Risk-Neutral Pricing, Real-World Returns, and Put Options
Summary
The document discusses whether an asset’s price, represented through risk-neutral valuation, must be below its expected discounted payoff under the real-world probability measure. The answer distinguishes how stocks and derivatives are priced: a stock’s market price reflects its risk, while an option is valued using risk-neutral discounting with the stock price as an input. It cautions that risk-premium intuition for a long position in a market-correlated asset does not transfer directly to every derivative position.
A put can benefit from falling stock prices, but that does not mean its risk-neutral value must be lower than its real-world expected discounted payoff. The response notes that the market as a whole holds the underlying asset and that the put’s short-stock exposure is only one part of its pricing context. It does not work through the Black–Scholes calculation or establish a universal inequality; it offers a conceptual clarification rather than a quantitative proof.
Key ideas
- The real-world measure incorporates risk premia, while risk-neutral valuation is used to price derivatives.
- Stock prices serve as inputs when valuing options under risk-neutral discounting.
- A put's exposure to falling stock prices does not by itself settle how its value compares across probability measures.
- Risk-premium intuition for a market-correlated stock position should not be applied mechanically to an option.
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Full text
# Is the market price of an asset always lower than the expected discounted value under the REAL WORLD measure? # Is the market price of an asset always lower than the expected discounted value under the REAL WORLD measure? The risk neutral measure is often said to reflect the risk aversion of investors. So intuitively, I would think that an asset's expected discounted value should be lower under the risk neutral measure (-> market price) than under the real world measure. In other words, a risk averse investor should always prefer a fixed payout equal to the expected discounted value (under the real world measure) to holding the risky asset. However, in a simple Black/Scholes model with constant volatility, I would expect the expected discounted value of a put option to be higher under the risk neutral measure than under the real world measure. This is because the stock, when modelled under the risk neutral measure, would take more negative paths, giving the put option a greater value. Is this correct? And how does that fit together with the intuition about investors being risk averse, if we imagine an investor who only wants to buy one put option and has no other assets in this portfolio? Interested in your thoughts, thanks :) ## Answer by dm63 (score 3) https://quant.stackexchange.com/a/70397 Your first paragraph is a bit confusing. The real world measure is the one that takes into account risk aversion (hence higher discount rates and lower price). The risk neutral measure is equivalent to assuming no risk aversion and therefore produces in general higher prices for assets. What I have just said is true for long positions in assets that are positively correlated with the overall market portfolio (thus, you only get paid for taking non diversifiable risk). Now you may ask, what if an investor has a short position in a stock (such as implicitly is that case if you are long a put option). That is a risky position so you should get paid for that. But it is actually not the case because overall, the market is long that asset and it is priced according to investors' overall position. I detect in your question some possible confusion about how options are priced versus how assets in general are priced. Remember, assets such as stocks are priced according to how risky they are, in a real world measure. Options and other derivatives are priced using risk neutral discounting, using the price of the stock as an input. Hope that clears up a few things.
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