**Proposition 1.**

$$\log \mathbb{P}[M^{\delta} \ge \xi] = -\int\_{0}^{\xi} \frac{\Phi(u)}{\int\_{u-\delta}^{u} \Phi(s) ds} du, \quad \xi \ge 0. \tag{4}$$

Caution is needed when interpreting the original paper Lehoczky (1977): Lehoczky uses the letter "a" for three different objects: The drift *μ*(*x*) is denoted as *<sup>a</sup>*(*x*), while −*a* is the left endpoint of the interval of the support of *X*; third, the threshold *δ* in his paper is also called *a*. An inspection of Lehozky's proof reveals that our more general version with *δ* ≤ *a* 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 *σ*(*x*) of order no worse than 1/2, 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 <sup>−</sup>*a*, because the process *X* cannot hit −*a* before it reaches a strictly positive maximum, due to strict positive volatility *σ*(0) > 0.

From Equation (4), it can be seen that when *μ*/*σ*<sup>2</sup> is constant, *M<sup>δ</sup>* 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)). Mijatovi´c and Pistorius (2012) extended this result to spectrally negative Lévy processes: For those, *M<sup>δ</sup>* 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:

**Theorem 2.** *The following are equivalent:*


**Proof of the Theorem.** Sufficiency of the first condition for the second one follows directly from Proposition 1. Suppose, therefore, that for each 0 < *δ* ≤ *a*, there exists Λ(*δ*) > 0 such that *M<sup>δ</sup>* is exponentially distributed with parameter <sup>Λ</sup>(*δ*). Then, due to Equation (4),

$$\int\_{0}^{\tilde{\varsigma}} \frac{\Phi(u)}{\int\_{u-\delta}^{u} \Phi(s) ds} du = \Lambda(\delta) \xi, \quad \tilde{\xi} \ge 0, \quad \delta \le a. \tag{5}$$

By this particular functional form, and, since *μ*/*σ*<sup>2</sup> is continuous, it follows that the functions Λ(*δ*) and <sup>Φ</sup>(*x*) are continuously differentiable. By differentiating Equation (5) with respect to *ξ*, we have

$$\Phi(\xi) = \Lambda(\delta) \int\_{\overline{\xi}-\delta}^{\overline{\xi}} \Phi(u) du, \quad \xi \ge 0, \quad \delta \le a,\tag{6}$$

and differentiating with respect to *δ* yields, in conjunction with the previous identity,

$$\frac{\Phi(\xi-\delta)}{\Phi(\xi)} = -\frac{\Lambda'(\delta)}{\Lambda^2(\delta)}, \quad \xi \ge 0, \quad \delta \le a.$$

Therefore, also

$$\frac{\Phi(\frac{x}{\delta})}{\Phi(\frac{x}{\delta} + \delta)} = -\frac{\Lambda'(\delta)}{\Lambda^2(\delta)}, \quad \xi \ge 0, \quad \delta \le a\_\nu$$

and dividing the last two equations yields Lobacevsky's functional equation<sup>2</sup>

$$
\begin{aligned}
\Phi(\mathfrak{J}-\delta)\Phi(\mathfrak{J}+\delta)&=\Phi(\mathfrak{J})^2, \quad \mathfrak{J}\geq 0, \quad \delta \leq a,\\
\Phi(0)&=1.
\end{aligned}
\tag{7}
$$

Note, Φ is continuously differentiable, and strictly positive. Hence, by taking derivatives with respect to *δ*, we ge<sup>t</sup>

$$\frac{\Phi'(\xi-\delta)}{\Phi(\xi-\delta)} = \frac{\Phi'(\xi+\delta)}{\Phi(\xi+\delta)}\sqrt{\}$$

and by setting *ξ* = *δ*, we thus have

$$\Phi'(2\xi) = \kappa \Phi(2\xi), \quad \Phi(0) = 1, \quad 0 < \xi \le a\_{\prime}$$

where *α* = Φ(0)/Φ(0) ∈ R. We conclude that for some *β* ∈ R,

$$\Phi(\xi^{\mathbf{x}}) = \mathfrak{e}^{\beta \tilde{\xi}}, \quad 0 \le \xi^{\mathbf{x}} \le 2a. \tag{8}$$

By Equation (7), we can extend the exponential solution to −*a* ≤ *ξ* < 0: By setting *ξ* = 0, we indeed have

$$\Phi(-\delta) = \frac{\Phi^2(0)}{\Phi(\delta)} = \frac{1}{e^{\delta \delta}} = e^{-\delta \delta}, \quad 0 < \delta \le a.$$

Similarly, we can successively extend the validity of Equation (8) to the right, using the functional Equation (7). Now that Φ(*ξ*) = *eβξ* for all *ξ* ∈ [−*a*, ∞) we have, by taking the logarithmic derivative of Φ, that *μ*(*x*)/*σ*<sup>2</sup>(*x*) is indeed a constant on [−*a*, <sup>∞</sup>).

Examples of processes for which the running maximum at drawdown is exponentially distributed are the following:


However, there are processes that do not satisfy Theorem 2, even though they may exhibit exponentially distributed gains before *δ* drawdowns for specific choices of *δ*. One can, for instance, let *μ*/*σ*<sup>2</sup> be constant only on [−1, <sup>∞</sup>), and modify *μ*, *σ*<sup>2</sup> on [−2, −<sup>1</sup>) in such a way, that the SDE Equation (3) has unique global strong solution. Then, by Proposition 1, for any *δ* < 1 the maximum at

<sup>2</sup> See (Aczél (1966) p. 82, Chapter 2 Equation (16)) and the references therein.

drawdown of size *δ* is exponentially distributed. It goes without saying, that there must exist *δ* > 1, for which this is not the case.

Similar, but more sophisticated, examples can be constructed by solving delay differential equations for <sup>Φ</sup>(·) = *e*<sup>−</sup><sup>2</sup> ·0 *μ*(*u*)/*σ*<sup>2</sup>(*u*)*du*, such that only for a specific threshold *δ*, *M<sup>δ</sup>* is exponentially distributed. Equation (6) reads in differential form:

$$\Phi'(\xi) = \Lambda(\delta) \left( \Phi(\xi) - \Phi(\xi - \delta) \right), \quad \xi \ge 0,$$

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 Λ(*δ*) = *δ* = 1, and we choose a strictly positive continuous function *g*(*x*) on [−1, 0] satisfying *g*(0) = 1. To obtain Φ on [0, <sup>∞</sup>), we solve

$$
\Phi'(\xi) = \Phi(\xi) - \Phi(\xi - 1), \quad \xi \ge 0,
$$

subject to Φ(*ξ*) = *g*(*ξ*) for *ξ* ∈ [−1, 0]. This problem has a unique solution with exponential growth. However, if *g* is not an exponentially linear function (that is, of the form *eλ<sup>x</sup>* for some *λ* > 0), then Φ is not, and therefore *μ*/*σ*<sup>2</sup> is not constant. An underlying diffusion process *X* with *M*<sup>1</sup> being exponentially distributed with parameter one can for instance be constructed, by solving SDE Equation (3), where *σ* = 1 and *μ* = − <sup>Φ</sup>(*x*) <sup>2</sup><sup>Φ</sup>(*x*) on [−1, <sup>∞</sup>). Due to Theorem 2, *M<sup>δ</sup>* is, in general, not exponentially distributed.

### **4. Lehoczky's Proof for Spectrally Negative Lévy Martingales**

We study in this section the distribution of maximal gains<sup>3</sup> of processes, prior to the occurrence of a fixed loss *δ* > 0. 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 *δ*; thus, on average, one gains *δ* before experiencing a loss of size *δ*. 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.<sup>4</sup>

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 Mijatovi´c 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.

<sup>3</sup> 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.

<sup>4</sup> 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.

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

$$\Psi(\theta) := \frac{1}{t} \log \mathbb{E}[e^{\theta \mathcal{X}\_t}], \quad \theta > 0,$$

which is of the form

$$\Psi(\theta) = \mu \theta + \frac{\sigma^2 \theta^2}{2} + \int\_{(-\infty, 0)} \left( e^{\theta \frac{x}{\theta}} - 1 - \theta \xi^x \mathbf{1}\_{[-1, 0)}(\xi) \right) \nu(d\xi), \quad \theta > 0,$$

with Lévy-Khintchine triplet *μ* ∈ R, *σ* ∈ R and a measure *ν*(*dξ*) supported on (−∞, <sup>0</sup>), integrating min(*ξ*2, <sup>1</sup>).

The scale function *W* is the unique absolutely continuous function [0, ∞) → [0, ∞) with Laplace transform

$$\int\_0^\infty e^{-\theta \ge} W(\mathbf{x}) d\mathbf{x} = \frac{1}{\Psi(\theta)'} \quad \theta > 0.$$

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.

**Theorem 3.** *(Bertoin 1996, Theorem VII.8) Let x*, *y* > 0*, the probability that X makes its first exit from* [−*x*, *y*] *at y is*

$$\mathbb{P}[\mathfrak{r}\_y < \mathfrak{r}\_{-x}] = \frac{W(x)}{W(x+y)}.$$

For a threshold *δ* > 0, we define *M<sup>δ</sup>* as the supremum of *X*, prior to a drawdown of size *δ*, that is

$$M^\delta = M(\tau^\delta), \quad \text{where} \quad M(t) := \sup\_{s \le t} X\_s, \quad \text{and} \quad \tau^\delta := \inf \{ t > 0 \mid M\_l - X\_l \ge \delta \}.$$

We are ready to state and proof the main theorem:

**Theorem 4.** *For a spectrally negative Lévy process, the maximal gain M<sup>δ</sup> before a δ-loss is exponentially distributed with parameter equal to the logarithmic derivative of the scale function, that is,*

$$\mathbb{P}[\mathcal{M}^\delta \ge \xi] = e^{-\frac{\mathcal{W}^\delta(\delta+)}{\mathcal{W}(\delta)}\xi}.$$

**Proof of 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 *Ak*,*<sup>n</sup>* be the event that *X* reaches *kξ*/2*n* before −*δ* + (*k* − 1)/2*nξ* (*k* = 1, ... , 2*n*). The set {*M<sup>δ</sup>* ≥ *ξ*} can be approximated by 3*nk*=<sup>1</sup>*Ak*,*n*, which are decreasing for increasing *n*. In other words,

$$\{\mathcal{M}^\delta \ge \xi\} = \bigcap\_{n=1}^\infty \bigcap\_{k=1}^{2^n} A\_{k,n}.$$

Therefore,

$$\mathbb{P}[\mathcal{M}^\delta \ge \xi] = \lim\_{n \to \infty} \mathbb{P}\left[\bigcap\_{k=1}^{2^n} A\_{k\mathcal{M}}\right].$$

<sup>5</sup> This is the natural non-degeneracy condition of Bertoin (1996), Chapter VII to ensure that the process creeps up to any level.

Due to state-independence of the process (translation invariance) and the Markov property

$$\mathbb{P}\left[\bigcap\_{k=1}^{2^n} A\_{k,n}\right] = \mathbb{P}[A\_{1,n}] \times \prod\_{k=2}^{2^n} \mathbb{P}[A\_{k,n} \mid A\_{k-1,n}] = \left(\mathbb{P}[A\_{1,n}]\right)^{2^n} = \left(\frac{\mathcal{W}(\delta)}{\mathcal{W}(\delta + \xi/2^n)}\right)^{2^n},$$

where the last identity follows from Theorem 3. Since *W* is differentiable from the right at *δ*, applying L'Hospital's rule yields

$$\log \mathbb{P}[\mathcal{M}^{\delta} \ge \xi] = \lim\_{n \to \infty} \log (\mathbb{P}[A\_{1,n}])^{2^n} = -\xi \frac{\mathcal{W}^{\prime}(\delta +)}{\mathcal{W}(\delta)}.$$

**Remark 2.** *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 <sup>C</sup>*<sup>1</sup>*-regularity is guaranteed (cf. the characterization given by (Kuznetsov et al. (2012), Lemma 2.4)).*
