Watermark options

Abstract

We consider a new family of derivatives whose payoffs become strictly positive when the price of their underlying asset falls relative to its historical maximum. We derive the solution to the discretionary stopping problems arising in the context of pricing their perpetual American versions by means of an explicit construction of their value functions. In particular, we fully characterise the free-boundary functions that provide the optimal stopping times of these genuinely two-dimensional problems as the unique solutions to highly nonlinear first order ODEs that have the characteristics of a separatrix. The asymptotic growth of these free-boundary functions can take qualitatively different forms depending on parameter values, which is an interesting new feature.

When to sell an asset amid anxiety about drawdowns

“…Different from the majority of existing literature on optimal stopping problems involving the maximum process, we do have a boundary condition at for the ODE, which helps us to obtain a unique solution as the candidate down‐crossing sell order. When such a boundary condition is not available, an appropriate (unique) candidate must be chosen from the set of infinitely many solutions of the ODE, by relying on various different methods; for example, using the transversality condition (see Guo & Zervos, 2010; Rodosthenous & Zervos, 2017, among others), or the maximality principle from Peskir (1998) (see Obłój, 2007 for diffusion models, Kyprianou & Ott, 2014 for Lévy models, among others).…”

“…In all aforementioned studies (Imkeller & Rogers, 2014; Leung & Zhang, 2019; Zhang, 2001, 2018), the use of trailing stops is exogenously imposed. On the other hand, Russian options and their extensions (see, e.g., Rodosthenous & Zervos, 2017) involve the use of trailing stops, but as in regret theory, their objective is to protect against drawdowns, and is not concerned about the utility realized from an asset sale. The purpose of this paper is thus to provide a framework that rationalizes the use of these types of orders from the perspective of selling an asset.…”

Discounted Optimal Stopping Problems for Maxima of Geometric Brownian Motions With Switching Payoffs

“…Moreover, since the right-hand sides of the corresponding normal form of the system of first-order nonlinear ordinary differential equations in (3.14) with (3.20)-(3.21) and (3.22) are (locally) Lipschitz in s, for s > K 1−i /L 1−i and every i = 0, 1, one can deduce by means of Gronwall's inequality that the functions g j,l (s), l ∈ N, are continuous, so that the functions g * j (s), j = 0, 1, are continuous too. The corresponding maximal admissible solutions of first-order nonlinear ordinary differential equations and the associated maximality principle for solutions of optimal stopping problems which is equivalent to the superharmonic characterisation of the payoff functions were established in [35] and further developed in [22], [34], [24], [16], [6], [25], [37]- [38], [21], [33], [30], [7], [17]- [19], and [40] among other subsequent papers (see also [39; Chapter I; Chapter V, Section 17] for other references).…”

“…For the case of non-switching payoffs with L 0 = L 1 and K 0 = K 1 , the problems of (1.1) and (1.2) were solved by Pedersen [34], Guo and Shepp [24], and Beibel and Lerche [8], for models with geometric Brownian motions, and in [16], for a geometric model driven by a Brownian motion and a compound Poisson process with exponential jumps. More recently, Guo and Zervos [25] and Rodosthenous and Zervos [40] derived solutions for discounted optimal stopping problems related to the pricing of perpetual American options with more general payoff functions depending on the current values of the process X and its running maximum. In the case of a Russian option with L 0 = L 1 and K i = 0, i = 0, 1, the problems of (1.1) and (1.2) were explicitly solved by Guo [23] for a model with geometric Brownian motions with switching coefficients.…”

Solving the dual Russian option problem by using change‐of‐measure arguments

“…Peskir () presented a solution to the Russian option problem for a geometric Brownian motion with a finite time horizon (see also Duistermaat, Kyprianou, and van Schaik () for a numeric algorithm of solving the corresponding free‐boundary problem and Ekström () for a study of asymptotic behavior of the optimal stopping boundary near the expiration). The problems of pricing of perpetual American lookback and other options with more complicated structure of payoffs depending on the running maxima of the underlying processes were studied in Baurdoux and Kyprianou (), Gapeev (), Guo and Zervos (), Ott (), Kyprianou and Ott (), and Rodosthenous and Zervos () among others (see also Gapeev () and Kitapbayev () for the finite time horizon American lookback options on the running maxima of geometric Brownian motions). Along with the article of Dubins, Shepp, and Shiryaev (), the papers of Shepp and Shiryaev (, ) also made a crucial contribution to the optimal stopping problems arising in the proofs of maximal inequalities for the continuous time processes further developed in Graversen and Peskir (,b) and Peskir () among others (see also Peskir and Shiryaev (); Chapter V for an extensive overview of the optimal stopping problems related to maximal inequalities).…”