Derivatives Flashcards
FP (equity security)
(S0 – PVD ) * (1 + rf)^T
FP (security without cf)
FP=S0*(1+rf)^t
FP (Equity index continuous dividend)
Soe^((cont.compoundend rf-cont compounded div yield)T)
Forward P. on coupon paying bond
(S0 – PVC ) * (1 + rf)^T –>T è la durata del contratto
Dove PVC = (coupon r * FV)/(1+r)^(t/360)
FV non è lo spot ma il par price (1000)
Coupon r –> se semiannual diviso due, il rate al denominatore rimane sempre intero
Value on forward on coupon paying bond
( St – PVC ) – (FP/( 1 + rf )^(T-t) )
QFP=Price of bond futures contract = forward price (using full price)
(full price (1+rf)^T-AI-FVC)(1/CF)
full price = clean price + accrued interest at t = 0
AI = accrued interest at futures contract maturity
–> AIT = (Days between last coupon and end of contract / days between coupon) * coupon $
T= tempo rimanente all’expire–> se finisce tra 90 giorni 90/360
FVC = Coupon(1+rf)^(tempo rimanente - 0.5) se semiannual coupon
Full price = dirty price
Clean price + Accrued Interest at t0
Accrued interests at t =0
(Days since last coupon / days between coupon) * coupon $
If coupons have just been paid AI =0
Accrued interest at T
(Days between last coupon and end of contract / days between coupon) * coupon $
FRA
Long FRA = I am a fixed payer
((1+long rate)/(1+short rate)-1)(360/(durata loan))
Dove
rates=MRR (t/360)
Value of FRA
Step 1:
((1+long rate)/(1+short rate)-1)(360/(durata loan))
Step 2:
(FRA-r)((t-→durata forward)/(360))(value$)
Step 3
((value step 2)/(1+MRRlong((t-→t long)/(360)) ))=value to short
Swap fixed rate
((1-Zn)/(Z1+Z2+⋯+Zn))
Dove
Zn = PV factors–> 1/(1+MRR*(t/T) )=1/(1+r)
Interest rate swap –> value to payer
∑Z *(SFR old – SFR new) * (days / 360) * notional
Dove Z sono tutti I periodi rimanenti
SFR va calcolato come swap fixed rate = ((1-Zn)/(Z1+Z2+⋯+Zn))
Equity swap
Notional((ST/S0)-pv of coupons+Pv of principal)
St puo anche essere ottenuto come 100*(1+ performance) dell’equity
Pv of principal = 1/ last discount factor
Currency swap
1) Calcolo PV cf $
2) Calcolo PV cf €-
3) trasformo 2 in $ con lo spot rate
Faccio punto 1 – punto 3
Se ho coupon–> calcolo PV “coupon rate” nella currency €, lo trasformo con lo spot exchange rate
Notional amount = Notional * “coupon rate”
Long FRA
Call - put
pay fix, receive floating
Short FRA
Put - call
Payer fixed swap
long cap + short floor= Cap – Floor
Receiver fixed swap
short cap + long floor = Floor – Cap
Put call parity
C + PV (x) = P + S
Put call parity when stock pays dividend
Put + Stocke-div yieldT=Call+ X * e -rT
Prob of up move for stock price tree
((1 + rf – D))/((U – D) )
Dove
U =(S+)/S and D=(S-)/S
Delta call
((Change in price of call))/((change in price of stock) )
* Delta of at the money call option = 0.5
* Delta of in the money call option = 1
- Delta put
- Delta of in the money put option = -1
of short call options
(# shares hedged)/(delta of call option)
of long put options
(# shares hedged)/(delta of put option)
Hedge ratio calls
(C+ -C-) / (S+ - S-)
BSM call
S0e^(-div yieldT) N(d1)-e^((-rT) ) KN(d2)
La prima parte è lo stock e la seconda è lo ZCB
Se non ci sono dividend
S0*N(d1)-e^((-rT) ) KN(d2)
BSM put
e^((-rT) ) KN(-d2)- S0e^(-div yieldT)N(-d1)
Dove N(-d1) = 1 – N(d1)
N (-d2)=1 – N(d2)
f given the current forward price (FPt) on the same underlying and with the same maturity–> V long position
(FPt-FP)/(1+rf)^T
FP (on a fixed–income security)
(S0 – PVC) × (1 + Rf)T^T
What is the position (Long short ) of a bond that is equivalent to fixed receiver / Payer?
RICEVO FISSO = LONG BOND –> compro il bond ricevo i coupon fissi
* fix payer –>issue bond perchè deve pagare I coupon (fissi)
o Equity return payer cosa riceve?
receive interest payment and pay the return on the portfolio if return is negative the payer receive a net payment greater than the lposs of the equity pf
