First Drawdown Epoch in Lévy Processes

• First Drawdown Epoch is the first time the reflection of a Lévy process at its running supremum exceeds a set threshold, indicating a significant adverse event.
• The derived sextuple law explicitly characterizes the joint distribution of the stopping time, supremum, infimum, and overshoot using scale functions and Lévy measures.
• Applications extend to portfolio risk management and derivative pricing, where understanding drawdown events enables accurate modeling of path-dependent financial instruments.

The first drawdown epoch is a concept central to the fluctuation theory of completely asymmetric (spectrally one-sided) Lévy processes, with particular significance for risk management, option pricing, and path-dependent modeling. For a Lévy process $X$, the drawdown process $Y$ is defined as the reflection of $X$ at its running supremum $\bar{X}$; that is, $Y_t = \bar{X}_t - X_t$ where $\bar{X}_t = \sup_{0 \leq s \leq t} X_s$. The first drawdown epoch, denoted $\tau_a$, is the first-passage time when the drawdown process $Y$ exceeds a predetermined threshold $a > 0$: $\tau_a = \inf\{ t \ge 0 : Y_t > a \}.$ This epoch encapsulates not only the occurrence of a significant adverse event (relative to historical highs) but, when analyzed jointly with other path functionals, provides profound insight into both the fine structure and probabilistic behavior of the process at the moment of drawdown.

1. Formal Definition and Structure of the Drawdown Process

In a completely asymmetric (e.g., spectrally negative) Lévy process, the drawdown process $Y$ serves as a path-functional tracking the loss from the historical maximum up to time $t$. The precise definitions are: $$\bar{X}_t = \sup_{0 \le s \le t} X_s, \qquad Y_t = \bar{X}_t - X_t, \qquad t \ge 0.$$ Here, $Y_t$ quantifies "how far down" $X_t$ is from its all-time high. This reflection structure is critical because the process $\bar{X}_t$ "remembers" the peak, inducing tractable properties leveraged in exit, overshoot/undershoot, and insurance computations.

The first drawdown epoch is the random time

$\tau_a = \inf\{t \ge 0 : Y_t > a\}$

which, for a process $X$ without monotone paths and jumps in one direction, is almost surely finite. In risk and financial modeling, $\tau_a$ is a key stopping time, marking events like position liquidations, option triggers (e.g., drawdown/Russian options), or sudden shifts in risk exposure.

2. Joint Law at the Drawdown Epoch: The Sextuple Law

The paper derives, via excursion-theoretic and compensation arguments, an explicit joint law for the sextuple of random variables observed at the first drawdown epoch: $$(\tau_a,\; \bar{G}_{\tau_a},\; \bar{X}_{\tau_a},\; \underline{X}_{\tau_a},\; a - Y_{\tau_a^-},\; Y_{\tau_a} - a)$$ where:

• $\tau_a$: the first time $Y$ exceeds $a$,
• $\bar{G}_{\tau_a}$: the last time prior to $\tau_a$ when $X$ attained its supremum,
• $\bar{X}_{\tau_a}$: the running supremum at $\tau_a$,
• $\underline{X}_{\tau_a}$: the running infimum at $\tau_a$,
• $a - Y_{\tau_a^-}$: the undershoot (distance just before exceeding $a$),
• $Y_{\tau_a} - a$: the overshoot (by how much $Y$ exceeds $a$ at $\tau_a$).

The law of this sextuple is expressed explicitly via the scale function $W^{(q)}$, its right-derivative $W^{(q)\prime}_+(a)$, the Lévy measure $\Lambda$, and the resolvent measure: $$\lambda(a, q) = \frac{W^{(q)\prime}_+(a)}{W^{(q)}(a)}, \qquad R_a^{(q)}(dy) = [\lambda(a,q)^{-1} W^{(q)}(dy) - W^{(q)}(y)dy ].$$ In particular, the joint Laplace transform for events like a jump-induced crossing (overshoot nonzero) is: $$\mathbb{E}_x [e^{-q \tau_a - r \bar{G}_{\tau_a}}; A_o] = \frac{W^{(q+r)}((x-u) \wedge a)}{W^{(q+r)}(a)} \cdot F_{q+r, q, a}(v - (x \vee (u+a))) \cdot R_a^{(q)}(dy)\; \Lambda(y-a-dh),$$ with $$F_{p,q,a}(y) = \lambda(a, q) \exp(-\lambda(a, p) y)$$, and $A_o$ specifying the appropriate region in path-space for a jump crossing. Alternative formulas are given for the creeping case (no overshoot).

This explicit, high-dimensional law enables not only the timing but also a detailed reconstruction of the process geometry at the drawdown—critical for applications requiring path-level information (option payoffs, risk attribution).

3. Scale Functions and Lévy Measure: Analytic Foundations

The scale function $W^{(q)}$ is central to the representation of exit-related quantities in spectrally negative Lévy processes. For Laplace exponent $\psi(\theta)$, it is defined via

$$\int_0^{\infty} e^{-\theta x} W^{(q)}(x) dx = \frac{1}{\psi(\theta) - q},\quad \theta > \Phi(q)$$

with right-inverse $\Phi(q)$. The $W^{(q)}$ is strictly increasing and continuous on $[0, \infty)$, yielding explicit solutions to boundary crossing and first passage problems.

The Lévy measure $\Lambda$ determines the jump intensity and is essential to capturing the statistics of overshoots in jump crossings. Both $W^{(q)}$ and $\Lambda$ are combined in the law for the full sextuple at $\tau_a$, as detailed above.

Relatedly, other functionals such as the resolvent $R_a^{(q)}$ and $Z^{(q)}(x) = 1 + q \int_0^x W^{(q)}(z)dz$ appear in these path-dependent distributions, providing a complete toolkit for analytic evaluation.

4. Significance of $\tau_a$ in Risk, Option Pricing, and Portfolio Management

The first drawdown epoch $\tau_a$ demarcates the first time an underperformance relative to the running maximum exceeds $a$. This is fundamentally important for several reasons:

• Risk Management: $\tau_a$ can trigger automatic trading responses, such as liquidation or position rebalancing upon reaching a prescribed loss from a peak.
• Performance Measurement: Pathwise knowledge at $\tau_a$ (i.e., the sextuple) provides insight into the severity and abruptness of losses, discriminating between jump and diffusive drawdown mechanisms.
• Option Pricing: In path-dependent, drawdown-sensitive options (e.g., Russian or drawdown options), $\tau_a$ is the relevant stopping time for payoff computation.
• Portfolio Constraints: Investors or funds may include drawdown-based constraints, making the distribution of $\tau_a$ and associated path functionals core to compliance and expected loss analysis.

The explicit formulas for the sextuple inform the probability that a drawdown occurs before a rally and are directly applicable in semi-analytic option pricing and risk quantification.

5. Application to Exponential Lévy Models and Practical Computation

In exponential Lévy models for asset prices, $S_t = S_0 \exp(X_t)$, the relative drawdown is simply: $\frac{\bar{S}_t}{S_t} = \exp(Y_t)$ so a relative drawdown crossing level $a$—or equivalently, a percentage drop $\alpha$ from the peak—has $a = -\log(1-\alpha)$. The first drawdown epoch is then

$\tau_a = \inf\{ t \ge 0 : Y_t > a \}.$

The sextuple law delivers, in closed form (by substitution of $W^{(q)}$ and $\Lambda$ appropriate for the model), probabilities for drawdown preceding a rally, the distribution of losses at stopping, and expected payouts for various financial derivatives. For example, in Carr and Wu’s S&P 500 model with negative jumps, the scale function can often be written in terms of Mittag-Leffler functions, yielding almost fully analytic pricing of drawdown-contingent features.

6. Implementation, Extensions, and Impact

From an implementation perspective, the scale function must be computed (e.g., via Laplace inversion or explicit formula, if available) for the chosen Lévy model. Given $W^{(q)}$ and $\Lambda$, all terms in the sextuple law are directly evaluable. Models with stable or compound Poisson jumps are tractable, and the theory extends naturally to more complex spectrally one-sided processes.

These formulas underpin simulation schemes, closed-form evaluation for exotic options, and the modeling of risk measures directly linked to historical peaks and troughs.

7. Summary and Theoretical Integration

The theory in "On the drawdown of completely asymmetric Lévy processes" (Mijatovic et al., 2011) provides a unified and explicit analytic framework for:

• describing the sample path structure of drawdowns at first passage,
• computing joint distributions of stopping times and associated functionals,
• rigorously linking fluctuation theory (scale function analysis) with practical risk, option pricing, and portfolio management applications in finance.

The approach connects deep probabilistic structure—excursion theory, compensation, and Lévy process fluctuation identities—with fully implementable formulas, ensuring both theoretical rigor and applied utility for high-frequency risk management and advanced financial engineering.