Basis
𝑏 =𝑆−𝑓
Hedge ratios
ℎ = 𝜎∆𝑆∆𝑓
=
2
𝜎
cov(∆𝑆,∆𝑓)
∆𝑓
var(∆𝑓)
Pricing
𝜎
= 𝜌∆𝑆∆𝑓 𝜎∆𝑆
∆𝑓
Bonds and Duration
D =
=
sum of ( years to payment  PV of payment )
sum of ( PV of payment )
2n
i
i =1


CFi
  2  (1 + y / 2) 
i
2n
CFi

i
i =1 (1 + y / 2 )
D* =

for semi-annual bonds
B  − D*  B  y
(1 + 𝐸𝐴𝑅)𝑇 = 𝑒 𝑟𝑐 𝑇
Interest rates and derivatives
𝐴𝑃𝑅 𝑀
𝐸𝐴𝑅 = (1 +
) −1
𝑀
R B TB − R A TA
TB − TA
𝑟𝑐 = ln (1 + 𝐸𝐴𝑅)
𝐴𝑃𝑅 = 𝑀[(1 + 𝐸𝐴𝑅)1/𝑀 − 1]
1
 (1 + RN + K ) N + K  K
F
=

 −1
N K
N
 (1 + RN )

FAB =
D
1 dB
= −
1+ y / 2
B dy
𝑉=
NP × (R − F) × (fraction of year)
1 + R × (fraction of year)
dirty price = settlement price × CF + AI
𝐶 − 𝑃 = 𝑆 − 𝐾𝑒 −𝑟𝑇
Options
𝒖 = 𝒆𝝈√∆𝑻
𝒅 = 𝒆−𝝈√∆𝑻
𝑪 = [𝒑 × 𝑪𝒖 + (𝟏 − 𝒑) × 𝑪𝒅 ] × 𝒆−𝒓.∆𝑻
𝑷 = [𝒑 × 𝑷𝒖 + (𝟏 − 𝒑) × 𝑷𝒅 ] × 𝒆−𝒓.∆𝑻
𝒑 =
𝒆𝒓.∆𝑻 − 𝒅
𝒖−𝒅
𝐶𝑢 − 𝐶𝑑
𝑆×𝑢 − 𝑆×𝑑
∆ =
(
S
  t ,  2 t
S
−qT
)
2
S
ln   + (r +
)T
K
2

d =
1
 T
2
S
ln   + (r −
)T
K
2

d =
= d − T
2
1
 T



2 
T ,  2T 
ln ST   ln S0 +   −
2 



. N (d ) − K .e−rT . N (d )
1
2
−qT
P = K .e−rT . N (−d ) − S0 .e
.N ( − d )
2
1
− qT
− rT
p + S0 .e = c + K .e
C = S0 .e
C = S. N (d ) − Ke−rT . N (d )
1
2
−
rT
P = Ke
. N ( − d ) − S .N ( − d )
2
1
(
(
)
C = e−rT F0 . N (d ) − K . N (d )
1
2
P = e−rT K . N (−d ) − F0 .N (−d )
2
1
− rT
− rT
p + F0 .e = c + K .e
)
For x < 0 use N(x) = 1 – N(-x)