early exercise, and the binomial value applies at all nodes. This makes European options
less valuable to the buyer. Knowing the value of your stock options can help you evaluate

your compensation package and make decisions about how to handle your stock options.
There are a wide range of valuation techniques for stock options but in this study, we are
going to apply the valuation method using binomial tree.
The risk neutral probability
The
cornerstone of option pricing is the risk neutral probability valuation principle. This
principle relates the current price of a stock to its expected value of financial product at a
future time. Basically, option valuation is done via the application of the risk neutrality
probability assumption over the life of the option, as the price of the underlying
instrument evolves (Chriss A.N., 2008).
This principle assumes a continuous compounding interest rates that takes the form of
discounting factor
e^-rt
where
t
is time in (months or years) followed the up and down
movements in stock price, and
r
is a continuous compounded interest rate. Suppose, the
following notations are used just to represent the probabilities of up and down
movements
{q, 1-q}.
There are two things we should consider when thinking of
{q, 1-q}
.
Either the asset moves forward to its future values or its payoffs move backward via
(discounting factor) to the present value of the stock option ((Dalton B., 2008). The
expected value of a stock under the compounded discounting factor,
E(St) = S0 e^rt.
According to Dalton, to get the values of
q
and
1-q,
we take the true results using
continuously compounded interest rate r and time t of the up and down movements in
the share price. The risk neutral probabilities of
q = e^rt - d/u-d
and 1
-q = u-e^rt/u-d
and the value of the stock S is
S0 = e^-rt*(qSu + (1-q)Sd)
. Therefore, if we have all the
parameters i.e. the compounding interest r and the current stock price at S0, then the
risk neutral probability can be used to create a binomial model for price movement and
subsequently a method to value option (Dalton B., 2008).
Example 1:
The price of one share of Dexia Bank at Brussels Stock Exchange is €15.50. The up and
down factors are 1.2 and 0.93 respectively. Find the price of a three-month put option with a strike
price of €15 on Dexia Bank shares at Brussels Stock Exchange. Assume that €1 today will be worth
1.015 in three months’ time (Dalton B., 2008).
Solution: S = 15.50, u = 1.2, d = 0.93, discount factor (1+ r) = 1.015 and strike price = 15
The payoff (X) = Max(15 – 18.60,0)= 0
The payoff (Y) = Max(15 – 14.42)= 0.58
t = 0
Figure1: A one-period binomial tree
q = (1.015 – 0.93)/1.2 – 0.93 = 0.085/O.27 = 0.315
and
1 – q = 0.685
15.50
18.60
14.42

Value = (1/1.015)*(0.315*0 + 0.685*0.58) = 0.985*0.397 = € 0.39
Binomial Tree Approach
The binomial tree pricing approach is widely used as it is able to handle a variety of
conditions for which other models cannot easily be applied. This is due to the fact that
the model underlies instrument over time as opposed to a particular point. For instance,
the American options which can be exercised at any point, the European options which
can be exercised at point of maturity or expiration and Bermudan options which can be
exercised at various points in time (Wikipedia, 2009).

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- Fall '10
- Bart Vinck
- Options, Wikipedia, Strike price