Chapter 6
Commodity Forwards and Futures
Question 6.1
The spot price of a widget is $70.00. With a continuously compounded annual risk-free rate of
5%, we can calculate the annualized lease rates according to the formula:
F0,T = S0 × e( r −δl )×T
⇔
F0,T
S0
= e( r −δl )×T
⇔
⎛F ln ⎜ 0,T
⎝ S0
⎞
⎟ = ( r − δ l) × T
⎠
⇔
δl = r −
1 ⎛ F0,T ⎞ ln ⎜
⎟
T ⎝ S0 ⎠
Time to expiration Forward price Annualized lease rate
3 months
$70.70
0.0101987
6 months
$71.41
0.0101147
9 months
$72.13
0.0100336
12 months
$72.86
0.0099555
The lease rate is less than the risk-free interest rate. The forward curve is upward sloping, thus the prices of exercise 6.1 are an example of contango.
Question 6.2
The spot price of oil is $32.00 per barrel. With a continuously compounded annual risk-free rate of 2%, we can again calculate the lease rate according to the formula: δl = r −
Time to expiration Forward price Annualized lease rate
3 months
$31.37
0.0995355
6 months
$30.75
0.0996918
9 months
$30.14
0.0998436
12 months
$29.54
0.0999906
The lease rate is higher than the risk-free interest rate. The forward curve is downward sloping, thus the prices of exercise 6.2 are an example of backwardation.
Question 6.3
The question asks us to find the lease rate such that F0,T = S0. We take our pricing formula, F0,T =
S0 × e( r −δl )×T , and immediately see that the sought equality is established if e( r −δl )×T = 1, which is guaranteed for any T > 0 if and only if r = δ.
If the lease rate were 3.5%, the lease rate would be higher than the risk-free interest rate.
Therefore, a graph of forward prices would be downward sloping, and thus
