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Lévy Processes in Finance: Theory, Numerics, and Empirical Facts
, 2000
"... Lévy processes are an excellent tool for modelling price processes in mathematical finance. On the one hand, they are very flexible, since for any time increment ∆t any infinitely divisible distribution can be chosen as the increment distribution over periods of time ∆t. On the other hand, they have ..."
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Cited by 32 (2 self)
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Lévy processes are an excellent tool for modelling price processes in mathematical finance. On the one hand, they are very flexible, since for any time increment ∆t any infinitely divisible distribution can be chosen as the increment distribution over periods of time ∆t. On the other hand, they have a simple structure in comparison with general semimartingales. Thus stochastic models based on Lévy processes often allow for analytically or numerically tractable formulas. This is a key factor for practical applications. This thesis is divided into two parts. The first, consisting of Chapters 1, 2, and 3, is devoted to the study of stock price models involving exponential Lévy processes. In the second part, we study term structure models driven by Lévy processes. This part is a continuation of the research that started with the author's diploma thesis Raible (1996) and the article Eberlein and Raible (1999). The content of the chapters is as follows. In Chapter 1, we study a general stock price model where the price of a single stock follows an exponential Lévy process. Chapter 2 is devoted to the study of the Lévy measure of infinitely divisible distributions, in particular of generalized hyperbolic distributions. This yields information about what changes in the distribution of a generalized hyperbolic Lévy motion can be achieved by a locally equivalent change of the underlying probability measure. Implications for
A Survey and Some Generalizations of Bessel Processes
 Bernoulli
, 1999
"... Bessel processes play an important role in financial mathematics because of their strong relation to financial processes like geometric Brownian motion or CIR processes. We are interested in the first time Bessel processes and more generally, radial OrnsteinUhlenbeck processes hit a given barrier. ..."
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Cited by 27 (1 self)
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Bessel processes play an important role in financial mathematics because of their strong relation to financial processes like geometric Brownian motion or CIR processes. We are interested in the first time Bessel processes and more generally, radial OrnsteinUhlenbeck processes hit a given barrier. We give explicit expressions of the Laplace transforms of first hitting times by (squared) radial OrnsteinUhlenbeck processes, i. e., CIR processes. As a natural extension we study squared Bessel processes and squared OrnsteinUhlenbeck processes with negative dimensions or negative starting points and derive their properties. Keywords: First hitting times; CIR processes; Bessel processes; radial Ornstein Uhlenbeck processes; Bessel processes with negative dimensions 1 Introduction Bessel processes have come to play a distinguished role in financial mathematics for at least two reasons, which have a lot to do with the models being usually considered. One of these models is the CoxI...
Asset Price Bubbles in Complete Markets
"... This paper reviews and extends the mathematical finance literature on bubbles in complete markets. We provide a new characterization theorem for bubbles under the standard no arbitrage (NFLVR) framework, showing that bubbles can be of three types. Type 1 bubbles are uniformly integrable martingale ..."
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Cited by 15 (4 self)
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This paper reviews and extends the mathematical finance literature on bubbles in complete markets. We provide a new characterization theorem for bubbles under the standard no arbitrage (NFLVR) framework, showing that bubbles can be of three types. Type 1 bubbles are uniformly integrable martingales, and these can exist with an infinite lifetime. Type 2 bubbles are nonuniformly integrable martingales, and these can exist for a finite, but unbounded, lifetime. Last, type 3 bubbles are strict local martingales, and these can exist for a finite lifetime only. When one adds a no dominance assumption (from Merton [24]), only type 1 bubbles remain. In addition, under Merton’s no dominance hypothesis, putcall parity holds and there are no bubbles in standard call and put options. Our analysis implies that if one believes asset price bubbles exist and are an important economic phenomena, then asset markets must be incomplete.
Strict local martingales, bubbles, and no early exercise
, 2007
"... We show pathological behavior of asset price processes modeled by continuous strict local martingales under a riskneutral measure. The inspiration comes from recent results on financial bubbles. We analyze, in particular, the effect of the strict nature of the local martingale on the usual formula ..."
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Cited by 5 (0 self)
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We show pathological behavior of asset price processes modeled by continuous strict local martingales under a riskneutral measure. The inspiration comes from recent results on financial bubbles. We analyze, in particular, the effect of the strict nature of the local martingale on the usual formula for the price of a European call option, especially a strong anomaly when call prices decay monotonically with maturity. A complete and detailed analysis for the archetypical strict local martingale, the reciprocal of a three dimensional Bessel process, has been provided. Our main tool is based on a general htransform technique (due to Delbaen and Schachermayer) to generate positive strict local martingales. This gives the basis for a statistical test to verify a suspected bubble is indeed one (or not).
On collisions of Brownian particles
 Annals of Applied Probability
, 2010
"... Abstract. We examine the behavior of n Brownian particles diffusing on the real line with bounded, measurable drift and bounded, piecewise continuous diffusion coefficients that depend on the current configuration of particles. Sufficient conditions are established for the absence, and for the prese ..."
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Cited by 5 (2 self)
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Abstract. We examine the behavior of n Brownian particles diffusing on the real line with bounded, measurable drift and bounded, piecewise continuous diffusion coefficients that depend on the current configuration of particles. Sufficient conditions are established for the absence, and for the presence, of triple collisions among the particles. As an application to the Atlas model for equity markets, we study a special construction of such systems of diffusing particles using Brownian motions with reflection on polyhedral domains. 1.
ON THE MARTINGALE PROPERTY OF CERTAIN LOCAL MARTINGALES: CRITERIA AND APPLICATIONS
, 905
"... Abstract. The stochastic exponential Zt = exp{Mt − M0 − (1/2)〈M, M〉t} of a continuous local martingale M is itself a continuous local martingale. We give a necessary and sufficient condition for the process Z to be a true martingale in the case where Mt = R t b(Yu) dWu and 0 Y is a onedimensional d ..."
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Cited by 1 (0 self)
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Abstract. The stochastic exponential Zt = exp{Mt − M0 − (1/2)〈M, M〉t} of a continuous local martingale M is itself a continuous local martingale. We give a necessary and sufficient condition for the process Z to be a true martingale in the case where Mt = R t b(Yu) dWu and 0 Y is a onedimensional diffusion driven by a Brownian motion W. Furthermore, we provide a necessary and sufficient condition for Z to be a uniformly integrable martingale in the same setting. These conditions are deterministic and expressed only in terms of the function b and the drift and diffusion coefficients of Y. We also classify, via deterministic necessary and sufficient conditions, when the process Z is a.s. strictly positive, when its limit Z ∞ is a.s. strictly positive, and when Z∞ is a.s. zero. This allows us to obtain a deterministic necessary and sufficient condition in the onedimensional setting for a discounted stock price to be a true martingale under the riskneutral measure, and for it to be a uniformly integrable martingale. These results enable us to ascertain the existence of financial bubbles in diffusionbased models. Finally, we
STOCHASTIC DISCOUNT FACTORS CONSTANTINOS KARDARAS
, 1001
"... Abstract. The valuation process that economic agents undergo for investments with uncertain payofftypicallydependson theirstatistical views on possible futureoutcomes, theirattitudes toward risk, and, of course, the payoff structure itself. Yields vary across different investment opportunities and t ..."
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Abstract. The valuation process that economic agents undergo for investments with uncertain payofftypicallydependson theirstatistical views on possible futureoutcomes, theirattitudes toward risk, and, of course, the payoff structure itself. Yields vary across different investment opportunities and their interrelations are difficult to explain. For the same agent, a different discounting factor has to be used for every separate valuation occasion. If, however, one is ready to accept discounting that varies randomly with the possible outcomes, and therefore accepts the concept of a stochastic discountfactor, thenan economically consistent theorycan bedeveloped. Asset valuationbecomes a matter of randomlydiscountingpayoffs underdifferentstates ofnature and weighing themaccording to the agent’s probability structure. The advantages of this approach are obvious, since a single discounting mechanism suffices to describe how any asset is priced by the agent. Within active and liquid financial markets, economic agents are able to make investment decisions. Capital is allocated today in exchange for some future income stream. If there is no uncertainty regarding the future payoff of an investment opportunity, the yield that will be asked on the investment will equal the riskfree interest rate prevailing for the time period covering the
A note about conditional OrnsteinUhlenbeck processes
, 2008
"... In this short note, the identity in law, which was obtained by P. Salminen [6], between on one hand, the OrnsteinUhlenbeck process with parameter γ, killed when it reaches 0, and on the other hand, the 3dimensional radial OrnsteinUhlenbeck process killed exponentially at rate γ and conditioned to ..."
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In this short note, the identity in law, which was obtained by P. Salminen [6], between on one hand, the OrnsteinUhlenbeck process with parameter γ, killed when it reaches 0, and on the other hand, the 3dimensional radial OrnsteinUhlenbeck process killed exponentially at rate γ and conditioned to hit 0, is derived from a simple absolute continuity relationship. Keywords: OrnsteinUhlenbeck process, Doob’s htransform, absolute continuity relationship.
TimeChanged Bessel Processes and Credit Risk ∗
, 2006
"... The Constant Elasticity of Variance (CEV) model is mathematically presented and then used in a CreditEquity hybrid framework. Next, we propose extensions to the CEV model with default: firstly by adding a stochastic volatility diffusion uncorrelated from the stock price process, then by more genera ..."
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The Constant Elasticity of Variance (CEV) model is mathematically presented and then used in a CreditEquity hybrid framework. Next, we propose extensions to the CEV model with default: firstly by adding a stochastic volatility diffusion uncorrelated from the stock price process, then by more generally time changing Bessel processes and finally by correlating stochastic volatility moves to the stock ones. Properties about strict local and true martingales in this study are discussed. Analytical formulas are provided and Fourier and Laplace transform techniques can then be used to compute option prices and probabilities of default. 1