Question 5 (3 points)
A bank has a portfolio of options on an asset. The delta of the options is -30
and the gamma is -5. The asset price is 20 and its volatility is 1% per day.
(a)Using the Quadratic model calculate the first three moments of the
change in the portfolio value.
E
(
∆ P
)
=.5
(
20
) (
−5
) (
0.01
)
2
=−0.1
E
(
∆ P2
)
=
(
20
)
2¿
E
(
∆ P
3
)
=4.5
(
20
)
4
¿
Use the Cornish Fisher Expansion to arrive at the three moments of the
change in portfolio value given by the quadratic model.
up=E
(
∆ P
)
=−0.1
σp=
√
¿¿
ξp=−32.415−3×36.03 ×
(
−0.1
)
+2¿¿
(b)Calculate a 1-day 99% VaR using the first two moments.
1Day 99 % VaR=u
p
+z
q
× σ
p
=
(
−0.1
)
−2.33 ×6.002=14.08
(c) Calculate a 1-day 99% VaR using the first three moments.
w
q
=−2.33+1
6
(
(
−2.33
)
2
−1
)
×
(
−0.1
)
=−2.404
1Day 99 % VaR=up+wq× σ p=
(
−0.1
)
+
(
−2.404
)
×6.002=14.529
