The stochastic exponential Z t = exp{M t − M 0 − (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 M t = t 0 b(Y u ) dW u and Y is a one-dimensional 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 . As an application we provide a deterministic criterion for the absence of bubbles in a one-dimensional setting.

In this paper, we present an algorithm for pricing barrier options in one-dimensional Markov models. The approach rests on the construction of an approximating continuous-time Markov chain that closely follows the dynamics of the given Markov model. We illustrate the method by implementing it for a range of models, including a local Lévy process and a local volatility jump-diffusion. We also provide a convergence proof and error estimates for this algorithm.

In this paper, we prove an approximate formula expressed in terms of elementary functions for the implied volatility in the Heston model. The formula consists of the constant and first-order terms in the large maturity expansion of the implied volatility function. The proof is based on saddlepoint methods and classical properties of holomorphic functions.

The drawdown process Y of a completely asymmetric Lévy process X is equal to X reflected at its running supremum X: Y = X − X. In this paper we explicitly express in terms of the scale function and the Lévy measure of X the law of the sextuple of the first-passage time of Y over the level a > 0, the time Gτ a of the last supremum of X prior to τa, the infimum X τa and supremum Xτ a of X at τa and the undershoot a − Yτ a− and overshoot Yτ a − a of Y at τa. As application we obtain explicit expressions for the laws of a number of functionals of drawdowns and rallies in a completely asymmetric exponential Lévy model. 2000 Mathematics Subject Classification. 60G51, 60G17.1 2 ALEKSANDAR MIJATOVIĆ AND MARTIJN R. PISTORIUS process as an investment performance criterion. Checkhlov et al. [14] introduced a one-parameter family of portfolio risk measures that was called conditional drawdown and defined to be equal to the mean of a percentage of the worst portfolio drawdowns. Pospisil et al. [32] proposed the probability of a drawdown of a given size occurring before a rally of a given size as risk measure, and calculated this probability in the setting of one-dimensional diffusion models; the finite horizon case for Brownian motion was treated by Zhang & Hadjiliadis [37]. A number of papers has been devoted to a distributional study of functionals of the drawdown process. The joint Laplace of the time to a given drawdown and the running maximum of a Brownian motion with drift was derived by Taylor [36]; this joint law was obtained by Lehoczky [24] in the case of a general diffusion. An explicit expression for the expectation and the density of the maximum drawdown of Brownian motion was derived by Douady et al. [16]; the case of Brownian motion with drift was covered by Magdon et al. [26] where also the large time asymptotics of the expectation were derived. The drawdown process also features in the solution of a number of optimal investment problems. Under the geometric Brownian motion model the optimal time to exercise the Russian option, which pays out the largest historical value of the stock at the moment of exercise, was shown by Shepp & Shiryaev [34] to be given by the first-passage of a drawdown process over a certain constant level. Such a first-passage time is also optimal when linear cost is included (Meilijson [27]), or under a spectrally negative Lévy model for the stock price (Avram et al. [3]). Although still widely used as benchmark, mainly on account of its analytical tractability, it is by now well established that many features of Samuelson's classical geometric Brownian motion model for the price of a stock are not supported by empirical data. A class of tractable models that captures typical features of stock returns data such as fat tails, asymmetry and excess kurtosis is that of exponential Lévy processes. This class has received considerable attention in the literature -we refer to Cont & Tankov [15] and Boyarchenko & Levendorskii [8] for background and references.By restricting ourselves to Lévy pr...