Derivative Flashcards
FP =
S0 × (1 + Rf)^T
V0 of long forward contract
0
V (long) during holding
St - [FP/(1+RF)^(T-t)]
short position = negative of long position
VT (at maturity)
St -FP
FP (on a stock) =
(S0 − PVD) × (1 + Rf)T
= [(S0 × (1 + Rf)T] − FVD
days=365
Vt(long position on a stock)
(St- PVDt) - [FP / (1+rf)^(T-t)]
FP(on equity index)
S0 * e^(Rfc-dvd yld) *T
FP(on fixed income security)
(S0 - PVC)*(1+Rf)^T
t=365
Vt(long position on fixed income security)
(St-PVCt) - [(FP/(1+rf)^(T-t)]
QFP
= FP/CF
=full (1+Rf)^T - AIT-FVC) / CF
AIT =accrued interest at future contract maturity
=(days since last coupon ptm nav / days b/w coupon ptm NTD ) * Counpon ptm —-frequent of coupon payment
FRAs forward rate agreements - long
= # of months until the contract expires
- # of month until the underlying loan is settled
= maturity of the loan
rights to borrow money
libor rates used in
FRA
swap
caps/ floors
* days/360
Equity, bond, currencies, stock option use 365
(1+r)^(days/365) periodic compounding
equity index
e^r*days/365 continuous compounding
Synthetic call
put + stock - riskless discount bond
synthetic put
call - stock + riskless discount bond
SFR= rfixed
(1-ZN)
/ (SUM OF Z)
z= 1/ (1+libor *(days/360))
value to payer =
Sum (z) * (SFR new - SFP old) * days/360 * notional amount
put-call parity must hold for arbitrage
P+S = C+Xe^(-rt)
value of equity swap
= (FPt -FP0)NA
/ (1+SFR)(t-t)
Ft = Currency forward price
price = so * (1+rpc)^t/365 / (1+rbc)^t/365
Vt = FPt -FP0
/ (1+rpc)^(T-t)
if continously compounded =Ft = S0^(Rprice-Rbase)*T
probability of upward movement
πU
(1+Rf-D)/ (U-D)
downward movement = 1- πU
assumptions underlying the Black-Scholes-Merton (BSM) model
The return on the underlying asset follows a lognormal distribution and the price change is smooth.
The (continuous) risk-free rate is constant and known.
The volatility and yield of the underlying asset is constant and known.
Markets are frictionless.
The options are European.
no cash flows on the underlying asset
number of call options needed to delta hedge
=
number of shares hedged /
delta of call option
long fra
pay fixed, receive floating
=long call + short put
short fra
pay float, receive fixed
value of fra =
(long rate-short rate)*days/360 * notional / [(1+long rate * days/360)]
FRA (60)= (new rate)
(1+long * days/360)/(1+short * days/360) * days/360 -1
SFR(swap fixed rate)
(1-last DF)/(SUM OF DF) * settlement periods per year
value of payer swap=
sum df *(SFR new - SFR old)/(#settlements/year) * Notional
= pv(cash inflow) - pv(cash outflow)
pv(cash flow) = sum DF* cash flow i
equity swap pv=
(current index value - index level at last settlement)
*notional
value = pv(cash inflow) - pv(cash outflow)
Hedge ratio
(C+-C-)/ (S+-S-)
# OF LONG POSITION = # OF OPTION * HEDGE RATIO * S+ # OF SHORT option=# OF OPTION * C+ call price =#shares hedged / delta
fractional units of stock =
(C+-C-)/ (S+-S-)
Interest rate call
PMT for interest rate calls
long receives pmt if reference rate > the strike (fixed)
= max(0,NP*(reference rate - strike) * days/365)
interest rate put
PMT for interest rate puts
long receives payment if reference rate fall below the strike rate
= max(0,NP*(strike-reference rate) * days/365)
cap
= series of Interest rate calls
floor =
series of interest rate puts
payer swap =
long cap + short floor with same strike
payer swaption
right to enter swap as fixed-rate payer
iwin if rates increase
payer swap = long payer swaption + short receiver swaption
receiver swaption
right to enter swap as fixed-rate receiver
win if rates fall
receiver swap = long receiver swaption + short payer swaption
BSM European call value C0 =
So*N(d1) - xe^-rft * N(d2) long N(d1) unit of stock short N(d2) units of zbc
BSM currency option C0=
S0e^-rbc * N(d1) - Xe^-rpc*T * N(d2)
Delta S
most sensitive when option is at the money
asset price = (C1-C0) / (S1-S0)
> 0 positive related to call
< 0 negative related to puts
Vega Sigma
Volatility
> 0 positive related to call/put
Rho
Rf interest rate
> 0 positive related to call
< 0 negative related to puts
Theta T
Time to expiration
< 0 value goes to 0
value goes to 0
X
exercise price
negative related for calls
positive related for puts
change in Co=
delta call * change in stock price
=e^(-signa*T) *N(d1) * change in stock price
for no dividend paying stock
put delta = call delta-1
change in p = (N(d1)-1) * change in S
OF SHORT option
=#shares hedged / delta
call delta <1, need more calls than shares
Gamma
rate of change in delta as stock price changes
change in call = call delta * change in S + 1/2 gamma * (change in S)^2
A swap is equivalent to a series of:
off-market FRAs.
the higher the gamma
the more delta changes as the asset price changes
the worse a delta hedge will perform over time
A payer swaption gives its holder:
the right to enter a swap in the future as the fixed-rate payer.
delta hedging
a long position in a stock with a short position in call option, so value of the portfolio does not change with the value of the stock
of short calls required
of shares/call delta
Over the life of a swap, the price of the swap:
The price of a swap, quoted as the fixed rate in the swap, is determined at contract initiation and remains fixed for the life of the swap.
The price of a forward contract is
established at the initiation of the contract and is expressed in different terms, depending on the underlying assets. It is the price that makes the contract value zero, and depends on current interest rates through the cost-of-carry calculation.
Backwadation
Spot > FP
The two fundamental rules of the arbitrageur
(a) do not use your own money and
(b) do not take any price risk.
