Why a Time-Dependent Market Price of Risk Can Fail at Maturity

Summary

The document raises a modeling issue for a time-dependent market price of risk defined using the excess asset drift divided by time remaining to maturity and volatility. As maturity approaches, the denominator shrinks, and at the maturity time the expression is undefined. The author wants a similar form that remains defined and continuous at that endpoint.

No solution, derivation, or empirical evidence is provided; the text is a question seeking feedback. The issue highlights the need to examine the limiting behavior of the numerator and denominator together, and to specify how the process behaves at maturity. Whether a continuous extension exists depends on the model assumptions and parameter paths, which the document does not state. It therefore frames a mathematical modeling question rather than establishing a general remedy.

Key ideas

• A time-dependent risk premium is expressed using excess drift, time to maturity, and volatility.
• The stated expression is undefined when current time reaches maturity.
• A continuous endpoint value depends on how the numerator and denominator behave near maturity.
• The document poses the problem but does not provide a solution or supporting analysis.

Tags

# Issues with a time-dependent market price of risk


# Issues with a time-dependent market price of risk












I have a time-dependent market price of risk of an asset as: $$ \lambda(t) = \frac{\mu(t)-r(t)}{(T-t)\sigma} $$ where $t$ is the current time and $T$ is a constant maturity time of an asset. Here, $\mu(t)$ is the drift of the asset, $r(t)$ is the risk free rate and $\sigma(T-t)$ is basically the time-dependent volatility of the asset.

However, as you can see, at $t=T$, the market price of risk is undefined because of the denominator. In my calculations, I need the market price of risk to be of this form or similar. Hence, do you have any input on how I can keep a similar form off $\lambda$ but still have it defined and continuous at the maturity time. I'm a bit stuck with this and thus, any insight/feedback would be greatly appreciated!

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.