and (7) that, in the case of zero recovery, this leads to a modified discounting factors, which are no longer functions of the sum of the interest rate and the default intensity rate. In addition, the diversion from the immersion hypothesis leads to the appearance of an adjusted dividend rate. Finally, if we were to study the finite horizon problem from a point of view of the backward stochastic differential equations (or BSDEs) as in [9; 10], then it could be shown that the dynamics of the no-arbitrage (pre-default) price will no longer satisfy a linear reflected BSDE but rather a linear reflected generalised BSDE where the generalised driver is related to the local time of the underlying asset at \(h\) or \(g\).