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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