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

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?

• Market price of risk is only a theoretical concept to highlight how the risk-neutral measure might differ from the real word measure. It turns out that for diffusion models it is just a drift. Black, Scholes and Merton got their Nobel prize for showing that a consistent arbitrage free pricing model can be formulated entirely within a risk-neutral world. Then drift of a stock price is unambiguously riskless rate minus dividend yield, and vol is whatever the trader decides. It boils down to: practitioners are rarely interested in MPR. May 13, 2022 at 8:31

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.