QuantoTermStructure

[Andrew Kolesnikov]

I guess there is an error during zeroYieldImpl valuation. As i know, dividend rate in quanto process should be equal to:
   
    (foreignRiskFreeRate - riskFreeRate) + originalDivRate - quantoAdjustment.

For the derivation just check Hull's book, it should be there. So there are two sign mismatch with QL code in this formula.

[Marcin Pawlik]

2009/9/2 Andrew Kolesnikov <[hidden email]>:
>
> I guess there is an error during zeroYieldImpl valuation. As i know, dividend
> rate in quanto process should be equal to:
>
>    (foreignRiskFreeRate - riskFreeRate) + originalDivRate -
> quantoAdjustment.
>
> For the derivation just check Hull's book, it should be there. So there are
> two sign mismatch with QL code in this formula.

I'm not so sure...

Please, correct if I'm wrong but in case of quantos the underlying
should evolve with a drift equal to:

r_for - dividend - quantoAdjustment

and should be discounted with r_dom. At least this is my simplified
understanding of the explanation presented by Musiela and Rutkowski in
"Martingale Methods in Fin. Mod.", chap. 4.5, p. 168-170.

If you use  QuantoTermStructure is as a "modified dividend curve" and
r_dom as risk free rate in an instance fo the
GeneralizedBlackScholesProcess then you get:

r_dom - (dividend + r_dom - r_for + quantoAdjustment)

which evaluates to:

r_for - dividend - quantoAdjustment

I spent some time thinking over it and I think that looking in the
ql/pricingengines/quantoengine.hpp also may be hepful.
I am not aware how this concept was reached but I arrived to the
conlusion that using r_dom as risk free rate instead of r_for has to
do with discounting. If you used r_for as risk free rate and
constructed dividend curve as you say than you would have problem with
discounting since you should discount at r_dom which in this case
wouldn't be the risk free rate (r_for would be).


M.

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[Andrew Kolesnikov]

Hi.
All that we need to create quanto instrument is to change numeraire to foreign money market (that's the meaning of quantos in traditional risk-neutral measure). So, as we know, numeraire changing involves adjustment to the expected growth rate, in our case it would be quanto adjustment. Futher conslusions are very simple:

   mu_dom - growth rate in domestic ccy
   mu_for - growth rate in foreign ccy
   q_dom - div. rate in dom. ccy
   q_for - div. rate in for. ccy

   mu_for = mu_dom + quantoAdj = r_dom - q_dom + quantoAdj
   
   q_for = r_for - mu_for = r_for - r_dom + q_dom - quantoAdj ,

so in my point of view exactly this formula should be used in QuantoTermStructure.