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Time-Varying Market Price of Risk in the Heston Model

Article Quant Q&A · Author: Mr Frog
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Summary

The document raises a notation and interpretation question about the market price of risk in the Heston stochastic-volatility model. In the expression discussed, one component depends on the instantaneous variance through its square root, while another component is unspecified because volatility is not itself traded. The question is whether this dependence means the market price of risk should carry a time subscript and be treated as stochastic.

This is a framing question rather than a worked answer: it provides no resolution, derivation, calibration, or empirical evidence. It points to the distinction between constant parameters in a model and quantities whose values vary with the stochastic state. Readers should not infer from the question alone how the risk premium is specified under a particular measure or model convention.

Key ideas

  • The Heston model includes a stochastic variance state that can affect the market price of risk.
  • The document questions whether a risk-price component dependent on variance should be written as time varying.
  • It notes that volatility is not directly traded, leaving one component of the risk price unspecified in the stated setup.
  • The document poses the issue but does not supply a derivation or answer.

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# Is the market price of risk deterministic or stochastic in the Heston model?


# Is the market price of risk deterministic or stochastic in the Heston model?












I am recently digging into the Heston model and I have noticed that every author refers to the market price of risk simply as $\lambda$, or sometimes it is more clearly specified to be bi-dimensional in a form such as $\lambda=(\lambda_1, \lambda_2)^T$ where $\lambda_1=\frac{\mu-r}{\sqrt{v_t}}$ and $\lambda_2$ is unknown as a result that volatility is not traded.

This notation seems confusing to me, and I wonder whether $\lambda$ should actually carry a subscript $t$, and therefore being written as $\lambda_t$, as a result of being dependant on $\sqrt{v_t}$. I mean, differently than what was true for the framework of B&S, even though $r$ and $\mu$ are constant, the market price of risk is now even stochastic, right?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.