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1
CHAPTER FIVE: Options and Dynamic No-Arbitrage
2
A Brief Introduction of Options
An option is the right of choice exercised in future. The holder (buyer, or longer) of the option has a right but not an obligation to buy or sell a special amount of the asset with a special quality at a pre-determined price.
• Call and put
• Exercise price
• Expiration date
• American options (C and P) vs. European options ( c and p)
X
T
3
The payoff profiles of call and putCall Put
Long Short
+ +
__ X XST ST
0 0
Long Short
+ +
__ X XST
ST
0 0
In-the-money, out-of-the-money, at-the-money, intrinsic value and time value
— A Brief Introduction of Options (Cont.)
4
The Basic No-Arbitrage1) , tStCtc XtPtp
2) tTrfXetp
3) , 0 tctC 0 tptP
4) If , then , tTtT 21 tCtC 21 tPtP 21
5)
C t S t X
P t X S t
max ,
max ,
0
0
0,max
0,max
TSXTp
XTSTc
5
The Basic No-Arbitrage (Cont.)The underlying is a non-dividend-paying stock
0,max tTrfXetStc
Suppose , then tTrfXetStc
Arbitrage Immediate Cash Flow Position Cash Flow on the expired date
Short a stock
Long an European call
Long riskless security
Net cash flows
S t S T
c t max ,S T X 0 tTrfXe X
tcXetS tTrf max ,S T X S T X 0
0 0Arbitrage Opportunity !
6
The Basic No-Arbitrage (Cont.) tStcXetS tTrf 0,max
T 0 tTrfe c t S tProposition
If the period to expiration is very long, the value of an European call is almost equal to its underlying.
0,max0,max XtSXetStC tTrf
Proposition
An American call on a non-dividend-paying stock should never be exercised prior to the expiration date.
C t c t
7
• The relationship between American options and European options
0,max tSXetp tTrf 0,max tSXetP tTrf
0,max tSXtP
0,max tSXtP ?
C t c t P t p tand
Conclusion:
8
The Parity of Call and Put• The underlying is a non-dividend-paying stock
tTrfXetptctS S can be replicated by c, p and riskless security
Suppose tTrfXetptctS
Position Cash flow at Cash flow at time T
time t
Buy a share
Short a call
Long a put
Short treasury
Net cash flow
S t X XtS S t S T S T
c t S T X
p t X S T tTrfXe X X
tTrfXetptctS
00
00
Arbitrage!
9
• Relationship between exercise and forward price
tTrfFetS F XXF
XF
tptc
c t p t
tptc
• Non-dividend-paying stock’s American call and put
C t c t
P t p t tTrfXetPtCtS
tTrfXetStPtC
XtStPtC ?
10
• Non-dividend-paying stock’s American call and put (Cont.)
Position Cash flow at Cash flow at time when put exercised
time t
Short a share
Long an Amer. call
Short an Amer. put
Long treasury
Net cash flow
XtS XtS
t
S t
C t
P t
X XtPtCtS
S t
C t
X S t
ttrfXe
tCXXe ttrf
S t
C t
0 ttrfXe
tStCXe ttrf
0 tTrtTr ff XetSXetStC 0,max
0 tTrttr ff XeXe
0
t t T XtStPtC ProveTo
11
• Non-dividend-paying stock’s American call and put (Cont.)
tTrfXetStPtCXtS
• Underlying is dividend-paying stock
DPVtSXetptP
XeDPVtStctCtTr
tTr
f
f
Present value of dividends at time t
Present value of a long stock forward position
Present value of a short stock forward position
12
• Underlying is dividend-paying stock
DPVXetptctS tTrf
For European call and put
For American call and put
tTrfXetStPtCXDPVtS
Holds for non-dividend-paying stock underlying
Dividend paid
tC
P t
Proved!
How to prove it?
Please see the next page!
13
• Proof of
Position Cash flow at Cash flow at time when put exercised
time t
Short a share
Effect of dividends
Long an Euro. call
Short an Amer. put
Long treasury
Net cash flow
XtS XtS t
S t
tc
P t
X
S t
tc
X S t
ttrfXe
S t
tc
0 ttrfXe
PV DT t PV D
T t PV DT t
tPtc
XDPVtStt
DPV
tcXXe
tT
ttrf
DPV
tStcXe
tT
ttrf
0
tTrfXeDPVtStc 0
ttrttr
tT
ttr fff XeXeDPVtStcXe
00
tctC
14
Proposition!For an American call, when there are dividends with big amount, the call may be early exercised at a time immediately before the stock goes ex-dividend.
Question:
If there are n ex-dividend dates anticipated, what’s the optimal strategy to early exercise an American call?
Answer:Please read the last paragraph of page 74 of the textbook.
15
Dynamic No-Arbitrage
100AP
107
98
49.114
86.104
04.96
?BP
uBP
dBP
67.107uuBP
97.102udBP
46.98ddBP
t=0 t=1 t=2
rf 2%
Bond A
Bond B
16
107
49.114
86.104
u
BP
67.107uuBP
97.102udBP
• Replication step by step
Using Bond A and riskless security with market value to replicate Bond B’s value in the above step
u Lu
PBu
uuuB LP 107
67.10702.149.114 uuuuB LP
97.10202.186.104 uuudB LP
u 0 488.
Lu 50 78.103107 uuu
B LP
103uBP
17
• Replication step by step (Cont.)
97.102udBP
46.98ddBP
dBP
Replicating the blow binomial tree by using Bond A and riskless security with market value
ddL
d 0 509.
Ld 48 62.50.9898 ddd
B LP
Replicating the left binomial tree by using Bond A and riskless security with market value
L
103uBP
5.98dBP
BP5.0
53.48L53.98100 LPB
18
• Self-financing
103uBP
5.98dBP
BP
uuuB LPL 10702.1107
103
Notes:
1. Dynamic replication is forward while the procedure of pricing is backward
2. Short sale is available for self-financing
19
Option Pricing—Binomial Trees
— One-Step Binomial Model• Non-dividend-paying stock’s European call
S 60
Su 90
Sd 30?c
cu 30
cd 0
Using the underlying stock and riskless security with market value to replicate the European call
L
05.L 14 71.
?
c
S
30 0
90 3005.
Sensitivity of the replicating portfolio to the change of the stock.
20
• Is probability relevant to option pricing?
S 60
Su 90
Sd 30
q
q1
Probability distribution
Answer:
1. Directly: No!
2. Indirectly: Yes! q S
Probability distribution is not relevant to No arbitrage pricing
21
— One-Step Binomial Model (Cont.)
• Notation
)(1
1
1
rateresklessrr
downmovespricethewhendecreaselpropotionad
upmovespricethewhenincreaselpropotionau
f
d r u
No Arbitrage S uSu S dSd
uS rL c uS X
dS rL c dS X
u
d
max ,
max ,
0
0
c c
S u d
Ldc uc
r u d
u d
u d
Replicating :
Short sale of riskless security
0
22
Risk-Neutrality — Risk-Aversion
• A Mini Case — Tossing a Coin
5$10$
0
Head
Tail
Fair Game Fair Game4$
Risk premium Risk discount
Investment Gambling
Investors: risk-averse Gamblers: risk-prefer
From real economy
be charged by casino
risk-neutral
6$
23
— Risk-Neutral Pricing
c S Lc c
u d
dc uc
r u d r
r d
u dc
r
u r
u dc
r pc p c
u d u du d
u d
1 1
11
risk-neutral probability mean or expectation on risk-neutral probability
discounted by risk-free rate
Analysis becomes very simple!
pr d
u d
1
pu r
u dand
In an imaginary world
A risk-neutral world
24
— What Kind of Problems Can Be Resolved in an Imaginary Risk-Neutral World?• Proposition :
If a problem with its resolving procedure is fully irrelevant to people’s risk-preference, then it can be resolved in an imaginary risk-neutral world and the solution would be still valid in the real world.
• Proposition :
No-Arbitrage equilibrium in financial markets is fully irrelevant to people’s risk-preference. Therefore, risk-neutral pricing is valid equilibrium pricing. Risk-neutral pricing and no-arbitrage pricing must be equivalent to each other.
25
— Risk-Neutral Pricing (Multi-Step Binomial Model )
S
uS
dS
Su2
udS
Sd 2
?c
uc
dc
0,max 2 XSucuu
0,max XudScud
0,max 2 XSdcdd
t=0 t=1 t=2
The Underlying Stock
The Call
26
— Risk-Neutral Pricing (Cont. )
c r pc p c
c r pc p c
u uu ud
d ud dd
1
1
1
1
c r p c p p c p cuu ud dd 2 2 2
2 1 1
0,max1!!
!
1!!
!
0
0
XSduppjnj
nr
cppjnj
nrc
jnjn
j
jnjn
n
jdu
jnjn
jnj
Generalizing:
27
— A Mini Case The Underlying Stock The Call
10
12
8
40.14
60.9
40.6
?c
uc
dc
40.6uuc
60.1udc
0ddc
t=0 t=1 t=2 t=0 t=1 t=2
r 102.
8.0
2.1
d
u 45.01,55.08.02.1
8.002.1
pdu
drp
c r p c p p c p cuu ud dd
2 2 2
2
2 2
2 1 1
1
102055 6 40 2 055 0 45 160 0 45 0 2 62
.. . . . . . .
• Risk-Neutral Pricing:
28
— A Mini Case (Cont.)• Dynamic No-Arbitrage Pricing:
u uu ud
u uu ud
c c
uuS udS
Ldc uc
r u d
6 40 160
14 4 9 610
0 8 6 40 12 160
102 12 0 87 84
. .
. ..
. . . .
. . ..
157.4843.7120.1 uu
u LuSc
cd 0862.
616.5
824.0
L
c S L 2 62.
29
— Implication of Risk-Neutral Pricing
n
tt
t
t
r
CEPV
1 1
Mean or mathematical expectation with probability in the real world
Discount rates with risk premium
n
tt
ft
t
r
CEPV
1 1
Risk-free rate used as discount rates without risk premium
Question:
Does risk-neutral probability exist and is it unique?
Mean or mathematical expectation with risk-neutral probability in the imaginary world
30
Fundamental Theorems of Financial Economics
The First Financial Economics Theorem:Risk-neutral probabilities exist if and only if there are no riskless arbitrage opportunities.
The Second Financial Economics Theorem:The risk-neutral probabilities are unique if and only if the market is complete.
The Third Financial Economics Theorem:Under certain conditions, the ability to revise the portfolio of available securities over time can dynamically make up for the missing securities and effectively complete the market.
31
— Problem and Inverse Problem• many investors make portfolio changes
• each portfolio’s change is limited
• the aggregation creates a large volume of buying and selling to restore equilibrium
• implying arbitrage opportunity exists
• each arbitrageur wants to take as large position as possible
• a few arbitrageurs bring the price pressures to restore equilibrium
Inverse Problem:
Knowing the market prices of securities, determine the market’s risk-neutral probabilities.
Problem:
Knowing the market’s risk-neutral probabilities, determine the market prices of securities.
Unfortunately, are actual securities markets like this ? Are they incomplete ? So it would seem that we will not be able to solve the inverse problem; that is, although risk-neutral probabilities may exist, they are not unique. However, in 1954, economist Kenneth Arrow saved the day by stating the third fundamental theorem of financial economics, the critical idea behind modern securities pricing theory.
32
— Equivalent Martingale• Definition:
ifonly and if ,0 ; ,
arbitraryany for , iesprobabilit lconditiona certain with
process stochastic A , tureinfrastruc ninformatio the
with any timeAt process. stochasticdrift -zeroa is Martingale
tsts
P
tS
E S t S ss
.martingalea is tS
The risk-neutral valuation approach is sometimes referred to as using equivalent martingale measure, i.e., the risk-neutral probability is referred to an equivalent martingale measure (probability distribution).
33
Summary of Chapter Five1. No-Arbitrage The Key of Finance Theory,
Especially For Derivatives Such as Options.
2. Dynamic No-Arbitrage Pricing Risk-Neutral Pricing.
3. Does Risk-Neutral Probability Exist and Is It Unique?
4. The Core of Finance Theory — The Fundamental Theorems of Financial Economics
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