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First, we give a closed-form formula for first passage time of a reflected Brownian motion with drift. This corrects a formula by Perry et al. (2004). Second, we show that the maximum before a fixed drawdown is exponentially distributed for any drawdown, if and only if the diffusion characteristic is constant. This complements the sufficient condition formulated by Lehoczky (1977). Third, we give an alternative proof for the fact that the maximum before a fixed drawdown is exponentially distributed for any spectrally negative Lévy process, a result due to Mijatović and Pistorius (2012). Our proof is similar, but simpler than Lehoczky (1977) or Landriault et al. (2017).
In Section 2, we compute the Laplace transform of the first hitting time of a fixed upper barrier for a reflected Brownian motion with drift. This expands on and corrects a result by Perry et al. (2004).
In Section 3, we show, by using an intrinsic delay differential equation, that for a diffusion process, the maximum before a fixed drawdown threshold is generically exponentially distributed, only if the diffusion characteristic is constant. This complements the sufficient condition formulated by Lehoczky (1977). By solving discrete delay differential equations, we further construct diffusions, where the exponential law only holds for specific drawdown sizes.
Section 4 uses Lehoczky (1977)’s argument to show that the maximum before a fixed drawdown threshold is exponentially distributed for any spectrally negative Lévy process, the parameter being the right-sided logarithmic derivative of the scale function. This yields an alternative proof to the original one in Mijatović and Pistorius (2012) and is also similar to the one in Landriault et al. (2017).
2. The First Hitting Time for a Reflected Brownian Motion with Drift
Let X be a reflected Brownian motion on , with drift and volatility . Then X can be written as
where is a standard Brownian motion, and is an inon-decreasing process, such that the induced random measure is supported on . Itô’s formula implies that for any satisfying , the process
is a martingale, where is the differential operator, defined by .1
For , we define the first hitting time:
Since, before reaching the boundary 0, the process cannot be distinguished from a Brownian motion with drift, we may confine ourselves to computing for barriers , where . Our aim is to compute the Laplace transform:
For , the Laplace transform of the first hitting time of a reflected Brownian motion with drift μ and volatility σ is given by
Pick , such that for . Furthermore, let ; then for any and , the function
satisfies and . According to the introductory notes of this section, the process is a uniformly bounded martingale; therefore, the stopped process
is also a true martingale, which starts at zero, - almost surely. Using the fact that , we find that the stopped process satisfies for any ,
Letting , we thus get by optional sampling,
For the two choices , where
we thus obtain two equations, for two unknown moments,
Solving this linear system for the involved moments yields the Laplace transform of , Equation (1). ☐
This result can also be obtained from a more general result for spectrally negative Lévy processes, reflected at an upper barrier (Avram et al. 2017, Proposition 4.B and Section 10.1). In fact, the distribution of is equal in distribution to the first hitting time 0 of the Brownian motion , starting at , reflected at . Its Laplace transform is therefore given by
For another “sanity check” of Theorem 4, we compute the Laplace transform Equation (1) independently when and . In this case, the reflected Brownian motion is equal to in law, where B is a standard Brownian motion. But then is equal in distribution to
Now, it is well known that the Laplace transform of is given by
which indeed coincides with Equation (1) for and .
Perry et al. (2004), Formula (5.2), state a different Laplace transforms than our Theorem 4. Letting in Perry et al. (2004), Formula (5.2) indeed yields for and ,
which contradicts Equation (2). The proof of Perry et al. (2004), Lemma 5.1, cannot be rectified, however, by merely fixing the (obviously) missing factor of for in the second line of their proof. Indeed, in the same line, they forget a factor in the second integrand; thus, by inserting special values of into the process in line 2, one does not get rid of the local-time term, as claimed.
3. Diffusions with Exponentially Distributed Gains Before Fixed Drawdowns
Let X be a diffusion process on the interval , satisfying the SDE
where and are locally Lipschitz continuous functions of linear growth on , and thereon.
For a threshold , we define as the maximum of X, prior to a drawdown of size , that is
We use the abbreviation , where . The following is due to Lehoczky (1977):
Caution is needed when interpreting the original paper Lehoczky (1977): Lehoczky uses the letter “a” for three different objects: The drift is denoted as , while is the left endpoint of the interval of the support of X; third, the threshold in his paper is also called a. An inspection of Lehozky’s proof reveals that our more general version with holds.
In terms of diffusion characteristics, Lehoczky’s result holds in a more general context. First, the assumption of locally Lipschitz coefficients are too strong, and can be relaxed. For example, we can relax to Hölder regularity of of order no worse than , due to Yamada et al. (1971). In addition, we can allow reflecting or absorbing boundary conditions, thus include reflected diffusions. For instance, Proposition 1 holds for a Brownian motion with drift, starting at 0 and being reflected at , because the process X cannot hit before it reaches a strictly positive maximum, due to strict positive volatility .
From Equation (4), it can be seen that when is constant, is exponentially distributed (the special case for for a Brownian motion with drift is due to Taylor (1975), and independently discovered by Golub et al. (2016)). Mijatović and Pistorius (2012) extended this result to spectrally negative Lévy processes: For those, is also exponentially distributed, with the parameter being the right-sided logarithmic derivative of the scale function, evaluated at the drawdown threshold.
This section characterizes the exponential law for diffusions:
The following are equivalent:
is a constant on .
For each , is exponentially distributed.
Proof ofthe Theorem.
Sufficiency of the first condition for the second one follows directly from Proposition 1. Suppose, therefore, that for each , there exists such that is exponentially distributed with parameter . Then, due to Equation (4),
By this particular functional form, and, since is continuous, it follows that the functions and are continuously differentiable. By differentiating Equation (5) with respect to , we have
and differentiating with respect to yields, in conjunction with the previous identity,
and dividing the last two equations yields Lobacevsky’s functional equation2
Note, is continuously differentiable, and strictly positive. Hence, by taking derivatives with respect to , we get
and by setting , we thus have
where . We conclude that for some ,
By Equation (7), we can extend the exponential solution to : By setting , we indeed have
Similarly, we can successively extend the validity of Equation (8) to the right, using the functional Equation (7). Now that for all we have, by taking the logarithmic derivative of , that is indeed a constant on . ☐
Examples of processes for which the running maximum at drawdown is exponentially distributed are the following:
(): Brownian motion with drift .
): Reflected Brownian motion with drift, reflected at .
Similar examples as in 1 and 2 can be constructed, where is constant. These include reflected diffusions.
However, there are processes that do not satisfy Theorem 2, even though they may exhibit exponentially distributed gains before drawdowns for specific choices of . One can, for instance, let be constant only on , and modify on in such a way, that the SDE Equation (3) has unique global strong solution. Then, by Proposition 1, for any the maximum at drawdown of size is exponentially distributed. It goes without saying, that there must exist , for which this is not the case.
Similar, but more sophisticated, examples can be constructed by solving delay differential equations for , such that only for a specific threshold , is exponentially distributed. Equation (6) reads in differential form:
which is the simplest non-trivial (discrete) delay differential equation. To construct a diffusion process for which the maximum before a drawdown of size 1 is exponentially distributed with parameter one, we set , and we choose a strictly positive continuous function on satisfying . To obtain on , we solve
subject to for . This problem has a unique solution with exponential growth. However, if g is not an exponentially linear function (that is, of the form for some ), then is not, and therefore is not constant. An underlying diffusion process X with being exponentially distributed with parameter one can for instance be constructed, by solving SDE Equation (3), where and on . Due to Theorem 2, is, in general, not exponentially distributed.
4. Lehoczky’s Proof for Spectrally Negative Lévy Martingales
We study in this section the distribution of maximal gains3 of processes, prior to the occurrence of a fixed loss . Golub et al. (2016, 2018) claim that for a Brownian motion (the toy model of a fair game), this gain is exponentially distributed, with parameter ; thus, on average, one gains before experiencing a loss of size . This result is independent of the volatility of the Brownian motion. In private communication, Golub (2014) raised the question of whether similar scaling laws hold for other processes, e.g., other diffusion models, or processes with jumps. Such models are useful as benchmark models in the context of certain event-based high-frequency trading algorithms, where the Brownian motion is used as a proxy for an asset, and the location of the maximum suggests the beginning of a trend reversal.4
The conjecture that a fair game on average experiences the exact same gain as is lost later on may appear intuitive. And this is indeed the case for many continuous-time martingales, those who are time-changed Brownian motions, with a quadratic variation tending to infinity, along almost every path (because the timing is not relevant here). But it is not true for Lévy martingales, as can be seen from Theorem 4. Nevertheless, the (exponential) distribution of gains, not its parameter, is universal within the class of spectrally negative Lévy processes. Besides, the martingale property is not needed to arrive at this result.
After Theorem 4 was proved in the summer of 2019, F. Hubalek kindly pointed out that the result is, in identical form, preceded by Mijatović and Pistorius (2012). Our proof is, however, similar to the one of Lehoczky (1977), and is therefore an alternative, and simpler one. Finally, we also found a replication of Lehoczky’s proof in Landriault et al. (2017), Lemma 3.1, however, this proof is also more difficult than ours due the more general discretization used therein.
We assume, that a Lévy process X is given with downward jumps only but not equal to the negative of a Lévy subordinator and not being a deterministic drift5. Such a process is defined by its Lévy exponent
which is of the form
with Lévy-Khintchine triplet , and a measure supported on , integrating .
The scale function W is the unique absolutely continuous function with Laplace transform
Since the processes lack positive jumps, they can only creep up. This assumption is essential to obtain exit probabilities from compact intervals and also for the main Theorem 4.
(Bertoin 1996, Theorem VII.8) Let , the probability that X makes its first exit from at y is
For a threshold , we define as the supremum of X, prior to a drawdown of size , that is
We are ready to state and proof the main theorem:
For a spectrally negative Lévy process, the maximal gain before a δ-loss is exponentially distributed with parameter equal to the logarithmic derivative of the scale function, that is,
Proofof Theorem 4.
The proof is inspired by Golub et al. (2016), however, the exact same idea can be traced back to Lehoczky (1977) in the general context of univariate diffusions processes. Let be the event that X reaches before (). The set can be approximated by , which are decreasing for increasing n. In other words,
Due to state-independence of the process (translation invariance) and the Markov property
where the last identity follows from Theorem 3. Since W is differentiable from the right at , applying L’Hospital’s rule yields
Theorem 4 implicitly requires right-differentiability of the scale functions, which is for free, because it can be rewritten as an integral of the tail of some finite measure, see (Bertoin (1996), Chapter VII). However, in many models, full -regularity is guaranteed (cf. the characterization given by (Kuznetsov et al. (2012), Lemma 2.4)).
The scale functions for the below processes are taken from review article of Hubalek and Kyprianou (2011). Throughout this section, denotes the exponential distribution with parameter .
(Compound Poisson Process).Assume we have a compound Poisson process with negative exponentially distributed jumps,
Therefore, by Theorem 4
Unlike the previous example, the following two examples exhibit the same qualitative dependence on the threshold , as the standard Brownian motion, where : when , the average maximum at drawdown of size tends to 0, and when , this average goes to infinity.
(Brownian motion with drift).A Brownian motion with drift and volatility σ,
The process exhibits Infinite variation jumps, and drifts to , because . The scale function is
Using Theorem 4, we thus get
The asymptotic behaviour of the logarithmic derivative of the scale function of a spectrally negative Lévy process can be characterized using the asymptotic behaviour of W and , cf. (Kuznetsov et al. 2012, Chapter 3). For instance, , if and only if the process is of infinite variation. In the case of finite variation, we can write the process as , where J is a subordinator; and then . Furthermore, , if a diffusion component is present, or if the Lévy measure is infinite. These general findings are consistent with the three examples.
This research received no external funding.
I thank John Appleby, Florin Avram, Huayuan Dong, Friedrich Hubalek, Andreas Kyprianou and two anonymous referees for useful comments.
Conflicts of Interest
The author declares no conflict of interest.
Aczél, János. 1966. Lectures on Functional Equations and Their Applications. Waltham: Academic Press, vol. 19. [Google Scholar]
Avram, Florin, Danijel Grahovac, and Ceren Vardar-Acar. 2017. The W, Z scale functions kit for first passage problems of spectrally negative Lévy processes, and applications to the optimization of dividends. arXiv arXiv:1706.06841. [Google Scholar]
Golub, Anton, Gregor Chliamovitch, Alexandre Dupuis, and Bastien Chopard. 2016. Multi-scale representation of high frequency market liquidity. Algorithmic Finance 5: 3–19. [Google Scholar] [CrossRef][Green Version]
Golub, Anton, James B. Glattfelder, and Richard B. Olsen. 2018. The alpha engine: Designing an automated trading algorithm. In High-Performance Computing in Finance. London: Chapman and Hall/CRC, pp. 49–76. [Google Scholar]
Hubalek, Friedrich, and Andreas E. Kyprianou. 2011. Old and new examples of scale functions for spectrally negative Lévy processes. In Seminar on Stochastic Analysis, Random Fields and Applications VI. Basel: Springer, pp. 119–45. [Google Scholar]
Kuznetsov, Alexey, Andreas E. Kyprianou, and Victor Rivero. 2012. The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters II. New York: Springer, pp. 97–186. [Google Scholar]
Landriault, David, Bin Li, and Hongzhong Zhang. 2017. On magnitude, asymptotics and duration of drawdowns for Lévy models. Bernoulli 23: 432–58. [Google Scholar] [CrossRef]
Lehoczky, John P. 1977. Formulas for stopped diffusion processes with stopping times based on the maximum. The Annals of Probability 5: 601–7. [Google Scholar] [CrossRef]
Mijatović, Aleksandar, and Martijn R. Pistorius. 2012. On the drawdown of completely asymmetric Lévy processes. Stochastic Processes and their Applications 122: 3812–36. [Google Scholar] [CrossRef]
Perry, David, Wolfgang Stadje, and Shelemyahu Zacks. 2004. The first rendezvous time of Brownian motion and compound Poisson-type processes. Journal of Applied Probability 41: 1059–70. [Google Scholar] [CrossRef]
Taylor, Howard M. 1975. A stopped Brownian motion formula. The Annals of Probability 3: 234–46. [Google Scholar] [CrossRef]
Yamada, Toshio, and Shinzo Watanabe. 1971. On the uniqueness of solutions of stochastic differential equations. Journal of Mathematics of Kyoto University 11: 155–67. [Google Scholar] [CrossRef]
See (Aczél (1966) p. 82, Chapter 2 Equation (16)) and the references therein.
This random gain is called “overshoot” in Golub et al. (2016). In this section, we refrain from using this terminology due to its established meaning in the field of Lévy processes—it is the discrepancy between a certain threshold, and a jump processes’ value, passing beyond that threshold.
It goes without saying that the first time this maximum is attained is not a stopping time; otherwise, one could devise arbitrage strategies that short-sell the asset at the maximum.
This is the natural non-degeneracy condition of Bertoin (1996), Chapter VII to ensure that the process creeps up to any level.
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