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?

  • 1
    $\begingroup$ It’s a dot product, and there is no correlation since it’s standard Brownian motion $\endgroup$ – starovoitovs Jun 19 '19 at 11:36
  • $\begingroup$ Answering the second question, that is simply "the foreign asset valued in domestic currency" in formula. $\endgroup$ – Vitomir Jun 19 '19 at 15:02

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