Estimating the Market Price of Risk (Hull's Section 36.3)

Question

I'm currently trying to understand risk-neutral valuation and transforming real-world stochastic processes to their risk-neutral version. If I understood it correctly, the main point of risk-neutral valuation is to not have to deal with real-world drifts which are very difficult to estimate, but instead it requires to estimate the market price of risk (as in section 36.3 of Hull). The formula for the market price of risk is the following:

$\lambda=\frac{\rho}{\sigma_m}(\mu_m-r)$

where

• $\lambda$: Market price of risk of the variable
• $\rho$: Instantaneous correlation between the percentage changes in the variable and returns on a broad index of stock market prices
• $\mu_m$: Expected return on broad index of stock market prices
• $\sigma_m$: Volatility of return on the broad index of stock market prices
• $r$: Short-term risk-free rate

So if I have a time-series that e.g. correlates with a stock I can use this formula. I can calculate the correlation $\rho$ e.g. with the Pearson correlation coefficient but how do I get the $\mu_m$? Isn't that again a real-world drift of a stock which is so difficult to estimate?

Answer 1

Not sure about my answer but I would think that the expected return of an index is likely to be less volatile than the expected return of a single stock and so you could use some historical return as an estimate of the expected return of the index (I think values around 6-7% are often considered)

Answer 2

The market price of risk just a name for a process arising from mathematics of changing a measure. So in that regard, I think you have missed the main point of risk neutral valuation.

It is not entirely clearly to me what you refer by "variable" in this context. A stock price perhaps?

Your main question is about estimating $\mu_m$. You can do this by averaging realizations of the index returns, using the time period you are considering. And if we where to model the index evolution by a SDE, e.g. geometric brownian motion, then you are right, $\mu_m$ would be the real-world drift parameter of the index.