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Fair valuation of L\'evy-type drawdown-drawup contracts with general insured and penalty functions
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Abstract
In this paper, we analyse some equity-linked contracts that are related to drawdown and drawup events based on assets governed by a geometric spectrally negative L\'evy process. Drawdown and drawup refer to the differences between the historical maximum and minimum of the asset price and its current value, respectively. We consider four contracts. In the first contract, a protection buyer pays a premium with a constant intensity p until the drawdown of fixed size occurs. In return, he/she receives a certain insured amount at the drawdown epoch, which depends on the drawdown level at that moment. Next, the insurance contract may expire earlier if a certain fixed drawup event occurs prior to the fixed drawdown. The last two contracts are extensions of the previous ones but with an additional cancellable feature that allows the investor to terminate the contracts earlier. In these cases, a fee for early stopping depends on the drawdown level at the stopping epoch. In this work, we focus on two problems: calculating the fair premium p for basic contracts and finding the optimal stopping rule for the polices with a cancellable feature. To do this, we use a fluctuation theory of L\'evy processes and rely on a theory of optimal stopping.
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In this paper we consider some insurance policies related with drawdown and drawup events of log-returns for an underlying asset modeled by a spectrally negative geometric L\'evy process. We consider four contracts among which three were introduced in Zhang et al. (2013) for a geometric Brownian motion. The first one is an insurance contract where protection buyer pays a constant premium until the drawdown of fixed size of log-returns occurs. In return he/she receives certain insured amount at the drawdown epoch. Next insurance contract provides protection from any specified drawdown with a drawup contingency. This contract expires early if certain fixed drawup event occurs prior to fixed drawdown. The last two contracts are extensions of the previous ones by additional cancellable feature which allows an investor to terminate the contract earlier. We focus on two problems: calculating the fair premium p for the basic contracts and identifying the optimal stopping rule for the polices with cancellable feature. To do this we solve some two-sided exit problems related with the drawdown and the drawup of spectrally negative L\'evy processes which is of own scientific interest. We also heavily rely on a theory of optimal stopping.
In this article, we study the concept of maximum drawdown and its relevance to the prevention of portfolio losses. Maximum drawdown is deflned as the largest market drop during a given time interval. We show that maximum drawdown can serve as an additional tool for portfolio managers on top of already existing contracts, such as put or lookback options.
The drawdown of an asset is a risk measure defined in terms of the running maximum of the asset's spot price over some period [0, T]. The asset price is said to have drawn down by at least K below its maximum-to-date. We introduce insurance against a large realization of maximum drawdown and a novel way to hedge the liability incurred by underwriting this insurance. Our proposed insurance pays a fixed amount should the maximum drawdown exceed some fixed threshold over a specified period. The need for this drawdown insurance would diminish should markets rise before they fall. Consequently, we propose a second kind of cheaper maximum drawdown insurance that pays a fixed amount contingent on the drawdown preceding a drawup. We propose double barrier options as hedges for both kinds of insurance against large maximum drawdowns. In fact for the second kind of insurance we show that the hedge is model-free. Since double barrier options do not trade liquidly in all markets, we examine the assumptions under which alternative hedges using either single barrier options or standard vanilla options can be used.
In this article, we define new 'Greeks' for financial derivatives: sensitivities to the running maximum and the running maximum drawdown of an underlying asset. Some types of portfolios, such as the net asset value of a hedge fund or performance fees, are sensitive to these parameters. In order to illustrate the concept of the new 'Greeks', we derive probabilistic representations of sensitivities for two classes of financial contracts: forwards on the maximum drawdown and lookback options. These results allow us to interpret the delta-hedge of the contracts in a novel way.
In this work we study drawdowns and drawups of general diffusion processes. The drawdown process is defined as the current drop of the process from its running maximum, while the drawup process is defined as the current increase over its running minimum. The drawdown and the drawup are the first hitting times of the drawdown and the drawup processes respectively. In particular, we derive a closed-form formula for the Laplace transform of the probability density of the drawdown of a units when it precedes the drawup of b units. We then separately consider the special case of drifted Brownian motion, for which we derive a closed form formula for the above-mentioned density by inverting the Laplace transform. Finally, we apply the results to a problem of interest in financial risk-management and to the problem of transient signal detection and identification of two-sided changes in the drift of general diffusion processes.
The present monograph, based mainly on studies of the authors and their - authors, and also on lectures given by the authors in the past few years, has the following particular aims: To present basic results (with proofs) of optimal stopping theory in both discrete and continuous time using both martingale and Mar- vian approaches; To select a seriesof concrete problems ofgeneral interest from the t- ory of probability, mathematical statistics, and mathematical ?nance that can be reformulated as problems of optimal stopping of stochastic processes and solved by reduction to free-boundary problems of real analysis (Stefan problems). The table of contents found below gives a clearer idea of the material included in the monograph. Credits and historical comments are given at the end of each chapter or section. The bibliography contains a material for further reading. Acknowledgements.TheauthorsthankL.E.Dubins,S.E.Graversen,J.L.Ped- sen and L. A. Shepp for useful discussions. The authors are grateful to T. B. To- zovafortheexcellenteditorialworkonthemonograph.Financialsupportandh- pitality from ETH, Zur ¨ ich (Switzerland), MaPhySto (Denmark), MIMS (Man- ester) and Thiele Centre (Aarhus) are gratefully acknowledged. The authors are also grateful to INTAS and RFBR for the support provided under their grants. The grant NSh-1758.2003.1 is gratefully acknowledged. Large portions of the text were presented in the “School and Symposium on Optimal Stopping with App- cations” that was held in Manchester, England from 17th to 27th January 2006.
The scientific study of complex systems has transformed a wide range of disciplines in recent years, enabling researchers in both the natural and social sciences to model and predict phenomena as diverse as earthquakes, global warming, demographic patterns, financial crises, and the failure of materials. In this book, Didier Sornette boldly applies his varied experience in these areas to propose a simple, powerful, and general theory of how, why, and when stock markets crash.Most attempts to explain market failures seek to pinpoint triggering mechanisms that occur hours, days, or weeks before the collapse. Sornette proposes a radically different view: the underlying cause can be sought months and even years before the abrupt, catastrophic event in the build-up of cooperative speculation, which often translates into an accelerating rise of the market price, otherwise known as a "bubble." Anchoring his sophisticated, step-by-step analysis in leading-edge physical and statistical modeling techniques, he unearths remarkable insights and some predictions--among them, that the "end of the growth era" will occur around 2050.Sornette probes major historical precedents, from the decades-long "tulip mania" in the Netherlands that wilted suddenly in 1637 to the South Sea Bubble that ended with the first huge market crash in England in 1720, to the Great Crash of October 1929 and Black Monday in 1987, to cite just a few. He concludes that most explanations other than cooperative self-organization fail to account for the subtle bubbles by which the markets lay the groundwork for catastrophe.Any investor or investment professional who seeks a genuine understanding of looming financial disasters should read this book. Physicists, geologists, biologists, economists, and others will welcomeWhy Stock Markets Crashas a highly original "scientific tale," as Sornette aptly puts it, of the exciting and sometimes fearsome--but no longer quite so unfathomable--world of stock markets.
The purpose of this review article is to give an up to date account of the theory and applications of scale functions for spectrally negative Lévy processes. Our review also includes the first extensive overview of how to work numerically with scale functions. Aside from being well acquainted with the general theory of probability, the reader is assumed to have some elementary knowledge of Lévy processes, in particular a reasonable understanding of the Lévy-Khintchine formula and its relationship to the Lévy-Itô decomposition. We shall also touch on more general topics such as excursion theory and semi-martingale calculus. However, wherever possible, we shall try to focus on key ideas taking a selective stance on the technical details. For the reader who is less familiar with some of the mathematical theories and techniques which are used at various points in this review, we note that all the necessary technical background can be found in the following texts on Lévy processes; (Bertoin, Lévy Processes (1996); Sato, Lévy Processes and Infinitely Divisible Distributions (1999); Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes and Their Applications (2006); Doney, Fluctuation Theory for Lévy Processes (2007)), Applebaum Lévy Processes and Stochastic Calculus (2009).
Lèvy processes are the natural continuous-time analogue of random walks and form a rich class of stochastic processes around which a robust mathematical theory exists. Their mathematical significance is justified by their application in many areas of classical and modern stochastic models.This textbook forms the basis of a graduate course on the theory and applications of Lèvy processes, from the perspective of their path fluctuations. Central to the presentation are decompositions of the paths of Lèvy processes in terms of their local maxima and an understanding of their short- and long-term behaviour.The book aims to be mathematically rigorous while still providing an intuitive feel for underlying principles. The results and applications often focus on the case of Lèvy processes with jumps in only one direction, for which recent theoretical advances have yielded a higher degree of mathematical transparency and explicitness. Each chapter has a comprehensive set of exercises with complete solutions.
This paper studies the stochastic modeling of market drawdown events and the
fair valuation of insurance contracts based on drawdowns. We model the asset
drawdown process as the current relative distance from the historical maximum
of the asset value. We first consider a vanilla insurance contract whereby the
protection buyer pays a constant premium over time to insure against a drawdown
of a pre-specified level. This leads to the analysis of the conditional Laplace
transform of the drawdown time, which will serve as the building block for
drawdown insurance with early cancellation or drawup contingency. For the
cancellable drawdown insurance, we derive the investor's optimal cancellation
timing in terms of a two-sided first passage time of the underlying drawdown
process. Our model can also be applied to insure against a drawdown by a
defaultable stock. We provide analytic formulas for the fair premium and
illustrate the impact of default risk.
In this paper, we introduce new techniques how to control the maximum drawdown (MDD). One can view the maximum drawdown as a contingent claim, and price and hedge it accordingly as a derivative contract. Trading drawdown contracts or replicating them by hedging would directly address the concerns of portfolio managers who would like to insure the market drops. Similar contracts can be written on the maximum drawup (MDU). We show that buying a contract on MDD or MDU is equivalent to adopting a momentum trading strategy, while selling it corresponds to contrarian trading. We also discuss more complex products, such as plain vanilla options on the maximum drawdown or drawup, or related barrier options on MDD and MDU which we call crash and rally options, respectively.
The purpose of this note is twofold. Firstly to complete a recent accumulation of results con- cerning extended version of Ito's formula for any one dimensional Levy processes, X. Secondly, we use the latter to characterise the parabolic generator of X
This paper studies drawdown and drawup processes in a general diffusion model. The main result is a formula for the joint distribution of the running minimum and the running maximum of the process stopped at the time of the first drop of size a. As a consequence, we obtain the probabilities that a drawdown of size a precedes a drawup of size b and vice versa. The results are applied to several examples of diffusion processes, such as drifted Brownian motion, Ornstein-Uhlenbeck process, and Cox-Ingersoll-Ross process.
We develop a stochastic calculus on the plane with respect to the local times of a large class of Lévy processes. We can then extend to these Lévy processes an Itô formula that was established previously for Brownian motion. Our method provides also a multidimensional version of the formula. We show that this formula generates many "Itô formulas" that fit various problems. In the special case of a linear Brownian motion, we recover a recently established Itô formula that involves local times on curves. This formula is already used in financial mathematics.
We analyze the optimal risky investment policy for an investor who, at each point in time, wants to lose no more than a fixed percentage of the maximum value his wealth has achieved up to that time. In particular, if "M"t is the maximum level of wealth W attained on or before time "t", then the constraint imposed on his portfolio choice is that Wtα"M"t, where α is an exogenous number betweenα O and 1. We show that, for constant relative risk aversion utility functions, the optimal policy involves an investment in risky assets at time "t" in proportion to the "surplus""W"t - α"M"t. the optimal policy may appear similar to the constant-proportion portfolio insurance policy analyzed in Black and Perold (1987) and Grossman and Vila (1989). However, in those papers, the investor keeps his wealth above a "nonstochastic" floor "F" instead of a stochastic floor α"M"t. the "stochastic" character of the floor studied here has interesting effects on the investment policy in states of nature when wealth is at an all-time high; i.e., when Wt ="M"t. It can be shown that at "W"t="M"t, α"M"t is expected to grow at a faster rate than "W"t, and therefore the investment in the risky asset can be expected to fall. We also show that the investment in the risky asset can be expected to rise when "W"t is close to α"M"t. We conjecture that in an equilibrium model the stochastic character of the floor creates "resistance" levels as the market approaches an all-time high (because of the reluctance of investors to take more risk when "W"t="M"t). Copyright 1993 Blackwell Publishers.
In this note we consider first exit problems of completely asymmetric (reflected) Lvy processes and present an alternative derivation of their Laplace transforms essentially based on potential theory of Markov processes.
The {\em drawdown} process Y of a completely asymmetric L\'{e}vy process
X is equal to X reflected at its running supremum : . In this paper we explicitly express in terms of the scale function and the
L\'{e}vy measure of X the law of the sextuple of the first-passage time of
Y over the level , the time of the last supremum of
X prior to , the infimum \unl X_{\tau_a} and supremum \ovl
X_{\tau_a} of X at and the undershoot and
overshoot of Y at . As application we obtain explicit
expressions for the laws of a number of functionals of drawdowns and rallies in
a completely asymmetric exponential L\'{e}vy model.
We show how the change-of-variable formula with local time on curves derived recently in Peskir (2002) can be used to prove that the optimal stopping boundary for the American put option can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premium representation. This settles the question raised in Myneni (1992) and dating back to McKean (1965).
A Note on a Change of Variable Formula with Local Time-Space for Lévy Processes of Bounded Variation
- A E Kyprianou
- B Surya
Kyprianou, A.E. and Surya, B. (2006) A Note on a Change of Variable Formula with Local Time-Space for Lévy Processes
of Bounded Variation. Séminaire de Probabilites, XL:97-104.