Changing Numeraires Requires More Than a Marginal Implied Density

Summary

The document poses a pricing question involving an option on an asset denominated in one currency when the available risk-neutral implied density was derived under another currency’s measure. It writes the desired expectation under a domestic pricing measure and considers changing to a foreign measure. That change introduces an exchange-rate conversion factor inside the expectation, so the available asset density alone may not determine the desired price.

The author notes that a joint Black–Scholes model for the asset and foreign exchange rate produces a covariance adjustment when changing measures, then asks whether pricing with fewer assumptions is possible using only the implied density. The text provides no answer or worked resolution. Its useful point is the distinction between an asset’s marginal implied distribution and the joint information needed to value a payoff under a different numeraire; the required dependence on FX is left unresolved.

Key ideas

• A risk-neutral density is associated with a particular pricing measure and numeraire.
• Changing currency numeraire introduces an exchange-rate factor into the pricing expectation.
• An asset’s marginal implied density alone may not capture its dependence with the FX rate.
• A joint asset and FX model can supply dependence assumptions, but the question seeks a less assumption-heavy method.
• The document poses the problem without providing a solution.

Tags

# How is the implied risk neutral density affected when changing numeraire?


# How is the implied risk neutral density affected when changing numeraire?












For example i would like to price \begin{equation*} E^{Q} \left[ e^{-\int_{0}^{T}r_{s}^{cur}ds} f \left( S_{T_f}^{cur_1} \right) | \mathcal{F}_{0} \right] = B_{cur}(0,T)E^{Q^{cur}_{T}}[ f(S_{T_f}^{cur_1})|\mathcal{F}_{0}] \end{equation*}

I have at my disposition the risk neutral currency implied density for $S_{T_{f}}^{cur_1}$ obtained under $Q^{cur_1}$ from Breenden-Litzenberg theorem , how can I then value my option ? At most I can say that it is equal to $B^{cur_1}(0,T) E^{Q^{cur_{1}}_{T}}[\frac{FX(cur_1,cur)(T)}{FX(cur_1,cur)(0)}f(S_{T_{f}}^{cur1})|\mathcal{F}_{0} ]$ but then i need to modelize the FX.

If i suppose my asset as well as my FX follows a BS model , I have the typical $e^{\int_{0}^{T_{f}} \rho_{S,FX}(t) \sigma_{S}(t) \sigma_{FX}(t) dt }$ factor that appears from changing my measure from foreign to domestic but how can i do the same thing with only some risk neutral implied density? I want to price with the least assumptions made.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.