Absolutely! Or even better, you can use HJM or BGM, if you are lucky enough
to manage the art of calibration.
Gilbert
----- Mensaje original -----
De: Chris Shortland <[log in to unmask]>
Para: 'Gilbert Peffer' <[log in to unmask]>;
<[log in to unmask]>; <[log in to unmask]>
Enviado: 28 June 2000 11:59
Asunto: RE:
>
> Gilbert,
>
> There is always the alternative of ditching the Black-Scholes model
> altogether and adopting a specific interest rate model. For example, you
> could use a Hull-White model which would naturally give you a bond price
> volatility tending to zero as the redemption date approached.
>
> Chris
>
>
>
>
> -----Original Message-----
> From: Gilbert Peffer [mailto:[log in to unmask]]
> Sent: 28 June 2000 10:30
> To: [log in to unmask]; [log in to unmask]
> Subject: RE:
>
>
> Hi Philip,
>
> Since I am far from being an expert on options pricing, tell me how you
deal
> with the fact that since price vol goes to zero at maturity of the
> underlying bond, yield vol will as well in BS. Thus using the spot yield
of
> the bond and the spot yield vol leaves you with the same problem (in my
> view) as when you use prices. What I actually was saying is that using
> forward price/yield and the respective forward vols allows you to
correctly
> price the extreme case of:
>
> 1. Expiry of option = Maturity of underlying bond
> 2. In that case, the bond price vol is zero at expiry
> (Unfortunately there is no corresponding yield, so the yield
> approach doesn't seem to work)
> 3. Using the spot price vol of the bond will definitely give you
the
> wrnog result since it states that the vol of the bond at expiry is
non-zero,
> whereas it is equal to zero.
>
> I am glad you agree about the difference between share/FX option pricing
on
> one hand and interest rate option pricing on the other hand. It is
> interesting to see that even in the simple case of hedging a basket of
> forwards on FX or shares (Forwards with different expiry dates), you could
> happilly just use the underlying in a spot hedge, i.e. just one
instrument.
> But this is not the case in a hedge of a basket of FRAs (again with
> different expiries) on, say, US LIBOR. Here you need in general 2N bonds
in
> a spot hedge, where N is the number of different FRA forward periods.
> Correlation and vol term structures become imensly important here.
>
> Hope this makes sense,
>
> Best regards,
>
> Gilbert
>
>
>
> ----- Mensaje original -----
> De: Philip E. Bennett <[log in to unmask]>
> Para: <[log in to unmask]>
> Enviado: 27 June 2000 21:43
> Asunto: RE:
>
>
> > 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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