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Bond Futures Option Model
We present a pricing model for bond futures options. Assuming that the bond futures price at the
maturity of the option is lognormal, the model adopts the Black’s analytical closed-form solution.
For bond futures options, the futures price is taken directly from the market instead of being
calculated from the bond futures calculator.
Let
)t(BF
be a price process of a given bond futures, and T be a payoff maturity date. The bond
futures option with the underlying BT is a European type derivative security whose matured
payoff at the settlement date is given by
)X)T(BT(,0max −
(1)
where
(+1 for call, -1 for put) is the call-put index, X is the strike. Assuming that the bond
futures price at the maturity of the option is lognormal, the prices of bond futures options at time
zero are given by using the Black’s formula, which are
)]d(XN)d(NBF)[T,0(Pc 210 −=
(2)
)]d(NBF)d(XN)[T,0(Pp 102 −−−=
(3)
where c and p are the call price and the put price,
)T,0(P
is the discount factor, and
1
d
and
2
d
are given by
T
2/T)X/BFln(
d2
0
1
+
=
(4)
T
2/T)X/BFln(
d2
0
2
−
=
. (5)
The volatility
is defined so that
T
is the standard deviation of the logarithm of the bond
futures price at the maturity of the option.
The above pricing formulae are based on the assumption that the futures price is a martingale,
i.e., there is no drift term in the futures price dynamics.
There are closed-form solutions available for the Greek values for options described above. We
adopt closed-form solutions for Deltas and Gammas, which are given by
)d(Ne 1
rT−
=
for calls and
)1)d(N(e 1
rT −= −
for puts, (6)
TBF
)d(N
e
0
1
rT
= −
for both calls and puts. (7)
For Vega, Theta and Rho values, We employ numerical differentiation. Vega is calculated as
0v PPV −=
(8)
where
v
P
is the option price corresponding to an upper 1% shift of futures price volatility, and
0
P
is the initial option price. Theta is given by
0t PP −=
(9)
where
t
P
is the option value assuming one-day shift of value time while all else remains the
same. And rho is calculated as
0r PPrho −=
(10)
where
r
P
is the option value assuming 10 bp parallel shift of interest rates.
We test the bond futures option model in several cases. The first case is a call option, with the
value date of 20031212 10:54, the futures price of 111.0, the risk free interest rate of 0.01, the
volatility of the futures price of 0.05, and the option maturity of 20040221 10:54.
The second case is a put option with the same inputs as the first case except that the volatility is
0.10.
The testing results are shown in Table 1.
Table 1. Results for Bond Futures Options
Case
Price
Delta
Gamma
Vega
Theta
rho
1
1.5498
0.6620
0.1488
0.1809
-0.006257
-0.000301