Risk – Options, Futures, and Other Derivatives

The notion of “arbitrage-free” pricing is important not only in Finance 4366 but also in your other finance studies. In Finance 4366, we take it as given that investors are risk averse. However, it turns out that we don’t need to invoke the assumption of risk aversion to price risky securities such as options, futures, and other derivatives; all we need to assume is that investors are “greedy” in the sense that they prefer more return to less return, other things equal. Through a variety of trading strategies, we can synthetically replicate any security we want and do so in such a way that we (in theory anyway) can take no risk, incur zero net cost of investment, and yet earn positive returns.

Without question, the notion of arbitrage-free pricing is THE key concept in Finance 4366. However, it is also important in corporate finance. For example, the famous Modigliani-Miller Capital Structure Theorem; i.e., that the value of a firm’s shares is unaffected by how that firm is financed, is based upon this principle. For your personal enjoyment and intellectual edification, I attach a copy of a short (3-page) teaching note that provides a “no-arbitrage” proof based on a simple numerical example:

The Modigliani-Miller Capital Structure Theorem – A “No-Arbitrage” Proof

Besides reviewing the course syllabus during the first day of class on Tuesday, January 16, we will also discuss a particularly important “real world” example of financial risk. Specifically, we will study the relationship between realized daily stock market returns (as measured by daily percentage changes in the SP500 stock market index) and changes in forward-looking investor expectations of stock market volatility (as indicated by daily percentage changes in the CBOE Volatility Index (VIX)):
As indicated by this graph (which also appears in the lecture note for the first day of class), daily percentage changes in closing prices for the SP500 (the y-axis variable) and the VIX (the x-axis variable) are strongly negatively correlated with each other. The blue dots are based on 8,574 contemporaneous observations of daily returns for both variables, spanning 34 years starting on January 2, 1990, and ending on January 12, 2024. When we fit a regression line through this scatter diagram, we obtain the following equation:

${R_{SP500}} = .00062 - .1147{R_{VIX}}$ ,

where ${R_{SP500}}$ corresponds to the daily return on the SP500 index and ${R_{VIX}}$ corresponds to the daily return on the VIX index. The slope of this line (-0.1147) indicates that on average, daily closing SP500 returns are inversely related to daily closing VIX returns. Furthermore, nearly half of the variation in the stock market return during this period (specifically, 48.87%) can be statistically “explained” by changes in volatility, and the correlation between ${R_{SP500}}$ and ${R_{VIX}}$ during this period is -0.70. While a correlation of -0.70 does not imply that daily closing values for ${R_{SP500}}$ and ${R_{VIX}}$ always move in opposite directions, it does suggest that this will be the case more often than not. Indeed, closing daily values recorded for ${R_{SP500}}$ and ${R_{VIX}}$ during this period moved inversely 78% of the time.

You can also see how the relationship between the SP500 and VIX evolves prospectively by entering http://finance.yahoo.com/quotes/^GSPC,^VIX into your web browser’s address field.

Category: Risk

On the importance of “arbitrage-free” pricing in finance

On the relationship between the S&P 500 and VIX