Calculations of Greeks
in the Black and Scholes Formula
Claudio Pacati
May 15, 2013
1
Non-dividend paying stock
In the Black and Scholes model the price of an European call option on a non-dividend
paying stock is
C = S N(d1 ) − K e−rτ N(d2 ) ,
(1)
where S is the stock’s price at valuation date, K is the strike price, r is the (constant) spot
rate, τ = T − t is the time to maturity, T the expiry, t the valuation date and
S
log K
+ (r + 12 σ 2 )τ
√
,
σ τ
S
√
log K
+ (r − 12 σ 2 )τ
√
d2 =
= d1 − σ τ ,
σ τ
d1 =
(2)
(3)
where σ is the stock’s volatility.
Theorem 1. The greeks for the call option are:
delta:
∆C =
gamma:
ΓC =
theta:
ΘC =
rho:
ρC =
vega:
VC =
∂C
= N(d1 ) ,
∂S
∂2C
N0 (d1 )
K e−rτ N0 (d2 )
√
√
=
=
,
∂S 2
Sσ τ
S2σ τ
σSN 0 (d1 )
∂C
σ N0 (d2 )
√
√
= −rK e−rτ N(d2 ) −
= −K e−rτ r N(d2 ) +
,
∂t
2 τ
2 τ
∂C
= τ K e−rτ N(d2 ) ,
∂r
√
√
∂C
= τ S N0 (d1 ) = τ K e−rτ N0 (d2 ) .
∂σ
In order to prove the theorem we collect some common calculations in the following
Lemma 1. It holds
S N0 (d1 ) − K e−rτ N0 (d2 ) = 0 ,
∂d2
1
∂d1
√ ,
=
=
∂S
∂S
Sσ τ
√
∂d1
∂d2
τ
=
=
,
∂r
∂r
σ
∂d2 ∂d1
σ
−
= √ ,
∂t
∂t
2 τ
∂d1 ∂d2 √
−
= τ .
∂σ
∂σ
1
(4)
(5)
(6)
(7)
(8)
Proof. First of all, we remember that
1
2
N0 (x) = √ e−x /2 .
2π
Statement (4) holds if and only if
S N0 (d1 ) = K e−rτ N0 (d2 )
⇐⇒
S rτ
N0 (d2 )
e = 0
K
N (d1 )
⇐⇒
log
S
d2 − d22
+ rτ = 1
.
K
2
Notice that the right hand side of the last condition is
√
√
d21 − d2
1
1
S
1
1
= (d1 + d2 )(d1 − d2 ) = (2d1 − σ τ )σ τ = log
+ (r + σ 2 )τ − σ 2
2
2
2
K
2
2
S
= log
+ rτ
K
and this completes the proof of (4).
The proofs of the other statements are straightforward calculations.
Proof of theorem 1. For the delta, we have that
∂C
∂d1
∂d2
= N(d1 ) + S N0 (d1 )
− K e−rτ N0 (d2 )
∂S
∂S
∂S
∂d1 0
−rτ 0
= N(d1 ) +
S N (d1 ) − K e
N (d2 )
∂S
= N(d1 )
∆C =
by (5)
by (4).
(9)
Using (9) and (5) the gamma is
ΓC =
∂2C
∂∆C
∂d1
N0 (d1 )
0
√ .
=
=
N
(d
)
=
1
∂S 2
∂S
∂S
Sσ τ
By (4) it can be also written in the form
ΓC =
K e−rτ N0 (d2 )
S√
Sσ τ
=
K e−rτ N0 (d2 )
√
.
S2σ τ
The theta is
∂C
∂d1
∂d2
= S N0 (d1 )
− rK e−rτ N(d2 ) − K e−rτ N0 (d2 )
∂t
∂t
∂t
∂d
σK
e−rτ N0 (d2 )
1
√
= −rK e−rτ N(d2 ) +
S N0 (d1 ) − K e−rτ N0 (d2 ) −
∂t
2 τ
σSN 0 (d1 )
√
= −rK e−rτ N(d2 ) −
2 τ
σK e−rτ N 0 (d2 )
√
= −rK e−rτ N(d2 ) −
2 τ
σ N0 (d2 )
−rτ
√
= −K e
r N(d2 ) +
.
2 τ
ΘC =
by (7)
by (4)
by (4)
For the rho we have
∂C
∂d1
∂d2
= S N0 (d1 )
+ τ K e−rτ N(d2 ) − K e−rτ N0 (d2 )
∂r
∂r
∂r
∂d1 −rτ
0
−rτ 0
= τK e
N(d2 ) +
S N (d1 ) − K e
N (d2 )
∂r
= τ K e−rτ N(d2 )
ρC =
2
by (6)
by (4).
Finally, the vega is
∂C
∂d1
∂d2
= S N0 (d1 )
− K e−rτ N0 (d2 )
∂σ
∂σ
∂σ
√
∂d
1
S N0 (d1 ) − K e−rτ N0 (d2 )
= τ K e−rτ N0 (d2 ) +
∂σ
√
= τ K e−rτ N0 (d2 )
√
= τ S N0 (d1 )
VC =
by (8)
by (4)
by (4).
Consider now a forward contract, with strike K and maturity T , i.e. with payoff at time
T given by F (T ) = S(T ) − K. Denote by F = F (t) = S(t) − K e−r(T −t) = S − K e−rτ its
price at time t.
Exercise. The Greeks of the forward contract are
delta:
∆F =
gamma:
ΓF =
theta:
ΘF =
rho:
ρF =
vega:
VF =
∂F
=1 ,
∂S
∂2F
=0 ,
∂S 2
∂F
= −rK e−rτ ,
∂t
∂F
= τ K e−rτ ,
∂r
∂F
=0 .
∂σ
By using the put-call parity relation C − P = F and the previous exercise it is straightforward to compute the Greeks for a put option.
Exercise. The Greeks of the put option are
delta:
∆P =
gamma:
ΓP =
theta:
ΘP =
rho:
ρP =
vega:
VP =
∂P
= − N(−d1 ) ,
∂S
∂2P
K e−rτ N0 (d2 )
N0 (d1 )
√
√
=
,
=
∂S 2
Sσ τ
S2σ τ
∂P
σSN 0 (d1 )
σ N0 (d2 )
−rτ
−rτ
√
√
= rK e
N(−d2 ) −
=Ke
r N(−d2 ) −
,
∂t
2 τ
2 τ
∂F
= −τ K e−rτ N(−d2 ) ,
∂r
√
√
∂C
= τ S N0 (d1 ) = τ K e−rτ N0 (d2 ) .
∂σ
(In order to better interpret the formaulae, recall that for every x, N 0 (x) = N 0 (−x)).
2
Dividend paying stock
Assume now the stock pays dividends at a constant dividend yield δ. We know that the
call option price Black and Scholes formula becomes
C = S e−δτ N(d1 ) − K e−rτ N(d2 ) .
Exercise. It holds
S e−δτ N0 (d1 ) − K e−rτ N0 (d2 ) = 0
and formulae (5), (6), (7) and (8) remain the same in the non-dividend paying case.
3
(10)
Exercise. The greeks for the call option are:
∂C
= N(d1 ) e−δτ ,
∂S
K e−rτ N0 (d2 )
∂2C
N0 (d1 ) e−δτ
√
√
=
,
=
=
∂S 2
Sσ τ
S2σ τ
∂C
σN 0 (d1 )
√
=
− rK e−rτ N(d2 )
= S e−δτ δ N(d1 ) −
∂t
2 τ
σ N0 (d2 )
−δτ
−rτ
√
r N(d2 ) +
= δS e
N(d1 ) − K e
,
2 τ
∂C
=
= τ K e−rτ N(d2 ) ,
∂r
√
√
∂C
=
= τ S e−δτ N0 (d1 ) = τ K e−rτ N0 (d2 ) .
∂σ
delta:
∆C =
gamma:
ΓC
theta:
ΘC
rho:
ρC
vega:
VC
We know the forward price in the dividend paying case to be F = S e−δτ − K e−rτ .
Exercise. Deduce the Greeks of the forward contract.
Put call parity relation remains formally the same: C −P −F ; of course all the quantities
involved have to be computed by the formulae for the dividend paying case.
Exercise. Using put-call parity and the previous results, obtain the Greeks of the put option.
4