Adesi, Giannopoulos, and Vosper (1999). The basic idea is to adopt a GARCH-type model
of portfolio variance. Taking a series of n past returns, one can estimate the GARCH
model with past data and then standardize each past return, R
tt
, based on the estimated
standard deviation for the same day; in this way a series of past standardized returns
(which do not necessarily follow a normal distribution or any other known distribution)
are derived. At the end of the sample, one could obtain the volatility estimate for the next
day, s
t+1
. Rather than generating a distribution of future returns by assuming a certain
return distribution, the risk manager can extract, with replacement, past values of histori-
cal standardized returns. In this way the ability to generate as many scenarios as needed
and to account for time-varying volatility can be combined while avoiding any formal
assumption on return distribution. If the risk manager needs to generate a distribution of
returns over a T-day horizon, he could simply build n paths of T daily returns where the
same GARCH-type model is used in each of the T days to estimate next-day volatility
given the random return of the preceding day.
3.3 Value at Risk for Option Positions
3.3.1 Problems in Option VaR Measurement
Option contracts comprise a wide variety of fi nancial contracts. In the simpler case of a
plain-vanilla call (or put) option, the holder pays a fi xed premium to the options writer
and then retains the right to buy (or sell) a given amount of the underlying asset at a given
price — defi ned as strike price in the future, either on a single date (for European
options) or on any date before maturity (for American options). Therefore, potential gains
and losses for the holder of the option are asymmetric. In fact, maximum loss is equal to
the premium paid (plus interest, if the premium is paid up front) if the option is not exer-
cised at or before maturity, while gains are almost unlimited if the underlying price rises
(for the call option) or falls (for the put option). The opposite is true for the writer. This
asymmetric feature intuitively implies that options’ reaction to changes in underlying
asset prices would also not be symmetric and is one of the clear sources of problems when
estimating options’ VaR. More precisely, the three key issues are the following.
1. The relationship between the price of the underlying asset and the value of the
option is nonlinear (and may be convex or concave, depending on whether the
trader is long or short the option).
2. Moreover, the same relationship may also be nonmonotonic, so extreme losses do
not always occur as a consequence of extreme movements of the underlying asset.
3. The price of the option is also exposed to other risk factors, such as time decay
and the level of implied volatility.
The fi rst and the third problems are very easy to explain. Let us consider a simple
European call option on a non-dividend-paying stock, whose value c can be evaluated
through the Black–Scholes–Merton (BSM) formula,
c SNd e XNd T
rT
=−
() ( )
σ
where S is the spot price of the underlying asset, r is the risk-free rate, T is the
options time to maturity, N(x) represents the cumulative distribution function of a
VALUE AT RISK FOR OPTION POSITIONS 51

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