Transcript When t=T

第三章
期权定价的离散模型--------二叉树方法
Chapter 3
Binomial Tree
Methods
------ Discrete Models
of Option Pricing
An Example
S  $45
u
T
S0  $40

STd  $35
Question: When t=0, buying a call option of the
stock at with strike price $40 and 1 month
maturity. If the risk-free annual interest rate is 12%
throughout the period [0, T], how much should the
premium for the call option（看涨期权） be?
Example cont.1

c

(
S

K
)
T
 （到期日收益）payoff = T
STu  $45, cT  (45  40)   $5
S0  $40
 Consider
STd  $35, cT  (35  40)   $0
a portfolio（投资组合）
  S  2c
Example cont.2
 When


t=T,
45  2*5  35, if S ,
VT ()  
 $35
35  2*0  35, if S .
 has fixed value $35, no matter S is up or
down
Example cont.3
 If
risk free interest r =12%, a bank
deposit of B=35/(1+0.01) after 1 month
35
1
VT ( B) 
 (1  12%)  35  VT ()
1  0.01
12

By arbitrage-free principle
V0 ( B)  V0 ().
Example cont.4
 That
is
S0  2c0  B0 
 Then
 This
35
 34.65
1  0.01
40  34.65
c0 
 2.695
2
is the investor should pay $2.695 for
this stock option.
Analysis of the Example
①
②
the idea of hedging: it is possible to
construct an investment portfolio  with S
and c such that it is risk-free.
The option price thus determined
(c_0=$2.695) has nothing to do with any
individual investor's expectation on the
future stock price.
One-Period & Two-State
 One-period:
assets are traded at t=0 & t=T
only, hence the term one period.
 Two-state: at t=T the risky asset S has two
possible values (states): STu & STd , with
their probabilities satisfying
0  Prop ST  STu  , Prop ST  STd   1
Prop ST  S
u
T
  Prop S
T
S
d
T
 1
One-Period & Two-State Model
 The
model is the simplest model.
 Consider a market consisting of two assets:
a risky S and a risk-free B
 If: risky asset St and risk free asset Bt
 known S0 , B0, when t=0,
 t=T, 2 possibilities
 S  S0u ,
ST :  d
u  d.
 ST  S0 d ,
Option Price at t=0?
(for strike price K, expired time T)
u
T
Analysis of the Model
St - Stock Price, is a stochastic variable
S  S 0u
Up, with probability p
STd  S0 d
Down, with probability 1-p
u
T
S0
V  ( S0u  K )
u
T
V0

VTd  ( S0 d  K ) 
where
Vt
is a stochastic variable.
Question & Analysis
If known VT  ( ST ) at t=T,
 how to find out V0 when t=0?
 Assume the risky asset to be a stock. Since the
stock option price is a random variable, the seller
of the option is faced with a risk in selling it.
However, the seller can manage the risk by buying
certain shares (denoted asΔ) of the stocks to
hedge the risk in the option.
 This is the idea!

Δ- Hedging Definition
 Definition:
for a given option V, trade Δ shares of the
underlying asset S in the opposite direction,
so that the portfolio
  V  S
is risk-free.
Analysis of Δ- Hedging
free asset BT   B0 ,   1  rT
 If Π is risk free, then, on t=T,
 risk
T  VT  ST
is risk free. i.e.
T   0
so that
VT  ST   0
Analysis of Δ- Hedging cont.

VT , ST are random variables, when t=T,
both of them have 2 possible values
u
VT  S0u   (V0  S0 ),
V  S0 d   (V0  S0 ),
d
T
where  &V0 are unknown, solve them:
VTu  VTd

S0 (u  d )
Analysis of Δ- Hedging
(Probability Measure)
1  d u u   d 
V0  T  S0  
VT 
VT 

 u d
ud

1

Define a new Probability Measure
 d
u
qu  ProbQ ST  ST  
ud
u
d
qd  ProbQ ST  ST  
ud
 Obviously
,
0  qu , qd  1, qu  qd  1.
Solution of Premium
 From
the discussion above,
1 Q
V0  E (VT ),

Q
E
(VT ) denotes the expectation of the
where
random variable VT under the probability
measure Q.
Definition of Discounted Price
 Let
U be a certain risky asset, and B a
risk-free asset, then U t / Bt is called
the discounted price贴现价格 (also
known as the relative price相对价格)
of the risky asset U
at time t. V
V 
 E  .
B0
 BT 
0
Q
T
Theorem 3.1
 Under
the probability measure Q, an
option's discounted price is its expectation
on the expiration date. i.e.
Q


E
(
S

K
)
/
B
,
call


T
T
V0 

B0  E Q  ( K  ST )  / BT  , put

Remark
 In
order to examine the meaning of the
probability measure Q, consider S is an
underlying risky asset. Calculate
1
Q  ST 
u
d
E  
qu ST  qd ST
 BT   B0


1   d
u
 S0

S 0u 
S0 d  

 B0  u  d
ud
 B0
Risk-Neutral World
 Under
the probability measure Q, the
expected return of a risky asset S at t=T is
the same as the return of a risk-free bond. A
financial market possessing this property is
called a Risk-Neutral World
 In a risk-neutral world, no investor demands
any compensation for risks, and the
expected return of any security is the riskfree interest rate.
Definitions
 the
probability measure Q defined by
qu  ProbQ ST  S
u
T
 d
 ud ,
u
qd  ProbQ ST  S  
ud
d
T
is called by risk-neutral measure.
 The option price given under the riskneutral measure is called the
risk-neutral price.
Definition of Replication

In a market consisted of a risky asset S and a riskfree asset B, if there exists a portfolio
  S   B
(where α,β are constants, Φ, V are both random
variables) such that the value of the portfolio Φ is
equal to the value of the option V at t=T, ST   BT  VT
then Φ is called a replicating portfolio of the option
V, then option price
V0  0   S0   B0
Theorem 3.2
 In
a market consisted of
a risky asset S
and
a risk-free asset B,
 d<ρ<u
is true if and only if
the market is arbitrage-free.
Proof of Theorem 3.2 (1st dir.)
 1)
arbitrage-free
d<ρ<u
 Suppose ρ>= u, consider the following
portfolio:
  S  (S0 / B0 ) B
Its values at t=0 and at t=T are:
 0   S0  ( S0 / B0 ) B0  0

T   ST  ( S0 / B0 ) BT
Proof of Theorem 3.2 (1st dir.) cont.

T is a random variable with two possible
values:
 u
 S0 
T   S0u     B0  (   u ) S0  0,
u

 B0 
for ST  ST
T  
 d   S d   S 0   B  (   d ) S  0
  0
0
0
 T
B
 0

d

for ST  ST
Proof of Theorem 3.2 (1st dir.) cont. 
So that, for the portfolio Φ
T  0,
& Prob T  0  Prob ST  STd   0.
That shows that there exists arbitrage opportunity
for portfolio Φ, contradiction! to that the market is
arbitrage-free.
 Same to ρ<=d.

Proof of Theorem 3.2 (2nd dir.)
 If
market is arbitrage-free, for any portfolio
  S   B
 If
0   S0   B0  0, T   ST   BT  0,
 then
T   ST   BT  0.
 In fact, define a risk-neutral measure Q
qu  ProbQ ST  S
u
T
 d
 ud ,
u
qd  ProbQ ST  S  
ud
d
T
Proof of Theorem 3.2 (2nd dir.) cont.
 Then
 Consider
0  qu , qd  1, qu  qd  1.
the expectation of the random
variable
T
Q
u
d
E (T )  qu T  qd T
 According
to the definition of the risk-neutral
measure
Q,
 d
u
Q
E ( T ) 
( S0u   B0 ) 
ud
  ( S0   B0 )   0  0.
ud
( S0 d   B0 )
Proof of Theorem 3.2 (2nd dir.) cont. That
qu   qd   0.
u
T
is,
 But
 Then
 i.e.
 There
d
T
Tu  0, Td  0
    0.
u
T
d
T
Prob T  0  0.
exists no arbitrage opportunity.
Theorem 3.2
 If
the market is arbitrage-free, then there
exists a risk-neutral measure Q defined by
qu  ProbQ ST  S
u
T
 d
 ud ,
u
qd  ProbQ ST  S  
ud
d
T
such that
Q


E
(
S

K
)
/ BT  , call
V0   T

B0  E Q  ( K  ST )  / BT  , put

Binomial Tree Method
 Divide
the option lifetime [0, T] into N
intervals: 0  t0  t1  ...  t N  T .
 Suppose the price change of the
underlying asset S in each interval
[tn , tn1 ](0  n  N  1)
can be described by the one-period twostate model, then the random movement
of S in [0, T] forms a binomial tree
Binomial Tree Method cont.

This means that if at the initial S
time
price of the
 Sthe
0
underlyingS asset is
, then at
T
t=T,
will have N+1N possible
values
 
S u
0

d

  0,1,... N
Take call option as example,
VT  ( ST  K )

the option value at t=T, is also a random variable,
with corresponding possible
values
N  
( S 0u d  K ) 


 0,1,... N
Binomial Tree Method Notation
 Denote
S  S u
n
n 

d , V  V ( S , tn ),
n
n
(0  n  N , 0    N ),
ˆ  max  | S u N  d   K  0, 0    N 
Binomial Tree
S 0u
S0
S0 d
S 0u
2
……
S0ud
S0 d
……
2
……
……
S 0u N
S 0u N 1d
…
S0u N  d 
…
S0ud
N 1
S0 d N
Possible Values of Option at t=T
V0N  S0u N  K  S0N  K ,
V  S0u
N
N
ˆ 1
V
N
N
V
 0,
 0.
N 
ˆ
d  K  S  K,
N
ˆ
Problem Option Pricing by BTM
V (0    N )
N
 If
are given, how can we
determine
N 1
V

in particular
0
0
(0    N  1)
V  V ( S 0 , t0 ) ?
Answer to the Problem
 With
the one period and two-state model,
and using backward induction, we can
determine
N h
V
step by step.
Induction Steps
 When
 to
VN (0    N )
N 1
V
are given,
(0    N  1)
find
 consider the following one period and twoSN 1u  SN
state model.
SN 1
and
VN 1
SN 1d  SN1
VN  ( SN  K )
VN1  ( SN1  K ) 
Induction Steps cont.
 Define
a risk-neutral measure Q
 d
u
qu 
 q, qd 
 1 q
ud
ud
N 1
V
 Then,
 So

1

[qVN  (1  q)VN1 ] (0    N  1).
that for any ˆ 
h(1  h  N )
 h  h l
V  h   q (1  q)l ( SNl  K )
 l 0  l 
N h
ˆ
when    , V  0.
N h
1
Induction Steps cont. But
 Denote

SNl  SN  hu h l d l , qu  (1  q )d  
qˆ  uq / 
Thus

VN  h
d

(1  q)  1  qˆ
 N  h ˆ   h  h l
l
S
q
(1

q
)
   
l 0  l 


K ˆ   h  h l
   h   q (1  q )l ,   ˆ ,
 l 0  l 

0,
  ˆ .

European call option valuation formula
 m  m l
l
(n, m, p)     p (1  p)
l 0  l 
n
 Denote
 Then
the European call option valuation
K
formula isN h
N h
V  S (   , h, qˆ )  h (ˆ   , h, q)

 Especially,
h=N, α=0,
V ( S0 , 0)  S0( , N , qˆ ) 
K

N
(ˆ , N , q)
Discount Factor
 Discount
Factor
BT  e  r (T t )
satisfies
dBt
 rBt , (0  t  T ),
dt
 The
BT  1.
financial meaning of the discount factor:
to have $1 at t=T (including continuous
compound interests), B
one needs to deposit
t
in bank at t (t<T).
Discount Factor in BTM
 in
the binomial tree method, ttrading
 tn (0 occurs
n  N)
at discrete times
 the compound interests should also be
calculated for the discrete case.
Bn (n  0,1,...N )
 Let
denote the discrete
discount factors. They satisfy the difference
equations
Bn1  Bn  r tBn , (0  n  N  1), BN  1.
Discount Factor in BTM cont.
 That
is
1
 1 

B 
Bn 1  

1  r t
 1  r t 

n
N n

N
B 
1
 N n
where ρis the [growth
t , t  t ] of the risk-free bond in
i.e.


Bn 1   Bn
Call---Put Parity in discrete form
 for
the binomial tree method,
the call---put parity (in discrete form)
becomes
N h
c
 K /   p
h
N h
N h
 S
European put option valuation formula
 Using
European call option valuation
formula and put---call party, we have
N h
p

K
h
SN  h  (   , h, q )  SN  h  (ˆ   , h, qˆ )
0  h  N , 0    N  h.
Investment vs. Gambling Game
 Investing
in options can be compared to a
gambling game.
U0
 Initial stake be
. After
U T one game, the
stake becomes .
UT

is a random variable.
 If the expectation
E (UT )  U 0
then the gamble is said to be fair
Fair Gambling Game
- the bet at n-th game, U n 1 - the next
bet.
 If under the condition that complete
information of all the previous n-games
are available, the expectation of U n 1
equals the previous stake U n i.e.,
 Un
E (U n1 |  (U1
U n ))  U n
then we say the gamble is fair.
σ-Algebra
 (U1
Un )
denotes
information of
U1 complete
Un
the bets
up to n-th game,
E( X | Y )
and
denotes the conditional
expectation of X under
condition
Y.
 (U1 U n )
 In mathematics,
is called σalgebra in stochastic theory

Martingale
 Martingale
is often used to refer to a fair
gamble.
Un : 0  n  N
 The bet sequence
that satisfies condition
E (U n1 |  (U1
U n ))  U n
as a discrete random process, is called a
Martingale.
Mathematical Definition of
Martingale
A sequence
Y  Yn : n  0
is a
Martingale with respect to sequence
X   X n : n  0
if for all n ≥0

E | Yn | 

E (Yn1 | X 0 , X 1
X n )  Yn
Risk-neutral measure vs. Martingale
 Under
the risk-neutral measure Q, the
discount prices
 S  of an underlying
  , ( n  0,1 N )
asset S,
as a discrete
 B t
random process, satisfy the equation:


S
Q  S 
   E     ( S0 Sn )  , (0  n  N )
B
 B tn

tn 1


n
Martingale Measure
 Hence
the discount price sequence of an
underlying asset is a martingale.
 The risk-neutral measure Q is called the
martingale measure
 Q equivalent to the probability measure P.
Definition of Equivalence

Probability measure P and probability
measure Q are said to be equivalent if and
only if for any probability event
A
(set)
there is
Prob P ( A )  0  ProbQ ( A )  0
 i.e.
the probability measures P and Q have
the same null set.
European option under
Martingale
 The
European option valuation formula
under the sense of equivalent Martingale
measure Q, can be written as

S
Q  S 
   E     ( S0 ,
 B t N h
  B t N
 or
VtN h   E
 Especially
h
Q
 S
V0  

, St N h ) 


 K  |  ( S0 ,

N
N
E
Q
 S
N
K
, StN h )



Relation of the arbitrage-free
principle & Martingale measure
What is the relation between the arbitragefree principle and the existence of equivalent
Martingale measure?
 Arbitrage-free principle  d< ρ <u


existence of equivalent
Martingale measure Q

European option pricing
in a risk-neutral world
Theorem 3.3
-
the fundamental theorem of asset
pricing
 If an underlying asset price moves as a
binomial tree, there exists an equivalent
Martingale measure if and only if the
market is arbitrage-free.
Proof of Theorem 3.3
a risky asset S and a risk-free asset B,
 “sufficiency” by Theorem 3.2.
 “necessity”- A portfolio

 if 0  0,  t *  0, s.t. ProbP (t*  0)  1
then what we need to prove is that there must be
Prob P ( t*  0)  0
where $P$ denotes an objective measure.
Proof of Theorem 3.3 cont.
 In
fact, let Q- equi. Mart. Meas. of P, then
  / B  0  E Q    / B t
 Thus E ( * )  0 B * / B0  0
t
t
*

Q
P  Q , it implies ProbQ ( t  0)  1
 we have ProbQ ( t  0)  1
 therefore ProbQ ( t  0)  0
 Since measure P and measure Q are
equivalent, this means
Prob P ( t*  0)  0
 Since
*
*
*
Dividend-Paying
 An
underlying asset pays dividends in two
ways:
 1. Pay dividends discretely at certain times
in a year;
 2. Pay dividends continuously at a certain
rate.
 This section, the continuous model is
considered only
Reason for Studying the Continuous
Model 1

Asset -- foreign currency.
 exchange rate changes randomly
 the foreign currency is a risky asset .
 If it is deposited in a bank in its native country, it
would accrue interests according to the local int.rate
 The interest be regards dividend of the "security"
 this dividend is paid continuously.
 Therefore, the "dividend rate" is the risk-free interest
rate of the foreign currency in its native country.
Reason for Studying the Continuous
Model 2
Suppose the underlying asset is a portfolio of a
large number of risky assets.
 Since each risky asset in the portfolio pays
dividend at a certain rate at certain times, the
number of dividend payments for the portfolio
would be large, and we can approximate it as
continuous payment (dividend rate can be timedependent).

Example

A company needs to buy M Euro at time
t=T to pay a German company. To avoid
any loss if Euro goes up, the company buys
a call option of M Euro with expiration date
t=T at rate K. How much premium should
the company pay?
Example cont.
 Over
the same period, due to the risk-free
interest ("dividend"),
 1Euro in the local bank can grow to
1Euro   Euro
  1  qt ,
 where
q is the risk-free interest
rate in a German bank.
Example cont.
[t , t  t ]
Therefore the value of 1 Euro in
changes as
S u ( RMB / Euro)
St ( RMB / Euro)

t
St d ( RMB / Euro)
Let B be a risk-free
[t , t  Bank
t ] of China bond. Its
change in
is
Bt ( RMB)   Bt ( RMB)
  1  r t
where
and r is the risk-free interest rate
in BOC bank.
Example cont.-[t , t  t ]
each interval
, apply Δ-hedging
strategy, i.e. to construct a portfolio
 In
  V  S
and select Δ, such that
is risk-free.
t t
Example cont.-- t t
 t t
 Solve

V  S u
 Vt t  St t   d
d
V


S
 t t
t t d
  t   (Vt  St )
the system,
u
t t
u
t t
d
Vt ut  Vt 
1
d
t

, Vt  [q uVt ut  q dVt 
t]
 (u  d ) St

Example cont.--

 /  d
u   /
qu 
, qd 
ud
ud
We assume dη<ρ<uη, so that
0  qu , q d  1, qu  q d  1
Example cont.--- Since
the price of the option at t=T ( in
RMB) is
VT  M ( ST  K )  M ( S0u N  d   K ) ,(0    N )
 where
M is the required amount of Euro,
and K is the agreed exchange rate.
 let M=1, similar to before, using
backward induction, we can get:
Option Pricing (Dividend, call)
 The
pricing formula for dividend-paying
European call option:
N h
V

SN h

h
K
~
ˆ
(   , h, q )  h (   , h, q )
^

 Where
q 
~

 /  d
ud
, q 
^
uq

~
Option Pricing (Dividend, put)
 The
pricing formula for dividend-paying
European put option:
N h
p

K
h
(   , h, q ^ ) 
SN h
h
(ˆ   , h, q ~ )
Binomial Tree Method of
American Option Pricing
 American
option pricing is different from
European option pricing.
 At each node
SN  h (1  h  N , 0    N  h)
for American option, the price must satisfy
the constraint
N h
V
N h 
 ( K  S
)
Backward Induction
- American option pricing
 Therefore
for American option pricing
(taking put option as example), its backward
induction process is:
VN  ( K  SN )  , 0    N
n=N
n=N-1
N 1
V
1
N
N
N 
 max  [qV  (1  q)V 1 ], ( K  S )  ,


0   N
Backward Induction cont.
N h
(0    N  1) is given, then

If V

1
N h
N h
N  h 1  
V
 max  [qV  (1  q)V 1 ], ( K  S
) 


A
B
(0    N  h  1)
 d
q
ud
where
N  h 1

in each step, after A is calculated, it must be
compared with the payoff function B, VN  h 1
be the larger of the two, and so on, until V 0
0
is arrived at.
Another View of American Option


Suppose ud=1
the underlying
n
nasset
  price
S  S0u d
(0    n)
 can
 For
be nwritten jas
S j  S0u
( j  n, n  2,
S0  1.
simplicity, let
we construct a grid:
, n  2, n)
In the plane (S,t)
American Option Grid
0

 S j  S j 1 
0  t0 
 where
tn 
 tN  T
Sj  u j,
j  0, 1,
tn  nt (t  T / N ), n  0,1,
,
, N.
V  V (S j , tn ).
n
j
Notice
 Then American put option pricing:

1
n 1
n

V  max   qV j 1  (1  q)V j 1  , ( K  S j ) 


n
j
Theorem 3.4
 If
V jn (n  0,1,
N , j  0, 1, )
option price, then

V V , V
n
j
n
j 1
is American put
n 1
j
V
n
j
Theorem 3.5

tn (0  n  N  1),  j  jn
when j  jn  1, V   j
n
j
when j  jn , V   j
n
j
when j  jn  1, V   j
n
j
s.t.
Optimal Exercise Boundary
– Free Boundary
t
t=T
Stopping Region
2
Continuation Region
2
S
Optimal Exercise Boundary
Continuation & Stopping Region


In region Σ1, the option value is greater than the payoff
from exercising the option, the option holder should
continue to hold the contract rather than early exercising it.
Therefore Σ1 is called the continuation region.
In region Σ2, since
V  1/ [qV
n
j
n 1
j 1
 (1  q)V
n 1
j 1
]
which means the option's expected return is less than the
risk-free interest rate, the holder should stop the contract,
i.e. early exercise the option immediately. Therefore Σ2 is
called the stopping region.
Optimal Exercise Boundary
S  S (t )
is of great importance in finance,
as the interface of the continuation and
stopping, and is called the optimal exercise
boundary.
 Theoretically, American option holder should
choose a suitable exercise strategy
according to the above analysis to avoid
loss.

American Call-Put Symmetry
 Call-put
parity does not hold for American
options.
 One naturally asks whether there exists
other kind of relation between American call
and put options.
American Call-Put Symmetry Example
 An
option as a contract gives its holder the
right to exchange cash for stock (call
option), or to exchange stock for cash (put
option), at the strike price on the expiration
date.
American Call-Put Symmetry Example

We may regard the cash as a risk-free bond earning
interests according to the risk-free interest rate, and regard
the stock as a risky bond earning risk-free interests
according to the dividend rate. Then we can see a certain
symmetry exists between the call and put options:
C ( S , K ;  , ; t )  P( K , S ; ,  ; t )

i.e. for options (European or American) with the same
expiration date, if the positions of S and K, and the
positions of η and ρ are both swapped, the call option price
and put option price should be equal.
Theorem 3.6
 (call-put
symmetry)
 If ud=1, then for American options with the
same expiration date, relation
C ( S , K ;  , ; t )  P( K , S ; ,  ; t )
is true, where
t  tn ,(0  n  N ).
Theorem 3.7
 For
American options with the same
expiration date, let Sc ( S p ) & K c ( K p )
denote the underlying asset price and
strike price for the call (put) option
respectively. If K p / S p  Sc / K c
then
C ( Sc , K c ;  , ; t ) P( S p , K p ; ,  ; t )
Sc K c
where

t  tn ,(0  n  N ).
SpKp
Summary
1.
Have Introduced a discrete model---BTM to
describe the underlying asset price movement,
and have priced its derivatives (options) using this
model.
2.
Based on the arbitrage-free principle, using the Δhedging technique, have introduced a risk-neutral
equivalent martingale measure. The BTM of
option pricing has produced a fair valuation that is
independent of any individual investor's risk
preference.
Summary cont.
3.
Using the BTM of American option, we have
shown that there exist two regions for American
put option: the continuation region and the
stopping region, which are separated by the
optimal exercise boundary.
4.
For American options, although there is no call--put parity, there exists call-put symmetry, as for
European options.
作业：P22、1，2，3，4