116 research outputs found
The continuous behavior of the numeraire portfolio under small changes in information structure, probabilistic views and investment constraints
The numeraire portfolio in a financial market is the unique positive wealth
process that makes all other nonnegative wealth processes, when deflated by it,
supermartingales. The numeraire portfolio depends on market characteristics,
which include: (a) the information flow available to acting agents, given by a
filtration; (b) the statistical evolution of the asset prices and, more
generally, the states of nature, given by a probability measure; and (c)
possible restrictions that acting agents might be facing on available
investment strategies, modeled by a constraints set. In a financial market with
continuous-path asset prices, we establish the stable behavior of the numeraire
portfolio when each of the aforementioned market parameters is changed in an
infinitesimal way.Comment: 16 pages; revised versio
Free Lunch
The concept of absence of opportunities for free lunches is one of the
pillars in the economic theory of financial markets. This natural assumption
has proved very fruitful and has lead to great mathematical, as well as
economical, insights in Quantitative Finance. Formulating rigorously the exact
definition of absence of opportunities for riskless profit turned out to be a
highly non-trivial fact that troubled mathematicians and economists for at
least two decades. The purpose of this note is to give a quick (and,
necessarily, incomplete) account of the recent work aimed at providing a simple
and intuitive no-free-lunch assumption that would suffice in formulating a
version of the celebrated Fundamental Theorem of Asset Pricing.Comment: 3 pages; a version of this note will appear in the Encyclopaedia of
Quantitative Finance, John Wiley and Sons In
Valuation and parities for exchange options
Valuation and parity formulas for both European-style and American-style
exchange options are presented in a general financial model allowing for jumps,
possibility of default and "bubbles" in asset prices. The formulas are given
via expectations of auxiliary probabilities using the change-of-numeraire
technique. Extensive discussion is provided regarding the way that folklore
results such as Merton's no-early-exercise theorem and traditional parity
relations have to be altered in this more versatile framework.Comment: 19 page
On the stochastic behaviour of optional processes up to random times
In this paper, a study of random times on filtered probability spaces is
undertaken. The main message is that, as long as distributional properties of
optional processes up to the random time are involved, there is no loss of
generality in assuming that the random time is actually a randomised stopping
time. This perspective has advantages in both the theoretical and practical
study of optional processes up to random times. Applications are given to
financial mathematics, as well as to the study of the stochastic behaviour of
Brownian motion with drift up to its time of overall maximum as well as up to
last-passage times over finite intervals. Furthermore, a novel proof of the
Jeulin-Yor decomposition formula via Girsanov's theorem is provided.Comment: Published in at http://dx.doi.org/10.1214/13-AAP976 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Stochastic discount factors
The valuation process that economic agents undergo for investments with
uncertain payoff typically depends on their statistical views on possible
future outcomes, their attitudes 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 discount
factor, then an economically consistent theory can be developed. Asset
valuation becomes a matter of randomly discounting payoffs under different
states of nature and weighing them according 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.Comment: 12 pages. To appear in the "Encyclopedia of Quantitative Finance
Finitely additive probabilities and the Fundamental Theorem of Asset Pricing
This work aims at a deeper understanding of the mathematical implications of
the economically-sound condition of absence of arbitrages of the first kind in
a financial market. In the spirit of the Fundamental Theorem of Asset Pricing
(FTAP), it is shown here that absence of arbitrages of the first kind in the
market is equivalent to the existence of a finitely additive probability,
weakly equivalent to the original and only locally countably additive, under
which the discounted wealth processes become "local martingales". The
aforementioned result is then used to obtain an independent proof of the FTAP
of Delbaen and Schachermayer. Finally, an elementary and short treatment of the
previous discussion is presented for the case of continuous-path semimartingale
asset-price processes.Comment: 14 pages. Dedicated to Prof. Eckhard Platen, on the occasion of his
60th birthday. This is the 2nd part of what comprised the older arxiv
submission arXiv:0904.179
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