Risk-Neutral Measures as a No-Arbitrage Pricing Tool
Summary
The document explains why derivative pricing uses risk-neutral measures even though real investors are risk-averse. It presents the fundamental theorem of asset pricing: under the relevant framework, absence of arbitrage is equivalent to the existence of an equivalent risk-neutral measure. This measure lets discounted asset prices be represented as martingales, so a derivative with a payoff determined by the terminal asset price can be valued as a discounted expected payoff under that measure.
A call option is given as an example, and the response emphasizes that risk-neutral valuation is a mathematical device for applying no-arbitrage arguments, rather than a claim that market participants literally have neutral risk preferences. It points to foundational work on martingales and arbitrage, and mentions the separate question of recovering physical probabilities. The explanation does not classify real-option pricing approaches or develop a model for estimating investors' actual beliefs, so its scope is the conceptual rationale for risk-neutral pricing.
Key ideas
- Risk-neutral valuation does not assert that actual market participants are indifferent to risk.
- In an arbitrage-free market, an equivalent risk-neutral measure supports derivative pricing under stated assumptions.
- Discounted asset prices are represented as martingales under the risk-neutral measure.
- A derivative can be priced as the discounted expected value of its payoff under that measure.
- Physical probability measures describe a different question from no-arbitrage valuation.
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Full text
# How literature come up with risk-neutrality problem, considering that market is not really risk-neutral?
# How literature come up with risk-neutrality problem, considering that market is not really risk-neutral?
I am searching on real-option pricing deficiencies to encounter risk-neutrality. As we know risk-neutrality assumption, is not hold in real situations. The problem is that I could not classified literature solutions to this problem. In financial market and real market. I really appreciate each piece of information.
## Answer by fni (score 4)
https://quant.stackexchange.com/a/15177
This goes back to the so-called First Fundamental Theorem of Asset Pricing saying that markets are arbitrage free if and only if there exists at least an equivalent risk neutral measure. So the reason why we are using risk neutral measures to price options is because it allows us to represent discounted stock diffusions as martingales and therefore express the price of any derivative, whose payoff is a deterministic function of the final price of the stock, as an expected value under the risk neutral measure, for instance for a call option: $$C(S_t,t) = \mathbb{E}_t^{\mathbb{Q}}[(S_T - K)^+e^{-r(T-t)}]=S_tN(d_1) - Ke^{-r(T-t)}N(d_2) $$ $$d_1=\left[log\frac{S_t}{K} + (r+\frac{\sigma^2}{2})(T-t)\right]\frac{1}{\sigma\sqrt{T-t}}$$ $$d_2 = d_1 - \sigma\sqrt{T-t}$$ So risk neutral valuation is just a trick to use no arbitrage arguments to price derivatives, but of course it’s well known markets participants are risk averse and it would be actually particularly interesting to know the physical measures they use.
Some references:
-Harrison, Michael J. and Kreps, David M, Martingales and arbitrage in multiperiod securities markets, Journal of Economic Theory, Volume 20, Issue 3, June 1979, Pages 381–408
-Harrison, Michael J. and Pliska, Stanley R., Martingales and stochastic integrals in the theory of continuous trading, Stochastic Processes and their Applications, Volume 11, Issue 3, August 1981, Pages 215–260
-Ross, Steve , The Recovery Theorem, Journal of Finance, forthcomingShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.