Optionen

Option value

We consider long positions as the result of buying options.

Call option

The (non mandatory) option of exercising an option is aquitted by paying an option premium. The premium increases with increasing uncertainty of a worthwile exercise. The latter in turn depends on price volatility, time to maturity and the ratio of underlying price to strike. Hence the option value decreases as:

• volatility decreases
• time to maturity dimishes
• the option goes further out of money

In real options such as hydraulic power plants, the last point is important: When the reservoir is empty or production runs at baseload, the plant has no  option value.

The value of an option

We base the discussion on the model by Black-Scholes. Input values for a given underlying asset are:

• Curent underlying price
• Strike price (price at which the option is exercised, if at all)
• Volatility of the market price (in % per year)
• Time to maturity (in years)

It is important to use the same time units for volatility and time to maturity!

Given the market price, volatility and time to maturity, the seller ("underwriter") may calculae the option's value depending on the specific strike. For the following example:

• Market price for underlying = 20
• Time to maturity = 1 year
• yearly volatility = 30%

Note: The call option value (c) is shown for a range of strikes (X).

The buyer ("holder") of an option is interested in the payoff, which is the Result of the total operation of buying the option and maybe exercising it and, in that case, balancing the position on the market. The buyer will exercise the option only if balancing the position on the market is beneficial. If he does not exercise the option he loses the option premium. Payoff is positive only if balancing the position from exercising the option pays the premium. In the following example:

• Strike = 20
• Time to maturity = 1 year
• Yearly volatility = 30%

Note: The value of the call option (red graph) is, unlike on the graph above, shown as a function of the underlying price for a fixed strike.