Transcript שקופית 1 - Bar-Ilan University

Introduction to Options
Option – Definition
An option is a contract that gives the holder the right but
not the obligation to buy or sell a defined asset called the
underlying asset at a predetermined price within a specified
period of time.
There are two fundamental types of options – call option
and put option
A call option gives the holder the right to buy the asset at a
predetermined price within a specified period of time.
A put option gives the holder the right to sell the asset at a
predetermined price within a specified period of time.
Option – Definitions
Underlying asset – the asset on which the option contract is
written.
Strike price (or Exercise price) – the fixed, predetermined price at
which the holder of the option can buy or sell the underlying asset.
Option Premium – the price paid for the option
Expiration date – the lat day on which the option can be exercised
There are two types of option:
European option – An option that can be exercised only on the
expiration date itself.
American option - An option that can be exercised at any time up
to and including the expiration date.
Option – Definition
We say the seller writes an option.
 If you buy an option , we say you have a long position in the
option; and when you write an option, we say you have a short
position in the option
Denoting
S – the value of the underlying asset
c – the value of a call option
p – the value of a put option
X – the exercise price
T – the expiration date
t – current time
 - T-t: time to expiration
Gain/Loss for a Buyer of a Call Option
at the Expiration Date
Profit / Loss
S
c
X
Gain/Loss for a writer of a Call Option
at the Expiration Date
Profit / Loss
c
X
S
Gain/Loss for a Buyer of a Put Option
at the Expiration Date
Profit / Loss
X
p
S
Gain/Loss for a Seller of a Put Option
at the Expiration Date
Profit or Loss
p
S
X
Intrinsic and Time Value
Option prices can be broken down into two components:
intrinsic value and time value.
The Intrinsic value is the value of the option if it is immediately
exercised or zero.
For calls: IVc = max(0, S0-X) and for puts: IVc = max(0, X- S0),
as the stock price change IV may change as well.
The time value is difference between the option current value
and the intrinsic value, reflecting the possibility that the option
will create further gains in the future.
Intrinsic and Time Value
Call options also classified into:
•At the money – when the current stock price is close to the strike
price.
In the money – when the current stock price is much above the
strike price.
Out of the money – when the current stock price is much below
the strike price.
Option value
Curren
t Value
Time value
Intrinsic
Value
S
X
Out of the
money
At the
money
In the
money
Option value
Curren
t Value
Time value
Intrinsic
Value
S
X
In the
money
At the
money
Out of the
money
The Range of A Call Option’s Values

X 
c 0  max 0, S0 
τ 
1  r  

At the expiration
date:
cT  max0, ST  X
Cash Flows
At Expiration
ST  X
Today (0) ST  X
Trading Strategy
Buy one call option
 c0
ST  X
Sell short one share of stock
 S0
 ST
 ST
X
X
Lend $X 1  r 
τ
Net cash flow
 X 1  r 
τ
 c0  S0  X 1  r 
τ
X
c 0  S0 
τ
1  r 
0
0
X  ST (positive)
A Call Option’s Upper and Lower
Boundaries
Call Price
S0
Value Range
S0  X 1  r 
τ
X 1  r 
τ
S0
The Range of A Put Option’s Values


X
p0  max0,
 S0 
τ
 1  r 

At the expiration
date:
pT  max0, X  ST 
Cash Flows
Trading Strategy
Today (0)
At Expiration
ST  X
ST  X
Buy one put option
 p0
0
Buy one share of stock
 S0
 ST
 ST
X
X
Borrow $X 1  r τ
Net cash flow
 X 1  r 
τ
 p0  S0  X 1  r 
τ
X
p0 
 S0
τ
1  r 
X  ST
ST  X(positive) 0
A Put Option’s Upper and Lower
Boundaries
put Price
X 1  r 
τ
Value Range
X 1  r   S0
τ
τ
X 1  r 
S0
The Factors that Affect on the Option Value
Call
S0
X
τ
r
Put
Option Strategies
Bull Spread
A bull spread is an option strategy designed to allow investors
to profit if prices rise but to limit his losses if prices fall.
A bull spread is employed by buying a call option with low
strike price (XL) and writing a call option with high strike price
(XH)
Numerical Example
Consider buying a call option X($45) at 8$ and writing a call
option X($55) at $3.
ST
Long Call
X=$45
Short Call
X=$55
Bull Spread
40
-8
3
-5
45
-8
3
-5
50
-3
3
0
55
2
3
5
60
7
-2
5
70
17
-12
5
Profit or Loss
XL
cH-cL
XH
s
Bull Spread
There is more than one way to implement a bull spread
strategy:
Buy a Call at XL and write a call at XH.
Buy a put at XL and write a call at XH.
Buy a put at XL, write a call at XH and buy the stock.
Option Strategies
Bear Spread
A bear spread is an option strategy designed to allow
investors to profit if prices fall but to limit his losses if prices
rise.
A bear spread is employed by writing a call option with low
strike price (XL) and buying a call option with high strike price
(XH)
Numerical Example
Consider writing a call option X($45) at 8$ and buying a call
option X($55) at $3.
ST
Short Call
X=$45
Long Call
X=$55
Bear Spread
40
8
-3
5
45
8
-3
5
50
3
-3
0
55
-2
-3
-5
60
-7
2
-5
70
-17
12
-5
Profit or Loss
cL-cH
XL
XH
s
Option Strategies
Long Straddle
A long straddle is an option strategy designed to investor who
believes that something dramatic will happen to the stock price
but he has no sure exactly which direction it will go .
A long straddle is employed by buying both put and call
options at the same strike price.
Numerical Example
Consider buying a call option X($50) at $5 and buying a put
option X($50) at $3.
ST
Long Call
X=$50
Long put
X=$50
Straddle
Spread
30
-5
17
12
40
-5
7
2
45
-5
2
-3
50
-5
-3
-8
55
0
-3
-3
60
5
-3
2
70
15
-3
12
Profit or Loss
X
- (p+c)
S
Short Straddle
A short straddle is an option strategy designed to investor who
believes that stock price will be stable.
A short straddle is employed by writing both put and call options
at the same strike price.
Profit or Loss
p+c
S
X
Long Butterfly
A Long Butterfly is an option strategy designed to investor who
believes that stock price will be stable but to limit his losses if the
price will be volatile.
A Butterfly is employed by buying a Call option with low strike
price (XL) and a Call option with high strike price (XH), and write
two Call options with medium strike price (XM).
Long Butterfly
Because of the non-linear relationship between the Call price and
the strike price:
C(XL )  C(XH )  2  C(XM )
C
(C(XL)+ C(XL))/2
C(XM)
X
XL
XM
XH
Long Butterfly
This implies that the premium balance is negative.
Profit or Loss
S
2CM-(CL+CH)
XL
XM
XH
Put-Call Parity
The value of a call option and a put option on the same
underlying asset, with the same exercise price and maturity, are
related by simple formula called put-call parity.
The formula is derived by the no-arbitrage argument, using a
strategy composed of the underlying asset, a put option, a call
option, and a riskless asset.
At the expiration date, this strategy’s cash flow is expected to be
zero in each event, and therefore, its value must be zero.
Cash Flows
At Expiration
Trading Strategy
Today (0) S  X
T
Buy one call option
Sell short one share of stock
Lend $X 1  r τ
ST  X
0
 ST
 X 1  r 
 ST
X
p0
0
 S0
τ
Write (Sell) one put option
Net cash flow
 c0
ST  X
 c0  S0  X 1  r   p0
X
 c0  S0 
 p0  0 
τ
1  r 
τ
0
X
 (X  ST )
0
X
c0  S0 
 p0
τ
1  r 
Numerical Example
The stock price is $100 and the risk-free interest rate is 5%. A
Call and a put options with a strike price of $100 and 6 month to
maturity are traded at $5 and $4, respectively. Show arbitrage
strategy!
X
100
c  p S
 4  100
 6.4  5
0.5
τ
(1  r )
1.05
Cash Flows
At Expiration
Trading Strategy
Buy one call option
Sell short one share of stock
Write (Sell) one put option
Lend $100/1.050.5=97.6
Net cash flow
Today (0) S  X
T
5
ST  X
ST  100
0
 ST
4
 ST
0
 (100 ST )
 97.6
100
100
1 .4
0
 100
0
Hedging with Options and Forward Contracts
A U.S. firm has been promised a payment of 1M£ in one month
The spot price is 1.8$/£. A Call and a Put options with a strike
price of 1.82$/£ and one month to maturity are traded at $0.05 and
$0.02, respectively. A forward contract with one month to maturity
is traded at a forward rate of 1.83$/£.
The firm wants to ensure that it will not get less than $1.78 per
one pound, but for each cent that the spot price will above 1.82 it
wants to gain one cent.
Buying a Put Option
Long £
1.82
-0.02
1.8
1.82
Selling a Forward and Buying a Call Option
Long £
1.83
-0.05
1.78
1.83
1.82
Binomial Option Pricing Model (BOPM)
The BOPM is a relatively simple way to price options and it is
based on the following assumptions:
An efficient market.
Short Selling is allowed with a full used of the proceeds.
Borrowing and lending at the risk-free interest rate is
permitted.
Future stock price will have one of two possible values.
The BOPM is developed in four steps
Step 1: Determine Stock Price Distribution
The two possible future values of the stock are Su and Sd,
where:
Su  u  S0
Sd  d  S0
u and d are constants and satisfy:
d  1 r  u
Su  u  S0
S0
Sd  d  S0
0
T
Step 2: Determine Option Price Distribution
Given the stock price distribution we can calculate the value of
the call option at expiration date.
cu  max(0, Su  X)
c0
cd  max(0, Sd  X)
0
T
Numerical Example
S0  100
X  100
d  0 .9
u  1 .2
Su  1.2 100  120, cu  120100  20
S0  100
Sd  0.9 100  90, cd  0
0
T
Step 3: Create A Riskless Portfolio
As the stock and the option’s values are fully correlated, we
can construct a riskless portfolio by holding the stock and the
option in opposite direction with some proportion:
Writing one call option
Buying h shares of stock such that the portfolio's future
cash flow will be identical in each state of nature:
 c u  hSu  c d  hSd
c u  cd
c u  cd
h

Su  Sd S0 u  d 
c u  cd
20
2
h


S0 (u  d) 100(1.2  0.9) 3
h is the number of shares we must buy for one call option we
write.
Cash Flows
At Expiration
Trading Strategy
Today (0)
Writing 3 call option
2 c0
Buy 2 shares of stock
 2 100
Net cash flow
2c0  200
ST  120
 3  20  60
ST  90
0
2 120  240
2  90  180
180
180
Step 4: Solve for the Call Using NPV
The portfolio's value is the present value of its expected cash
flows
As the portfolio’s future cash flows is known for certainty, the
appropriate discount rate should be the risk-free interest rate.
CFT
I0 
τ
1  r 
hSu  c u
 c 0  hS0 
τ
1  r 
c u  hSu
c 0  hS0 
τ
1  r 
Numerical Example
Consider the pervious example and suppose that the time to
maturity is =1/4 year and the risk-free interest rate is 5%
2
20  120
2
3
c 0  100
 7.4
0.25
3
1  0.05
Arbitrage Opportunity
Suppose that the call option is traded at $8, which implies that
it is overpriced
Cash Flows
At Expiration
Trading Strategy
Write (sell) 3 call option
Buy 2 shares of stock
Today (0)
Su  120
3  8  24
 3  (120 100)  60
 2 100  200
2 120  240
Borrow (200-24)=176
176
Net cash flow
0
Sd  90
0
2  90  180
176×(1. 05)0.25  178.2
 178 .2
1 .8
1 .8
Arbitrage Opportunity
Suppose that the call option is traded at $6, which implies that
it is underpriced
Cash Flows
At Expiration
Trading Strategy
Buy (sell) 3 call option
Today (0)
3  (120 100)  60
 3  6  18
Net cash flow
 182
0
Sd  90
0
 2 120  240
2  90  180
 182 (1.05)0.25  184.2
184 .2
Sell short 2 shares of stock
2 100  200
Lend (200-18)=182
Su  120
4 .2
4 .2
The analytical Solution of the BOPM
Substituting the hedging (h) equation in the pricing
equation:
1  R d
uR
c  c u
 cd

R  u d
u d 
R  1  r 
τ
Substituting the Call pricing equation in the Put-Call-Parity:
1  R d
uR
p  p u
 pd

R  u d
u d 
The Multi-Period BOPM
The number of choices in period n is equal to n+1:
Suu
Su
S0
Su d  Sd u
Sd
Sd d
The recursive solution n=2
τ 1
r  5 % u  1 .2
d  0.9 X  100 S0  100
Suu  144
Su  120
Su d  108
S0  100
Sd  90
Sd d  81
cuu  44
cu
cu d  8
c0
cd
cd d  0
1  R d
uR
c u  c uu
 c ud
R
u d
u  d 
1  1.05  0.9 1.2  1.05

44

8

24
.
76
1.05  1.2  0.9
1.2  0.9 
1  R d
uR
cd  cdu
 cdd
R
u d
u  d 
1  1.05  0.9
1.2  1.05

8

0

3
.
81
1.05  1.2  0.9
1.2  0.9 
1  R d
uR
c 0  c u
 cd

R  u d
u d 
1 
1.05  0.9
1.2  1.05

24.76
 3.81
 13.6


1.05 
1.2  0.9
1.2  0.9 
The analytical solution n=2

1
2
2
c 0  2 c uu Q u  2c ud Q u Q d  c dd Q d
R
R d
uR
Qu 
, Qd 
u d
u d
Qu  Qd  1

R  d 1.05  0.9
Qu 

 0.5
u  d 1.2  0.9
Q d  1  0.5  0.5


1
2
2
c0 
44  0.5  2  8  0.5  13.6
2
1.05
Black-Scholes Option Pricing Model (B&S)
The B&S Model is based on a creation of a fully hedged
portfolio; thus we can you a risk-free interest rate to discount the
cash flow.
The model assumptions are:
An efficient market.
Short Selling is allowed with a full used of the proceeds.
Borrowing and lending at the risk-free interest rate is
permitted.
The stock price is Log-Normal distributed, which implies
that the stock returns are Normally distributed.
B&S Formula
 rτ
c  SN (d1 )  Xe
N(d 2 )
N(.) is the cumulative area of the standard normal
distribution at the value d.
e is the base of natural logarithms and is approximately
equal to 2.7128.
r is the continuously compounded, annual risk-free interest
rate.
N(d)
d
B&S Formula
 rτ
c0  S0 N(d1 )  Xe
N(d 2 )
N(d1) is also known as delta  - the sensitivity of the option
value to changes in the stock value,
or the hedging ratio - the number of shares we must buy for
each one call option we write to create a riskless portfolio (a
fully hedged portfolio).
N(d2) is the probability of exercising the option:
The deeper is the option in the money the higher the option
exercising probability.
S
σ
ln( )  (r 
)τ
X
2
d1 
σ τ
2
d 2  d1  σ τ
 is the standard deviation of stock returns reflecting the
volatility of stock price.
The higher the stock price volatility the higher the option
value.
Thus, we need five variables – X, S, , r,  - to calculate the
price of a call option.
p0  Xert N(d 2 )  S0 N(d1 )
Numerical Example
Consider the following data:
S0  $100, X  $100, σ  30%, r  5%, τ  1/ 4
What is the value of a call option?
S
σ2
ln( )  ( r 
)τ
X
2
d1 

σ τ
100
0.32
ln(
)  (0.05 
)0.25
100
2
 0.1583
0.3 0.25
d2  d1  σ τ  0.1583 0.3 0.25  0.0083
Numerical Example
N(d1 )  N(0.1583)  0.563
N(d 2 )  N(0.0083)  0.503
c 0  S0 N(d1 )  Xe rτ N(d 2 )
c 0  100 0.563 100 e 0.050.25  0.503  $6.63
B&S Formula for Currency Options
c 0  S0 e
 rF τ
 rL τ
N(d1 )  Xe
N (d 2 )
p 0  Xe rL τ N(d 2 )  S0 e  rF τ N(d1 )
rL – The local risk-free interest rate
rF – The foreign risk-free interest rate
S
σ2
ln( )  (rL  rF 
)τ
X
2
d1 
σ τ
d 2  d1  σ τ
Numerical Example
Consider the following data: the spot $/£ exchange rate is
S0= $1.6, the US interest rate is rL= 2%, the UK interest rate is
rF = 4%, the exchange rate volatility is  =10%, and the time to
maturity is =1/4. What is the value of a call option and a put
option with strike price of X=$1.55?
S
σ2
ln( )  ( rL  rF 
)τ
X
2
d1 

σ τ
1 .6
0.12
ln(
)  (0.02  0.04 
)  0.25
1.55
2
 0.56
0.1  0.25
d2  d1  σ τ  0.56  0.1 0.25  0.51
Numerical Example
N(d1 )  N(0.56)  0.712
N(d 2 )  N(0.51)  0.695
c  SNe  rF τ (d1 )  Xe rL τ N(d 2 )
c  1.6  e 0.040.25 0.712 1.55 e 0.020.25  0.695  $0.056