Binomial Trees Chapter 11
Options, Futures, and Other Derivatives, 7th Edition, Copyright Š John C. Hull 2008
1
A Simple Binomial Model
A stock price is currently $20 In 3 months it will be either $22 or $18
Stock Price = $22 Stock price = $20 Stock Price = $18
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
2
A Call Option (Figure 11.1, page 238) A 3-month call option on the stock has a strike price of 21. Stock Price = $22 Option Price = $1 Stock price = $20 Option Price=? Stock Price = $18 Option Price = $0
Options, Futures, and Other Derivatives 7th Edition, Copyright Š John C. Hull 2008
3
Setting Up a Riskless Portfolio
Consider the Portfolio: long ∆ shares short 1 call option 22∆ – 1
18∆
Portfolio is riskless when 22∆ – 1 = 18∆ or ∆ = 0.25 Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
4
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 4.5e – 0.12×0.25 = 4.3670
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
5
Valuing the Option
The portfolio that is long 0.25 shares short 1 option is worth 4.367 The value of the shares is 5.000 (= 0.25 × 20 ) The value of the option is therefore 0.633 (= 5.000 – 4.367 )
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
6
Generalization (Figure 11.2, page 239) A derivative lasts for time T and is dependent on a stock
S0 ƒ
S0u ƒu S0d ƒd Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
7
Generalization (continued)
Consider the portfolio that is long ∆ shares and short 1 derivative S0u∆ – ƒu S0d∆ – ƒd
The portfolio is riskless when S0u∆ – ƒu = S0d∆ – ƒd or
ƒu − f d ∆= S 0u − S 0 d Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
8
Generalization (continued)
Value of the portfolio at time T is S0u∆ – ƒu
Value of the portfolio today is (S0u∆ – ƒu)e–rT
Another expression for the portfolio value today is S0∆ – f
Hence ƒ = S0∆ – (S0u∆ – ƒu )e–rT Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
9
Generalization (continued)
Substituting for ∆ we obtain ƒ = [ pƒu + (1 – p)ƒd ]e–rT where
e −d p= u −d rT
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
10
p as a Probability
It is natural to interpret p and 1-p as probabilities of up and down movements The value of a derivative is then its expected payoff in a risk-neutral world discounted at the risk-free rate
S0 ƒ
(1 – p)
p
S0u ƒu S0d ƒd
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
11
Risk-Neutral Valuation
When the probability of an up and down movements are p and 1-p the expected stock price at time T is S0erT This shows that the stock price earns the risk-free rate Binomial trees illustrate the general result that to value a derivative we can assume that the expected return on the underlying asset is the riskfree rate and discount at the risk-free rate This is known as using risk-neutral valuation
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
12
Original Example Revisited p
S0 ƒ
(1 – p)
S0u = 22 ƒu = 1 S0d = 18 ƒd = 0
Since p is the probability that gives a return on the stock equal to the risk-free rate. We can find it from 20e0.12 ×0.25 = 22p + 18(1 – p ) which gives p = 0.6523 Alternatively, we can use the formula
e rT − d e 0.12×0.25 − 0.9 p= = = 0.6523 u −d 1.1 − 0.9 Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
13
Valuing the Option Using RiskNeutral Valuation
S0 ƒ
3 2 5 6 0. 0.34 77
S0u = 22 ƒu = 1 S0d = 18 ƒd = 0
The value of the option is e–0.12×0.25 (0.6523×1 + 0.3477×0) = 0.633 Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
14
Irrelevance of Stock’s Expected Return
When we are valuing an option in terms of the price of the underlying asset, the probability of up and down movements in the real world are irrelevant This is an example of a more general result stating that the expected return on the underlying asset in the real world is irrelevant Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
15
A Two-Step Example Figure 11.3, page 242
24.2 22 19.8
20 18
16.2
Each time step is 3 months K=21, r=12% Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
16
Valuing a Call Option Figure 11.4, page 243
D
22 20 1.2823
A
B
2.0257 18 0.0
E
24.2 3.2 19.8 0.0
C F
16.2 0.0
Value at node B is e–0.12×0.25(0.6523×3.2 + 0.3477×0) = 2.0257 Value at node A is e–0.12×0.25(0.6523×2.0257 + 0.3477×0) = 1.2823
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
17
A Put Option Example Figure 11.7, page 246
K = 52, time step =1yr r = 5%
D
60 50 4.1923
A
B
1.4147 40
72 0 48 4
E
C
9.4636
F
32 20
Options, Futures, and Other Derivatives 7th Edition, Copyright Š John C. Hull 2008
18
What Happens When an Option is American (Figure 11.8, page 247) D
60 50 5.0894
A
B
1.4147 40
72 0 48 4
E
C
12.0
F
32 20
Options, Futures, and Other Derivatives 7th Edition, Copyright Š John C. Hull 2008
19
Delta
Delta (∆) 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 ∆ varies from node to node
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
20
Choosing u and d One way of matching the volatility is to set u =e σ
∆t
d =1 u =e −σ
∆t
where σ is the volatility and ∆t is the length of the time step. This is the approach used by Cox, Ross, and Rubinstein
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
21
The Probability of an Up Move p=
a−d u−d
• a = e r∆t for a nondividen d paying stock • a = e ( r − q ) ∆t for a stock index wher e q is the dividend yield on the index •a = e
( r − r f ) ∆t
for a currency where rf is the foreign
risk - free rate • a = 1 for a futures contract
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008
22