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Shared Brownian Risk Factors in Stock and HJM Models

Article Quant Q&A · Author: Bubblethan
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Summary

The discussion explains how Brownian drivers are interpreted when stock prices and forward rates are written using the same indexed Wiener processes. If both equations appear under one probability measure and use the same notation, the response treats the drivers as the same processes, representing common market risk factors. Different asset exposures arise through their volatility vectors.

It also describes how correlation between a stock and a forward rate can be represented even when the underlying Brownian motions are independent: choose volatility vectors with the desired relative orientation while preserving each variable’s total volatility, given by the vector’s Euclidean norm. The answer offers a conceptual clarification rather than a derivation or numerical example. It does not explain the relationship between the risk-neutral measures asked about in the question, so that part remains unresolved; the shared notation assumption also depends on the models being stated under the same measure.

Key ideas

  • Brownian processes with the same notation in equations under one measure are interpreted as shared risk drivers.
  • Volatility vectors determine how each modeled variable loads on the Brownian risk factors.
  • Correlation between variables can be encoded through the relative orientation of their volatility vectors.
  • Changing vector orientation can alter correlation while preserving each variable’s total volatility.
  • The response does not address the relationship between the two risk-neutral measures in the question.

Tags

Fixed incomeDerivatives pricingStatisticsRisk management
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# Answer by siou0107 (score 0)


# What's the relationship between the risk-neutral probability in HJM and the risk-neural probability under domestic money market?












In shreve's book, we model the stock price dynamics as: $$S_i(t) = \alpha(t)S_i(t)dt +S_i(t)\sum ^d_{j=1}\sigma _{ij}(t)dW_j(t)$$ and the forward rate can be written as : $$df(t,T) = \gamma(t,T)dt + \sum ^d_{j=1}\beta _{ij}(t,T)dW_j(t)$$ I'm wondering are we assuming the risk factors $W_j$ to be the same for stock price and forward rate, and they could be viewed as the risk factors for the financial market. or they are different? And what's the relationship between the two risk neutral probability measures in two models?

## Answer by siou0107 (score 0)

https://quant.stackexchange.com/a/50093

A priori, if those two $W_j$ notations are in the same paragraph, they identify the same Wiener processes, under the same probability measure. So yes, they are seen as the different risk factors in your financial market.

If you assume that your $W_j$'s are independent of one another, you can introduce correlation between your two variables (stock price and forward rate) by changing the volatility vectors, as long as you keep the Euclidean norm of those vectors (which is the total volatility of your variables) unchanged.

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.