Question

1. Explain the principle of risk-neutral valuation.

2.. Explain the meaning of mean reversion.

All assignments must include at least one full page in APA format with at least two outside references.

Answer 1

Explain the principle of risk-neutral valuation: Risk neutral valuation refers that you can value options in terms of their expected payoffs, discounted from expiration to the present, assuming that they grow on average at the risk-free rate.

Option value = Expected present value of payoff (under a risk-neutral random walk).

• Principle of risk neutral valuation exploits the perfect correlation between changes in the value of the stocks and options. Because of this if an options value rises because of the rise in the value of the stock, then a long option and a short option shouldn't have any random fluctuations, therefore the stock hedges the option. Thus the resulting portfolio is risk-free.
• In that case you need to know the correct number of stocks to sell short, that's called the "delta" which usually comes from a model.
• Delta is constantly changing so you must always be buying or selling stock to maintain a risk-free position. Obviously, this is not possible in practice. Second, it hinges on the accuracy of the model. The underlying has to be consistent with certain assumptions, such as being Brownian motion without any jumps, and with known volatility.

• Here are some further explanations of risk-neutral pricing:

Explanation 1: If you hedge using the Black-Scholes world then all risk is eliminated. If there is no risk then we should not expect any compensation for risk. We can therefore work under a measure in which everything grows at the risk-free interest rate.

Explanation 2: If the model for the asset is dS = ixSdt + oSdX then the ixs cancel in the derivation of the Black-Scholes equation.

Explanation 3: Two measures are equivalent if they have the same sets of zero probability. Because zero probability sets don't change, a portfolio is an arbitrage under one measure if and only if it is one under all equivalent measures. Therefore a price is non-arbitrageable in the real world if and only if it is non-arbitrageable in the risk-neutral world. The risk-neutral price is always non-arbitrageable.

Explain the meaning of mean reversion: Mean reversion holds that asset prices and historical returns eventually will revert to the long run mean or average of the entire dataset. This theory has evolved as many investing strategies that involves purchasesn and sale of shares and other securities whose recent performances have differed greatly from their historical means. However, a change in returns also could be a sign that a company no longer has the same prospects it once did, in which case it is less likely that mean reversion would occur.

Because of its quantitave applications mean reversion theory is used as a part of the market statistical analysis and it can also used as a part of overall trading strategy.

1. Outside example: Let's assume that on an average day Microsoft moves $1.00 either up or down. If one day Microsoft's share price increases $7.00 and there was no significant news or announcement, than a mean revisionist would most likely believe that the stock would decrease the next day.
2. From the year 2000 until 2009, emerging market stocks were up by over 160% or more than 10% per year. During that same time frame, the S&P 500 was actually down by a total of almost -10% or nearly -1% per year.

Emerging markets have been one of the few bright spots for economic and stock market growth in the past decade.