Transcript Binomial Option Pricing Model

Binomial Option
Pricing Model
Finance (Derivative Securities) 312
Tuesday, 3 October 2006
Readings: Chapter 11 & 16
Simple Binomial Model
Suppose that:
• Stock price is currently $20
• In three months it will be either $22 or $18
• 3-month call option has strike price of 21
Stock Price = $22
Option Price = $1
Stock price = $20
Option Price = ?
Stock Price = $18
Option Price = $0
Option Pricing
Consider a portfolio:
• long D shares, short 1 call option
22D – 1
18D
Portfolio is riskless when 22D – 1 = 18D
D = 0.25
Option Pricing
Riskless portfolio:
• long 0.25 shares, short 1 call option
Value of the portfolio in three months:
• 22 x 0.25 – 1 = 4.50
Value of portfolio today (r = 12%):
• 4.5e–0.12(0.25) = 4.3670
Value of shares: 0.25  20 = 5
Value of option: 5 – 4.367 = 0.633
Generalisation
A derivative lasts for time T and is
dependent on a stock
S
ƒ
Su
ƒu
Sd
ƒd
Generalisation
Consider the portfolio that is long D shares
and short 1 derivative
SuD – ƒu
SdD – ƒd
The portfolio is riskless when SuD – ƒu =
ƒu  f d
SdD – ƒd or
D
Su  Sd
Generalisation
Value of portfolio at time T:
• SuD – ƒu
Value of portfolio today:
• (SuD – ƒu)e–rT
Cost of portfolio today:
• SD – f
Hence ƒ = SD – (SuD – ƒu )e–rT
Generalisation
Substituting for D we obtain
ƒ = [ pƒu + (1 – p)ƒd ]e–rT
where
e d
p
ud
rT
Risk-Neutral Valuation
Variables p and (1 – p) can be interpreted
as the risk-neutral probabilities of up and
down movements
Value of a derivative is its expected payoff
in a risk-neutral world discounted at the
risk-free rate
Expected stock price: pS0 u + (1 – p)S0 d
• Substitute for p, gives S0erT
Risk-Neutral Valuation
Su = 22
ƒu = 1
S
ƒ
Sd = 18
ƒd = 0
Since p is a risk-neutral probability
20e0.12(0.25) = 22p + 18(1 – p)  p = 0.6523
Alternatively, using the formula:
e rT  d e 0.120.25  0.9
p

 0.6523
ud
1.1  0.9
Risk-Neutral Valuation
Su = 22
ƒu = 1
S
ƒ
Sd = 18
ƒd = 0
Value of option:
• e–0.12(0.25) (0.6523 x 1 + 0.3477 x 0) = 0.633
Two-Step Tree
D
22
20
1.2823
A
B
2.0257
18
E
19.8
0.0
C
0.0
Value at node B
24.2
3.2
F
16.2
0.0
• e–0.12(0.25)(0.6523 x 3.2 + 0.3477 x 0) = 2.0257
Value at node A
• e–0.12(0.25)(0.6523 x 2.0257 + 0.3477 x 0)
= 1.2823
Valuing a Put Option
D
60
50
4.1923
A
B
1.4147
40
72
0
48
4
E
C
9.4636
F
32
20
Valuing American Options
D
60
50
5.0894
A
B
1.4147
40
72
0
48
4
E
C
12.0
F
32
20
Delta
Delta (D) is the ratio of the change in the
price of a stock option to the change in the
price of the underlying stock
The value of D varies from node to node
Determining u and d
 Determined from
stock volatility
u  e
Dt
d  1 u  e 
Dt
ad
p
ud
a  e rDt
for a nondividend payingstock
a  e ( r  q ) Dt for a stockindex where q is thedividend
yield on theindex
ae
a 1
( r  r f ) Dt
for a currency where rf is theforeign
risk - free rate
for a futurescontract
Tree Parameters
Conditions:
erDt = pu + (1 – p)d
2Dt = pu2 + (1 – p)d 2 – [pu + (1 – p)d ]2
u = 1/ d
Where Dt is small: u  e  Dt
d  e 
Dt
ad
p
ud
a  e r Dt
Complete Tree
S0u3
S0u2
S0u
S0
S0d
S0u
S0
S 0d
S0d 2
S0d 3
S0
Example: Put Option
Parameters
• S0 = 50; K = 50; r = 10%;  = 40%;
• T = 5 months = 0.4167; Dt = 1 month = 0.0833
Implying that:
u = 1.1224; d = 0.8909;
a = 1.0084; p = 0.5076
Example: Put Option
89.07
0.00
79.35
0.00
70.70
0.00
62.99
0.64
56.12
2.16
50.00
4.49
70.70
0.00
62.99
0.00
56.12
1.30
50.00
3.77
44.55
6.96
56.12
0.00
50.00
2.66
44.55
6.38
39.69
10.36
44.55
5.45
39.69
10.31
35.36
14.64
35.36
14.64
31.50
18.50
28.07
21.93
Effect of Dividends
For known dividend yield:
• All nodes ex-dividend for stocks multiplied by
(1 – δ), where δ is dividend yield
For known dollar dividend:
• Deduct PV of dividend from initial node
• Construct tree
• Add PV of dividend to each node before exdividend date