Deri Formula Sheet
Content
FINA0301 Derivative Securities 2010 Fall Final Formula Sheet Forward Price: 1. 2. Discrete dividend: F0,T = S0e
rT
− ∑i =1 e r (T − ti )Dt i
n
Continuous dividend: F0,T = S0e
( r −δ )T
Zero Coupon Bond Price: P (0, t ) =
1 [1 + r (0, t )]t
Implied Forward Rate (non‐annualized): r0 (t , t + s ) = Coupon Bond Price: Bt (t , T , c, n) = Coupon Rate of Par Coupon: c = Duration and convexity 1. Modified Duration = −
P(0, t ) −1 P(0, t + s )
∑
n
n i =1
cPt (t , ti ) + Pt (t , T )
∑
1 − Pt (t , T )
P (t , ti ) i =1 t
Change in bond price 1 × change in yield B( y )
=
⎡n i ⎤ 1 1 C/m n M + ⎢∑ i n⎥ B( y ) 1 + y / m ⎣ i =1 m (1 + y / m) m (1 + y / m) ⎦
Macaulay Duration = Modified Duration × (1 + y/m) Duration Matching: N = −
2. 3.
D1B1 ( y1 ) /(1 + y1 ) where D is the Macaulay Duration D2 B2 ( y2 ) /(1 + y2 )
4.
Convexity = ⎢ B( y ) ⎣ i =1
1 ⎡
∑
n
⎤ i(i + 1) C/m n(n + 1) M + i+2 n+2 ⎥ 2 2 m (1 + y / m) m (1 + y / m) ⎦
Bond Price Approximation:
B( y + ε ) = B( y ) − [D × B( y ) × ε ] + 0.5 × convexity × B( y ) × ε 2 , where D is the Modified
Duration.
∑ Swap Formula: R =
n
i =1
P(0, t i ) f 0 (t i )
n
∑i=1 P(0, ti )
Put Call Parity: C ( K , T ) − P ( K , T ) = PV0 ,T ( F0 ,T − K )
Parity for Options on Currencies: C ( K , T ) − P ( K , T ) = x0e
− rf T
− Ke− rT
P P
Generalized Option Parity: C ( St , Qt , T − t ) − P ( S t , Qt , T − t ) = Ft ,T ( S ) − Ft ,T (Q ) Different Strike Price: K1 < K2 < K3
C ( K1 ) − C ( K 2 ) ≤ K 2 − K1 ; P ( K 2 ) − P ( K1 ) ≤ K 2 − K1
C ( K 1 ) − C ( K 2 ) C ( K 2 ) − C ( K 3 ) P ( K 2 ) − P ( K1 ) P ( K 3 ) − P ( K 2 ) ≤ ≥ ; K 2 − K1 K3 − K 2 K 2 − K1 K3 − K 2
Binomial Solution:
u = e ( r −δ ) h +σ Δ = e −δh
h
;d = e
( r −δ ) h −σ h
Cu − C d − rh uC d − dCu ; B = e S (u − d ) u−d
⎛ e ( r −δ ) h − d u − e ( r −δ ) h ⎞ ⎟ + Cd C = e − rh ⎜ Cu ⎜ u−d u−d ⎟ ⎝ ⎠
e ( r −δ ) h − d Risk‐Neutral Probability: p* = u−d
The Black‐Scholes Formulas:
C ( S , K , σ , r , T , δ ) = Se −δT N (d1 ) − Ke − rT N (d 2 ) P ( S , K , σ , r , T , δ ) = Ke − rT N (−d 2 ) − Se −δT N (− d1 )
Δ call = e −δT N (d1 ) Δ put = Δ call − e −δT
where
d1 =
ln( S / K ) + (r − δ + σ 2 / 2)T and d 2 = d1 − σ T σ T
Prepaid Forward: F0,T = S 0 −
P
∑
n i =1 ti
P Dti
rT
− ∑i =1 e r (T − ti )Dt i
n
Continuous dividend: F0,T = S0e
( r −δ )T
Zero Coupon Bond Price: P (0, t ) =
1 [1 + r (0, t )]t
Implied Forward Rate (non‐annualized): r0 (t , t + s ) = Coupon Bond Price: Bt (t , T , c, n) = Coupon Rate of Par Coupon: c = Duration and convexity 1. Modified Duration = −
P(0, t ) −1 P(0, t + s )
∑
n
n i =1
cPt (t , ti ) + Pt (t , T )
∑
1 − Pt (t , T )
P (t , ti ) i =1 t
Change in bond price 1 × change in yield B( y )
=
⎡n i ⎤ 1 1 C/m n M + ⎢∑ i n⎥ B( y ) 1 + y / m ⎣ i =1 m (1 + y / m) m (1 + y / m) ⎦
Macaulay Duration = Modified Duration × (1 + y/m) Duration Matching: N = −
2. 3.
D1B1 ( y1 ) /(1 + y1 ) where D is the Macaulay Duration D2 B2 ( y2 ) /(1 + y2 )
4.
Convexity = ⎢ B( y ) ⎣ i =1
1 ⎡
∑
n
⎤ i(i + 1) C/m n(n + 1) M + i+2 n+2 ⎥ 2 2 m (1 + y / m) m (1 + y / m) ⎦
Bond Price Approximation:
B( y + ε ) = B( y ) − [D × B( y ) × ε ] + 0.5 × convexity × B( y ) × ε 2 , where D is the Modified
Duration.
∑ Swap Formula: R =
n
i =1
P(0, t i ) f 0 (t i )
n
∑i=1 P(0, ti )
Put Call Parity: C ( K , T ) − P ( K , T ) = PV0 ,T ( F0 ,T − K )
Parity for Options on Currencies: C ( K , T ) − P ( K , T ) = x0e
− rf T
− Ke− rT
P P
Generalized Option Parity: C ( St , Qt , T − t ) − P ( S t , Qt , T − t ) = Ft ,T ( S ) − Ft ,T (Q ) Different Strike Price: K1 < K2 < K3
C ( K1 ) − C ( K 2 ) ≤ K 2 − K1 ; P ( K 2 ) − P ( K1 ) ≤ K 2 − K1
C ( K 1 ) − C ( K 2 ) C ( K 2 ) − C ( K 3 ) P ( K 2 ) − P ( K1 ) P ( K 3 ) − P ( K 2 ) ≤ ≥ ; K 2 − K1 K3 − K 2 K 2 − K1 K3 − K 2
Binomial Solution:
u = e ( r −δ ) h +σ Δ = e −δh
h
;d = e
( r −δ ) h −σ h
Cu − C d − rh uC d − dCu ; B = e S (u − d ) u−d
⎛ e ( r −δ ) h − d u − e ( r −δ ) h ⎞ ⎟ + Cd C = e − rh ⎜ Cu ⎜ u−d u−d ⎟ ⎝ ⎠
e ( r −δ ) h − d Risk‐Neutral Probability: p* = u−d
The Black‐Scholes Formulas:
C ( S , K , σ , r , T , δ ) = Se −δT N (d1 ) − Ke − rT N (d 2 ) P ( S , K , σ , r , T , δ ) = Ke − rT N (−d 2 ) − Se −δT N (− d1 )
Δ call = e −δT N (d1 ) Δ put = Δ call − e −δT
where
d1 =
ln( S / K ) + (r − δ + σ 2 / 2)T and d 2 = d1 − σ T σ T
Prepaid Forward: F0,T = S 0 −
P
∑
n i =1 ti
P Dti
