Binomial trees are frequently used to approximate the movements in the price of a stock or other asset
In each small interval of time the stock price is assumed to move up by a proportional amount u or to move down by a proportional amount d
A Simple Binomial Model
Stock Price = $22
Stock Price = $18 Stock price = $20 probabilities can’t be 50%-50%, unless you are risk-neutral A stock price is currently $20
In three months it will be either $22 or $18
A Call Option
Stock Price = $22
Option Price = $1 Stock Price = $18
Option Price = $0 Stock price = $20
Option Price=? if you were risk-neutral (and r=0), you could say that the option is worth: 0.5=50%*1$+50%*$0 A 3-month call option on the stock has a strike price of 21.
18 22 20 U(18) U(22) U(20) Expcted U (50% prob) prob(22) has to be > prob (18), because otherwise Utility function is linear Then, in order to know prob we need to know the Utility function. But this is an impossible task, and we have to find a shortcut .... i.e. we have to find a way of “linearizing” the world
Setting Up a Riskless Portfolio
22D – 1 18D Consider the Portfolio: long D shares short 1 call option
Portfolio is riskless when 22D – 1 = 18D or D = 0.25
Valuing the Portfolio
risk-fre rate=12% p.a. ---> 3% quarterly ---> disc. factor=exp(-0.12*0.25)=0.970446
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
Note that this pay-off is deterministic, so its PV is obtained by simple discounting
Valuing the Option
The value of the portfolio today is 4.5e – 0.12´0.25 = 4.3670
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 )
Valuing the Option
note that the value of the option has been obtained without knowing the shape of the utility function
but if the solution is independent of preferences functional form, then it is valid also for all utility function
Then, it is valid also for risk-neutral preferences .....
... eureka !!! let’s imagine a risk-neutral world ---> derive risk-neutral probabilities
Summing up ... Movements in Time Dt
Su
Sd S
p 1 – p
Risk-neutral Evaluation
Su = 22
ƒu = 1 Sd = 18
ƒd = 0 S
ƒ p (1 – p ) hyp: risk-free rate=12% p.a.; t = 3m
Since p is a risk-neutral probability 20e0.12 ´0.25 = 22p + 18(1 – p ); p = 0.6523
p is called the risk-neutral probability
show simple_example.xls
Tree Parameters for a Nondividend Paying Stock
We choose the tree parameters p, u, and d so that the tree gives correct values for the mean & standard deviation of the stock price changes in a risk-neutral world
er Dt = pu + (1– p )d
s2Dt = pu 2 + (1– p )d 2 – [pu + (1– p )d ]2
A further condition often imposed is u = 1/ d
We know the value of the option at the final nodes
We work back through the tree using risk-neutral valuation to calculate the value of the option at each node, testing for early exercise when appropriate
Valuing the Option
Su = 22
ƒu = 1 Sd = 18
ƒd = 0 S
ƒ 0.6523 0.3477
The value of the option is e–0.12´0.25 [0.6523´1 + 0.3477´0]
= 0.633
A Two-Step Example
20 22 18 24.2 19.8 16.2
Each time step is 3 months
Valuing a Call Option
20
1.2823 22 18 24.2
3.2 19.8
0.0 16.2
0.0 2.0257 0.0 A B C D E F
Value at node B = e–0.12´0.25(0.6523´3.2 + 0.3477´0) = 2.0257
Value at node C =0
Value at node A = e–0.12´0.25(0.6523´2.0257 + 0.3477´0)
= 1.2823
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