Hedging Quantity Risks with Standard Power Options - UC Berkeley ...

706 Naval Research Logistics, Vol. 53 (2006)
where
B 1 (p) ≡
g(p) (
f p (p)
m2 − m 1
E Q[ g(p)
] = exp log p + m2 1 − m2 2
− (m 1 − m 2 ) 2 )
s 2 2s 2 s 2 f p (p)
= e − (m 1 −m 2 )(m 1 −3m 2 )
2s 2 p m 2 −m 1
s 2
B 2 (p) ≡ E[y(p, q)|p] =E[(r − p)q|p] =(r − p)
(
m + ρ u )
s (log p − m 1)
B 3 ≡ E Q [E[y(p, q)|p]]
(
= (r − E Q [p]) m − ρ u )
s m 1 + ρ u s (rEQ [log p]−E Q [p log p])
= ( r − e m ) ( 2+ 1 2 s2 m − ρ u )
s m 1 + ρ u (
rm2 − (m 2 + s 2 )e m )
2+ 1 2 s2 .
s
We have used the following formulas in the calculation.
E Q [log p] =m 2
E Q [p] =e m 2+ 1 2 s2
E Q [p log p] =(m 2 + s 2 )e m 2+ 1 2 s2
E Q [p 2 ]=e 2m 2+2s 2
1
(−
g(p)
f p (p) = ps √ sπ exp 1 2
1
ps √ sπ exp (− 1 2
( ) ) 2 log p−m2
s
(
log p−m1
s
) 2
) = exp
(
m2 − m 1
log p + m2 1 − )
m2 2
s 2 2s 2
[ ] ( g(p)
E Q m2 − m 1
= exp m
f p (p)
s 2 2 + m2 1 − m2 2
+ (m 2 − m 1 ) 2 ) ( (m1 − m 2 ) 2 )
= exp
2s 2 2s 2 s 2
We’ve also used q|p ∼ N ( m + ρ u s (log p − m 1), u 2 (1 − ρ 2 ) )
to obtain
Then, we can get the explicit functions for the optimal payoff
for the mean-variance utility:
ln E[e −ay(p,q) |p] ≡ln E[e −a(r−p)q |p]
(
=−a(r − p) m + ρ u )
s (log p − m 1)
+ 1 2 a2 (r − p) 2 u 2 (1 − ρ 2 ).
3.1.6. Bivariate Lognormal Distribution
for Price and Load
Suppose the marginal distributions of p and q, on the other
hand, follow bivariate lognormal distributions as follows:
Under P : log p ∼ N(m 1 , s 2 ), log q ∼ N ( m q , uq) 2 ,
Corr(log p, log q) = φ
Under Q : log p ∼ N(m 2 , s 2 ).
Naval Research Logistics DOI 10.1002/nav
where
x ∗ (p) = 1 a (1 − B 1(p)) − B ′ 2 (p) + B′ 3 B 1(p), (15)
B 2 ′ (p) ≡ E[y(p, q)|p] =E[(r − p)q|p]
= (r − p)e m q+φ uq s (log p−m 1)+ 1 2 u2 q (1−φ2 )
since log q|p ∼ N(m q + φ u q
s (log p − m 1), u 2 q (1 − φ2 )), and
B ′ 3 ≡ EQ [E[y(p, q)|p]]
= re m q+φ uq s (m 2−m 1 )+ 1 2 u2 q (1−φ2 )+ 1 2 φ2 u2 q
s 2 s2
− e m 2+m q +φ uq s (m 2−m 1 )+ 1 2 u2 q (1−φ2 )+ 1 2 (φ uq s +1)2 s 2 . (16)