Guide to Risk Neutral Probability

By Kenny Zhu · March 14, 2023 · 6 minute read

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Guide to Risk Neutral Probability

“Risk neutral,” in the context of investing, means that an investor focuses on the expected gains of a potential investment rather than its accompanying risks. This concept comes up frequently in options trading, as it’s one of the core tenets in how options are valued.

Risk neutrality is more of a conceptual focus for valuation than a strategy that’s applied on a daily basis. It’s often used as a conceptual framework for the valuation of options and other complex derivatives by sophisticated investment firms.

What Is Risk Neutral?

Risk-neutral investors are solely concerned with the expected returns of an investment, regardless of its underlying risks. When confronted with a gamble versus a sure thing, risk neutral investors are indifferent as long as the expected value of both options balance out.

Risk Neutral vs Risk Averse

Contrast risk neutrality with “risk aversion,” which does consider risk and strongly prefers certainty when comparing investment alternatives. While risk averse investors consider expected value, they will also demand a “risk premium,” or additional benefit, for taking on any additional risk in a transaction. This is what leads to their preference for the more “certain” option, even when the mathematical expected value of two alternative investments is the same.

Risk neutral investors are indifferent between investment options with the same expected values, regardless of the accompanying risk factors. The concept of risk does not play into a risk-neutral investor’s decision-making process, and no risk premium is demanded for uncertain outcomes with equal expected values.

In reality, most retail investors are risk averse, e.g. they have a low risk tolerance, rather than risk neutral. It’s easy to spot this investor preference, given the incessant focus of financial firms on mitigating risk. Terms like “risk-adjusted returns” are frequently used, and entire doctrines in behavioral economics and game theory are built around the cornerstones of loss or risk-aversion.

The difference between risk-neutral vs. risk-averse investors can be illustrated using an example comparing separate sets of probabilities.

Example of Risk Neutrality

To illustrate risk neutrality, consider a hypothetical situation with two investment options: one which involves a guaranteed payoff of $100, while the other involves a gamble, with a 50% chance of a $200 payoff and a 50% chance you receive nothing.

In our hypothetical scenario, the risk neutral investor would be indifferent between the two options, as the expected value (EV) in both cases equals $100.

1.    EV = 100% probability X $100 = $100

2.    EV = (50% probability X $200) + (50% probability X $0) = $100 + 0 = $100

However, a risk averse investor would introduce the added variable of risk into their decision, thereby unbalancing the alternatives above. Given that the 2nd option involves uncertainty, and therefore risk, the risk averse investor would demand an added payoff to justify taking on any added risk.

Reframing the problem above, the risk averse investor would choose option 1, given a) both options return the same expected value, and b) option 1 involves the greatest certainty.

On the other hand, the risk neutral investor would remain indifferent, as risk does not factor into their decision-making process.

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Risk Neutral Pricing and Valuation

Conceptually, risk neutrality is used extensively in valuing derivative securities because it establishes a theoretical basis for finding the equilibrium price between buyers and sellers in any transaction. It’s therefore an important aspect of options trading strategies.

Given that risk-averse investors demand a premium for taking on additional risk, while each individual investor’s risk tolerance can differ. This risk premium can present a problem from an analytical perspective; it introduces “noise” and analytical complexity which can complicate the pricing of derivatives and other investments.

Conceptually, the value of an investment is calculated as the present value of all its current and future cash flows. Future cash flows are discounted using its expected rate of return, which factors in the risk-neutral rate of return along with any added risk premia.

While the risk-neutral rate of return can be assumed to be the same for a given set of investments, the risk premium can vary according to the risk tolerance of individual investors, which complicates the present value calculation, additionally it often skews the calculated value of a security below what the expected future benefit might imply.

To adjust for this complexity in derivatives trading, mathematicians and financial professionals often find it useful to apply risk-neutral measures when pricing derivatives.

Understanding Risk Neutral Probability

The concept of risk neutrality is used to find objective pricing for derivatives; risk neutral probability therefore removes the noisy risk factor from calculations when finding fair value.

This differs from real-world risk-based pricing, which introduces any number of security-specific or market-based factors back into the calculation. The downside of this “real-world probability” is that it makes calculating value an exceedingly complex exercise, as you would need to make fine-tuning adjustments for almost every unique factor that might affect your investment.

Ultimately, risk-neutral probabilities allow you to apply a consistent single rate towards the valuation of all assets for which the expected payoff is known. This allows for ease and simplicity when approaching the valuation process.

However, that’s not to say that risk-aversion and other costs are not factored into calculations, as risk-averse investors would never choose to accept trades that don’t offer risk premiums over the long run.

Instead, risk-neutral probabilities represent the basis on which to build your investment valuation thesis, allowing you to selectively layer on any number of other risk factors later in the process.

Investing Today

Identifying what type of investor you are is important before diving in. If you’re a risk-neutral investor, choosing between risky and non-risky investments will be based on expected values.

If you are risk averse, your investment opportunities will need to be assessed based on whether you are receiving a risk premium commensurate with the risk you perceive.

If you want to learn more about risk and investing, SoFi Invest is a great place to get started. When you set up an online stock trading account, you have access to a range of self-directed options for both brokerage and retirement investing.

Remember that options trading is complex and can entail significant risk for new investors. It’s important to establish a solid investing foundation before moving onto more advanced trading strategies like options trading.

With SoFi, user-friendly options trading is finally here.


Is risk neutral the same as risk free?

Risk neutral does not imply risk free. Risk neutral is simply a conceptual approach for evaluating trade offs without the impact of risk-factors.

Risk continues to exist in the context of each investment when evaluating tradeoffs; risk neutral simply suspends risk as a factor in the evaluation process.

What makes some companies risk neutral?

From a theoretical perspective, companies behave in a risk-neutral manner because firms have the means to hedge their risks away. They can do this by purchasing insurance, buying financial derivatives, or transferring their risk to other parties. This allows them to focus on expected outcomes rather than the risk-related costs of those decisions.

Conceptually, shareholders also want firms to make decisions in a risk-neutral manner, as individual investors can hedge risk exposure themselves by buying the shares of any number of other firms to diversify and offset these risk factors.

What is an example of risk neutral?

An example of risk neutral would be an individual who’s indifferent between 1) a 100% chance of receiving $1,000, versus 2) a 50% chance of receiving $2,000, and a 50% chance of receiving nothing.

In both cases, the expected value would be $1,000, after calculating for both probability and return. This expected value would be what risk-neutral investors would focus on. By contrast, a risk-averse individual would choose option 1, as the outcome has more certainty (and less risk).

Photo credit: iStock/Szepy

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Options involve risks, including substantial risk of loss and the possibility an investor may lose the entire amount invested in a short period of time. Before an investor begins trading options they should familiarize themselves with the Characteristics and Risks of Standardized Options . Tax considerations with options transactions are unique, investors should consult with their tax advisor to understand the impact to their taxes.
Disclaimer: The projections or other information regarding the likelihood of various investment outcomes are hypothetical in nature, do not reflect actual investment results, and are not guarantees of future results.

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