Interpreting Dot Notation in Quanto Adjustment Derivations

Summary

The document asks how to read the dot between volatility vectors and Brownian motion in a derivation of the quanto adjustment. It also asks why multiplying the foreign asset price by the exchange rate produces a domestic currency value and whether that relationship is assumed. The question points to two linked ideas: vector notation for stochastic shocks and the currency conversion of an asset payoff.

No answer or derivation is included, so the document does not establish how the dot product is interpreted or work through the relevant multidimensional Itô terms. Readers should treat it as a prompt for clarification rather than a completed explanation. The key issue raised is that correlated Brownian components can contribute covariance terms when applying Itô's lemma, while the asset conversion itself follows from expressing a foreign-currency price in domestic units.

Key ideas

• The document asks whether the dot between volatility and Brownian motion denotes a vector inner product.
• It questions how covariance terms enter multidimensional Itô calculations.
• It asks why multiplying a foreign asset price by an exchange rate gives its domestic currency value.
• The text poses these questions without providing a derivation or answer.

Tags

# Symbol "." in the derive of Quanto Adjustment


# Symbol "." in the derive of Quanto Adjustment












I am reading "Analysis, Geometry and Modeling in Finance". In section 2.10.2 which derives the quanto adjustment, it states that (in page 46) by definition the process $S_t^{d/f}S_t^f$ is the foreign asset valued in the domestic currency and therefore should be driven under $\mathbb{P}^d $ by $\frac{dS_t^{d/f}S_t^f}{S_t^{d/f}S_t^f} = r_ddt+\sigma_S.dW^d_t+\sigma_{d/f}.dW_t^d$

My question is:

1) What is the "$.$" in $\sigma_S.dW^d_t$ and $\sigma_{d/f}.dW_t^d$? I cannot find it in "Symbol Description" of the book.

If it is just multiplication, why there is no correlation in the the multi-dimension Ito's lemma? (theorem 2.2 in page 21) It say $dW^i_t.dW^j_t=\delta_i^jdt$

2) Why can we formulate the $S_t^{d/f}S_t^f$ like this? Is it an assumption?

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