文档介绍：Numerical Procedures Chapter 18
Binomial Trees
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
Movements in Time dt (Figure )
Su
Sd
S
p
1 – p
1. 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
2. Tree Parameters for a Nondividend Paying Stock (Equations to )
When dt is small, a solution to the equations is
plete Tree (Figure )
S0
S0u
S0d
S0
S0
S0u2
S0d2
S0u2
S0u3
S0u4
S0d2
S0u
S0d
S0d4
S0d3
Backwards Induction
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
Example: Put Option
S0 = 50; X = 50; r =10%; s = 40%;
T = 5 months = ;
dt = 1 month =
The parameters imply
u = ; d = ;
a = ; p =
Example (continued) Figure
Calculation of Delta
Delta is calculated from the nodes at time dt