Estimating Short-Rate Risk Premia in the Hull-White Framework
Summary
The note distinguishes risk-neutral pricing from estimation of the market price of risk for short interest rates. Bond and bond-option prices are used to calibrate risk-neutral dynamics, but those parameters alone do not identify the physical-measure risk premium. The response recommends estimating model dynamics under both the physical and risk-neutral measures and interpreting their differences as compensation for risk.
It describes two approaches. Kim and Orphanides, and Kim and Wright, use Kalman filtering in generalized affine term structure models, with survey forecasts anchoring expectations. Adrian, Crump, and Moench instead use linear regressions, with realized bond excess returns helping identify ex-ante premia. Hull-White is presented as a special case of the broader affine framework. These are methodological references rather than a worked estimation or comparison, and results depend on model assumptions and the information used to identify physical expectations.
Key ideas
- Risk-neutral calibration from bond prices alone does not identify the physical-measure short-rate risk premium.
- The risk premium can be inferred by estimating dynamics under both physical and risk-neutral measures.
- Kalman-filter approaches use survey forecasts to anchor rate expectations.
- Regression approaches use realized bond excess returns to estimate ex-ante risk premia.
- Hull-White fits within the broader family of affine term structure models.
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# Estimation of market price of risk of short interest rate under the Hull-White model # Estimation of market price of risk of short interest rate under the Hull-White model I think I am a bit confused. I intend to estimate the market price of risk the short interest rate, say, under the Hull-White model. I have the following two questions. - Is it correct to state state that the (zero coupon) bond price and the (zero coupon) bond European option price are both valued in the risk-neutral measure, because they are both derivatives of the short rate? If so, the parameters calibrated from these prices would not provide the market price of risk since the latter could only be estimated in the real world measure. Is this correct? - If the answer to the above question is affirmative, how does one estimate the market price of risk (risk premium) of the short interest rate? ## Answer by Helin (score 1, accepted) https://quant.stackexchange.com/a/41289 There's a large literature dedicated to this topic. The following two methods are my preferred (they also happen to be the most popular): - Kim and Orphanides's Term Structure Estimation with Survey Data on Interest Rate Forecasts or Kim and Wright's An Arbitrage-Free Three-Factor Term Structure Model and the Recent Behavior of Long-Term Yields and Distant-Horizon Forward Rates. The method described in these two papers are pretty generic – the authors calibrate model parameters under both the physical and risk neutral measure by means of Kalman filtering. Difference in interest rates calculated under the two measures then reflect the market price of risk. The innovation of these papers is the introduction of survey data to anchor down the expectation component, so the results are more robust and realistic. Both papers are based on generalized affine term structure models, of which Hull White is a special case. - A more recent method is proposed by Adrian, Crump, and Moench in Pricing the Term Structure with Linear Regressions. Instead of Kalman filtering, they use simple regression techniques to obtain model parameters in the two measures. Instead of using survey data to pin down rate expectations, they use realized bond excess returns to pin down ex-ante risk premium.
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