Page 91 - DCOM510_FINANCIAL_DERIVATIVES

Financial Derivatives

Notes Assumptions

1. The current selling price of the stock (S) can only take two possible values i.e., an upper
value (S ) and a lower value (S ).
u d
2. We are operating in a perfect and competitive market, i.e.

(a) There are no transaction costs, taxes or margin requirements.
(b) The investors can lend or borrow at the riskless rate of interest, r, which is the only
interest rate prevailing.

(c) The securities are tradable in fractions, i.e. they are divisible infinitely.
(d) The interest rate (r) and the upswings/downswings in the stock prices are predictable.
3. The value of (1+r) is greater than d, but smaller than u i.e., u<1+r<d. This condition or
assumption ensures that there is no arbitrage opportunity.
4. The investors are prone to wealth maximisation and lose no time in exploiting the
arbitrage opportunities.

6.3.2 Single Period Binomial Model

The single period binomial model is also known as a one-step binomial model. We shall assume
a unit period of option’s life, while BOPM can be used for deriving the value of multi-period
options also. ‘Unit period’ of option’s life is implied that the option’s stock price will move
either up or down by the date of expiration of the option. On the other hand, in the multi-period
model, the stock price may move many times between a given date and the expiration date of
the option. Logically the unit period case is unrealistic and the multi-period case is more likely
to happen in real situation.

!
Caution However, for the purpose of simplicity and understanding, we shall restrict
ourselves to the Unit (single) period model.
Although BOMP can be used for dividend paying stocks, however, again for simplicity we shall
be assuming non-dividend stocks.

Use of the Model

The Binomial options pricing model approach is widely used as it is able to handle a variety of
conditions for which other models cannot easily be applied. This is largely because the BOPM
models the underlying instrument over time — as opposed to at a particular point. For example,
the model is used to value American options which can be exercised at any point and Bermudan
options which can be exercised at various points. The model is also relatively simple,
mathematical, and can therefore be readily implemented in a software (or even spreadsheet)
environment.

Although slower than the Black-Scholes model (to be discussed later in this chapter), it is considered
more accurate, particularly for longer-dated options, and options on securities with dividend
payments. For these reasons, various versions of the binomial model are widely used by
practitioners in the options markets.
For options with several sources of uncertainty (e.g. real options), or for options with complicated
features (e.g. Asian options), lattice methods face several difficulties and are not practical.

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