B4
Black Model
Value European options on futures, caplets and floorlets, caps and floors, and payer and receiver swaptions with Black's formula on a forward price or rate, with the parity check for each.
Learning focus: the Black model prices an option on a forward quantity (a futures price, a forward MRR or a forward swap rate) and then discounts the payoff from when it is paid, so no carry term is needed.
Instrument
Option on futures
yrs
%
%
Call value
9.9785
Put value
6.0772
Put–call parity
3.9012
c − p = e^(−rT)(F − K)
Delta
0.6104
Gamma
0.0178
Vega (per 1% vol)
0.2195
Theta (per day)
−0.0197
Rho (per 1% rate)
−0.0499
Value against the underlying
The gap between the lines is time value. Deep in the money the value line runs below the payoff, because the payoff is received only at expiry and is discounted at r.