Estimating the Market Price of Risk from Index Returns
Summary
The document asks how to estimate the expected return of a broad stock index when calculating the market price of risk in a change from real-world to risk-neutral dynamics. It presents the relationship between that price and an asset’s correlation with the index, the index’s expected return and volatility, and the risk-free rate. The central point is that the index expected return remains a real-world quantity and may be estimated by averaging observed index returns over a chosen period.
One response suggests that index returns may offer a more stable estimate than returns on an individual stock, while another clarifies that the market price of risk arises in the mathematics of changing probability measures. The discussion does not provide a worked estimation procedure, data frequency, or evidence that one historical window is reliable. Historical averages are estimates, and the result depends on the selected period and model assumptions.
Key ideas
- The market price of risk relates an asset’s index correlation to the index’s excess expected return per unit of volatility.
- The index expected return is a real-world expectation, even when used in risk-neutral valuation.
- A historical average of index returns can serve as an estimate over a chosen period.
- The discussion gives no method for selecting a reliable estimation window.
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Full text
# Estimating the Market Price of Risk (Hull's Section 36.3)
# 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?
## Answer by Grégoire Courtois (score 1)
https://quant.stackexchange.com/a/66451
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 by j4bert0 (score 1)
https://quant.stackexchange.com/a/66457
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.