Derivatives -- the View from the Trenches
Derivatives -- the View from the Trenches
Derivatives -- the View from the Trenches
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Theorem 2 and <strong>the</strong> Equity <strong>Derivatives</strong> Markets<br />
The implied volatility has to satisfy<br />
Q<br />
0 E [ V( T) V(0)]<br />
1<br />
E u S u V u du<br />
2<br />
Q<br />
T<br />
2 2 2<br />
[ ( ( ) ) ( ) ( )|<br />
0<br />
SS , 0<br />
]<br />
<br />
T<br />
Q<br />
E [ V( u)| dN( u)]<br />
du<br />
0<br />
, 0<br />
where Q in this case is <strong>the</strong> "market" risk neutral measure.<br />
Time for a few (very) rough calculations: Set<br />
we hold a log-contract. We get<br />
V<br />
ln S, that is<br />
1 1<br />
<br />
E S E S<br />
2<br />
S<br />
2 2 Q<br />
Q<br />
0 ( ) ( [ ln ] [ ])<br />
1 (<br />
2 2 ) (<br />
Q [ln(1 )]<br />
Q<br />
E I E [ I ])<br />
<br />
2<br />
1 2 2 1 2<br />
( ) m<br />
2 2<br />
2 2 2<br />
m<br />
( <br />
)<br />
So if for S&P 500<br />
- Implied volatility is 0.20<br />
- Historical volatility is 0.17<br />
<strong>the</strong>n we have<br />
Quiz: which one<br />
m<br />
0.1 +/-33%<br />
1 +/-11%<br />
10 +/-3%<br />
16
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