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Risk-Neutral Dynamics for an Independent Geometric Brownian Asset

Article Quant Q&A · Author: zjo892
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Summary

The document poses a change-of-measure question for two independent geometric Brownian assets and a bank account. A measure is specified under which the first asset, discounted by the bank account, is a martingale; the question asks what this implies for the second asset’s dynamics and how to price the product of the two assets.

No answer or derivation is included, so the post does not resolve whether the stated measure determines the second asset’s drift or whether the product must be priced under another measure. The setup raises issues about which assets are traded and what constraints define an admissible pricing measure. It is a useful prompt for studying risk-neutral measure selection and product pricing, but supplies no evidence, formula, or conclusion to apply directly.

Key ideas

  • The setup specifies risk-neutral dynamics for one asset while leaving the second asset’s dynamics under that measure in question.
  • The assets are driven by independent Wiener processes in the stated model.
  • The post asks whether the product asset should be priced under the same measure or another measure.
  • No response is provided, so the measure-selection and pricing questions remain unresolved.

Tags

Derivatives pricingStatisticsRisk management
Full text
# Dynamics of independent Geometric Brownian Motions under risk-neutral measure Q


# Dynamics of independent Geometric Brownian Motions under risk-neutral measure Q












Suppose I have two Geometric Brownian motions and a bank account: $$dB_t=rB_tdt$$ $$ dS=S(\alpha dt + \sigma dW_t) $$ $$ dY = Y(\beta dt + \delta dV_t) $$

Where $dW_t$ and $dV_t$ are independent Wiener processes. Now suppose that I have a martingale measure $Q$ such that $d(S_t/B_t)$ is a martingale. Then we have: $$dS = S(rdt+\sigma dW_t^Q)$$

Is it possible write the dynamics of $Y_t$ under $Q$?

Suppose also I am interested in pricing $Z_t=S_tY_t$. Would I still use the measure $Q$ to price $Z$ or am I using a separate measure $\tilde{Q}$ such that $d(Z_t/B_t)$ is a martingale?

Thanks!

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