Risk-Neutral Pricing and Expected Discounted Cash Flows

Summary

The response corrects the premise that ordinary, or real-world, valuation avoids taking expectations of future cash flows. It states that asset valuation in either framework involves expected discounted cash flows; the key distinction is how risk is handled in the discounting or probability measure. For a derivative, applying a real-world discount rate is difficult because the appropriate risk premium may not be readily known.

Risk-neutral pricing offers a way around that difficulty when a derivative can be combined with other assets, such as its underlying stock, to form a portfolio with fixed future cash flows. Because the hedged portfolio has no uncertainty in the described argument, it can be valued using the risk-free rate. The derivative’s value is then recovered after accounting for the known assets in the portfolio. This is a conceptual explanation, not a worked valuation; it does not state the assumptions or limits required for constructing such a hedge.

Key ideas

• Both real-world and risk-neutral valuation use expected discounted future cash flows.
• Risk treatment differs between the approaches, and a derivative’s real-world risk premium may be hard to determine.
• A hedge can combine a derivative with other assets to create fixed cash flows.
• The response values the hedged cash flows at the risk-free rate and then isolates the derivative’s value.

Tags

# Risk neutral pricing vs real world pricing


# Risk neutral pricing vs real world pricing












Could you please explain why to calculate the asset price under risk-neutral probability, we have to take the expectation of the future cashflow, while in the normal world, we don't have to take the expectation of future cashflow ? (i.e. we keep the same amount of future cashflow then discounted for its required rate of return)

Thank you very much.

## Answer by Aksakal almost surely binary (score 3)

https://quant.stackexchange.com/a/55000

> while in the normal world, we don't have to take the expectation of future cashflow

Grossly incorrect. The formula or approach are exactly the same: the value of an asset is the expectation of its discounted cash flow. The only difference is the discount rate. If you have a nonlinear instrument such as a vanilla Euro call option, it's not clear what should be the discount factor in real world. The discount factor should include the risk premium, so what is the risk premium of this option?

That's where the risk neutral trick comes handy. It turns out that you can form a portfolio of the derivative instrument and some other instruments with known values (such as underlying stocks) in such a way that the cash flow of this portfolio is fixed. If it is fixed then there is no uncertainty, there is no risk. Hence, there is no need to think of the risk premium. Therefore, you can discount at risk free rate. Then you remove the value of known assets, and the remaining is the value of the derivative.

Shown 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.