Bill & Gilbert,
At the risk of muddying the water still more, you both have valid points.
Bill, the formula you're refering to is what I refer to as the 'general'
Black-Scholes/Black formula. The RHS of your "C=" is actually the Black
formula for futures options. You are correct in stating that it can be
simple to calculate as setting y=0 produces the no-dividend equity option
model(i.e. traditional Black-Scholes) while y=continuous dividend yield
produces equity index option model and y=futures price/yield produces the
Black futures option model.
As for Gilbert and fixed income option pricing, you are correct in noting
the convergence problem in pricing bond options. Using the bond price as
the underlying (and thus using price volatility) is a very restricted
approach. For example, you can't use that approach to price a one-year
option on a 3-year bond. A more general approach is to use yield as the
underlying and then use the bond yield formula to produce the price. This
totally eliminates the problem of price converging to par as that is
addressed through the bond yield formula. As you point out, the 'maths' are
very different in fixed income. For example, Euro-dollar futures options
are priced in this manner where the put on the futures price is really the
call on the futures rate.
Regards,
Phil Bennett
-----Original Message-----
From: [log in to unmask]
[mailto:[log in to unmask]]On Behalf Of Bill
Igoe
Sent: Tuesday, June 27, 2000 2:58 PM
To: Gilbert Peffer; [log in to unmask]
Subject: RE:
I prefer to look at the BS formula in the following equation.
This format makes the question very intuitive and makes reverting back to
the P-C parity very simple to calculate especially for a traders on the
floor.
C = exp(-r*t)*(F*N(d1) - K*N(d2));
F = S*exp((r-y)*t);
r =cost of carry for asset
y =yield on asset
if in the case of equities and no dividends you get....y = 0;
You can rewrite
to
c = S*N(d1) - exp(-r*t)*N(d1)
The formula typically shown in the text books.
If the forward is not implicit in the formula you will get you clock cleaned
by professional traders with 8th grade math skills.
-----Original Message-----
From: [log in to unmask]
[mailto:[log in to unmask]]On Behalf Of Gilbert
Peffer
Sent: Tuesday, June 27, 2000 1:28 PM
To: [log in to unmask]
Subject:
Bill Igoe wrote:
>
> > Know your formulas. The forward price is implicit in the BS option
> pricing
> > model.
> > Using simple algerba the user can use the forwardas well as the spot.
Why
> > is the forward important? The forward includes the cost/earnings
> associated
> > with hedging.
> >
> > Bill Igoe
We are not talking about BS for *shares* here, where the formula can be
manipulated as you wish, but about adjusting BS so it can deal with zero
vol
at *bond* maturity. Rearranging the standard BS won't help. For shares, the
forward price approach and spot price approach give you the same result,
because you assume the same (constant) vol in both cases. However, this is
obviously not in the case of *bond* options.
Also, saying that the forward price is implicit in the BS model is
misleading. The
standard model for BS has a stochastic process for the spot price of the
share, and a previsible process for the rolled-up money market account. You
can certainly multiply the dS process with exp(-r*(T-t)), but that is
neither here nor there.
From this it simply follows that in the case of share options the forward
price is not important for anything, also not for hedging.
Bond options are different, and one sees many stock market players making
the mistake of thinking fixed income investments are not much more
difficult to value and risk manage than stock. Fixed income is an other
world, the maths and the reasoning is much more involved. Simlpe algebra
won't help in most cases.
Best regards,
Gilbert
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