AN AT-THE-MONEY EUROPEAN CALL OPTION ON THE DJ EURO STOXX 50 INDEX...
12.
An at-the-money European call option on the DJ EURO STOXX 50 index with a strike of 2200 and maturing in
1 year is trading at EUR 350, where contract value is determined by EUR 10 per index point. The risk-free rate
is 3% per year, and the daily volatility of the index is 2.05%. If we assume that the expected return on the DJ
EURO STOXX 50 is 0%, the 99% 1-day VaR of a short position on a single call option calculated using the
delta-normal approach is closest to:
a.
EUR 8.
b.
EUR 53.
c.
EUR 84.
d.
EUR 525.
Correct Answer: d
Rationale:
Since the option is at-the-money, the delta is close to 0.5. Therefore a 1 point change in the index would
translate to approximately 0.5 * EUR 10 = EUR 5 change in the call value.
Therefore, the percent delta, also known as the local delta, defined as %D = (5/350) / (1/2200) = 31.4.
So the 99% VaR of the call option = %D * VaR(99% of index) = %D * call price * alpha (99%) * 1-day volatility = 31.4 *
EUR 350 * 2.33 * 2.05% = EUR 525. The term alpha (99%) denotes the 99th percentile of a standard normal
distribution, which equals 2.33.
There is a second way to compute the VaR. If we just use a conversion factor of EUR 10 on the index, then we can
use the standard delta, instead of the percent delta:
VaR(99% of Call) = D * index price * conversion * alpha (99%) * 1-day volatility = 0.5 * 2200 * 10 * 2.33 * 2.05% =
EUR 525, with some slight difference in rounding.
Both methods yield the same result.
Section:
Valuation and Risk Models
Reference:
Linda Allen, Jacob Boudoukh and Anthony Saunders, Understanding Market, Credit and Operational
Risk: The Value at Risk Approach, Chapter 3, “Putting VaR to Work.”
Learning Objective:
Compare delta-normal and full revaluation approaches for computing VaR.
2015 Financial Risk Manager (FRM®) Practice Exam