Asset Values Derived from Options

Risk-Neutral Probability Density Functions

Updated with data through May 3, 2012

• New developments
• Current asset reports
• About this report
• How to interpret the risk-neutral probability density functions
• Archive of past exhibits

New Developments

Commodity Markets
Trading was above median again in the exchange rates. Options on the real estate index also traded at the high end of their history, and options on S&P 500 and silver futures were average.

Market Notes:

Additional charts:

• Selected RNPD skews:

Large chart

• DJ Real Estate Index:

Large chart

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Current asset reports

This report contains exhibits which show the “6-month” risk-neutral probability density functions for the following assets:

The density functions presented in each exhibit are based on options that expire roughly six months from each pricing observation date. As such, they contain information about the distribution of future values of each asset class based on the expectations of market participants measured at two distinct points in time (the current date and one month ago).

Our reports continue to evolve in response to suggestions by readers and input from researchers and practitioners. We currently are providing the following information:

• Volatility as a function of strike price has been moved from the appendix to the main report for most markets
• Volumes associated with trading activity are included in the vol-strike graphs
• Risk-neutral PDFs for the exchange rate markets are now presented in log returns
• Time series charts of 20% return changes have been added for most markets

The volatility-strike price relationships are presented in the top panel of each markets report. We believe that interpretation of changing market dynamics can be facilitated by viewing these underlying data in combination with the RNPDs themselves.

To gauge the depth of the market that produced the prices used in estimating the RNPDs, we overlay the data presented in the volatility-strike price chart with associated volumes. At-the-money trading will tend to dominate the chart in every period so we show only volumes for out-of-the-money trading. The volumes are aggregated over the five-day period used to obtain the prices (see description of methodology here [PDF]).

We also present time series, risk-neutral probabilities of 20% price changes to get a longer view of changing market preferences. Looking at upside and downside probabilities together provides another view of the amount of skewness embedded in the RNPDs.

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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.

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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.

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Archive of past exhibits

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