Q.
2626 T wo managers - X and Y - are looking to establish the 1-day VaR for a long position in an at-
the-money call option on a non-dividend-paying stock with the following information: Current stock
price: USD 100
Estimated annual stock return volatility: 15%
Current Black-Scholes-Merton call option value: USD 4.80,
Call option delta: 0.5
To compute VaR, manager X uses the delta-normal model, while manager Y opts for the Monte Carlo
simulation method for full revaluation. Which manager will estimate a higher value for the 1-day 99%
VaR?
A. Manager Y.
B. Manager X.
C. Both managers will have the same VaR estimate.
D. Insufficient information to determine.
T he correct answer is B.
Manager X, who uses the delta-normal model, will estimate a higher value for the 1-day 99% VaR.
T his is because options are nonlinear derivatives, meaning their value is related to the market price
of the underlying variable in a convex, non-linear relationship. T he payoff of such products varies
not only with the value of the underlying, but also with other elements such as interest rates,
volatility, and dividends. T he option’s price function is convex with respect to the value of the
underlying. For such a non-linear portfolio, the delta-normal model provides only a linear
approximation which does not capture the positive effect of this curvature on the portfolio value. It
understates the probability of high option values and overstates the probability of low option values.
T herefore, for a long position in the call option, VaR and the expected shortfall under the delta
normal model will be extremely high. On the other hand, for a short position in the call option, the
VaR and the expected shortfall under the delta-normal model will be extremely low.
Choi ce A i s i ncorrect. Manager Y, who uses the Monte Carlo simulation method for full
revaluation, is not likely to estimate a higher value for the 1-day 99% VaR. T his method takes into
account all possible price paths and their associated probabilities, which includes both large and small
changes in the option's value. T herefore, it tends to produce a more conservative (lower) VaR
estimate compared to the delta-normal model.
Choi ce C i s i ncorrect. Both managers will not have the same VaR estimate because they are using
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