A Jump Diffusion Model for Option Pricing with Three Properties: Leptokurtic Feature, Volatility Smile, and Analytical Tractability

Brownian motion and normal distribution have been widely used, for example, in the Black-Scholes-Merton option pricing framework, to study the return of assets. However, two puzzles, emerged from many empirical investigations, have got much attention recently, namely (a) the leptokurtic feature that the return distribution of assets may have a higher peak and two (asymmetric) heavier tails than those of the normal distribution, and (b) an empirical abnormity called ``volatility smile'' in option pricing. To incorporate both the leptokurtic feature and ``volatility smile'', this paper proposes, for the purpose of studying option pricing, a jump diffusion model, in which the price of the underlying asset is modeled by two parts, a continuous part driven by Brownian motion, and a jump part with the logarithm of the jump sizes having a double exponential distribution. In addition to the above two desirable properties, leptokurtic feature and ``volatility smile'', the model is simple enough to produce analytical solutions for a variety of option pricing problems, including options, future options, and interest rate derivatives, such as caps and floors, in terms of the $Hh$ function. Although there are many models can incorporate some of the three properties (the leptokurtic feature, ``volatility smile'', and analytical tractability), the current model can incorporate all three under a unified framework.

A Conversation with Chris Heyde

Born in Sydney, Australia, on April 20, 1939, Chris Heyde shifted his interest from sport to mathematics thanks to inspiration from a schoolteacher. After earning an M.Sc. degree from the University of Sydney and a Ph.D. from the Australian National University (ANU), he began his academic career in the United States at Michigan State University, and then in the United Kingdom at the University of Sheffield and the University of Manchester. In 1968, Chris moved back to Australia to teach at ANU until 1975, when he joined CSIRO, where he was Acting Chief of the Division of Mathematics and Statistics. From 1983 to 1986, he was a Professor and Chairman of the Department of Statistics at the University of Melbourne. Chris then returned to ANU to become the Head of the Statistics Department, and later the Foundation Dean of the School of Mathematical Sciences (now the Mathematical Sciences Institute). Since 1993, he has also spent one semester each year teaching at the Department of Statistics, Columbia University, and has been the director of the Center for Applied Probability at Columbia University since its creation in 1993. Chris has been honored worldwide for his contributions in probability, statistics and the history of statistics. He is a Fellow of the International Statistical Institute and the Institute of Mathematical Statistics, and he is one of three people to be a member of both the Australian Academy of Science and the Australian Academy of Social Sciences. In 2003, he received the Order of Australia from the Australian government. He has been awarded the Pitman Medal and the Hannan Medal. Chris was conferred a D.Sc. honoris causa by University of Sydney in 1998. Chris has been very active in serving the statistical community, including as the Vice President of the International Statistical Institute, President of the Bernoulli Society and Vice President of the Australian Mathematical Society. He has served on numerous editorial boards, most notably as Editor of Stochastic Processes and Their Applications from 1983 to 1989, and as Editor-in-Chief of Journal of Applied Probability and Advances in Applied Probability since 1990.Comment: Published at http://dx.doi.org/10.1214/088342306000000088 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

How does the introduction of hidden orders affect limit order markets?

First author draf

An Annotated Bibliography of Recent Literature on Current Developments in Philanthropy

As philanthropic organizations play an increasingly important role in societies around the world, the research on philanthropy – from giving and volunteering practices to regulatory frameworks to digital innovations – has also evolved in recent decades. It is important to develop a thorough overview of the relevant scientific discourses and literature on current developments in philanthropy. This will allow researchers and practitioners to enhance the understanding of philanthropy and to improve its practice worldwide. This report provides new insights on current developments and important changes in the global philanthropic landscape, including trends in global philanthropy and its interaction with other sectors of society

Electrophysiologic Demonstration of an Atriofascicular Accessory Pathway

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/100317/1/j.1540-8159.1988.tb04538.x.pd

Concealed anterograde accessory pathway conduction during the induction of orthodromic reciprocating tachycardia

AbstractThe purpose of this study was to determine whether concealed anterograde accessory pathway conduction occurs during the induction of orthodromic tachycardia by an artrial extrastimulus (S2). Sixteen patients with an overt (n = 9) or concealed (n = 7) accessory pathway had inducible orthodromic tachycardia by S2during an atrial drive (S1) cycle length of 500 to 650 ms. A ventricular extrastimulus (S3) was introduced coincident with the His depolarization resulting from S2during the longest S1S2interval that reproducibly induced orthodromic tachycardia. The S1S3interval was decreased in 10 ms steps until S3reached ventricular refractoriness. Retrograde accessory pathway conduction of S3in the presence and absence of S2was compared at the same S1S3intervals.In the absence of S2there was retrograde accessory pathway conduction after S3in each patient. In the presence of S2, in patients with overt pre-excitation, retrograde accessory pathway conduction after S3was absent in one patient, prolonged in four patients and present only after long S1S3intervals in three patients. Only one patient had unchanged retrograde conduction regardless of the presence or absence of S2. In patients with a concealed accessory pathway, retrograde accessory pathway conduction after S3was absent in five patients and was prolonged in two. Thus, concealed anterograde accessory pathway conduction was present in 15 of 16 patients at the time of orthodromic tachycardia induction.In conclusion, concealed anterograde accessory pathway conduction occurs in a majority of patients with an overt or a concealed accessory pathway during induction of orthodromic tachycardia by an atrial extrastimulus. In some patients, the initiation of orthodromic tachycardia may depend on a critical interaction between the degree of concealed anterograde accessory pathway conduction and atrioventricular conduction delay after S2

Bitcoin mining and electricity consumption

We propose a dynamic industry equilibrium model for Bitcoin electricity consumption in a general framework, including Bitcoin miners’ optimal entry and exit with technology innovation. By adopting average operating costs as an approximation to the true operating costs, we overcome the difficulty of strong path-dependency due to the interaction among entry, exit, and technology innovation. The model can capture both the upside and downside co-movements of miners’ computing power, electricity consumption, and mining revenue. Our model shows that the Bitcoin electricity consumption will not grow indefinitely, with the ratio of Bitcoin electricity consumption to the miners’ revenue fluctuating within a range.First author draf

Robo-advising: a dynamic mean-variance approach

In contrast to traditional financial advising, robo-advising needs to elicit investors’ risk profile via several simple online questions and provide advice consistent with conventional investment wisdom, e.g., rich and young people should invest more in risky assets. To meet the two challenges, we propose to do the asset allocation part of robo-advising using a dynamic mean-variance criterion over the portfolio’s log returns. We obtain analytical and time-consistent optimal portfolio policies under jump-diffusion models and regime-switching models.Accepted manuscrip