Category Archives: Finance

Are markets efficient? Let the two Nobel Economics Laureates debate!

Two Nobel laureates in economics from the University of Chicago, Eugene Fama (2013) and Richard Thaler (2017) debate the efficient market hypothesis. This debate is required viewing for anyone with even a remote interest in finance! (spoiler alert – virtually all derivatives pricing models covered in Finance 4366 assume that the underlying asset follows a random walk, which corresponds to the so-called “weak form” of Fama’s efficient market hypothesis)…

Eugene F. Fama and Richard H. Thaler discuss whether markets are prone to bubbles.


Important empirical evidence on hedge fund performance…

An ongoing debate in finance is whether “active” investment strategies can outperform “passive” strategies. The empirical evidence in favor of passive strategies which appears in studies published by peer-reviewed scientific journals is overwhelming. For example, in studies of mutual fund performance, passive strategies almost always blow away active strategies. Similarly, the empirical evidence on frequency of trading by “retail” customers is that on average, portfolio performance is inversely related to trading frequency; i.e., the more people trade, the worse they do. Even hedge funds chronically underperform passive investment strategies. For example, the authors of a 2011 Journal of Financial Economics (JFE) article entitled “Higher risk, lower returns: What hedge fund investors really earn” find that hedge fund returns are on the magnitude of 3% to 7% lower than corresponding buy-and-hold fund returns, reliably lower than the return on the Standard & Poor’s (S&P) 500 index, and only marginally higher than the riskless rate of interest.

On the origins of the binomial option pricing model

In my previous posting entitled “Historical context for the Black-Scholes-Merton option pricing model,” I provide links to the papers in which Black-Scholes and Merton presented the so-called “continuous time” version of the option pricing formula. Both of these papers were published in 1973 and eventually won their authors (with the exception of Fischer Black) Nobel prizes in 1997 (Black was not cited because he passed away in 1995 and Nobel prizes cannot be awarded posthumously).

Six years after the Black-Scholes and Merton papers were published, Cox, Ross, and Rubinstein (CRR) published a paper entitled “Option Pricing: A Simplified Approach”. This paper is historically significant because it presents (as per its title) a much simpler method for pricing options which contains (as a special limiting case) the Black-Scholes-Merton formula. The reason why we began our analysis of options by first studying CRR’s binomial model is because pedagogically, this makes the economics of option pricing much easier to comprehend. Furthermore, such an approach removes much (if not most of) the mystery and complexity of Black-Scholes-Merton and also makes that model much easier to comprehend.

Historical context for the Black-Scholes-Merton option pricing model

Although we won’t get into the “gory” details on the famous Black-Scholes-Merton option pricing model until sometime later this semester when we cover Hull’s chapter entitled “The Black-Scholes-Merton Model” and my teaching note entitled “Derivation and Comparative Statics of the Black-Scholes Call and Put Option Pricing Equations“, I’d like to call your attention to the fact that the original papers by Black-Scholes and Merton are available on the web:

The Black-Scholes paper originally appeared in the Journal of Political Economy (Vol. 81, No. 3 (May – Jun., 1973), pp. 637-654). The Merton paper appeared at around the same time in The Bell Journal of Economics and Management Science (now called The Rand Journal). Coincidentally, the publication dates for these articles on pricing options roughly coincide with the founding of the Chicago Board Options Exchange, which was the first marketplace established for the purpose of trading listed options.

Apparently neither Black and Scholes nor Merton ever gave serious consideration to publishing their famous option pricing articles in a finance journal, instead choosing two top economics journals; specifically, the Journal of Political Economy and The Bell Journal of Economics and Management Science. Mehrling (2005) notes that Black and Scholes:

“… could have tried finance journals, but the kind of finance they were doing was outside the rubric of finance as it was then organized. There was a reason for the economist’s low opinion of finance, and that reason was the low analytical level of most of the work being done in the field. Finance was at that time substantially a descriptive field, involved mainly with recording the range of real-world practice and summarizing it in rules of thumb rather than analytical principles and models.”

Another interesting anecdote about Black-Scholes is the difficulty that they experienced in getting their paper published in the first place. Devlin (1997) notes: “So revolutionary was the very idea that you could use mathematics to price derivatives that initially Black and Scholes had difficulty publishing their work. When they first tried in 1970, Chicago University’s Journal of Political Economy and Harvard’s Review of Economics and Statistics both rejected the paper without even bothering to have it refereed. It was only in 1973, after some influential members of the Chicago faculty put pressure on the journal editors, that the Journal of Political Economy published the paper.”


Devlin, K., 1997, “A Nobel Formula”.

Mehrling, P., 2005, Fischer Black and the Revolutionary Idea of Finance (Hoboken, NJ: John Wiley & Sons, Inc.).

Delta hedging, replicating portfolio, and risk neutral valuation approaches to pricing options

The delta hedging, replicating portfolio and risk neutral valuation perspectives for pricing options are important in that they enable us to think carefully as well as deeply concerning the economics of option pricing. For example, the delta hedging approach illustrates that an appropriately hedged portfolio consisting of either long-short call-share positions or long-long put-share positions is riskless and consequently must produce a riskless rate of return. Similarly, the replicating portfolio approach reminds us that a call option represents a “synthetic” margined stock investment, whereas a put option represents a “synthetic” short sale of the share combined with lending money. As we saw in our study earlier this semester of forward contracts, if there is any difference between the value of a derivative and its replicating portfolio, then one can earn profits with zero net investment and no exposure to risk. Thus, the “arbitrage-free” price for the derivative (option or forward) corresponds to the value of its replicating portfolio. The arbitrage-free pricing principle further implies that a risk-neutral valuation relationship exists between the derivative and its underlying asset, which in turn enables us to calculate risk neutral probabilities. Once we know what the risk-neutral probabilities are for up and down price movements, we can price options by discounting the risk-neutral expected value of the option payoff (i.e., its certainty-equivalent) at the riskless rate of interest.

During our next class meeting tomorrow, we will spend a bit more time on the risk neutral valuation approach, as well as extend our analysis from a single timestep to a multiple timestep setting. To do this, we need to introduce an important concept called backward induction. Backward induction involves beginning at the very end of the binomial tree and working our way back to the beginning. We’ll continue with the numerical example introduced during class today and illustrate the backward induction method for the delta hedging, replicating portfolio and risk neutral valuation approaches. We’ll discover that multiple timesteps imply that the delta hedging and replicating portfolio methods imply “dynamic” trading strategies which require portfolio rebalancing as the price of the underlying asset changes over time.

Era of Calm Ends as Volatility Returns to Markets

With the return of volatility in stocks, those investors and trades that profit when markets are calm are suffering heavy losses.
The above referenced WSJ article (published yesterday) tells a very interesting story about volatility as an asset class. VIX exchange-traded products (such as Credit Suisse’s now infamous and soon-to-become-defunct) VelocityShares Daily Inverse VIX Short-Term exchange-traded note (XIV)) were originally conceived of in the aftermath of the global financial crisis as a form of insurance against against increases in market volatility.
As we have previously discussed (see “On the relationship between the S&P 500 and the CBOE Volatility Index (VIX)“), returns on the S&P 500 stock market index and VIX tend to be strongly negatively correlated with each other.  Thus, VIX exchange-traded products such as XIV offer investors the opportunity to hedge against increases in volatility.  Indeed, by reversing the letters in the VIX ticker symbol, the VelocityShares Daily Inverse VIX Short-Term exchange-traded note in particular effectively branded itself as a financial product which hedges volatility.  However, as market volatility subsided during recent months and years, many investors began to sell rather than buy products such as XIV in hopes of boosting portfolio returns.  With stocks trading at historically low volatility levels lately, this strategy seemed to be working pretty well for many investors; that is, until this past week when volatility made its comeback:
The next graph shows the time series for daily closing prices on XIV and on VIX, from 11/30/2010 (which is the first day for which daily data for XIV are available) through yesterday (2/6/2018):
 Within this date range, the correlation between XIV and VIX is -.5608.  Of course, the most interesting aspect of this graph corresponds to the enormous drop in XIV from its all-time closing high of 144.75 (on January 12, 2018) to 7.35 at the close yesterday.  On the same day that XIV reached its all-time closing high,  VIX closed at 10.16, but stood at 37.32 at the close on Monday, February 5.

A Random Walk Down Wall Street

In my opinion, if you were to read only one book about finance, it would have to be “A Random Walk Down Wall Street: The Time-Tested Strategy for Successful Investing” by Burton G. Malkiel. Malkiel’s book (now in its 11th edition) provides a compelling argument in favor of efficient markets theory and investing in (passively managed) index funds.

Efficient market theory implies that stock prices follow a random walk. These ideas were originally conceived of by Professors Paul Samuelson and Eugene Fama in the 1960’s, and subsequently popularized by folks like Professor Malkiel. In Finance 4366, we rely extensively upon the notion that prices of speculative assets (e.g., stocks, bonds, commodities, foreign exchange, etc.) follow random walks as we consider the technical details associated with pricing and hedging risk using financial derivatives.

The Index Fund featured as one of “50 Things That Made the Modern Economy”

Tim Harford also features the index fund in his “Fifty Things That Made the Modern Economy” radio and podcast series. This 9 minute long podcast lays out the history of the development of the index fund in particular and the evolution of so-called of passive portfolio strategies in general. Much of the content of this podcast is sourced from Vanguard founder Jack Bogle’s September 2011 WSJ article entitled “How the Index Fund Was Born” (available at Here’s the description of this podcast:

“Warren Buffett is the world’s most successful investor. In a letter he wrote to his wife, advising her how to invest after he dies, he offers some clear advice: put almost everything into “a very low-cost S&P 500 index fund”. Index funds passively track the market as a whole by buying a little of everything, rather than trying to beat the market with clever stock picks – the kind of clever stock picks that Warren Buffett himself has been making for more than half a century. Index funds now seem completely natural. But as recently as 1976 they didn’t exist.  And, as Tim Harford explains, they have become very important indeed – and not only to Mrs Buffett.”

Warren Buffett is one of the world’s great investors. His advice? Invest in an index fund