SYLLABUS Previous: 4.1.2 Parameters illustrated with
Up: 4.1 Plain vanilla stock
Next: 4.2 Exotic stock options
4.1.3 Application, time value and implied volatility
[ SLIDE
call(SMI) -
implied volatility -
smile -
time value ||
VIDEO
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We are ready now to use the VMARKET applet and compare the fair
value obtained from a model with the market price of products that are
sold by financial institutions. For example, take the
European call
on the
Swiss Market Index (SMI)
expiring on Dec 19, 2002 with a strike at 7000.
Nine months before the expiry, the underlying index was trading at
6610
with a market volatility around
18%
(follow the links to obtain the current market data).
Under reasonable assumptions in Switzerland of a 2% risk-free rate and
a 2% average dividend yield for the shares that constitute the SMI index,
the VMARKET applet below calculates the fair
price for this option according to the Black-Scholes model.
VMARKET applet: press Start/Stop
to run the simulation until it stops exactly nine months before the
option expires.
For an approximative solution, you may simply click inside the plot
area to measure the payoff V(S) around the coordinate 6610.
For a complete printout of the numerical solution, switch from
Double-click below in the applet to Print data to console,
set TimeStep=0. and press Step 1;
the number output can be now read from the Java-console (with Netscape
open Communicator->Tools->Java console) where x[]
is the price of the underlying, f0[] is the intrinsic
value in grey and f[] the solution in black.
Don't forget to switch back to Double-click below
avoiding to overflow the Java console...
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After interpolation, the fair price (CHF 251.6) is encouragingly close to
the offer from
Crédit Suisse First Boston
CSFB (500 x 0.51 = CHF 255.0).
Considering the crude approximations we made for the input parameters and
the limitations of the Black-Scholes model, this agreement may however well
be fortuitous: in fact, there is no guarantee that market prices coincide
with any model at all-the offer and demand from traders on the option
markets need not to be rational!
You may well ask now why people use financial modeling... It turns out
that the predicted values are nevertheless often in the right range.
Modeling is particularly useful to estimate what should be a fair
value when there are not a sufficient number of buyers / sellers to
make a market, for example when an option is offered for the first
time or when an exotic option is tailored by a financial institution
(the market maker) to meet the
specific needs of only two clients.
Simple products such as the vanilla call above have more than 100 million
options listed on the Swiss exchange: this is enough to set a price only
by offer and demand.
Instead of calculating the option price as a function of the volatility,
the Black-Scholes model is then often used as a market standard to
calculate an implied volatility, i.e.
the volatility that has to be used in the model to reproduce the price
from the market.
Keeping the same expiry date, the implied volatility can be measured for
different strike prices 
; this curve is traditionally called the
smile, but has a shape that really depends on
the market conditions and can equally well be a frown (exercise 4.04).
Although there is no guarantee to make a profit from the so-called
volatility trading, some investors buy options with a low- and short
options with a high implied volatility: their bet is that market forces
will eventually move the prices of options so as to make implied
volatilities comparable in the future.
To complete this analysis dealing with the European option payoff dynamics,
simply note that the difference between the present and the
intrinsic (or final) value of an option is
traditionally called time value.
The simulation using the VMARKET applet below
shows that the time value is usually negative for a put option in-the-money
and is sometimes strictly positive for a call option.
VMARKET applet: press Start/Stop
to simulate the value of a European put V(S,t) before the option
expires; observe the relative position of the present value (in black)
and the intrinsic or final value (in grey).
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SYLLABUS Previous: 4.1.2 Parameters illustrated with
Up: 4.1 Plain vanilla stock
Next: 4.2 Exotic stock options
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