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A risk-neutral measure is a probability measure that yields an expected present value (discounted at the risk-free rate) which is equal to the current market price. The risk-neutral measure is also called an equivalent martingale measure.

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### Market Price of Risk for Consumption Asset - Hull's Example 28.1

Note that just before Equation (28.9), Hull writes $-$ my emphasis: The market price of risk of [asset] $\theta$ measures the trade-offs between risk and return that are made for securities depend …
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### Extending an incomplete market to generate a complete one

I believe I have found an exact answer to my question in Thomas Bjork's book, Arbitrage Theory in Continuous Time, on page 122 (third edition): Meta-theorem 8.3.1 Let $M$ denote the number of underly …
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### Why discounted derivative price is a martingale?

Under a Black-Scholes framework, the dynamics of the stock price under the risk-neutral measure $\mathbb{Q}$ are given by ... $$S_t = r S_tdt +\sigma S_tdW^{\mathbb{Q}}_t$$ ... and those of the ri …
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### General Dynamics of a Tradable Asset under the Risk Neutral Measure

Our market has a tradeable asset $S$ and a risk-less money market account $B$, that is, the numéraire of the risk-neutral measure. We assume the following standard conditions, which are widely applica …
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### Formal proof for risk-neutral pricing formula

You have two main papers that show this result: In a finite framework and in a somewhat simplified continuous framework, see Harrison & Pliska (1981), Corollary on page 228, Proposition 2.9 and Prop …
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