lect6 options
DESCRIPTION
Lecture 6 for foutre and optionsTRANSCRIPT
Chapter 9
Mechanics of Options
Markets
1
Review of Option Types
A call is an option to buy
A put is an option to sell
A European option can be exercised only at
the end of its life
An American option can be exercised at any
time
2
Option Positions
Long call
Long put
Short call
Short put
3
Long Call (Figure 9.1, Page 195)
Profit from buying one European call option: option
price = $5, strike price = $100, option life = 2 months
4
30
20
10
0 -5
70 80 90 100
110 120 130
Profit ($)
Terminal
stock price ($)
Short Call (Figure 9.3, page 197)
Profit from writing one European call option: option
price = $5, strike price = $100
5
-30
-20
-10
0 5
70 80 90 100
110 120 130
Profit ($)
Terminal
stock price ($)
Long Put (Figure 9.2, page 196)
Profit from buying a European put option: option
price = $7, strike price = $70
6
30
20
10
0
-7 70 60 50 40 80 90 100
Profit ($)
Terminal
stock price ($)
Short Put (Figure 9.4, page 197)
Profit from writing a European put option: option price
= $7, strike price = $70
7
-30
-20
-10
7
0 70
60 50 40
80 90 100
Profit ($) Terminal
stock price ($)
Payoffs from Options What is the Option Position in Each Case?
K = Strike price, ST = Price of asset at maturity
Long call: max(0,ST-K) Short call: -max(0,ST-K)
Long put: max(0,K-ST) Short call: - max(0,K-ST) 8
Payoff Payoff
ST ST K
K
Payoff Payoff
ST ST K
K
Assets Underlying
Exchange-Traded Options Page 198-199
Stocks
Foreign Currency
Stock Indices
Futures
9
Terminology
Moneyness :
At-the-money option
In-the-money option
Out-of-the-money option
10
Terminology (continued)
Option class: all options of the same type (calls
or puts) traded on a certain asset
Option series: all the options of a given class with
the same expiration date and strike price
Intrinsic value: maximum of zero and the value of
exercise if it were exercised immediately
Time value: premium a rational investor would
pay over its current exercise value (intrinsic
value), based on the probability it will increase in
value before expiry.
11
Stock Splits (Page 202-204)
Example: before the stock split the value of a firm is 100$ and is divided into 100 shares worth 1$ each
After a 2 for 1 stock split the value of the firm is divided into 200 shares worth 0.5$ each
Suppose you own N options with a strike price of K :
When there is an n-for-m stock split,
• the strike price is reduced to mK/n
• the no. of options is increased to nN/m
Stock dividends are handled similarly to stock splits
12
Dividends & Stock Splits: example
Consider a call option to buy 100 shares
for $20/share
How should terms be adjusted:
for a 2-for-1 stock split? Call option to buy
200 shares for $10/share
for a 5% stock dividend? Equivalent to a 21
for 20 stock split (21/20=1.05): the holder of
the option has the right to buy 100*21/20=105
share for 20/21*20$=19.05$ each
13
Market Makers
Typically trading of options in an exchange market
takes place through a market maker.
A market maker quotes both bid and ask prices
when requested
The market maker does not know whether the
individual requesting the quotes wants to buy or
sell
The offer price is higher than the bid and
exchanges can set up upper bounds to the bid-
offer spread
Market makers make the market more liquid
14
Margins (Page 205-206)
Margins are required when options are sold
When a naked option is written the margin is the
greater of:
A total of 100% of the proceeds of the sale plus
20% of the underlying share price less the
amount (if any) by which the option is out of the
money
A total of 100% of the proceeds of the sale plus
10% of the underlying share price (call) or
exercise price (put)
For other trading strategies there are special rules
15
Employee Stock Options and
Convertible Bonds Employee stock options are a form of remuneration
issued by a company to its executives
They are usually at the money when issued
When options are exercised the company issues
more stock and sells it to the option holder for the
strike price
Convertible bonds are regular bonds that can be
exchanged for equity at certain times in the future
according to a predetermined exchange ratio
16
Chapter 10
Properties of Stock Options
17
Notation
18
c: European call option price
p: European put option price
S0: Stock price today
K: Strike price
T: Life of option
s: Volatility of stock price
C: American call option price
P: American put option price
ST: Stock price at option maturity
D: PV of dividends paid during life of option
r Risk-free rate for maturity T with cont.
comp.
Effect of Variables on Option
Pricing (Table 10.1, page 215)
Variable c p C P
S0 + − + −
K − + − +
T ? ? + +
s + + + +
r + − + −
D − + − +
19
American vs European Options
20
An American option is worth at least as much
as the corresponding European option
C c
P p
Calls: An Arbitrage Opportunity? Suppose that
Is there an arbitrage opportunity?
Short the stock and buy the call to get a cash flow of 20$ - 3$ = 17$. Invest for one year and get 17e0.1 = 18.79$
If ST > 18$, exercise the call and get a profit of 0.79$
If ST < 18$, for example, 17$, buy the stock back and make a profit of 18.79$ - 17$ = 1.79$
21
c = 3; S0 = 20; K = 18; T = 1; r = 10%; D = 0
71.371.31820 1.0
0 ceKeS rT
Lower Bound for European Call
Option Prices; No Dividends (Equation 10.4, page 220)
c S0 –Ke -rT
22
Puts: An Arbitrage Opportunity? Suppose that
Is there an arbitrage opportunity?
Borrow 38$ to buy the put and the stock, and repay after 6 months 38e0.05*0.5=38.96$
If ST < 40 the arbitrageur exercises the put and sells for 40$. The profit is 40$ - 38.96$ = 1.04$
If ST > 40, for example 42$, the arbitrageur discards the option, sells the stock and makes a profit of 42$ - 38.96$ = 3.04$
23
p= 1; S0 = 37; K = 40; T = 0.5; r =5%; D = 0
$01.2$01.2$37$40 5.005.0
0 peSKe rT
Lower Bound for European Put
Prices; No Dividends (Equation 10.5, page 221)
p Ke -rT–S0
24
Put-Call Parity: No Dividends
Consider the following 2 portfolios:
Portfolio A: European call on a stock + zero-
coupon bond that pays K at time T
Portfolio C: European put on the stock + the stock
25
Values of Portfolios
26
ST > K ST < K
Portfolio A Call option ST − K 0
Zero-coupon bond K K
Total ST K
Portfolio C Put Option 0 K− ST
Share ST ST
Total ST K
The Put-Call Parity Result (Equation
10.6, page 222)
Both are worth max(ST , K ) at the maturity of
the options
They must therefore be worth the same
today. This means that
c + Ke -rT = p + S0
c – p = S0 - Ke -rT
27
Suppose that
What are the arbitrage possibilities when p = 2.25? In this case
the put-call parity condition does not hold:
Portf. A: c + Ke–rT =3+ 30e-0.1*0.25=32.26$ and
Portf. C: p+S0=2.25+31=33.25$
Portfolio C is overpriced: short C and buy A.
Buy the call and short both the put and the stock to generate the
cash flow -3 + 2.25 + 31= 30.25$
Invest this cash flow at 10% to get 30.25e0.1*0.25=$31.02 in 3 months
If ST>30, exercise the call and buy stock for 30.00$. If ST<30 the put
is exercised and the counterparty sells at 30.00$.
So independently of ST the net profit is: =31.02-30=1.02$
28
Arbitrage Opportunities
c= 3; S0= 31; K =30; r =10%; T = 0.25; D = 0
Suppose that
What are the arbitrage possibilities when p = 1? In this case the
put-call parity condition does not hold:
Portf.A: c + Ke –rT =3+ 30e-0.1*0.25=32.26$ and
Portf C: p+S0=1+31=32$
Portfolio A is overpriced: short A and buy C.
Short the call to finance the purchase of the put and the stock which
lead to a cash outflow 31+1-3=29$.
Borrow 29$ at the 10%p.a. to repay 29e0.1*0.25=$29.73 in 3 months
If ST<30, exercise the put and sell at 30.00$. If ST>30, the call will
be exercised and the arbitrageur will sell at 30.00$.
So independently of ST the net profit is: = 30.00 - 29.73 =0.27$
29
Arbitrage Opportunities
c= 3; S0= 31; K =30; r =10%; T = 0.25; D = 0
Early Exercise
Usually there is some chance that an
American option will be exercised early
An exception is an American call on a non-
dividend paying stock
This should never be exercised early
30
Reasons For Not Exercising a Call
Early (No Dividends) Consider an American call option:
S0 = 100; T = 0.25; K = 60; D = 0
Suppose you think the stock is a good investment. Should you exercise immediately? No, for the following reasons:
– No income is sacrificed (the stock bears no dividends)
– You would forgo interest payment (time value of money)
– You would be worse off in the case that the spot price falls below 60 at maturity (holding the call provides an insurance against stock price falling below strike)
31
Reasons For Not Exercising a
Call Early (No Dividends) Suppose you do not feel that the stock is a good investment (it is overpriced). Should you exercise the option and sell the stock?
Doing so would generate a profit of S0 – K =40$ (intrinsic value of the option)
No, it is better to sell the call option because it has both intrinsic value and time value: C > S0 – K
Remember the lower bound for call prices:
c ≥ S0 – Ke-rT
. Because C ≥ c, C ≥ S0 – Ke-rT.
Because Ke-rT <K, it follows that C > S0 – K
32
Bounds for European or American Call
Options (No Dividends)
33
Should Puts Be Exercised
Early ?
• Are there any advantages to exercising an American put early?
• In contrast to call options it can be optimal to exercise put options early if deep in the money.
• Exercising the option earlier means getting money sooner than later (time value of money, i.e. interest rate)
• Because asset prices cannot be negative, if stock prices are sufficiently low the time value of holding the option is small
34
Bounds for European and American
Put Options (No Dividends)
35