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I need help with estimating market price of risk. Assume money market account and two risky assets which exposed to same two sources of risks follow process:

$dM(t)=rM(t)dt$
$dS_1(t)=S_1(t)(\mu_1dt+\sigma_{11 }dW_1(t)+\sigma_{12}dW_2(t))$
$dS_2(t)=S_2(t)(\mu_2dt+\sigma_{21 }dW_1(t)+\sigma_{22}dW_2(t))$

and market price of risks defined as:

$\theta_1 = \frac{\sigma_{22}(\mu_1-r)-\sigma_{12}(\mu_2-r)}{\sigma_{11}\sigma_{22}-\sigma_{12}\sigma_{21}}$

and $\theta_2$ is derived in similar manner.

Assume $S_1 and S_2$ risky securities exposed to same uncertainty and their values are derived from those uncertainties. For example they are GDP linked derivatives/bonds and $dW_1$ comes from interest rate $dW_2$ comes from GDP.

My question is, given these two (or probably more) security prices and short rate and GDP data, how can I estimate parameters such as $\sigma_{11}$ $\sigma_{12}$ etc. or $\theta_1$ and $\theta_2$ themselves.

Thanks in advance.

PS: I got the idea/equations from https://www3.ntu.edu.sg/home/achleon/FE6516/MFE6516_seminar_slide_04%20(2ppg).pdf page 47.

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  • $\begingroup$ Hi, do you know/assume W1 or W2; or must they be inferred as the statistically most dominant risk factors driving S1 and S2? $\endgroup$ – demully Nov 27 '19 at 21:50
  • $\begingroup$ Yes I assume those two are the most dominant factors. I’ve checked yuima package on R and closed form MLE package on Matlab of Yacine Ait. But couldn’t see any model fit my problem. $\endgroup$ – TryingtobeQuant Nov 27 '19 at 22:02

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