164 research outputs found
A Note on Delta Hedging in Markets with Jumps
Modelling stock prices via jump processes is common in financial markets. In
practice, to hedge a contingent claim one typically uses the so-called
delta-hedging strategy. This strategy stems from the Black--Merton--Scholes
model where it perfectly replicates contingent claims. From the theoretical
viewpoint, there is no reason for this to hold in models with jumps. However in
practice the delta-hedging strategy is widely used and its potential
shortcoming in models with jumps is disregarded since such models are typically
incomplete and hence most contingent claims are non-attainable. In this note we
investigate a complete model with jumps where the delta-hedging strategy is
well-defined for regular payoff functions and is uniquely determined via the
risk-neutral measure. In this setting we give examples of (admissible)
delta-hedging strategies with bounded discounted value processes, which
nevertheless fail to replicate the respective bounded contingent claims. This
demonstrates that the deficiency of the delta-hedging strategy in the presence
of jumps is not due to the incompleteness of the model but is inherent in the
discontinuity of the trajectories.Comment: 16 pages, 1 figur
Asymptotic independence of three statistics of maximal segmental scores
Let be an iid sequence with negative mean. The
-segment is the subsequence and its
\textit{score} is given by . Let be the
largest score of any segment ending at time , the largest score of
any segment in the sequence , and the overshoot of
the score over a level at the first epoch the score of such a size arises.
We show that, under the Cram\'er assumption on , asymptotic independence
of the statistics , and holds as
. Furthermore, we establish a novel Spitzer-type
identity characterising the limit law in terms of the laws of
-scores. As corollary we obtain: (1) a novel factorization of the
exponential distribution as a convolution of and the stationary
distribution of ; (2) if (where is the
Cram\'er coefficient), our results, together with the classical theorem of
Iglehart \cite{Iglehart}, yield the existence and explicit form of the joint
weak limit of .Comment: 13 pages, no figure
Arbitrage-free prediction of the implied volatility smile
This paper gives an arbitrage-free prediction for future prices of an
arbitrary co-terminal set of options with a given maturity, based on the
observed time series of these option prices. The statistical analysis of such a
multi-dimensional time series of option prices corresponding to strikes
(with large, e.g. ) and the same maturity, is a difficult task
due to the fact that option prices at any moment in time satisfy non-linear and
non-explicit no-arbitrage restrictions. Hence any -dimensional time series
model also has to satisfy these implicit restrictions at each time step, a
condition that is impossible to meet since the model innovations can take
arbitrary values. We solve this problem for any n\in\NN in the context of
Foreign Exchange (FX) by first encoding the option prices at each time step in
terms of the parameters of the corresponding risk-neutral measure and then
performing the time series analysis in the parameter space. The option price
predictions are obtained from the predicted risk-neutral measure by effectively
integrating it against the corresponding option payoffs. The non-linear
transformation between option prices and the risk-neutral parameters applied
here is \textit{not} arbitrary: it is the standard mapping used by market
makers in the FX option markets (the SABR parameterisation) and is given
explicitly in closed form. Our method is not restricted to the FX asset class
nor does it depend on the type of parameterisation used. Statistical analysis
of FX market data illustrates that our arbitrage-free predictions outperform
the naive random walk forecasts, suggesting a potential for building management
strategies for portfolios of derivative products, akin to the ones widely used
in the underlying equity and futures markets.Comment: 18 pages, 2 figures; a shorter version of this paper has appeared as
a Technical Paper in Risk (30 April 2014) under the title "Smile
transformation for price prediction
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