## 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.

## Keywords

Optimal stopping Running maximum process Variational inequality Two-dimensional free-boundary problem Separatrix## Mathematics Subject Classification (2010)

49L20 60G40## JEL Classification

G13 C61## 1 Introduction

Put options are the most common financial derivatives that can be used by investors to hedge against asset price falls as well as by speculators betting on falling prices. In particular, out-of-the-money put options can yield strictly positive payoffs only if the price of their underlying asset falls below a percentage of its initial value. In a related spirit, equity default swaps (EDSs) pay out if the price of their underlying asset drops by more than a given percentage of its initial value (EDSs were introduced by J.P. Morgan London in 2003, and their pricing was studied by Medova and Smith [16]). Further derivatives whose payoffs depend on other quantifications of asset price falls include the European barrier and binary options studied by Carr [2] and Večeř [27], as well as the perpetual lookback American options with floating strike that were studied by Pedersen [20] and Dai [5].

In this paper, we consider a new class of derivatives whose payoffs depend on asset price falls relative to their underlying asset’s historical maximum price. Typically, a hedge fund manager’s performance fees are linked with the value of the fund exceeding a “high watermark”, which is an earlier maximum. We have therefore named this new class of derivatives “watermark” options. Deriving formulas for the risk-neutral pricing of their European-type versions is a cumbersome but standard exercise. On the other hand, the pricing of their American-type versions is a substantially harder problem, as expected. Here, we derive the complete solution to the optimal stopping problems associated with the pricing of their perpetual American versions.

Watermark options can be used for the same reasons as the existing options we have discussed above. In particular, they could be used to hedge against relative asset price falls as well as to speculate by betting on prices falling relatively to their historical maximum. For instance, they could be used by institutions that are constrained to hold investment-grade assets only and wish to trade products that have risk characteristics akin to the ones of speculative-grade assets (see Medova and Smith [16] for further discussion that is relevant to such an application). Furthermore, these options can provide useful risk-management instruments, particularly when faced with the possibility of an asset bubble burst. Indeed, the payoffs of watermark options increase as the running maximum process \(S\) increases and the price process \(X\) decreases. As a result, the more the asset price increases before dropping to a given level, the deeper they may be in the money.

Watermark options can also be of interest as hedging instruments to firms investing in real assets. To fix ideas, consider a firm that invests in a project producing a commodity whose price or demand is modelled by the process \(X\). The firm’s future revenue depends on the stochastic evolution of the economic indicator \(X\), which can collapse for reasons such as extreme changes in the global economic environment (see e.g. the recent slump in commodity prices) and/or reasons associated with the emergence of disruptive new technologies (see e.g. the fate of DVDs or firms such as Blackberry or NOKIA). In principle, such a firm could diversify risk by going long in watermark options.

The applications discussed above justify the introduction of watermark options as derivative structures. These options can also provide alternatives to existing derivatives that can better fit a range of investors’ risk preferences. For instance, the version associated with (1.5) effectively identifies with the Russian option (see Remark 3.5). It is straightforward to check that if \(s \geq1\), then the price of the option is increasing as the parameter \(b\) increases, ceteris paribus. In this case, the watermark option is cheaper than the corresponding Russian option if \(b< a\). On the other hand, increasing values of the strike price \(K\) result in ever lower option prices. We have not attempted any further analysis in this direction because this involves rather lengthy calculations and is beyond the scope of this paper.

The parameters \(a, b > 0\) and \(K \geq0\) can be used to fine-tune different risk characteristics. For instance, the choice of the relative value \(b/a\) can reflect the weight assigned to the underlying asset’s historical best performance relative to the asset’s current value. In particular, it is worth noting that larger (resp. smaller) values of \(b/a\) attenuate (resp. magnify) the payoff’s volatility that is due to changes of the underlying asset’s price. In the context of the problem with value function given by (1.5), the choice of \(a\), \(b\) can be used to factor in a power utility of the payoff received by the option’s holder. Indeed, if we set \(a = \tilde{a} q\) and \(b = \tilde{b} q\), then \(S_{\tau}^{b} / X_{\tau}^{a} = ( S_{\tau}^{\tilde{b}} / X_{\tau}^{\tilde{a}} )^{q}\) is the CRRA utility with risk-aversion parameter \(1-q\) of the payoff \(S_{\tau}^{\tilde{b}} / X_{\tau}^{\tilde{a}}\) received by the option’s holder if the option is exercised at time \(\tau\).

From a modelling point of view, the use of geometric Brownian motion as an asset price process, which is standard in the mathematical finance literature, is an approximation that is largely justified by its tractability. In fact, such a process is not an appropriate model for an asset price that may be traded as a bubble. In view of the applications we have discussed above, the pricing of watermark options when the underlying asset’s price process is modelled by diffusions associated with local volatility models that have been considered in the context of financial bubbles (see e.g. Cox and Hobson [3]) presents an interesting problem for future research.

The Russian options introduced and studied by Shepp and Shiryaev [25, 26] are the special cases that arise if \(a=0\) and \(b=1\) in (1.5). In fact, the value function given by (1.5) identifies with the value function of a Russian option for any \(a, b > 0\) (see Remark 3.5). The lookback American options with floating strike that were studied by Pedersen [20] and Dai [5] are the special cases that arise for the choices \(a=b=1\) and \(a=b=K=1\) in (1.4), respectively (see also Remark 3.3). Other closely related problems that have been studied in the literature include the well-known perpetual American put options (\(a = 1\), \(b = 0\) in (1.4)), which were solved by McKean [15], the lookback American options studied by Guo and Shepp [10] and Pedersen [20] (\(a=0\), \(b=1\) in (2.1)), and the \(\pi\)-options introduced and studied by Guo and Zervos [11] (\(a < 0\) and \(b>0\) in (1.3)).

Further works on optimal stopping problems involving a one-dimensional diffusion and its running maximum (or minimum) include Jacka [13], Dubins et al. [8], Peskir [21], Graversen and Peskir [9], Dai and Kwok [6, 7], Hobson [12], Cox et al. [4], Alvarez and Matomäki [1], and references therein. Furthermore, Peskir [22] solves an optimal stopping problem involving a one-dimensional diffusion, its running maximum as well as its running minimum. Papers on optimal stopping problems with an infinite time horizon involving spectrally negative Lévy processes and their running maximum (or minimum) include Ott [18, 19], Kyprianou and Ott [14], and references therein.

In Sect. 2, we solve the optimal stopping problem whose value function is given by (1.3) for \(a =1\) and \(b \in(0, \infty) \setminus\{ 1 \}\). To this end, we construct an appropriately smooth solution to the problem’s variational inequality that satisfies the so-called transversality condition, which is a folklore method. In particular, we fully determine the free-boundary function separating the “waiting” region from the “stopping” region as the unique solution to a first order ODE that has the characteristics of a separatrix. It turns out that this free-boundary function conforms with the maximality principle introduced by Peskir [21]: it is the maximal solution to the ODE under consideration that does not intersect the diagonal part of the state space’s boundary. The asymptotic growth of this free-boundary function is notably different in each of the cases \(1< b\) and \(1>b\), which is a surprising result (see Remark 2.6).

In Sect. 3, we use an appropriate change of probability measure to solve the optimal stopping problem whose value function is given by (1.4) for \(a =1\) and \(b \in(0, \infty)\setminus\{ 1 \}\) by reducing it to the problem studied in Sect. 2. We also outline how the optimal stopping problem defined by (1.1)–(1.3) for \(a = b =1\) reduces to the problem given by (1.1), (1.2) and (1.4) for \(a = b =1\), which is the one arising in the pricing of a perpetual American lookback option with floating strike that has been solved by Pedersen [20] and Dai [5] (see Remark 3.3). We then explain how a simple re-parametrisation reduces the apparently more general optimal stopping problems defined by (1.1)–(1.4) for any \(a>0\), \(b>0\) to the corresponding cases with \(a=1\), \(b>0\) (see Remark 3.4). Finally, we show that the optimal stopping problem defined by (1.1), (1.2) and (1.5) reduces to the one arising in the context of pricing a perpetual Russian option that has been solved by Shepp and Shiryaev [25, 26] (see Remark 3.5).

## 2 The solution to the main optimal stopping problem

We make the following assumption.

### Assumption 2.1

### Remark 2.2

We prove the following result in the Appendix.

### Lemma 2.3

*Consider the optimal stopping problem defined by* (1.1), (1.2) *and* (2.1). *If the problem data is such that either* \(m+1 > 0\) *or* \(n+1-p < 0\), *then* \(v \equiv\infty\).

### Lemma 2.4

*Suppose that the problem data satisfy Assumption*2.1.

*Given any*\(\delta> 0\),

*there exist points*\(s_{\circ}= s_{\circ}(\delta)\)

*and*\(s^{\circ}= s^{\circ}(\delta)\)

*satisfying*

*where*\(s_{\dagger}(\delta) \geq\delta\)

*is the unique solution to*(2.19),

*such that the following statements hold true for each*\(s_{*} \in[s_{\circ}, s^{\circ}]\):

*If*\(p \in(0,1)\),

*the ODE*(2.15)

*has a unique solution*

*satisfying*(2.18)

*that is a strictly increasing function such that*

*where*\(c = \frac{m+1}{mK} \in(0, \varGamma)\) (

*see Fig*. 2).

*If*\(p>1\),

*the ODE*(2.15)

*has a unique solution*

*satisfying*(2.18)

*that is a strictly increasing function such that*

*where*\(c = \bigl( \frac{(m+1) (p-n-1)}{(n+1) (p-m-1)} \bigr) ^{1/(n-m)} \in(0,1)\) (

*see Fig*. 3).

(III) *The corresponding functions* \(A\) *and* \(B\) *defined by* (2.13) *and* (2.14) *are both strictly positive*.

### Remark 2.5

Beyond the results given in the last lemma, we can prove the following:

(a) Given any \(s_{*} \in(s_{\dagger}, s_{\circ})\), there exist a point \(\hat{s} = \hat{s} (s_{*}) \in(0, \infty)\) and a function \(H(\cdot) := H(\cdot; s_{*}) : (0, \hat{s}) \rightarrow{\mathcal {D}}_{H}\) that satisfies the ODE (2.15) as well as (2.18). In particular, \(H\) is strictly increasing and \(\lim_{s \uparrow\hat{s}} H(s) = (\varGamma\hat{s}^{p}) \wedge\hat{s}\).

(b) Given any \(s_{*} > s^{\circ}\), there exists a function \(H(\cdot) := H(\cdot; s_{*}) : (0, \infty) \rightarrow{\mathcal {D}}_{H}\) which is strictly increasing and satisfies the ODE (2.15) as well as (2.18).

Any solution to (2.15) that is as in (a) does not identify with the actual free-boundary function \(H\) because the corresponding solution to the variational inequality (2.6) does not satisfy the boundary condition (2.7). On the other hand, any solution to (2.15) that is as in (b) also does not identify with the actual free-boundary function \(H\) because we can show that its asymptotic growth as \(s \rightarrow\infty\) is such that the corresponding solution \(w\) to the variational inequality (2.6) does not satisfy (2.24), and the transversality condition, which is captured by the limits on the right-hand side of (2.27), is also not satisfied. To keep the paper at a reasonable length, we do not expand on any of these issues that are not really needed for our main results thanks to the uniqueness of the value function.

### Remark 2.6

### Lemma 2.7

*Suppose that the problem data satisfy Assumption*2.1.

*Also*,

*consider any*\(s_{*} \in[s_{\circ}(\delta), s^{\circ}(\delta) ]\),

*where*\(s_{\circ}(\delta) \leq s^{\circ}(\delta)\)

*are as in Lemma*2.4

*for some*\(\delta> 0\),

*and let*\(H(\cdot) = H(\cdot; s_{*})\)

*be the corresponding solution to the ODE*(2.15)

*that satisfies*(2.18).

*The function*\(w\)

*defined by*(2.20)

*and*(2.21)

*is strictly positive*,

*satisfies the variational inequality*(2.6)

*outside the set*\(\{ (x,s) \in{\mathbb {R}}^{2} : s > 0\textit{ and }x = H(s) \}\)

*as well as the boundary condition*(2.7),

*and is such that*(2.22)

*and*(2.23)

*hold true*.

*Furthermore*,

*given any*\(s>0\),

*there exists a constant*\(C = C (s) > 0\)

*such that*

*where*

We can now prove our main result.

### Theorem 2.8

*Consider the optimal stopping problem defined by* (1.1), (1.2) *and *(2.1), *and suppose that the problem data satisfy Assumption * 2.1. *The optimal stopping problem’s value function* \(v\) *identifies with the solution* \(w\) *to the variational inequality* (2.6) *with boundary condition* (2.7) *described in Lemma * 2.7, *and the first hitting time* \(\tau_{\mathcal {S}}\) *of the stopping region* \({\mathcal {S}}\), *which is defined as in* (2.8), *is optimal*. *In particular*, \(s_{\circ}(\delta) = s^{\circ}(\delta)\) *for all* \(\delta> 0\), *where* \(s_{\circ}\leq s^{\circ}\) *are as in Lemma * 2.4.

### Proof

## 3 Ramifications and connections with the perpetual American lookback with floating strike, and with Russian options

### Assumption 3.1

### Theorem 3.2

*Consider the optimal stopping problem defined by*(1.1), (1.2)

*and*(3.1)

*and suppose that the problem data satisfy Assumption*3.1.

*The problem’s value function is given by*

*and the first hitting time*\(\tau_{\mathcal {S}}\)

*of the stopping region*\({\mathcal {S}}\)

*is optimal*,

*where*\(v\)

*is the value function of the optimal stopping problem defined by*(1.1), (1.2)

*and*(2.1),

*given by Theorem*2.8,

*and*\({\mathcal {S}}\)

*is defined by*(2.11)

*with*\(\tilde{\mu}\), \(\tilde{r}\)

*defined by*(3.2)

*and the associated*\(\tilde{m}\), \(\tilde{n}\)

*in place of*\(\mu\), \(r\)

*and*\(m\), \(n\).

### Proof

### Remark 3.3

### Remark 3.4

### Remark 3.5

## Notes

### Acknowledgements

We are grateful to the Editor, the Associate Editor and two referees for their extensive comments and suggestions that significantly improved the paper.

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