Derivatives

Question 1

FP (equity security)

Answer 1

(S0 – PVD ) * (1 + rf)^T

Question 2

FP (security without cf)

Answer 2

FP=S0*(1+rf)^t

Question 3

FP (Equity index continuous dividend)

Answer 3

Soe^((cont.compoundend rf-cont compounded div yield)T)

Question 4

Forward P. on coupon paying bond

Answer 4

(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

Question 5

Value on forward on coupon paying bond

Answer 5

( St – PVC ) – (FP/( 1 + rf )^(T-t) )

Question 6

QFP=Price of bond futures contract = forward price (using full price)

Answer 6

(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

Question 7

Full price = dirty price

Answer 7

Clean price + Accrued Interest at t0

Question 8

Accrued interests at t =0

Answer 8

(Days since last coupon / days between coupon) * coupon $

If coupons have just been paid AI =0

Question 9

Accrued interest at T

Answer 9

(Days between last coupon and end of contract / days between coupon) * coupon $

Question 10

FRA

Long FRA = I am a fixed payer

Answer 10

((1+long rate)/(1+short rate)-1)(360/(durata loan))
Dove
rates=MRR (t/360)

Question 11

Value of FRA

Answer 11

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

Question 12

Swap fixed rate

Answer 12

((1-Zn)/(Z1+Z2+⋯+Zn))
Dove
Zn = PV factors–> 1/(1+MRR*(t/T) )=1/(1+r)

Question 13

Interest rate swap –> value to payer

Answer 13

∑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))

Question 14

Equity swap

Answer 14

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

Question 15

Currency swap

Answer 15

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”

Question 16

Long FRA

Answer 16

Call - put
pay fix, receive floating

Question 17

Short FRA

Answer 17

Put - call

Question 18

Payer fixed swap

Answer 18

long cap + short floor= Cap – Floor

Question 19

Receiver fixed swap

Answer 19

short cap + long floor = Floor – Cap

Question 20

Put call parity

Answer 20

C + PV (x) = P + S

Question 21

Put call parity when stock pays dividend

Answer 21

Put + Stocke-div yieldT=Call+ X * e -rT

Question 22

Prob of up move for stock price tree

Answer 22

((1 + rf – D))/((U – D) )
Dove
U =(S+)/S and D=(S-)/S

Question 23

Delta call

Answer 23

((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

Question 24

• Delta put

Answer 24

• Delta of in the money put option = -1

Question 25

of short call options

Answer 25

(# shares hedged)/(delta of call option)

Question 26

of long put options

Answer 26

(# shares hedged)/(delta of put option)

Question 27

Hedge ratio calls

Answer 27

(C+ -C-) / (S+ - S-)

Question 28

BSM call

Answer 28

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)

Question 29

BSM put

Answer 29

e^((-rT) ) KN(-d2)- S0e^(-div yieldT)N(-d1)
Dove N(-d1) = 1 – N(d1)
N (-d2)=1 – N(d2)

Question 30

f given the current forward price (FPt) on the same underlying and with the same maturity–> V long position

Answer 30

(FPt-FP)/(1+rf)^T

Question 31

FP (on a fixed–income security)

Answer 31

(S0 – PVC) × (1 + Rf)T^T

Question 32

What is the position (Long short ) of a bond that is equivalent to fixed receiver / Payer?

Answer 32

RICEVO FISSO = LONG BOND –> compro il bond ricevo i coupon fissi
* fix payer –>issue bond perchè deve pagare I coupon (fissi)

Question 33

o Equity return payer cosa riceve?

Answer 33

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