In last month’s issue of Treasury Today, we looked at the basic structure of the Black-Scholes model. This shows that the value of the call option on a stock is a function of the present values of both purchasing the stock outright and paying the exercise price on the day the option expires.

This is a very important model underpinning calculations valuing not just ‘plain vanilla’ options, but also barriers, compounds and other ‘exotic’ options. However, it is based on a number of assumptions, which must be understood.

There are five main assumptions:

1. A European-style option is modelled. A European-style option can only be exercised on the day of expiration. The alternative, American-style option can be exercised at any point up to the day of expiration. In practice, American-style options tend to be exercised close to expiration as otherwise the holder will lose the time value of the option (see Finance A to Z, Treasury Today October 2004). The exception is if the holder purchases an American call option because it is unsure when it will need to exercise. Because of the restrictions on the exercise of European-style option, Americanstyle options will usually be more expensive. However, because of the uncertainty over the date of exercise, there is no model to value American-style options.
2. The stock pays no dividends during the life of the option. This is relatively easy to overcome as any such dividends can be discounted to a present value and subtracted from the present value of the stock.
3. Stock markets are efficient.There are a number of aspects to this:
   1. There is continuous trading.This allows investors to continuously revalue their portfolios.
   2. There is no scope for arbitrage.As a result of the constant revaluation, the market is assumed to be in equilibrium, such that the price of an option is always the difference between the present values of purchasing the stock outright and paying the exercise price on the day the option expires.

   3. Market participants cannot predict future prices either of individual stocks or of the market as a whole.

4. No commissions are charged.In reality, small commissions are charged on every transaction. The size of these commissions will vary according to the nature of the investor and therefore will further distort the effect of the model.

Next month
In next month’s issue, we will examine how these assumptions are used in the Black-Scholes model.

5. The model uses a constant risk-free interest rate to discount future prices to a present value and volatility is assumed to be constant.Although models often use the discount rate on US Treasury Bills, in practice it is not possible to define a ‘risk-free’ rate. The problem is that rates can change over the life of an option, especially when markets are more volatile.