When a Relative Price Becomes a Martingale Without Changing Measure

Summary

The document discusses two securities driven by the same Brownian motion, each with drift expressed as the risk-free rate plus a shared market price of risk multiplied by its volatility. The question is whether choosing the parameter to match the second security's volatility connects the ratio of their prices becoming a martingale to Girsanov's theorem, and whether this is used in derivative pricing.

The answer distinguishes an Itô calculation from a change of probability measure: Itô's lemma is sufficient to show the ratio's martingale property in the stated setup, so Girsanov's theorem is not needed for that result. It also points to the no-arbitrage implication that securities exposed to the same Brownian risk must carry consistent risk premiums. The discussion is a narrow conceptual explanation; it does not develop a pricing procedure or address more general models with multiple sources of risk.

Key ideas

• Itô's lemma can establish the stated price-ratio martingale property directly.
• The martingale result described does not require a change of measure or Girsanov's theorem.
• No-arbitrage links the risk premiums of securities driven by the same Brownian risk.
• The explanation is limited to the shared single-factor setup in the question.

Tags

# Equivalent Martingale Measure result Hull?


# Equivalent Martingale Measure result Hull?












I've been reading Hull's chapter about Martingales and measures where he states that if you have the dynamics of two securities as follows:

\begin{align} \frac{df}{f} = (r + \lambda \sigma_f) dt + \sigma_f dW_t^{\mathbb{P}} \\ \frac{dg}{g} = (r + \lambda \sigma_g) dt + \sigma_g dW_t^{\mathbb{P}} \end{align}

and we choose $\lambda=\sigma_g$ (which he calls market price of risk) then the process $(\frac{f}{g})$ becomes a martingale. I understand why it becomes a martingale but I'd like to know if there's some relation between doing this and Girsanov's theorem? or is this approach commonly used when pricing derivatives?

## Answer by Antoine Conze (score 0, accepted)

https://quant.stackexchange.com/a/39189

To show that $f/g$ is a martingale you only need to use Ito's Lemma. There is no change of measure and thus no need to use the Girsanov theorem.

Also although its not directly related to your question, proving that two securities driven by the same Brownian motion must have the same risk premium $\lambda=(E^P[df/f]/dt -r)/\sigma_f = (E^P[dg/g]/dt -r)/\sigma_g$ is a classic case of applying the no arbitrage opportunity condition.

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