Page 99 - DCOM510_FINANCIAL_DERIVATIVES

Financial Derivatives

Notes 5. The underlying security pays no dividends during the life of the option, the higher the
yield of dividend, the lower the call premium as thus, and the market prices of the calls are
not likely to be the same.
6. The volatility of the underlying instrument (may be the equity share or the index) is
known and constant over the life of the option.

7. The distribution of the possible share prices (or index levels) at the end of a period of time
is log normal or, in other words, a share’s continuously compounded rate of return follows
a normal distribution. Essentially, this means that the share in question has the same
likelihood to double in value as it to halve with the added implication that the share prices
cannot become negative.
8. The price of the underlying instrument follows a geometric Brownian motion in particular
constant drift  (expected gain) and volatility :

9. The market is an efficient on. This implies that as a rule, the people cannot predict the
direction of the market or any individual stock.

6.4.2 Black-Scholes European Model

The original Black-Scholes option-pricing model was developed to value options primarily on
equities. This model has a number of restrictive assumptions including the limitation that the
underlying asset pays no dividends. The model has since been “modified” to value European
options on dividend paying equities, as well as on bonds, foreign exchange, futures and
commodities. This enhanced model is known as the Modified Black-Scholes European model. It
prices European options or options that may only be exercised at expiration.
The Modified Black-Scholes European model makes the following assumptions:
1. The option may not be exercised prior to its expiration date.
2. The price changes of the underlying asset are lognormally distributed.

3. The risk-free interest rate is fixed over the life of the option.
4. Dividend payments are not discrete; rather, the underlying asset yields cash flows on a
continuous basis.

6.4.3 Black-Scholes American Model

An American-style option is an option that may be exercised at any time during the life of the
option. The Modified Black-Scholes American option-pricing model is the same as the Modified
Black-Scholes European model except that it checks to see if the value returned is below the
intrinsic value of the option. If this is the case, then the Modified Black-Scholes model returns the
intrinsic value of the option.
Black-Scholes American = Max (Black-Scholes European, Intrinsic Value)
The Modified Black-Scholes American model makes the following additional assumptions:

1. The price the option may be exercised prior to its expiration date.
2. Changes of the underlying asset are lognormally distributed.
3. The risk-free interest rate is fixed over the life of the option.
4. Dividend payments are not discrete; rather, the underlying asset yields a continuous
constant amount.

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