4.1.2 Parameters illustrated with VMARKET experiments

[ SLIDE parameters - volatility - rate - dividend || VIDEO modem - LAN - DSL ]

Apart from the *terminal conditions* that specify
the value of an option when it expires (
for a vanilla option or
`StrikePrice` in the applet) and the
*numerical parameters* that specify the
precision of the calculation (such as the `TimeStep`,
`Walkers` and `MeshPoints`), the Black-Scholes model depends
on only four *financial parameters*:

- The
*time to the expiry*( or`RunTime`) is usually expressed in a fraction of a year, e.g. 0.25 for three months or a quarter of year to the expiry date. - The
*short term interest or spot rate*( or`SpotRate`) is specified as a annual fraction of the capital investment, e.g. 0.05 for a risk free return of five percent per year. - the
*dividend yield*( or`Dividend`), here modeled with a continuous payment that is proportional to the underlying price, e.g. 0.04 for a dividend paying four percent of the share value during one year, - the
*volatility of the underlying*( or`Volatility`) estimated as the standard deviation of closing prices in sect.1.5, e.g. 0.5 for 50 percent for a volatile share.

The evolution of the payoff should be far from obvious, but can be disentangled by investigating the effect of each parameter separately.

Have a look first at the volatility using the *VMARKET* applet above.

These experiments show that the main effect of the volatility is to
``smear out sharp edges'', i.e. where the vanilla call and put options
are at-the-money
.
This phenomenon, known as *diffusion* in engineering sciences, is
strongest at the begining of the simulation when the option is close
to expiry date. It is the result of unpredictable market fluctuations:
even if the value of the underlying share is below the strike price of
a call
before the expiry date, there is a finite chance that the
market price will suddenly rise above that value, which would allow the
holder of a call option to make a final profit
.
Such a right to make a potential profit without any obligation has of
course a finite value, which decreases as the time approaches the
expiry date.

Now play with the *VMARKET* applet below, using the default
parameters focusing on the effect of the interest rate.

The effect of the risk-free interest rate can be understood from the drift that affects any type of investment: to finally coincide with the exercise price on the expiry date, the strike price has to be discounted back in time (1.3#eq.6) to . This is clearly visible in the applet, where the value at-the-money shifts to lower prices as the simulation runs backward in time. With a drift that is proportional to the strike price, the interest rate appears to have its largest relative effect when the option is at-the-money while the underlying is kept fixed; this is somewhat misleading, since the underlying should also grow by the same amount but is here used as a parameter. In fact, the graph could be continuously renormalized with the same amplification factor for the share, strike and option value-e.g. introducing a new currency after every time step, so that the graph would not evolve anymore at all.

The following experiments can be carried out in the *VMARKET*
applet above.

Hopefully, these last experiments contribute more to your understanding
than your confusion: with payments that are proportional to the underlying,
the dividend yield continuously reduces the value of the share by the same
amount; this results in a drift along the horizontal axis (in the opposite
direction from the effect of interest rates) and appears as if the share
prices were *amplified* when the time runs backwards.
If the interest rate is equal to the dividend yield, the drifts in the
horizontal direction cancel out and all that remains is the effect from
the discounting at a risk-free interest rate.

With a good intuition for each parameter taken separately, it is a good exercise to now return to the first applet and discuss the main features that characterize an option payoff when all the parameters are combined into one calculation. Also, remember that unrealistically large parameters have been used in this section to exaggerate the effect from each parameter; realistic values will be used for an real option pricing calculation in the next section.

**SYLLABUS** ** Previous:** 4.1.1 The European Black-Scholes
**Up:** 4.1 Plain vanilla stock
**Next:** 4.1.3 Application, time value