The Probability Density Function of Interest Rates Implied in the Price of Options

44 Pages Posted: 4 May 2006

See all articles by Fabio Fornari

Fabio Fornari

European Central Bank (ECB)

Roberto Violi

Bank of Italy

Date Written: 1998


The paper contributes to the stochastic volatility literature by developing simulation schemes for the conditional distributions of the price of long term bonds and their variability based on non-standard distributional assumptions and volatility concepts; itillustrates the potential value of the information contained in the prices of options on long and short term lira interest rate futures for the conduct of monetary policy in Italy, at times when significant regime shifts have occurred. Risk-neutral probability density functions (PDFs) of interest rates are estimated for several episodes since 1992, using an extended version of the model developed by Söderlind and Svensson (1997), in which the true PDF of an asset price is approximated by a parametric mixture of gaussian distributions with displaced means and mean-reverting deterministic volatilities. This approach generalizes the familiar Black and Scholes option pricing model by letting the returns of the underlying asset follow a mean-reverting stochastic process governed by the existence of two regimes. The parameters of this functional form are obtained by minimising the squared deviations between predicted and true option prices. The ability of the model to fit observed option prices seems encouraging; a significant degree of skewness (so-called risk-reversal), large changes over time and fatness of the tails (which gives rise to the volatility smile effect) are the elements which characterize the estimated PDFs). The analysis of the information conveyed by PDFs of future interest rates begins in the wake of the 1992 EMS crisis, when a large negative skewness emerged in the distribution of lira-denominated bond futures, suggesting that the market factored in the possibility of a sharp decline in prices, triggered by a monetary tightening aimed at defending the exchange rate peg and by the mounting inflationary risk induced by a large devaluation. We subsequently examine a recent sequence of monetary easings, which provide an interesting variety of market participants' reactions. The official rate reduction of July 1996, which followed a sequence of monetary tightenings in 1994-5, appears to have been discounted by the market, since the estimated PDFs of bond and 3-month T-bill futures prices did not change and an almost gaussian shape was maintained. The subsequent reductions - October 1996 and January 1997 - showed instead a different kind of market reaction, especially for long rate PDFs which returned from negative skewness and fairly high kurtosis, observed before the policy move, to normality, with narrower interquartile range and less kurtosis after the interest rate reduction.

Keywords: Option Pricing, risk-neutral probability density function, interest rate, risk reversal, monetary policy

JEL Classification: E44, E58, G12, G13

Suggested Citation

Fornari, Fabio and Violi, Roberto, The Probability Density Function of Interest Rates Implied in the Price of Options (1998). Bank of Italy Economic Research Paper No. 339, Available at SSRN: or

Fabio Fornari

European Central Bank (ECB) ( email )

Sonnemannstrasse 22
Frankfurt am Main, 60314

Roberto Violi (Contact Author)

Bank of Italy ( email )

91 via Nazionale
Rome, rome 00184
+390647924108 (Phone)

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