Computations of Greeks in a market with jumps via the Malliavin calculus

Using the Malliavin calculus on Poisson space we compute Greeks in a market driven by a discontinuous process with Poisson jump times and random jump sizes, following a method initiated on the Wiener space in [5]. European options do not satisfy the regularity conditions required in our approach, however we show that Asian options can be considered due to a smoothing effect of the integral over time. Numerical simulations are presented for the Delta and Gamma of Asian options, and confirm the efficiency of this approach over classical finite difference Monte-Carlo approximations of derivatives.Received: July 2003, Mathematics Subject Classification (1991): 90A09, 90A12, 90A60, 60H07JEL Classification: C15, G12We thank M. Coutaud for contributions to the simulations.     

On the Malliavin approach to Monte Carlo approximation of conditional expectations

Given a multi-dimensional Markov diffusion X, the Malliavin integration by parts formula provides a family of representations of the conditional expectation E[g(X ₂)|X₁]. The different representations are determined by some localizing functions. We discuss the problem of variance reduction within this family. We characterize an exponential function as the unique integrated mean-square-error minimizer among the class of separable localizing functions. For general localizing functions, we prove existence and uniqueness of the optimal localizing function in a suitable Sobolev space. We also provide a PDE characterization of the optimal solution which allows to draw the following observation : the separable exponential function does not minimize the integrated mean square error, except for the trivial one-dimensional case. We provide an application to a portfolio allocation problem, by use of the dynamic programming principle.Mathematics Subject Classification: 60H07, 65C05, 49-00JEL Classification: G10, C10The authors gratefully acknowledge for the comments raised by an anonymous referee, which helped understanding the existence result of Sect. [4.2] of this paper.     

When and how consumers are willing to exchange data with retailers: An exploratory segmentation

Consumer data is a crucial resource for retailers. Yet accessing this data increasingly requires consumers to willingly participate in data exchange. This paper draws on social exchange theory and privacy calculus to investigate differences in consumer willingness to exchange data with retailers. Consumers are also profiled on their perceptions of retailer's use and abuse of data, along with the antecedents and outcomes of these perceptions. We employ a cross-sectional quantitative survey and collect data from a sample of 463 US consumers. For statistical analysis, we employ a latent class segmentation and identify six consumer segments which differ in their perceptions of the consumer-retailer data exchange. The key drivers of these segment differences include privacy concerns, technology readiness, and general trust of, and engagement with, retail brands. The segments also differ in their subsequent views towards the use and abuse of their data by retailers, including willingness to exchange data. Hence, when accessing and utilizing consumer data, it is important that brands identify different segments, and adapt their approach accordingly.