An example of a two-step option valuation of a European Call will be used here to demonstrate the general functioning of this pricing method. In order for this model be properly examined a few assumptions have to be made:
1. The direction and degree of the underlying asset’s fluctuation is given. Assume for each stage of the tree that the underlying asset’s price may either go up or down by 10%.
2. The risk free interest rate is known with certainty. In practice, this is commonly the government bond (i.e. T-Bills) that is closest in time to the maturity of the option. For this example assume 12%.
3. The current price of the stock is used to project forward possible movements through the binomial tree. In practice this information is readily available to all investors. For the purpose of this example a price of $10.50 is used, with the option having a strike price of $10.00. Thus, the option is in the money (it pays to exercise)
4. No arbitrage.
The following notation will be used in this
example:
So = Current price of the stock, Let So = $10.00
Co = Current price of the call on the stock, Ci = Call price at period i
X = Strike or Exercise price of option
r = Risk-free rate interest, Let r = 12%
T = time between periods in years. This example has two 3-month periods, making T = 3months/12months or .25. T = .25
U = Degree or percentage change in upward movement, Let U = 1.1
D = Degree or percentage change in downward movement, Let D = 0.9
Pu = Probability of an upward movement
1-P = Pd = Probability of a
downward movement
A two-step binomial tree model would, in theory,
look like this:
figure 1
U = 1 upward movement in the price of the underlying asset.
UU = 2 consecutive upward movements in the price of the underlying asset.
D = 1 downward movement in the price of the underlying asset
DD = 2 consecutive downward movements in the price of the underlying asset.
UD = An upward or downward movement in the price of the underlying asset followed by an upward or downward movement in the price of the underlying asset.
The present time, located at the first node on the tree, is the point in time in which the value of the option must be calculated. In simple terms, the price of the option at this time is equal to the expected value of the option at each node, and this expected value is discounted back to the previous node at the risk-free rate of interest to obtain its present-value.
The assumptions given above state that the present
price of the stock is $10.00, and this price may fluctuate either up or
down in increments of 10%. Thus the price of the stock at U is $11.00,
and the price at UU is $12.10.
The tree with all of the possible asset prices
would look like this:
figure 2
The next stage in the valuation is to calculate the price or value of the call assuming the second stage of the tree corresponds with the maturity of the option. This implies that only intrinsic value is priced into the option, there is no time value as the option has expired. It is assumed that the strike price of the option is $10.50, and the following possible option prices are then calculated (at maturity). Thus, the value of the call at maturity, at each node would be calculated as follows:
Cuu = 12.10 – 10.50 = 1.60
Cud = 10 – 10.50 = -.50, nil
Cdd = 8.10 – 10.50 = -2.40, nil
These prices are two periods forward,
and the valuation occurs at the present period. Thus, the prices of the
options may be calculated through working a path back through the tree,
eventually ending up at the first node. For example, the option price at
node U is calculated as the expected value of the two nodes that follow
it discounted by the risk-free rate
of interest.
However, to calculate the expected value one needs to calculate the probability of an upward and downward movement. This probability of an upward movement may be calculated through using the following formula:
Subbing in the values assumed at the beginning of this example, Pu is found to be .6523 and 1 – P is .3477.
This expected value equation can be found through simple mathematics, and is calculated as follows:
= $1.04
This expected value has to be discounted back one period to node U. This is the present value equation (see time value of money):
= $1.01
The tree would now look like this:
figure 3
Where the value of the call at time zero is found through this equation:
Therefore, the price of this call is $0.97.
This process is identical for all nodes on a given binomial tree. The value of the calls are found at maturity, their expected values are calculated using probabilities, and these probabilities are discounted back at the risk free rate back to the first period, or time zero. The value of the call at time zero results in the correct pricing of the option.
The following topics are discussed in greater detail, and are recommended for persons that have a more advanced understanding of derivatives.
This page borrowed heavily from the text, Introduction
to Options and Futures Markets, by John C. Hull. Further readings
from this book are recommended for a more in-depth understanding of the
material discussed in this section of the report.
