Risk Neutrality and Arbitrage-Free Pricing

The binomial model works on risk neutrality. In fact, the __risk-neutral probability__ is found from using the weighted sum of expected values of the underlying set equal to the current option price: $$\displaystyle \pi = \frac{1 + r - R^d}{R^u - R^d} $$ Do you think everyone is risk neutral?

Right.

No, not everyone.

Think about a risk-neutral person. Which payoff do you think he or she will select?

Right, a risk-neutral person will ignore risk and not care about a risk-adjusted return. He or she will simply select the investment with the highest expected return.

No, a risk-neutral person will ignore risk and not care about a risk-adjusted return. He or she will simply select the investment with the highest expected return.

The hypothetical portfolio constructed from the underlying and the derivative with a known future value is called a __hedge portfolio__. As the value of both the portfolio in the future and of the underlying today are known, the value of the derivative today must be the value that gives the hedge portfolio a return equal to the risk-free rate. What do you think happens if the hedge portfolio pays a return greater than the risk-free rate?

Correct.

Incorrect.

To summarize this lesson: [[summary]]

The fact that most people are risk-averse results in the need for riskier investments to offer a higher expected return to compensate for greater risk. However, in derivative pricing, the risk aversion level of investors doesn't matter. Based on the law of one price, the value of a derivative is the value which causes the hedge portfolio to have a return equal to the risk-free rate. This rule is fulfilled whether the investor is risk-averse or risk-neutral. For this reason, derivative pricing is sometimes referred to as __risk-neutral pricing__. The overall process of using the law of one price and risk neutrality is called __arbitrage-free pricing__.

$$\text{Asset} + \text{Derivative} = \text{Risk-free asset}$$ If the hedge portfolio yields a return greater than the risk-free rate, the price of the derivative is too low and investors will buy it until equilibrium is restored.

In fact, most people are risk averse, preferring not to pay expected values for the flip of a coin, for instance. Some are even risk-seeking, but that's rare.

Yes

No

Expected return of USD 110 with a standard deviation of 20%

Expected return of USD 100 with a standard deviation of 5%

Investors take a long position in the derivative

Investors take a short position in the derivative

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