Mean-Variance Pricing of a Call in an Incomplete Market

Summary

The document formulates a mean-variance approach to pricing a European call in an incomplete market. It seeks an equivalent martingale measure that minimizes the expected squared difference between the terminal value of a dynamically managed portfolio and the option payoff. This expresses the pricing problem as choosing the martingale measure, or an associated control, that gives the smallest quadratic hedging error.

The setup specifies a portfolio’s terminal value through its initial capital, risk-free growth, and a stochastic integral involving the underlying asset, volatility, and a strategy process. The target payoff is the call payoff under a geometric Brownian motion model. However, the text only states the optimization problem and model ingredients; it provides no derivation, solution, assumptions establishing existence or uniqueness, or numerical evidence. The framework therefore serves as a starting formulation rather than a complete pricing method, and its practical conclusions depend on how the control and admissible measures are defined.

Key ideas

• The proposed pricing rule selects an equivalent martingale measure by minimizing expected squared terminal hedging error.
• The portfolio value combines risk-free growth with gains from a controlled exposure to the underlying asset.
• The target is a European call payoff modeled from a geometric Brownian motion terminal price.
• The formulation alone does not establish a solution or uniqueness; additional assumptions and derivation are needed.

Tags

# Mean-variance minimizser


# Mean-variance minimizser












I am working on a project that involves pricing european call options in incomplete markets. Now I need to find a unique measure $Q^*$ such that

$$Q^* = \min_{M_e} E_Q [V(T)-F(w)]^2 = \min_{u} E_Q [V(T)-F(w)]^2$$

where $V(T)$ is the terminal value of a portfolio given

$$V(T) = V(0)e^{rt} + \int_0^T e^{r(t-u)}\beta(u)S(u)\sigma dB(u)$$

and

$$F(w)=(e^{\sigma B(T)+(r-\frac{1}{2}\sigma^2)T}-K)$$

$M_e$ is the set of all equivalent martingales.

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