Risk-Neutral Probability Density Functions

Updated with data through March 20, 2014

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March 20, 2014 [PDF]


The S&P 500 was down -30 basis points over the past two weeks. Options trading on the S&P 500 index was strong. The average CCAR bank price rose 260 basis points and the average insurance company price fell -70 basis points. Equity market related RNPD standard deviations (tail risks) generally rose and RNPD skews ticked lower (more negative). These measures do NOT incorporate stress test related information. We intend to follow up with separate analysis of RNPD CCAR reactions.

This time period DOES include market reaction to recent FOMC decisions and guidance. Our standard reports of RNPDs derived from inflation caps and floors show only small changes in the tails. In particular, the reports show the risk neutral probability of deflation changing very little.

We also looked more carefully at risk neutral probabilities in the middle of the distribution. Specifically, we examined changes in risk neutral probabilities at 1%, 2%, and 3% outcomes. The chart below presents the time series of risk neutral probabilities for these outcomes over the subsequent 2 years.

Chart 1

Large chart

We point out the interesting changes from March 17th to March 19th. The risk neutral probability of a 1% inflation rate over the next two years rose 345 basis points while the probabilities of 2% and 3% inflation fell 120 and 277 basis points respectively. For more information on the methodology for making these estimations see Kitsul and Wright.1

Banks & Insurance Companies
Options trading was also relatively brisk for the insurance companies we follow. Volumes were about average in the broader CCAR universe of banks. RNPD standard deviations and skews were mixed.

Banks with above average and increasing trading volume were BK, KEY, and FITB. Insurance companies with above average and increasing trading volume were HIG, LNC, and MET.

Additional Notes:

  • Shares of KEY rallied 5.3%. The RNPD skew derived from options on KEY share became less negative. (See KEY report.)
  • The behavior of FITB shares and its RNPD was similar to that of KEY. The share price rose 4.8% while its RNPD standard deviation and skew fell. This generally the short term relationship we observe between RNPD skew and price. (See FITB report.)

Chart 2

Large chart

  • Despite the volume, RNPDs were relatively unchanged for the insurance firms listed above. (See detail reports.)
  • AMP and PFG skew changes were large but occurred on very light trading. (See AMP and PFG reports.)

Other Commodity Markets
With the exception of wheat, spot prices in the other commodity markets we follow generally fell over the past two weeks. Corn fell -1.8%, gold fell -1.6%, WTI crude fell -2.0%, and the real estate ETF fell -2.1%. The dollar was stronger. Options trading volumes were light. Consistent with the direction of spot prices, RNPD standard deviations rose slightly.

Additional notes:

  • Trading in options on wheat futures was active. The RNPD for wheat shifted noticeably from two weeks ago. Wheat prices rose 8.4% and tail risks in as measured by RNPD standard deviation increased by 240 basis points. RNPD skews for all of the grains remain positive. (See corn, wheat, and soybean reports.)

Chart 3

Large chart

  • The RNPD standard deviation derived from options on cattle futures continues to increase. The RNPD skew is negative. (See cattle report.)
  • The RNPD skews derived from options on soybeans, gold, and WTI crude, are at the highest levels we have measured in the past 20 two week periods. (See soybeans, gold and oil reports.)


1 Kitsul, Yuriy and Jonathan H. Wright (2012): The Economics of Option-implied Inflation Probability Density Functions, NBER Working Paper 18195. We note that K-W have updated their methodology in the more recent version of the paper appearing the Journal of Financial Economics (2013).


Please send comments and suggestions to: option-report-feedback@mpls.frb.org.

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About this report

Large changes in asset values or commodities prices can be associated with significant changes in economic output and price and financial stability. For that reason, policymakers seek to monitor market concerns about extreme movements—so-called tail events—in key asset and commodities markets. The prices of options to buy (“call options”) or sell (“put options”) assets or commodities in the future at specified prices (“strike prices”) provide valuable information about potential tail events. The prices of call and put options with differing strike prices can be used to estimate a probability density function of the payoff of the underlying asset or commodity.

In this report, we use options prices to produce what are known as “risk-neutral probability density functions” (RNPDs) for future asset and commodity values. Interpreting RNPDs can be subtle, because they combine both investors’ expectations of the future price of the traded asset or commodity and the compensation they received for taking on related aggregate economic risks. The changes over time in RNPDs that we show reflect changes to both of these components, and decomposing the change into the constituent parts is not possible without making additional strong assumptions. Nonetheless, we regard the RNPDs shown here as valuable barometers of financial market sensitivity to possible extreme movements in the prices of key assets and commodities.

We welcome feedback on all aspects of the report and material on this web page, which we consider a “work in progress.” Please send comments and suggestions to: option-report-feedback@mpls.frb.org.


How to Interpret the Risk-Neutral Probability
Density Functions

The exhibits contained in this report show density functions at three different points in time, allowing viewers to see how risk-neutral expectations of market participants have evolved over time. Specifically, we think it isuseful to examine the following characteristics within each exhibit and note how they have changed over time:

  1. Central tendency—Has the center of the density function shifted toward higher or lower values?
  2. Dispersion—Is the majority of the density function more or less tightly concentrated around its midpoint? One possible reason for a density function that is becoming less concentrated (and is more widely spread out) would be that investors are more uncertain about the future value of the asset.
  3. Skewness—Is the density function largely symmetric around its midpoint, or is it skewed to the right or the left? The latter would suggest that investors are more concerned about changes in one direction than the other.
  4. Tail thickness—Do the density functions have roughly the same amount of mass in their tails? If the weight present in one or both tails suddenly increases, it could indicate that the probability of a large change has increased.
Alternatively, the features described above could be changing due to shifts in the compensation investors expect for bearing aggregate risks affecting the value of assets or commodities in the future.


Methodology for Estimating Risk Neutral Probability Density Functions

How the Minneapolis Fed estimates Risk Neutral Probability Density Functions [PDF]