2.2. MULTIPERIOD BINOMIAL TREE MODELS 87
from scratch because the parameters have special and more restricted inter-
pretations than the simple model. More sophisticated discretization proce-
dures from the numerical analysis of partial differential equations also lead to
additional discrete option pricing models which are hard to justify by build-
ing them from scratch. The discrete models derived from the Black-Scholes
model are used for simple and rapid numerical evaluation of option prices
rather than motivation.
Sources
This section is adapted from: “Chapter 2, Discrete Processes” inFinancial
Calculusby M. Baxter, A. Rennie [5] andQuantitative Modeling of Derivative
Securitiesby M. Avellaneda and P. Laurence [2].
Problems to Work for Understanding
- Consider a two-time-stage example. Each time stage is a year. A stock
starts at 50. In each year, the stock can go up by 10% or down by 3%.
The continuously compounded interest rate on a $1 bond is constant
at 6% each year. Find the price of a call option with exercise price
50, with exercise date at the end of the second year. Also, find the
replicating portfolio at each node. - Consider a three-time-stage example. The first time interval is a month,
then the second time interval is two months, finally, the third time
interval is a month again. A stock starts at 50. In the first interval, the
stock can go up by 10% or down by 3%, in the second interval the stock
can go up by 5% or down by 5%, finally in the third time interval, the
stock can go up by 6% or down by 3%. The continuously compounded
interest rate on a $1 bond is 2% in the fist period, 3% in the second
period, and 4% in the third period. Find the price of a call option with
exercise price 50, with exercise date at the end of the 4 months. Also,
find the replicating portfolio at each node. - A European cash-or-nothing binary option pays a fixed amount of
money if it expires with the underlying stock value above the strike
price. The binary option pays nothing if it expires with the underlying
stock value equal to or less than the strike price. A stock currently has
price $100 and goes up or down by 20% in each time period. What is