Stochastic Drawdowns

The following sections are included:

• Dedication
• Preface
• About the Author
• Contents

Financial turmoils are most often marked with series of sharp falls in asset prices triggered by certain market events, with the recent examples of the S&P downgrade of US debt, and default speculations of European countries. Many individual and institutional investors are wary of large market drawdowns as they not only lead to portfolio losses and liquidity shocks, but also indicate potential imminent recessions. As is well known, hedge fund managers are typically compensated based on the fund’s outperformance over the last record maximum, or the high-water mark. As such, drawdown events can directly affect the manager’s income. Also, a major drawdown may also trigger a surge in fund redemption by investors, and lead to the manager’s job termination. Hence, fund managers have strong incentive to seek insurance against drawdowns (see, e.g., Burghardt et al. (2003)).

We begin our journey to the subject of drawdown in Chapter 2, where we determine the probability that a drawdown of a units precedes a drawup of b units in a finite time-horizon. Formally, the drawup of a stochastic process X· is defined as

U_t := X_t − X_t, ∀t ≥ 0

where X_t := infs∈[0,t] Xs denotes the running minimum of X·. Thus, this probability assesses the relative strength of downside risk (drawdown) compared to upward momentum (drawup) over a finite time-horizon. To determine this probability, we first consider the simple case with equal-sized drawdown/drawup (i.e., a = b), and derive analytic formulas of this probability by drawing connections to the first exit problems under a simple random walk model and a Brownian motion with drift model. For the general case, we randomize the time-horizon with an independent exponential random variable — a technique known as Canadization, and reduce the probability of interest to the Laplace transform of the first passage time of the drawdown when it precedes a drawup. Using a classical approximation argument as in Lehoczky (1977), we derive analytical formulas for this Laplace transform under general linear diffusion models. Finally, we use Laplace inversion to evaluate the drawdown preceding drawup probability and the conditional density of the maximum relative drawup given a drawdown event, under a geometric Brownian motion (GBM) model.

Apart from the finite time-horizon properties of drawdowns and drawups, the issue of how fast a market crash occurred is also of vital importance to investors and portfolio or hedge fund managers. This motivates us to study the joint distribution of the drawdown and the speed at which it is realized in Chapter 3. In particular, we measure the speed of market crash by the time elapsed between the first passage time of the drawdown to level a > 0, τ+D(a), and the last reset time of the running maximum prior to that drawdown g_a := sup{t < τ+ D(a) : X_t = X_t}. The analysis involves studying of the last reset time of maximum ga, for which we make use of the technique of progressive enlargement of filtration and path decomposition at the random time ga. By deriving the joint Laplace transform of ga and τ+ D(a), we provide the analytical basis for the pricing of financial claims based on the drawdown and its speed, which can be used to hedge against dramatic market crashes.

While sustaining downside risk can be appropriately characterized using the drawdown process and its first passage time, economic turmoil and volatile market fluctuations are better described by quantities containing more path-wise information, such as the frequency of drawdowns. In Chapter 4, we directly address the problem on the frequency of drawdowns by determining the probability distributions of two sequences of drawdown first passage times, depending on whether the maximum is revisited between these first passage times. These stopping times characterize the fluctuations of the underlying through consecutive drawdowns. We illustrate how our analytical formulas can be used for evaluating options written on the number of drawdowns in a given time-horizon.

The amount of time when the underlying price is in distress can characterize investment risks from a different angle. In Chapter 5, we propose to study a number of occupation times that are related to the drawdowns and drawups of a linear diffusion process. To that end, we derive analytical formulas for the Laplace transform of the occupation time of the underlying below a fixed threshold y upon leaving a given finite interval. We then use Lehoczky’s approximation technique and progressive enlargement of filtration to extend these results to occupation times of the drawdown or drawup process. In special cases of drifted Brownian motion and threedimensional Bessel process, our results imply that the occupation time of the drawdown process has the same distribution as the first passage time of the drawdown process. We also demonstrate how our analytical results can be used for computing probabilities of drawdown events under Omega default models, and for evaluation of Parisian-like or α-quantile options written on the drawdowns.

An alternative way to model the time under distress for an asset price is through the length of excursions from its running maximum process, namely, the amount of time it takes for the asset price to revisit the previous peak. Such quantities are of interest to investors who are more concerned about the duration of drawdown than the actual magnitude of drawdowns experienced. In Chapter 6, we study the duration of drawdown for a wide class of L´evy models, by deriving the distribution of a Parisian time of the drawdown process. Given a time threshold b, this Parisian time is defined as the first time when the underlying is continuously below the running maximum for more than b units of time. Our analysis makes use of renewal equations, weak convergence, asymptotics of the magnitude of drawdowns, and some recent work on the asymptotic analysis of the running maximum of L´evy process. We illustrate how our results can be applied to popular models including Brownian motion with drift, spectrally negative α-stable process, spectrally negative Gamma process, and Kou’s jump diffusion.

Chapters 7–9 are dedicated to applications of drawdown and maximum drawdown in hedging, insurance and trading, respectively. As the building blocks for more sophisticated options on drawdown or maximum drawdown, in Chapter 7 we study two types of digital options written on them, namely, the digital call on drawdown preceding a drawup, and the digital call on maximum drawdown. Such options are of interest to an asset manager who knows in advance that her portfolio risk is being evaluated wholly or in part by the portfolio’s maximum drawdown or drawdowns. To determine the fair value of these digital options, we aim to develop self-financing replicating portfolios with the least possible time instances in which trading/rebalance is involved. Using results in Chapter 2, we develop a static model-free replication of the former digital option with one-touch knockouts. By using reflection principle and the expansion of the Canadized option price, we also develop semi-static replication strategies for both digital options with gradually more liquid instruments such as one-touches or path-independent options. We demonstrate that these strategies are valid for models such as Black model, geometric Brownian motion (GBM) model and models obtained from their continuous time-changes.

Our study on drawdown insurance in Chapter 8 approaches the protection against drawdowns differently. In particular, we derive the fair premium for a drawdown insurance that delivers a fixed compensation upon an drawdown event. In order to provide the investor with more flexibility in managing the path-dependent drawdown risk, we also incorporate the right to terminate the contract early. We formulate an optimal stopping problem that aims to maximize the early termination premium. By using the principle of smooth pasting, we solve the associated variational inequality and identify the optimal cancellation strategy as a first passage time of the drawdown process, which subsequently determines the fair premium of the contract. Moreover, we expand our analysis of drawdown insurance contracts by incorporating a drawup contingency, and the default risk of the underlying — a feature absent in other related studies on drawdown, and studying their impacts to the fair premium.

Trailing stops are a popular trade order widely used by proprietary traders and retail investors to provide downside protection for an existing position. A trailing stop is triggered when the prevailing price of an asset falls below a stochastic floor, which is often a percentage of the running maximum. In contrast, a limit sell order can be considered as a selling order that is triggered when the asset price reaches some target. Whether an asset holder should passively wait until a trailing stop is triggered or sell the asset more aggressively with a limit sell order is a question that remains open. In Chapter 9, we address this question by deriving the optimal trading strategy when a trailing stop is imposed. We develop an analytical tractable framework with linear diffusion models, and identify the early exercising premium that an asset holder can get by trading aggressively. Our approach is to first derive some key technical results by solving an auxiliary optimal trading problem with a fixed stop-loss level. Then, we study the optimal exit strategy for selling the asset no later than the trailing stop, and prove that the early exercising premium is maximized when a limit sell order is placed in conjunction with the trailing stop. In order words, an asset holder should only trade passively if the key price target is not reached; otherwise, an immediate sell is optimal and better than waiting for the trailing stop. Different from the variational inequality approach we used in Chapter 8, our solution method is to demonstrate that a proposed stopping rule yields strictly positive “time value”, then prove that the optimal continuation region cannot be larger than the one in the proposed stopping rule. Moreover, we also study the optimal entrance strategy for purchasing this asset and setting up the trailing stop. Using an exponential Ornstein–Uhlenbeck (OU) model, we numerically demonstrate our optimal trading strategy, and show that the optimal entrance region is of the form (0, A), while the optimal exit region is (B, ∞) for 0 <A <B < ∞.

The following sections are included:

• Appendix: Briefly on One-Dimensional Linear Diffusions
• Bibliography
• Index