Skip to content
All library documents

Risk-Neutral Measures with Multiple Stock Return Processes

Article Quant Q&A · Author: nearhome
Read the original

Summary

The document asks whether two stocks driven by the same Brownian motion can have different physical drift terms and still admit a risk-neutral measure. Its answer distinguishes between processes that are tradable assets and related return series that are not separately tradable. A change of measure can adjust the drift of the tradable asset to the risk-free rate, while a non-tradable representation derived from it need not independently satisfy the same pricing requirement.

The example contrasts a stock’s total return process with its price return process when distributions are paid. Under the risk-neutral measure, the drifts are changed in the stated way, but only one process represents a tradable security. The discussion is conceptual and illustrative: it does not establish general existence conditions for an equivalent martingale measure, which depend on the market’s full specification and no-arbitrage assumptions.

Key ideas

  • A risk-neutral measure may exist even when two modeled processes have different physical drifts.
  • The pricing requirement applies to tradable assets after accounting for the risk-free rate.
  • A total return process and a price return process can describe different quantities associated with one stock.
  • A non-tradable process need not independently behave as a priced asset under the risk-neutral measure.

Tags

EquitiesDerivatives pricingStatistics
Full text
# Is there a risk-neutral measure if there are two stocks with different drift terms?


# Is there a risk-neutral measure if there are two stocks with different drift terms?












There are two stocks: $S_t$ and $P_t$ $$dS_t = S_t(\mu dt + \sigma dB_t)$$ $$dP_t = P_t((\mu + \varepsilon) dt + \sigma dB_t)$$ Is there any risk-neutral measure? My thoughts are pretty simple: $μ$ is for the physical measure, so there's no risk-neutral measure. Please shed light on this question.

## Answer by Frido (score 3, accepted)

https://quant.stackexchange.com/a/75563

Well there can definitely be a risk-neutral measure but only one of the processes is a tradable. For instance, consider the total return process of a stock $S_t$ $$ dS_t = \mu S_t dt + \sigma S_t dW_t $$ and its price return $$ dP_t = (\mu - q)P_t dt + \sigma P_t dW_t, \quad P_0 := S_0 $$ Under the risk -neutral measure you have exactly the same SDEs with $r$ replacing $\mu$, but only one of them is tradable. In this example the total return process is tradable, the price return is not.

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.