1

CHAPTER FIVE: Options and Dynamic No-Arbitrage

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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.)

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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 !

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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

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• 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:

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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!

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• 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 ?

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• 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

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• 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

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• 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!

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• 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

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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.

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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

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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

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• 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

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• 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

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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.

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• 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

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— 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

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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$

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— 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

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— 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.

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— 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

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— 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:

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— 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:

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— 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.

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— 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

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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.

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— 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.

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— 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).

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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