Introduction to
Binomial Trees
th
(7
Ch 12
edition)
th
Ch 11 (pre 7 edition)
Valuation of stock options
•If I told you that a $25 stock today will, with 100% certainty,
jump to $30 tomorrow, what premium would you pay to own that
stock today?
•$10…..$12…..$2….perhaps a value < $5
•If I told you that a $25 stock today will, with 100% certainty,
jump to $30 in 3 months, what premium would you pay to own that
stock today?
•$5…..$7…..$12….perhaps a value < $5e – 0.12*(3/12)
•The trouble with 100% certainty in the real world is that it is nonexistant, especially in the stock market. Changes in stock prices
follow a random walk process which is consistent with the notion
of market efficiency since stock prices cannot be predicted
consistently over time.
Valuation of stock options
•This chapter takes a first look at valuation of stock options with the
assumption that stock prices can be predicted
•In this simple situation, the movements of stock prices during the life
of the option are governed by a binomial tree (different paths)
•In the binomial tree model, it is assumed that the option lasts until time
‘T’ and there are only two possible stock prices and option prices at
time ‘T’
•We consider a portfolio consisting of a short position in one call option
and a long position in ‘delta’ shares
•We determine that for some value of ‘delta’, the value of the portfolio
today is the present value of the portfolio at time T
•This is the risk-neutral valuation argument; it assumes that all market
participants expect to earn the risk-free rate (riskless investors)
A Binomial Tree
Stock Price = $22
Stock price = $20
Stock Price = $18
•diagram represents different paths for the stock price
•quite useful in valuing a stock option in a riskless world
•set up a portfolio of the stock and stock option with no uncertainty
over the portfolio value at option expiration date
•since the portfolio has no risk, the return it earns MUST equal the
risk-free interest rate
A Simple Binomial Model
• A stock price is currently $20
• Right now, it is known that in three
months, the stock price will be either
$22 or $18
Stock Price = $22
Stock price = $20
Stock Price = $18
A Call Option
A 3-month European call option on the stock has a
strike price of 21.
Stock Price = $22
Option Payoff = $1
Stock price = $20
Option Price=?
Stock Price = $18
Option Payoff = $0
Setting Up a Riskless Portfolio
• Consider the Portfolio: long D shares
short 1 call option
22D – 1
18D
• Portfolio is riskless when 22D – 1 = 18D or
D = 0.25
Valuing the Portfolio
(Risk-Free Rate is 12%)
• The riskless portfolio is:
long 0.25 shares
short 1 call option
• The value of the portfolio in 3 months is
220.25 – 1 = 4.50
• The value of the portfolio today is the
present value of the portfolio at time T
4.5e – 0.120.25 = 4.367
Valuing the Option
• The portfolio that is
long 0.25 shares
short 1 option
is worth 4.367
• The value of the shares today is
5.000 (= 0.2520 )
• The value of the option is therefore
0.633 (= 5.000 – 4.367 )
Generalization
• The options lasts for time T and is
dependent on a stock
S0
ƒ
S0u
ƒu
S0d
ƒd
Generalization
(continued)
• Stock can move up by a proportional increase of ‘u’
and down by a proportional decrease ‘d’
• Consider the portfolio that is long D shares and short
1 call option
S0 uD – ƒu
S0– f
S0dD – ƒd
• The portfolio is riskless when S0uD – ƒu = S0d D – ƒd or
ƒu  f d
D
S0 u  S0 d
 D is the ratio of the change in option price to the
change in stock price as we move between nodes
Generalization
(continued)
• Value of the portfolio at time T is
S0u D – ƒu
• Value of the portfolio today is
(S0u D – ƒu )e–rT
• But portfolio cost today is S0D – f
• Hence
ƒ = S0D – (S0u D – ƒu )e–rT
Generalization
(continued)
• Substituting for D we obtain
ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
where
e d
p
ud
rT
Risk-Neutral Valuation
• ƒ = [ p ƒu + (1 – p )ƒd ]e-rT
• The value of a derivative is its expected payoff in a
risk-neutral world discounted at the risk-free rate
• The variables p and (1 – p ) can be interpreted as the
risk-neutral probabilities of up and down movements
S0
ƒ
S 0u
ƒu
S 0d
ƒd
Valuing the Option
S0u = 22
ƒu = 1
S0
ƒ
S0d = 18
ƒd = 0
The value of the option is
e–0.120.25 [0.65231 + 0.34770]
= 0.633
A Two-Step Example
24.2
22
19.8
20
18
16.2
• Each time step is 3 months
• Strike price is $21 (call option)
• Risk free rate is 12%
Valuing a Call Option
D
22
20
1.2823
A
B
2.0257
18
24.2
3.2
E
19.8
0.0
C
0.0
F
16.2
0.0
• Value at node B
= e–0.120.25(0.65233.2 + 0.34770) = 2.0257
• Value at node A
= e–0.120.25(0.65232.0257 + 0.34770)
= 1.2823
A Put Option Example; K=52
T = 2 yrs, r = 5%
D
60
50
4.1923
A
B
1.4147
40
72
0
48
4
E
C
9.4636
F
32
20
What Happens When an Option
is American (page 252)
D
60
50
5.0894
A
B
1.4147
40
72
0
E
48
4
C
12.0
32
F
20
•Work backwards through the tree and test at each node to see
whether early exercise is optimal
•At each node, the value of the option is the greater between
the payoff from early exercise and the value given by our
equation
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
• It is the number of shares of the stock we should
buy for each stock option sold OR the number of
shares of stock we should sell for each stock
option bought to create a riskless hedge
• The construction of a riskless hedge is often
called delta hedging.
ƒu  f d
D
S0 u  S0 d
• The value of D varies from node to node; it
changes over time
Choosing u and d
•
Up to now we assumed these values
u  es
Dt
d  1/ u
where s is the volatility and dt is the
length of the time step.
Questions
5th and 6th Edition: 11.1, 11.4, 11.5,
11.6, 11.12, 11.13, 11.19, 11.20 (a)
and (b) only
7th Edition: 12.1, 12.4, 12.5, 12.6,
12.12, 12.13, 12.19, 12.20 (a) and (b)
only