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Unformatted text preview: 9). An option to buy is known as a call option and is usually purchased in the
expectation that the price of the stock will rise. Thus a call option may allow its
holder to buy a share in company ABC for $500 on or before June 2001. If the price
of the stock rises above $500 the holder of the option can exercise it (pay $500) and
retain the difference. The holder’s payoff is that sum minus the price paid for the
option. A put option is bought in the expectation of a falling price and protects
against such a fall. The exercise price is the price at which the option can be
exercised (in this case $500) (Bailey et al., in press).
The central problem with options is working out how much the owner of the contract
should pay at the outset. Basically, the price is equivalent to an insurance premium; it
is the expected loss that the writer of the contract will sustain. Clearly, the ability to
exercise the option at any time up to the maturity date makes American options more
valuable than European options. What is less obvious is that this apparently minor
difference necessitates different procedures for calculating option prices (Galli et al.,
1999).
112 The standard assumption made in option theory is that prices follow lognormal
Brownian motion. Black and Scholes developed the model in the early 1970s. Experimental studies have shown that it is a good approximation of the behaviour of
prices over short periods of time. If they obey the standard BlackScholes model,
then the spot price, psp, satisfies the partial differential equation:
d psp = σpWt + µ pspdt
where psp=spot price, Wt=Brownian motion, µ=drift in spot prices, p=price and
σ=spot price volatility.
Applying Ito’s Lemma, the explicit formula for psp can be shown to be:
µ  σ2)/2] t +σWt} psp = p0e{[( where psp=spot price, Wt=Brownian motion, µ=drift in spot prices, and σ=spot price
volatility, p0=oil price, t=time
It is relatively easy to evaluate European options (which can be exercised only on a
specified date), particularly when the prices follow BlackScholes model because
there is an analytic solution to the corresponding differential equation. This gives the
expected value of max[p exp (rt) – p k, 0] for a call or of max[p k – p exp(rt),0] for a
put. In general the simplest way of getting the histogram and the expected value is by
simulating the diffusion process and comparing each of the terminal values to the
strike price of the option. Solving the partial differential equation numerically gives
only the option price (Galli et al., 1999).
Evaluating the price of an American option is more difficult because the option can be
exercised at any time up to the maturity date. From the point of view of stochastic
differential equations, this corresponds to a freeboundary problem (see Wilmot,
Dewynne and Howison, 1994). The most common way of solving this type of 113 problem is by constructing a binomial tree. This is similar to a decision tree (for an
explanation see Galli et al., 1999).
In the formula for NPV given in section 5.2, the discount rate was used to account for...
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