Risk-Neutral Valuation and No-Arbitrage Pricing
Summary
The document seeks an intuitive introduction to risk-neutral valuation and arbitrage theory. Its response contrasts pricing derivatives under the historical probability distribution of the underlying asset with pricing under a risk-neutral measure. It explains that historical drift can lead to option prices inconsistent with the underlying asset and other traded instruments, creating apparent arbitrage opportunities for a market maker.
The risk-neutral measure is presented as a probability framework under which asset prices are consistent with the absence of arbitrage. This helps produce coherent prices across derivatives and their underlying assets. The explanation is qualitative rather than mathematical: it gives no valuation formula, derivation, assumptions, or worked numerical example. It recommends an introductory derivatives textbook, but does not develop the broader continuous-time theory or explain how to construct the measure in a particular market.
Key ideas
- Historical probabilities describe observed asset behavior but do not by themselves guarantee arbitrage-consistent derivative prices.
- Risk-neutral valuation uses a probability measure that supports consistent pricing under a no-arbitrage framework.
- Inconsistent option and underlying prices can expose a market maker to arbitrage risk.
- The explanation is intuitive and does not provide a formula or derivation.
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Full text
# Risk neutral valuation formula # Risk neutral valuation formula I am totally new to Finance and Arbitrage theory and I have started reading Björk (2018) Arbitrage theory in continuous time. Can anyone please explain to me what is the risk-neutral valuation formula in an intuitive manner? Moreover is there any introductory textbook on "Arbitrage theory"? ## Answer by Ezy (score 3, accepted) https://quant.stackexchange.com/a/42606 The absolute reference for starters which does not dwelve too much into mathematical details but enough to be accurate is Hull so i suggest you have a look at this book first. Options, Futures, and Other Derivatives That being said to give a simple illustration: sometimes people try to value derivative assets (like options eg) by looking at the historical distribution of the underlying asset. This is called working with the historical measure. While this can be informative by itself this can lead to so-called arbitrage opportunities because using the prices derived from the historical measure you would be able to construct a portfolio of these instruments which would be free of any market risk but would offer you a higher return than the risk-free rate. For instance if you imagine a market where you were the single market maker for options on a stock S which has say a pretty large drift (say 20% per year) and if you were using the historical measure to price options on S you would very quickly get out-of-business because you would most likely be bidding the otm calls way too high and offering the otm puts way too cheap which would make your pricing of implied forward completely out-of-line with reality (and make you out of business :) ). By contrast, the risk-neutral measure is another probability measure which has the property that when assets are priced using it then there can be no arbitrage opportunity. This leads to a consistent view of both options and the underlying spot price.
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