Contemporary Mathematics
Hedge Funds as Knock-Out Options
Marcos Escobar, Stefan Kr¨ amer, Florian Scheibl, Luis Seco,
and Rudi Zagst
Abstract. This paper introduces a new theoretical framework to price hedge
funds’ equity. It is inspired on the famous framework of Black and Cox for the
valuation of companies’ equity as call options. Our structural model describing
hedge funds uses barrier options (i.e. down-and-out call options as well as up-
and-out put options) to allow for the special structure of hedge funds’ debt
position. The quality of these models is evaluated by its capability to reproduce
a high range of historical hedge fund returns. Different variations of the model
are compared using this criteria. To fit the model parameter to real hedge
fund data the method of moments is used. The application of the models to a
set of over 1000 hedge funds showed that the model fits the first four moments
of the real returns quite well. Especially the documented stylized features of
high kurtosis and skewness are very well captured by this model.
1. Introduction
Although the beginnings of hedge funds are now more than 50 years ago they
had their breakthrough in the last decade. Especially big crashes like that of famous
LTCM led to more public attention on hedge
funds.1
Again, in the subprime crisis
in the summer of 2007 hedge funds got return to the public eye: Some for gaining
tremendous returns because they anticipated the crisis, others for being closed
because of
bankruptcy.2
It is very difficult to explain the past returns of hedge funds because of two
reasons, first a lack of information about the company’s portfolio, secondly the
the sparsity of data. This is the nature of hedge funds. The strategy is the core
asset of a hedge fund. Often anomalies in the capital markets or opportunities
for statistical arbitrage are exploited. Therefore it is obvious that predicting the
behaviour of hedge funds is an even more challenging task in the future. Most
models try to explain hedge fund returns using very sophisticated and complex
approaches. For example, several techniues have been developed to identify the
structure of the portfolio implicitly by observing the historical returns of the funds.
Needless to say this approach is doom in most cases because of lack of information.
Key words and phrases. Hedge fund replication, barrier options, leverage, fitting.
1cp.
Edwards (1999)
2cp.
Acharya (2007), cp. Teo (2007)
1
Contemporary Mathematics
Volume 515, 2010
1
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