Risk-Neutral Pricing and Arbitrage-Free Processes in Continuous Time

Summary

The document asks how to interpret a remark that, unlike in discrete-time models, risk-neutral price processes do not generally comprise every arbitrage-free price process in a continuous-time setting. It distinguishes a risk-neutral price process constructed as a conditional expectation from the broader class of price processes compatible with no arbitrage.

The discussion notes that a conditional-expectation construction yields discounted martingale prices under an equivalent local martingale measure, which is sufficient for no arbitrage. The converse need not hold: an arbitrage-free process may be a local martingale under such a measure without having the specific conditional-expectation form used in the proposition. The document offers this as the likely resolution, but gives no formal theorem, example, or precise assumptions. Its value is mainly clarifying the distinction between one construction of fair prices and all possible arbitrage-free processes.

Key ideas

• A risk-neutral conditional-expectation construction produces discounted martingale prices under an equivalent local martingale measure.
• No-arbitrage prices need not all have that particular conditional-expectation form.
• The distinction concerns the scope of a construction, rather than whether that construction implies no arbitrage.
• The discussion is conceptual and does not supply a formal proof or example.

Tags

# Why does risk-neutral price processes do not, in general, compose all arbitrage-free price processes?


# Why does risk-neutral price processes do not, in general, compose all arbitrage-free price processes?












I was reading reviewing my mathematical finance notes and I came across a remark I cant understand fully

Remark :Contrary to discrete time models, the risk-neutral price processes do not, in general, compose all arbitrage-free price processes in the sense of Proposition 4.3.2. given below

Here $\mathbb{Q}$ is an ELMM(A measure equivalent to $\mathbb{P}$ such that all discounted price processes are local martingales)

The word 'compose' is really confusing me since Prop 4.3.2 claims that any risk neutral price process given the proposition above implies the existence of an ELMM(since the discounted contingent claim $\frac{\pi_t}{X^0_t}$ is a martingale by construction and hence a local martingale) which implies no arbitrage. How do I make sense of the remark? For completeness i am also attaching a picture of the remark below

Edit: I think I understood. What the remark means is that there exists some arbitrage free processes (as a consequence of them being local martingales under $\mathbb{Q}$ which implies $\mathbb{Q}$ is ELMM) but not a risk neutral process of the particular form above described as a conditional expextation

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