Estimating Market Prices of Risk from Asset Prices

Summary

The document presents a two-asset, two-factor diffusion model with a risk-free money market account. It gives the market price of the first risk factor as a function of the assets' expected returns above the short rate and their factor loadings, with the second price defined analogously. The motivating example uses securities exposed to interest-rate and GDP uncertainty.

The main question is how to estimate the volatility or factor-loading parameters and the market prices of risk from observed security prices, short rates, and GDP data. No estimation procedure or answer is supplied, and the model is stated rather than empirically assessed. Applying the equations would require suitable return and factor observations, assumptions about the dynamics and identification of factor exposures, and an estimation design; these details and potential limitations are not developed in the document.

Key ideas

• The model relates excess expected returns to exposures to two Brownian risk factors.
• The market prices of risk can be obtained by solving a system involving asset drifts and factor loadings.
• The example associates the factors with interest rates and GDP uncertainty.
• The document asks how to estimate the model inputs from observed data but provides no estimator or empirical results.

Tags

# Estimating Market Price of Risk


# Estimating Market Price of Risk












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