Equity Related Products
Futures and Options
Professor Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 1
Futures
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 2
Spot – Future - Parity
Today, one (theoretical) Index-Future is sold at 4.090 €
(1€ per Index-point). Long and Short-positions can be
described by a profit and loss diagram:
Long Future
= Buyer
Profit
Index
4090
Short Future
= Seller
Loss
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
If you are Long-Future,
then you may claim for
delivery of „one index“ at
a price of 4090 € at the
maturity of the indexfuture. That means, if the
index at delivery is quoted
at more than 4090, you
will win from your futures
position.
slide no.: 3
Spot – Future - Parity
You hold an Index-Portfolio, currently valued at 5,500 €
(1 Index-point = 1 €). If the annual risk free rate rf is at
3.5 % and the expected dividends on your Index portfolio
are at 100 € (d = 100/5,500) , an Index – Future with
one year to maturity has a fair price of:
F0  S0  1  rF  d 
F0  5 ,500  1  0 ,035  0 ,0182
F0  5 ,592.40 €
To prevent our Index-Portfolio from losses, we could
hedge the price risk by taking a short – future position
(selling a future at 5,592.40).
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 4
Spot – Future - Parity
The total expected payoffs from your
portfolio will depend on the future
state of the environment (see below
payoffs 1-5). A decreasing stock
market will be compensated by
profits from the short future position,
increasing stock prices will be
outbalanced by losses due to
payment obligations from the future.
Assets
Stock
Portfolio
Payoff1
Payoff2
Profit
Index
5692,40
Loss
Payoff3
Payoff4
Payoff5
+4500,00
+5000,00
+5500,00
+6000,00
+6500,00
+100,00
+100,00
+100,00
+100,00
+100,00
Short Future
+1092,40
+592,40
+92,40
-407,60
-907,60
Total
+5692,40
+5692,40
+5692,40
+5692,40
+5692,40
Dividends
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 5
Spot – Future - Parity
Assets
Payoff1
Payoff2
Payoff3
Payoff4
Payoff5
Stock Portfolio
+4500,0 +5000,00 +5500,00 +6000,00
0
+6500,0
0
Dividends
+100,00
+100,00
+100,00
+100,00
+100,00
Short Future
+1092,4
0
+592,40
+92,40
-407,60
-907,60
Total
+5692,4 +5692,40 +5692,40 +5692,40
0
+5692,4
0
Initially you have paid 5,500 € for your stock portfolio. Taking the
short future position, the final outcome of your portfolio will be
5,692,40 €, whatever the stock price will be, i.e. you will earn 192,40
which equals 3.5%. Obviously, this profit is riskless:
F
0
 D   S0
 rF
S0
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
 F0  S0  1  rF  d 
Spot-FutureParity
slide no.: 6
Spot – Future - Parity
Rising future prices will – due to arbitrage trading - induce rising spot
prices. For example, a future traded at 6,000 € is clearly overpriced,
when the stock portfolio remains unchanged at 5,500 €. In this case,
„smart“ traders will make arbitrage profits of 407,50 € per contract and
bring back the market to equilibrium:
Action
t0
t1
Borrow money at rF (3,5%)
+ 5,500.00
- 5,692.50
Buy/Sell Stock Portfolio
- 5,500.00
+ Stock
Sell/Buy Future at 6,000
0
+ 6,000.00 - Stock
Total
0
+ 307,50
Note, that the arbitrage profit equals the difference between a
fair- and mispriced future (6,000 – 5,592,40) plus Dividends.
Higher Future prices will lead to massivly increased demand at
spot markets until spot prices and futures are back to
equilibrium.
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 7
Spot – Future – Parity
Financial Market Stability
• Spot Markets and Future (Forward) Markets are
interlinked.
• Mispriced spot or future market instruments will affect
both markets
• Future market speculations that drive futures prices will
also drive spot market prices due to arbitrage trading (et
vice versa)
• Speculation on futures markets, resulting in higher future
prices will induce higher spot market prices due to
arbitrage trading. Finally this may result in spot market
bubbles that jeopardizes the allocation mechanism of real
goods markets.
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 8
Options
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 9
Economic Benefits Provided
by Options
Derivative securities are instruments that derive
their value from the value of other assets.
Derivatives include options, futures, and swaps.
Options and other derivative securities have several
important economic functions:
• Help bring about a more efficient allocation of risk;
• Save transactions costs…sometimes it is cheaper to trade a
derivative than the asset underlying it, and
• Permit investment strategies that would not otherwise be
possible.
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 10
Options Vocabulary
Call option
• Gives the holder the right to
purchase an asset at a specified price
on or before a certain date
Put option
• Gives the holder the right to sell an
asset at a specified price on or before
a certain date
Strike price or exercise price: the price specified
for purchase or sale in an option contract
American or
European
option
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
• American options allow holders to
exercise at any point prior to
expiration
• European options allow holders to
exercise only on the expiration date
slide no.: 11
Options Vocabulary
Long position
• The buyer of an option has a long
position, and has the right to
exercise the option.
Short position
• The seller (or writer) of an option has
a short position, and must fulfill the
contract if the buyer exercises.
• As compensation, the seller receives
the option premium.
Neither trade usually has any connection to the
underlying firm.
Can trade options on an exchange (such as CBOE)
or in the over-the-counter market.
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 12
Moneyness of Options
S = current stock price
X = strike price
S>X
Call
In-the-money
S=X
At-the-money
S<X
Out-of-themoney
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
Put
Out-of-themoney
At-the-money
In-the-money
slide no.: 13
Option Quotations
Option quotations
OptiTech
30.00
• The price per share for an option
contract, which is a contract to buy
or sell 100 shares of the underlying
stock.
• CBOE options expire on the third
Saturday of the expiration month.
Expire Strik
Call
s
e
April 27.50 3.26
Put
0.67
30.00
May
27.50 3.91
1.23
30.00
April
32.50 0.85
3.24
30.00
May
32.50 1.55
3.83
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
In-the-money calls
Out-of-the-money
puts
In-the-money puts
Out-of-the-money
calls
slide no.: 14
Intrinsic and Time
Value of Options
Intrinsic
value
Time value
• For in the money options: the
difference between the current price of
the underlying asset and the strike
price (S-X for calls and X-S for puts).
• For out of the money options: the
intrinsic value is zero.
• The difference between an option’s
intrinsic value and its market price
(premium)
• Consider the May call with $27.50 strike price from previous
table:
•Intrinsic value = $30.00 - $27.50 = $2.50
•Time value = $3.91 - $2.50 = $1.41
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 15
Payoff Diagrams
Shows value of an option on the expiration date
Y-axis plots exercise value or “intrinsic value”
X-axis plots price of underlying asset
Long and short positions
Use payoff
diagrams for:
Gross and net positions (the net
positions subtract the option
premium)
Payoff: the price of the option at expiration date
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 16
Long Call Option Payoffs
Payoff at Expiration
x = $75, premium = $8
Payoff
slope = 1
-8
75
83
stock price
Net payoff
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 17
Short Call Option Payoffs
+8
75premium
83 = $8
x = $75,
Payoff at expiration
stock price
Payoff
Net payoff
slope = -1
• Both long and short positions have zero net payoff at a price of $83
• On net basis, buyer of the call makes a profit when the price exceeds $
83; seller of the call makes a profit when price is below $83.
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 18
Long Put Option Payoffs
75
x = 75, premium = $7
68
Payoff at expiration
Payoff
Price of stock
68
75
-7
Net payoff
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 19
Short Put Option Payoffs
Payoff at expiration
x = 75, premium = $7
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
Net payoff
7
68
75
Stock price
Payoff
-75
slide no.: 20
Portfolios of Options
Look at payoff diagrams for combinations of options
rather than just one
Shows the range of potential strategies made
possible by options
Some positions can be a form of portfolio insurance.
Some strategies allow investor to speculate on the
volatility (or lack thereof) of a stock rather than
betting on which direction it will move.
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 21
Portfolio Containing 1 Call
and 1 Put (Long Straddle)
Call x = 30, premium = $4.5, Put x = 30, premium = $3.5
Payoff
Net payoff
30
22
38
-8
• Buy a put and a call on the same
stock at the same strike price and the
same expiration date
• Profits come with large price
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 22
Option Strategies
(Straddle)
80
70
60
50
40
30
20
10
0
-10
-20
40
60
Stock Price
Long Call (Profit / Loss)
Long Put (Profit / Loss)
Straddle (Profit/Loss)
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
80
40
-10
75
65
100
60
-10
55
45
120
140
160
180
200
80 100 120 140 160 180 200
-10 -10 -10 10 30 50 70
35 15 -5
-5
-5
-5
-5
25 5 -15 5
25 45 65
slide no.: 23
Option Strategies
(Strangle)
Position Value
Strangle - Long call and long put (at different exercise prices)
Strategy for profiting from high volatility
Strangle
Share Price
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 24
Option Strategies
Synthetic Long Future
Position Value
Synthetic Long Future
(Long Call & Short Put)
Long Call
Short Put
Share Price
Exercise Price (Strike)
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 25
Option Strategies
Synthetic Short Future
Position Value
Synthetic Short Future(Short Call & Long Put)
Short Call
Share Price
Long Put
Exercise Price (Strike)
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 26
Option Strategies
(Short Butterfly)
100
80
60
40
20
0
-20
-40
-60
40
60
80
Stock Price
Long Call (Profit / Loss)
Long Put (Profit / Loss)
Short Call (Profit/Loss)
Short Put (Profit/Loss)
Butterfly
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
100
40
-10
75
5
-37
33
60
-10
55
5
-17
33
120
80
-10
35
5
3
33
140
100
-10
15
5
3
13
120
-10
-5
5
3
-7
160
140
10
-5
5
3
13
180
160
30
-5
5
3
33
180
50
-5
-15
3
33
200
200
70
-5
-35
3
33
slide no.: 27
Other Option Portfolio
Payoffs
Now look at portfolios containing options, stocks, and
bonds:
Looking at these payoffs will help lead us to an
important option pricing relationship: put-call
parity.
Construct portfolios that include options, stocks and
bonds:
Stock and put options
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
Bond and call options
slide no.: 28
Payoff at expiration
Gross Payoff of Stock + Put
$X = strike price of put
x
x
•
•
Stock price
Position allows investor to profit if stock price rises above
$X.
If stock price falls below $X, portfolio provides protection:
put option allows investor to sell at a price no lower than $X.
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 29
Payoff at expiration
Gross Payoff of Bond + Call
$X = strike price of call
and face value of bond
x
x
stock price
• The
bond
assuresand
a minimum
payoffone
of $X
This
payoff
diagram
the preceding
are
• The call allows for aidentical!
higher payoff if the stock price
rises
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 30
Put-Call Parity
Future payoffs of “stock+put” are identical to payoffs
of “bond+call” provided:
•
•
•
•
•
Put and call have same exercise price and expiration date;
Underlying stock pays no dividends during life of options;
Put and call are European options;
Bond is risk-free, zero-coupon, price at maturity = strike (X),
Bond matures when options expire.
If two assets A and B, have same future payoffs
with certainty, then they should sell for the
same price now
Price of put + price of stock = Price
of call + price of bond
P+S=C+B
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 31
Factors Affecting Option Prices
(holding other factors equal)
Price of
underlying
asset
• Asset price and call price are
positively related.
• Asset price and put price are
negatively related.
Time to
expiration
• More time usually makes options more
valuable.
Strike price
• Higher X means higher put price; lower
X means higher call price.
Interest rate
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
• Calls: higher rate means higher call
value.
• Puts: higher rate reduces put value.
slide no.: 32
Evaluation Framework
Assume that a stock is currently quoted at 150 €. If nothing happens
over the coming year, the stock‘s price will also be at 150 € in one
year. The one-year risk free rate is at 10%.
Under this assumptions, a Call – Option, maturing one year from now
with a strike of 120 € is easy to value:
1 year
Stock – price t0:
150 €
Strike Call t0 (PV):
108,58 €
= 120 X 2,7184 – 0,10
Stock – price t1:
150 €
Strike Call t1:
120 €
The Intrinsic Value of the Call (X-S) is 41,42 € !!
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 33
Evaluation Framework
Under this conditions, the Call Option Pricing Model is like:
PVCALL  S  X  e  r
41,42  150  120  2 ,71840 ,10
This price is a fair price, as it does not allow to gain risk-free profits
from arbitrage-trading:
One could also borrow money to buy the stock now. At a risk free rate
of 10% p.a. the cost of borrowing over one year will add to ((150€ x
(2,7184 0,10) - 1)) 15.7764 €.
The other way round – borrowing money to buy the option - and one
year later the stock leads to the same borrowing costs: Borrowing of
41,42 € at 10% means to pay back 41,42 x 2,7184 0,10 = 45,7764 €
after one year. Netted with the profit from the option‘s exercise at a
strike of 120 € (150 € - 120 € = 30 €), the costs add to 15.7764 €.
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 34
Determinants of Option Prices
Variable
Strike
Term to
Exercise
Price of the
Underlying
Volatility
of the
Interest Rate
Underlying
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
Direction
...the higher
the strike
...the longer
the duration
...the higher
the price
...the higher
...the
higher the
volatility
the rate
Option Price
...the smaller
the price
...the higher the
price
...the higher the
price
...the higher
...the
higherthe
the
price
price
slide no.: 35
Price & Value Chart
Option Price
Intrinsic Value
Time Value
„Greeks“ show the
sensitivity of the option
price referring to:
Option Price
Intrinsic Value
Time Value
Premium
Stock Price
Strike
Out of the
money
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
at the
money
in the
money
D = Delta:
Call Price and Spot
Price
G = Gamma:
Delta
 = Theta:
Call Price and Time to
Expiration
K = Kappa / Vega:
Call Price and Volatility
R = Rho:
Call Price and Int. Rate
slide no.: 36
Factors Affecting Option Prices
Volatility
Suppose a stock now worth $40 might increase or
decrease in value by $10:
Call option with X = $40 will pay $10 or $0.
Now suppose a stock worth $40 might increase or
decrease in value by $20:
Call option with X = $40 will pay $20 or $0.
The 2nd call option is more valuable…upside is
better, downside the same as the 1st option.
Prof. Dr. Rainer Stachuletz
Corporate Finance
Berlin School of Economics
slide no.: 37