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

t−t

, 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 option’s 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

option’s time to maturity, N(x) represents the cumulative distribution function of a

VALUE AT RISK FOR OPTION POSITIONS 51

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