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FRM一级练习题：估值与风险模型-Delta-normal Approach

来源：高顿网校 2016-03-04

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.

Answer:D

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.

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