Formulating a Minimum-Variance Martingale Measure Problem

Summary

The document presents an attempted Lagrangian formulation for finding a minimal martingale measure among equivalent martingale measures. It defines an option payoff, a terminal self-financing portfolio value, a candidate density process, and a constraint relating the control to the asset’s drift, volatility, and interest rate. The central quantity to evaluate is the expected squared hedging error, weighted by the change-of-measure density.

The author asks how to compute that expectation and differentiate the objective to solve for the control. No answer or worked derivation is included, so the document does not establish that the stated density or optimization setup is correct. In particular, the model assumptions, admissible controls, and dependence of terminal portfolio value on the control would need clarification before the expectation or first-order conditions could be derived. It is best read as a formulation question rather than a resolved method.

Key ideas

• The optimization aims to minimize a squared terminal hedging error under a candidate equivalent martingale measure.
• A Lagrange multiplier is introduced to impose a constraint on the measure-changing control.
• The objective weights the squared error by the likelihood ratio between the candidate measure and the physical measure.
• The document asks how to compute the expectation and derive the control but provides no solution.

Tags

# mean variance minimizer


# mean variance minimizer












I need to use the lagragian multiplier to find the minimal martingale measure from the set of equivalent martingale measures. i formed the lagragian as L = $L(u(t,S(t)),\lambda) = E_\mathbb{P}[\dfrac{d\mathbb{Q}}{d\mathbb{P}}(H - V(T))^2] + \lambda(\sigma(t,S(t))u(t,S(t)) - \alpha(t,S(t)) + r(t))$ where H is the payoff of an option and $V(T)$ is the terminal value of a self financing portfolio and $\dfrac{d\mathbb{Q}}{d\mathbb{P}} = \exp \{ \int_0^T udB(s) + \frac{1}{2} \int_0^T u^2 ds\}$ $\quad$ I need to slove for the above minimistion problem to get $u$. I have been trying this for more than a week now and i dont seem to yield any reasonable result.Can someone pls help me out with it especially how to compute this $E_\mathbb{P}[\dfrac{d\mathbb{Q}}{d\mathbb{P}}(H - V(T))^2] $ .I think when i get that i can now take the partial derivatives to get my $u$ .Thank you

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