152 research outputs found
Back to basics: historical option pricing revisited
We reconsider the problem of option pricing using historical probability
distributions. We first discuss how the risk-minimisation scheme proposed
recently is an adequate starting point under the realistic assumption that
price increments are uncorrelated (but not necessarily independent) and of
arbitrary probability density. We discuss in particular how, in the Gaussian
limit, the Black-Scholes results are recovered, including the fact that the
average return of the underlying stock disappears from the price (and the
hedging strategy). We compare this theory to real option prices and find these
reflect in a surprisingly accurate way the subtle statistical features of the
underlying asset fluctuations.Comment: 14 pages, 2 .ps figures. Proceedings, to appear in Proc. Roy. So
Hedged Monte-Carlo: low variance derivative pricing with objective probabilities
We propose a new `hedged' Monte-Carlo (HMC) method to price financial
derivatives, which allows to determine simultaneously the optimal hedge. The
inclusion of the optimal hedging strategy allows one to reduce the financial
risk associated with option trading, and for the very same reason reduces
considerably the variance of our HMC scheme as compared to previous methods.
The explicit accounting of the hedging cost naturally converts the objective
probability into the `risk-neutral' one. This allows a consistent use of purely
historical time series to price derivatives and obtain their residual risk. The
method can be used to price a large class of exotic options, including those
with path dependent and early exercise features.Comment: LaTeX, 10 pp, 3 eps figures (in text
Closure Theorem for Sequential-Design Processes
This chapter focuses on stochastic control and decision processes that occur in a variety of theoretical and applied contexts, such as statistical decision problems, stochastic dynamic programming problems, gambling processes, optimal stopping problems, stochastic adaptive control processes, and so on. It has long been recognized that these are all mathematically closely related. That being the case, all of these decision processes can be viewed as variations on a single theoretical formulation. The chapter presents some general conditions under which optimal policies are guaranteed to exist. The given theoretical formulation is flexible enough to include most variants of the types of processes. In statistical problems, the distribution of the observed variables depends on the true value of the parameter. The parameter space has no topological or other structure here; it is merely a set indexing the possible distributions. Hence, the formulation is not restricted to those problems known in the statistical literature as parametric problems. In nonstatistical contexts, the distribution does not depend on an unknown parameter. All such problems may be included in the formulation by the device of choosing the parameter space to consist of only one point, corresponding to the given distribution
Elements for a Theory of Financial Risks
Estimating and controlling large risks has become one of the main concern of
financial institutions. This requires the development of adequate statistical
models and theoretical tools (which go beyond the traditionnal theories based
on Gaussian statistics), and their practical implementation. Here we describe
three interrelated aspects of this program: we first give a brief survey of the
peculiar statistical properties of the empirical price fluctuations. We then
review how an option pricing theory consistent with these statistical features
can be constructed, and compared with real market prices for options. We
finally argue that a true `microscopic' theory of price fluctuations (rather
than a statistical model) would be most valuable for risk assessment. A simple
Langevin-like equation is proposed, as a possible step in this direction.Comment: 22 pages, to appear in `Order, Chance and Risk', Les Houches (March
1998), to be published by Springer/EDP Science
Growth Optimal Investment and Pricing of Derivatives
We introduce a criterion how to price derivatives in incomplete markets,
based on the theory of growth optimal strategy in repeated multiplicative
games. We present reasons why these growth-optimal strategies should be
particularly relevant to the problem of pricing derivatives. We compare our
result with other alternative pricing procedures in the literature, and discuss
the limits of validity of the lognormal approximation. We also generalize the
pricing method to a market with correlated stocks. The expected estimation
error of the optimal investment fraction is derived in a closed form, and its
validity is checked with a small-scale empirical test.Comment: 21 pages, 5 figure
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Hedging by sequential regressions revisited
Almost 20 years ago Föllmer and Schweizer (1989) suggested a simple and influential scheme for the computation of hedging strategies in an incomplete market. Their approach of local risk minimization results in a sequence of one-period least squares regressions running recursively backwards in time. In the meantime there have been significant developments in the global risk minimization theory for semimartingale price processes. In this paper we revisit hedging by sequential regression in the context of global risk minimization, in the light of recent results obtained by Černý and Kallsen (2007). A number of illustrative numerical examples is given
Compactness of the space of non-randomized policies in countable-state sequential decision processes
Ein warhafftige Historien von Zweyen Mewssen (1543)
Die "Warhafftige Historien" (1543) des Protestanten Burkard Waldis reiht sich in das konfessionspolemische Schrifttum des 16. Jahrhunderts ein. Berichtet wird von dem angeblichen Skandal, dass zwei Mäuse durch katholische Geistliche verbrannt worden seien, weil sie die Hostien in einem Kirchentabernakel aufgefressen hätten.
Diese online-Edition bietet eine kritische Textausgabe, eine ausführliche Einleitung sowie einen gründlichen Wort- und Sachkommentar
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