What discount rate to use when valuing binomial option with real probabilities

Question

We all know that we can use the argument of risk-neutrality and the law of one price, to get the option value without the real world probability.

However, suppose if we use the real world probability and discount the option value with the project's risky rate, are we supposed to get same results as the risk neutral valuation?

Today I came across a question (Krishnan 2006) with the following setup:

There is a project with the following estimated values. \$10 million by the end of the first year, if the things work out well. and \$2.7 million if things do not turn out well. in the latter case, the company can sell the assets for \$3 million. There is a 50 percent chance that the business will succeed. Assets of comparable risk carry a required return of 23 percent. Risk free rate is 5 percent.

If we use the risk neutral valuation, we calculate the PV of the project with the real rates, then calculate the risk neutral probabilities. P_up should equal 0.38. The option value is then 0.62 * 0.3 / 1.05.

However, when I use real rates, the option value becomes 0.5 * 0.3 / 1.23 which is not equal to the risk neutral option value.

I was wondering why is it the case? Shouldn't the law of one price give exactly one answer?

Reference:

V. Sivarama Krishnan, Study Guide for Use with Principles of Corporate Finance

Answer 1

The Pricing equations are derived from duplicating portfolios consisting of underlying and a risk free asset. This means that the price of your option is relative only to the price of the underlying.

In your case: Relative to the project, your option on the project does not command a risk-premium, which is basically the idea of risk-neutral pricing.

Now, if i understood your question correctly, you exchanged the risk-free rate for the 23% expected return of the project, thereby implying a higher risk-free rate of return (in comparison to the 5%) to value your option based on the underlying. Obviously, the present value of the option decreases, as you increase the risk-free rate (see for example the risk-neutral pricing formula for a call as shown below).