Page 94 - DCOM510_FINANCIAL_DERIVATIVES

Unit 6: Valuation and Pricing of Options

The solution of these two linear equations yields the value of Z and M as follows: Notes
C  C C  C
Z  t,u t,d or t,u t,d

(u d) S S  S
t 1 t,u t,d

More generally,
C  C
Z  u u ...(6.1)
S  S d
u

or more generally,

(1  d)C  (1 u)C

and M  t,u t,d ... (6.2)


(u d) (1 r)
The values of Z and M indicate respectively the number of shares to be bought and amount of
money to be borrowed at a riskless rate in order to perfectly replicate a call option.
Any difference in the price of the call and the levered portfolio would induce arbitrage
opportunities. Since we are operating in a perfect market, the arbitrage of opportunities will
phase out as the market forces come into play.
After having got the values of Z & M, let us get back to our basic portfolio structure.

i.e. ZS – M
Now as per the presumption, the value of this portfolio has to be equal to the value of the call,
i.e.
C = Z S – M
t-1 t–1
If we substitute the values of Z & M in this, we get


 ...(6.3)


r – d u – r
Where x = and1 - X=
u – d u – d
This is the Binomial Option Pricing Model (BOPM) equation for unit period non-dividend
paying stock’s call option. However, dividend paying stock’s call option value can be computed
through BOPM with suitable modifications.

Similarly, call option formulae could be developed for options having two or more periods
remaining before expiration.
As stated in the beginning, BOPM is the most flexible of option pricing models. All European
and American put and call options can be valued through it, irrespective of whether with or
without dividends. It can be used for single or multiple period options. The only handicap,
which this model suffers from, is that when the option life is subdivided into multiple trading
intervals, it becomes a complex and tedious mathematical exercise.

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