文档介绍：Chapter 20 Basic Numerical Procedures
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012
1
Approaches to Derivatives Valuation
Trees
Monte Carlo simulation
Finite difference methods
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012
2
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
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012
3
Movements in Time Dt (Figure , page 428)
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012
4
Su
Sd
S
p
1 – p
Tree Parameters for asset paying a dividend yield of q
Parameters p, u, and d are chosen so that the tree gives correct values for the mean & variance of the stock price changes in a risk-neutral world
Mean: e(r-q)Dt = pu + (1– p )d
Variance: s2Dt = pu2 + (1– p )d 2 – e2(r-q)Dt
A further condition often imposed is u = 1/ d
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012
5
Tree Parameters for asset paying a dividend yield of q (continued)
When Dt is small a solution to the equations is
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012
6
plete Tree (Figure , page 430)
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012
7
S0u4
S0u2
S0d2
S0d4
S0
S0u
S0d
S0
S0
S0u2
S0d2
S0u3
S0u
S0d
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
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012
8
Example: Put Option (Example , page 430)
S0