Research & Publications
Back-to-Basics

The following article is reprinted from the Jul/Aug, 1998 issue of
On the Edge, the Interactive Data Fixed Income Analytics bimonthly newsletter.

Back-to-Basics: Are Investors Really "Risk Neutral?"

Teri Geske
Senior Vice President, Product Development


I don't know about you, but I would not describe myself as "neutral" when it comes to risk - no, I definitely prefer less risk to more. Therefore, I found it difficult to accept that models for valuing financial assets rely so heavily upon this notion of being "risk-neutral". In fact, noted professor of financial economics and derivatives expert, John Hull (co-author of the Hull-White term structure model), states that "risk-neutral valuation is without doubt the single most important tool for the analysis of derivative securities."¹ Given the importance of this concept (and making the bold assumption that if something is confusing to me, there are others out there who are also somewhat perplexed), I thought I would write this month's Back-to-Basics column about what is meant by "Risk Neutral".

Let's begin with my basic premise, i.e. that some (or most) of us do not identify ourselves as being neutral with respect to risk. A group of us get together and agree that we are risk-averse investors. Imagine we then wish to determine the value (price) of some asset today whose future payoffs are contingent upon one or more things occurring (such as a callable bond whose future cashflows depend upon the level of interest rates when the bond's call date is reached, or an option on a stock, etc.). If the present value of that asset depends upon one's level of risk-aversion, this presents a dilemma. Assume I am the most risk-averse investor in the group. In order to satisfy my need for a high expected return, the present value of the asset must be lower for me than for everyone else. But if someone else in the group is slightly less risk-averse than I, that person would be willing to pay more for the same asset. Must we therefore evaluate everyone's level of risk aversion and take some average of the price each of us is willing to pay for the asset in question? Thankfully, no - the notion of risk-neutral pricing rescues us from that fate, allowing us to price options AS IF (underlined and bold) all investors had the same risk preference, such as being risk-neutral.

How can this be? Let's review the Black-Scholes option pricing formula (the equation that revolutionized the field of valuing derivative securities). If we examine this formula, we see that the variables affecting an option's value are: the price or level of the underlying asset (e.g. a stock price, exchange rate, and so on); the strike price of the option (the call or put price), the volatility of the underlying asset, the time to the option's expiration and the risk-free rate of return. What is easy to overlook here (I know, because I overlooked it), is that nowhere in the equation are we asked to specify the expected return on the underlying asset.² This eliminates the dilemma cited above where it seemed that each investor's risk preference would somehow affect the price of a contingent claim. In other words, the value of an (embedded) option is independent of investors' risk preferences.

(Before moving on, let us clarify an important distinction between risk-neutral pricing of options and the pricing of corporate bonds, mortgage-backed securities and other risky assets. When determining the present value of risky cashflows, such as the payments on a corporate bond where credit downgrades or defaults may occur, or for mortgage-backed securities where the timing of principal and interest payments is uncertain, we cannot assume that investors are indifferent to risk. We must adjust the discount rate applied to those cashflows to be greater than the risk-free rate; this is done by layering a spread (OAS) over the Treasury curve. The benefits of risk-neutral pricing apply only to options, because an option's payoff can be fully replicated with no uncertainty by combining other assets in the correct proportions. This was the vital insight of the Black-Scholes model).

Now for the critical line of reasoning: If the price of an option does not depend on risk preferences, we can make the simplifying assumption that all investors have the same tolerance for risk. In other words, we can assume that investors are risk-neutral, risk-averse or risk-seeking and obtain the same result. Of these three, it is most convenient to assume that investors are risk-neutral. Now, being risk-neutral is another way of saying that one does not require any premium for accepting risks (risk-averse means a premium is required, risk-seeking means one is willing to pay a premium to take on risk). If we assume investors are risk-neutral, the expected return on all assets is the risk-free rate; therefore, we can therefore use the risk-free rate of return to discount contingent cashflows to determine their present value. This is an extremely useful result, given that we can actually observe today's risk-free rate. (Of course, we must also model the risk-free rate of return into the future, which is the subject of term structure modeling).

If we prefer to assume that the world is risk-averse, we would increase the expected rate of return on the underlying asset (or interest rate) in question to be greater than the risk-free rate. At the same time, we would increase the discount rate used to value the payoffs from the option. The two effects would cancel each other out, leaving us with the same result as in the "risk-neutral" world. In other words, the assumption of risk-neutrality is a simplifying device that allows us to price options without knowing everyone's actual risk preferences. When we recall that a callable bond's (or prepayable mortgage's) price is a combination of the present value of an option-free series of cashflows minus (or plus, in the case of a put option) the value of the embedded call or prepayment option, we can see how risk-neutral valuation is applicable to pricing options embedded in fixed income securities.

We hope this discussion has been helpful in explaining why valuation models can be based on the notion of a "risk-neutral" world (while still allowing us to have different degrees of risk-aversion, so that you can play the lottery and I can hide my money under the mattress). If you have ideas for other topics to be covered in this Back-to-Basics column, we'd like to hear from you. Please contact marketing at (310) 479-9715 or via email - fia.marketing@interactivedata.com.


¹J. Hull, "Options, Futures and Other Derivative Securities", second edition; Prentice-Hall, 1993.

²Note that the value of the option is a function of the volatility of the underlying asset (see the Interactive Data Fixed Income Analytics Newsletter, August 1996) but not the expectation of an "up" move versus a "down" move.