Our discussion on the effects of decay lets you see
that decay is an important factor that needs to be considered
by an option trader. However, things get worse for the option
trader if volatility also changes over time. Using Option Tutor's
option calculator you can examine the term structure of implied
volatilities for equity options. Consider the set of American
call options trading on IBM at the close of November 22, 1996.
Assume for this example that the dividend yield
is at a constant continuous rate equal to 0.896% except for December
and January options which will be assumed to have a zero dividend
yield (because IBM does not go ex-dividend over this time). The
expiration is the Saturday after the 3rd Friday of
the contract month. The close of IBM was 158.625 and other relevant
numbers are in the table below:
* Computed from the Treasury strip market, remaining from T-bills.
From the short end of the term structure of implied volatilities
curve you can see that the decay for this example is stronger
than predicted under the constant volatility assumptions of the
Black-Scholes world. This is because at the short end implied
volatility is upward sloping and recall that the option premium
is increasing with volatility. Therefore as maturity declines
a declining volatility implies that the option premium declines
more than if volatility remained constant.
Note: In the above example by using the last traded price implies
that it is not known whether this was a bid or an ask that was
hit. Not having this information can introduce a source of bias
because the typical size of an option's spread is such that it
will have a significant impact upon implied volatilities. Thus,
you should use the examples as a learning device for understanding
the mechanics of how computations are made.
How were the numbers in the last column of the above table arrived
at?
Tutor Break: Applying the Option Calculator to Compute Implied
Volatility
Step 1: Compute the continuous compounded form of the
risk free rate of interest plus the annualized time to maturity.
Select the Option Calculator subject then click on the menu item
Options and select the sub-menu item Calculator. In the calculator
fix today's date to equal 11/22/96 (i.e., the closing market prices).
For the December option the time of maturity is 12/21/96. Similarly,
from the T-bill market the price is 4.83 for the December 26,
maturing T-bill. We will use this as the approximation for the
risk free rate covering the December 21, 1996 option. This needs
to be converted into a continuously compounded form. The Calculator
will let you perform both of these calculations as follows:
Step 2: Click on the transfer button for each calculation
(Maturity, and Interest Rate) to transfer this to the Option Calculator.
Then update the additional fields as follows. Being an American
option on a non-dividend paying stock (at least for the December
option) you can use the analytic Black Scholes model as follows:
Note: for longer maturity options on IBM you will fill in the
dividend yield field and compute the option as an American call
option.
Step 3: You can now test calibrate the numerical approximations.
You can do this by selecting American option. With zero dividends
early exercise is never desirable and thus numerically this give
the same value as for the European call option. As a result,
this provides a test of numerical accuracy for different methods.
For the case of a binomial computation with 1000 steps the result
is:
Step 4: Consider the same calculation using the Trinomial
tree approach. Click beside Trinomial and then click the Calculate
Button.
The trinomial is slightly more accurate for the same number of
steps. This is consistent with its advantage that it should converge
much faster.
Step 5: Now consider the two approaches for 100 steps:
Binomial
Trinomial
Again the Trinomial provides a more accurate estimate over fewer steps. This is the recommended method when using Option Tutor's calculator for numerical option work.
(C) Copyright 1997, OS Financial Trading System