Is there a good step-by-step reference to approach the quanto feature using the change of numeraire technique?Say S is a stock quoted in dollars and we want to quanto it in eurosThe dollar risk-neutral SDE for S is dS/S = r dt + \sigma dWThe dollar risk-neutral SDE for the exchange rate N is dN/N = q dt + \eta dZ, with correlation \rho between dW and dZI am trying to derive the euro risk-neutral SDE for SI understand that for X = S/N it must be dX/X = q dt + \sigma dW' since X is a euro-tradable assetBut for quanto we are interested in the dynamics of S, not X, and I'm not sure how to get them... Ito's lemma?e.

You know that for X = S/N it must be dX/X = q dt + \sigma dW' since X is a euro-tradable asset.You also know that 1/N is a euro-tradable asset with drift....Now you apply ITO lemma to S/N, this will give ... dS + ... dN + cross trem in dt (where dS is euro risk neutral, there is no change of measure in ITO). It must match dX so you will found quanto drift adjustment needed.

quantos

QuoteOriginally posted by: MarsYou know that for X = S/N it must be dX/X = q dt + \sigma dW' since X is a euro-tradable asset.You also know that 1/N is a euro-tradable asset with drift....Now you apply ITO lemma to S/N, this will give ... dS + ... dN + cross trem in dt (where dS is euro risk neutral, there is no change of measure in ITO). It must match dX so you will found quanto drift adjustment needed.ok I'm trying your approach:Nd(1/N) = (q - r)dt + \eta dZ'Ito on S/N: d(S/N) = 1/N*dS + S*d(1/N) + (dS)*d(1/N)My problem here is that we don't know yet the euro risk-neutral equation for dS, hence we can't really calculate the cross term (dS)*d(1/N) ?e.

When changing numeraire you will only change the drift (Girsanov theorem). So euro risk neutral will be dS / S = mu dt + \sigma dW' where W' is a brownian under euro risk neutral measure and mu is what we try to find.The cross term is then : S/ N rho sigma eta dt and we have mu = q - rho sigma eta

QuoteOriginally posted by: MarsWhen changing numeraire you will only change the drift (Girsanov theorem). So euro risk neutral will be dS / S = mu dt + \sigma dW' where W' is a brownian under euro risk neutral measure and mu is what we try to find.The cross term is then : S/ N rho sigma eta dt and we have mu = q - rho sigma etaI think I get it now: Ito doesn't know probability measures, so we can always evaluate the cross-term using the dollar SDEs...The rest is algebrathanks for your helpe.

You can also use the fact that is a domestic tradable asset where is the price of 1 unit of foreign currency in the domestic currency (ie the foreign/domestic FX rate). Then the drift of must be in the domestic cash numeraire. As the drift of is , you can deduce the drift of with little Itô calculus.

QuoteOriginally posted by: MarsWhen changing numeraire you will only change the drift (Girsanov theorem). So euro risk neutral will be dS / S = mu dt + \sigma dW' where W' is a brownian under euro risk neutral measure and mu is what we try to find.The cross term is then : S/ N rho sigma eta dt and we have mu = q - rho sigma etaI do understand that Girsanov says the dynamics of S in the euro RN measure will be dS / S = mu dt + \sigma dW', but somehow my step-by-step derivation doesn't quite fit:* Let S be the dollar asset price, S^e the euro asset price and X = S/S^e the exchange rate* Under Q^$ assume dS/S = r_$ dt + \sigma dW^$, and dX/X = (r_$ - r_e) dt + \eta dZ, with (dW)(dZ) = \rho dt* By Girsanov under Q^e we must have dS^e/S^e = r_e + \sigma dW^e [Eq. A]* We are looking for the dynamics of S under Q^e. By Girsanov this should have the form: dS/S = \mu dt + \sigma dW'* By Ito+Girsanov, under Q^e: d(1/X) / (1/X) = (r_e - r_$) dt - \eta dZ^e* By Ito: dS^e = d(S * 1/X ) = dS/X + Sd(1/X) + (dS)(d1/X) = ... = S^e * [ (\mu + r_e - r_$ - \rho \sigma \eta) dt + \sigma dW' + \eta dZ^e ] [Eq. B]Matching Eq. A and B does produce the correct drift \mu = r_$ + \rho \sigma \etaBUT we also get: \sigma dW^e = \sigma dW' + \eta dZ^e which is impossible...?e.