Derivative
Terms
Put call
Boundary
click to edit
Call
Put
Boundary
intrinsic value
Stock price, underlying price
click to edit
Exercise price
X
S
click to edit
Call
Put
C= S-X
P = X-S
S has no ceiling
S has floor = 0
If no excercise
Save money by saving EXERCISE price (X)
If no exercise
Not earning EXERCISE price (X)
put call parity
Time
Strike price
Interest rate
Time
Strike price
Interest rate
C = P + S - PV(X)
PV(X) = X - r
Binomial option Pricing Model
One period model
Two period
Hedge portfolio
American option vs European option
Beneficial of time value
Dividend vs no dividend
Put
Call
Factors
Note
Determine of the call can only be at expiration aka right to left ONLY
Stock price
Exercise price
Interest rate
d < 1+ r< u
time
Volatility of stock price
Term
h = head ratio
C = value of a call
p = % of portfolio going up
1- p = % of portfolio going down
page 8: risk neutral rate / pricing
S = Value of porfolio
Contract multiplier
aka how many call for 1 stock
u = value of S at p
d = value of S at 1-p
click to edit
Continuous approach
discount rate = e^ (r x t)
r = discreate rate, t = time
Condition:
same strike price
same expiration
Black Schole
Weiner process
Model how variable move overtime from time 0 to time t
Continue function in time
Increment are normally distributed
N(0,1)
Mean = 0
Independent of what happen prior
Black Scholes Model
Component
Formula
Meaning
C = the call option price
S = the current stock price
x = the strike price
r = the risk-free interest rate
t = the time to expiration (in years)
N(d1), N(d2) = the cumulative normal distribution function
d1 and d2 are intermediate variables calculated as:
e^(-rt) = Expected return periodic
European Call option
C = SN(d1) - X[e^(-rt)]N(d2)
d1= [ln(S/X) + (r + [o^2]/2)t ] / [o sqrt(t) ]
sigma=o= Standard deviation of log return
d2= [ln(S/X) + (r - [o^2]/2)t ] / [o sqrt(t) ]
OR d2 = d1- [o x sqrt(t)]
The right to buy the underlying asset
The right to sell the underlying asset
Markov Process
Formula
Change in Stock price
dS= S(t) - S(t-1)
week 4 page 7
Drift rate (up vs down)
volativity rate
Black Scholes differential equation
component
S =0
T=0
Ln(S/X) ->0
d1, d2->0
N(d1), N(d2) -> 0
r=0
click to edit
x = 0
Ln(S/X) -> 1
d1, d2 ->1
N(d1), N(d2) -> 1
C = S + 0 = S
Call is a stock, no longer an option
With Dividend payment
S0 =S0 - D(t) x e^(-rt)
t must be in year
WITH DIVIDEND YEILDS
When dividend accrual continuously
S0=S0- Dt x e^(-rt)
Accumulation index - price index = div yield
PSEUDO AMERICAN OPTION
t
Ex Div
T
t= ex Div
s'
x = x- div
click to edit
t = T
S= S'-PV(Div)
x = x'
PUT CALL PARITY
volatility
Historical vs Implied
historical = past information or past volatility
Implied volatility
St Deviarion of Ln(P/PP(t-1)
RISK
directional
delta
gama
Volatility
Vega
Time
Theta
Interest rate
Rho
Delta is the rate of change of an option's price relative to changes in the price of the underlying stock or other security
Gamma is the rate of change of delta in the change of the stock price
For ex, delta = 7 mean oif Stock price (S) increased for 1$, a call option (C) will go up 70 cent
risk of changes in implied volatility or the forward-looking expected volatility of the underlying asset price
change in call price /SMALL change in stock price
Change in option price / small change in risk free rate
volatility has the same impact on call and put (POSITIVE)
Volatility of call = volatility of put
Positive
Gamma for call = gamme for put
generally negative
Strategy
Protective put
+S +P
Long a stock
+S
Long a put
+P
Cover Call
+S -C
Additional terminology
C, P = current Call , Put price
x = strike (exercise) price
Π =PROFIT
N = number of...
Profit equation
CALL held to expiration
Π = 𝑁𝐶[Max(0, 𝑆𝑇 - X) - C]
Buyer of 1 call
NC= 1
+C
Π = Max(0, 𝑆𝑇 - X) - C
Seller of 1 call
NC = -1
-C
Π = -Max(0, 𝑆𝑇 - X) + C
puts held to expiration
Π =𝑁𝑃[Max(0,X - 𝑆𝑇) - P]
buyer of one put
𝑁𝑃 = 1
+P
seller of one put
𝑁𝑃 = -1
-P
Π = Max(0,X - ST) - P
Π = -Max(0,X - ST) + P
STOCK
Π = 𝑁𝑆[𝑆𝑇 - 𝑆0 ]
buyer of one share
(𝑁𝑆 = 1)
+S
Π = 𝑆𝑇 - 𝑆0
short seller of one share
𝑁𝑆 = -1
-S
Π = - 𝑆𝑇 + 𝑆0
Max loss = -S0
Max gain = S0
Max loss = infinitive
If ST>X
=>In the money
Π= ST-X-C
ST<X
ST-X max = 0
Π= -C
Max loss
Breakeven Π=0
ST= X+C
If ST>X
Π=C-ST+X
ST<X
Π =C
Max gain
Breakeven Π =0
ST= X+C
ST<X =>In the money
Π=X-ST-P
Breakeven Π=0
ST= X-P
ST<X
ST>X -> Max = 0
Π=-P
Max loss
Π= -X+ST +P
ST>X => Max =0
Π=P
Max gain
Breakeven Π=0
ST= X-P
Limit possible loss from writing a call/Buying a stock
Long a Stock
+S
Short a CALL
-C
Payoff
Π = 𝑆𝑇 - 𝑆0 -Max(0, 𝑆𝑇 - X) + C
ST>X
In the money
ST< X
Max =0
Π =X+C-S0
Π =ST-S0+C
Max loss/Min profit, ST =0
Π =C-S0
Max profit
Protect again stock price falling
Bearish
Π = 𝑆𝑇 - 𝑆0 + Max(0,X - ST) - P
ST<X or X>ST
ST> X or X<ST
Π = X-𝑆0 - P
Π = 𝑆𝑇 - 𝑆0 -P
Max profit
Min Profit/Max los
protect from S Price fall
Breakeven Π = 0
ST=S0+P
Synthetic option
Synthetic put
+C -S
𝑃 = 𝐶 − 𝑆0 + 𝑋𝑒^(−𝑟𝑇)
Long CALL
+C
Short S
-S
Π = 𝑁𝑆 𝑆𝑇 − 𝑆0 + 𝑁𝐶 [𝑀𝑎𝑥 0, 𝑆𝑇 − 𝑋 − 𝐶]
ST>X
In the money
ST< X
Max =0
MULTIPLE CALLS OR PUT
Bullspread
+C1 -C2
Bearspread
-C1 +C2
C2< C1
Timespread
Limits up and downside potential
Good for arbitrage if one option is overprice relative to another
Π = Max(0, 𝑆𝑇 - X1) - C1 - Max(0, 𝑆𝑇 - X2) + C2
ST> X1>X2
in the money
ST<X1<X2
Max = 0
All out of the money
X1<ST<X2
MAX2=0
Π = 𝑆𝑇 - X1 - C1 + C2
Π = X2- X1- C1+C2
Π = -C1 +C2
BUTTERFLY SPREAD
+C1 - 2xC2 +C3
Benefit from speculation of a stock price fall, but with limited up /
downward potential
Π = -Max(0, 𝑆𝑇 - X1) + C1 + Max(0, 𝑆𝑇 - X2) - C2
ST>X1>X2
All, in the money
X1<ST<X2
ST<X2<X1
All, out of the money
Π =X1 + C1 - X2 - C2
Π = 𝑆𝑇 - X1 - C1 + C2
Π = C1 - C2
BEARSPEAD WITH PUT
-P1+P2
click to edit
Π = -Max(0,X1 - ST) + P1 + Max(0,X2 - ST) - P2
ST < X1 < X2
Π = X2-X1+P1- P2
X1 < ST < X2
Π =P1 + X2 - ST - P2 if
X1 < X2 < ST
Π = P1 - P2
MAX PROFIT
Max loss
Breakeven Π =0
ST = P1-P2+X2
COLLAR
+S +P1 -C2
click to edit
Π = ST - S0 + Max(0,X1 - ST) - P1 - Max(0,ST - X2) + C2
ST < X1 < X2
X1 < ST < X2
X1 < X2 < S
Π= X1 - S0 - P1 + C2
Π= ST - S0 - P1 + C2
Π=X2 - S0 - P1 + C2
Speculate that large stock price movements are unlikely
Calendar Spread
Benefit from volatility and time value decay
Straddle, Strap and Strips
Straddle
+C +P
You think the stock will move (but don’t know in which direction)
Strip
Double bet on Stock going down
down
+P +P +C
Straps
Double bet on Stock going up
UP
+C + C +P
Π = Max(0,ST - X) - C + 2Max(0,X - ST) - 2P (assuming Nc = 1, Np = 2)
ST > X
PUT out of money
CALL in the money
Π = ST - X - C - 2P
if ST < X
PUT IN THE MONEY
CALL OPUT OF THE MONEY
Π = 2X - 2ST - C - 2P
Break even
Break even
ST*=X+C+2P
ST* = X - P - C/2
FUTURE
obligation
Forward
Agreement between two parties that call for delivery of an asset at a future point in time with a price agreed upon today
Expose to DEFAULT RISK
Future
is Forward contract
has standardised terms,
is traded in an organised exchange, and follows daily settlement procedures where losses from one party are paid to the other.
Delivery term
Delivery cycles
FUTURE AND FORWARD CONTRACT
FUTURE
FORWARD
Hedging with FUTURE
why
Reduce or Alter risk exposure
Reason for NOT HEDGING
Hedging can give a misleading impression of the amount of risk reduce
Hedging eliminates the opportunity to take advantage of favourable market conditions
Risk is not reduced or eliminate, it is transferred to another type/style
SHORT HEDGE AND LONG HEDGE
SHORT
LONG
Basis
= spot price - futures price
To cover any possible in the fall in price of the underlying
To cover any possible in the RISE in the price of the underlying
Π = (𝑆𝑇 − 𝑆0) − (𝑓𝑇 − 𝑓0)
Π = −𝑆𝑇 + 𝑆0 + (𝑓𝑇 − 𝑓0)
Spot price from the spot market
Future from the future market
𝑆𝑇 − 𝑆0
𝑓𝑇 − 𝑓0
Financial instrument (contract) that derives it value from the price of an underlying instrument
Use to
Hedge
Speculate
Arbitrage
S>X -> in the money
S<X out of the money
S< X -> In the money
S>X-> out of the money
WHEN HAVE EXPECTATION OF MARKET GOING UP OR DOWN
TAKE POSITION BASED ON THAT ASSUMPTION
Spot market
derivative market
no requirement for payment today
buy, receive today
value between 0 and 1
Dt = dividend at time t
click to edit
Higher r -> higher C
Stock volatility
Higher delta -> higher C,
Higher r -> lower P
Stock volatility
Similar to call