Page 117 - DMGT513_DERIVATIVES_AND_RISK_MANAGEMENT

Derivatives & Risk Management

Notes

Notes Assumptions Underlying Black-Scholes Model
The key assumptions of the Black-Scholes model are:
1. The risk-free interest rate exists and is constant (over the life of the option), and the
same for all maturity dates.
2. The short selling of securities with full use of proceeds is permitted.
3. There is no transactions cost and there are no taxes. All securities are perfectly
divisible (e.g. it is possible to buy 1/100th of a share).
4. There is no riskless arbitrage opportunities; security trading is continuous.
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, 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 St, in
particular with constant drift ì (expected gain) and volatility ó:
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.

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.

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

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