Implied Risk Neutral Densities From Option Prices: Hypergeometric, Spline, Lognormal, and Edgeworth Functions

• DOI: http://doi.org/10.1002/fut.21668
• Date: 01 July 2015
• Author: André Santos,João Guerra
• Published date: 01 July 2015

IMPLIED RISK NEUTRAL DENSITIES FROM

OPTION PRICES:HYPERGEOMETRIC,

SPLINE,LOGNORMAL,AND

EDGEWORTH FUNCTIONS

ANDR ´

ESANTOS and JO ˜

AO GUERRA*

This work examines the performance of four different methods to estimate the “true” Risk-

Neutral Density functions (RNDs) using European options. These methods are the Mixture

of Lognormal distributions (MLN), the Smoothed Implied VolatilitySmile (SML), the Density

Functional Based on the Confluent Hypergeometric function (DFCH), and the Edgeworth

expansions (EE). The “true” RND is unknown, so it was generated using the stochastic Heston

model and considering parameters that reflect the characteristics of the options market for

the US dollar and Brazilian real exchange rate (USD/BRL). Wefind that the DFCH and MLN

have the best performance in capturing the “true” RNDs. ©2014 Wiley Periodicals, Inc. Jrl

Fut Mark 35:655–678, 2015

1. INTRODUCTION

It is accepted by market participants that the prices of financial derivatives provide informa-

tion about future expectations of the underlying asset prices, especially forwards, futures,

and options. Forwards and futures only give us the expected value for the underlying as-

set under the assumptions of risk-neutrality, which makes using cross-sections of observed

option prices more attractive because they allow the estimation of an implied probability

density function that indicates the probabilities that market agents attribute to future asset

price movements. The reactions and expectations of the financial markets participants are

very important to risk managers, investors, and policy makers, so having an estimated Risk-

Neutral Density Function (RND) that is robust and consistent with the true one is crucial

and can attenuate the danger of taking wrong decisions.

It is known that the standard methods in options pricing, the Black and Scholes model,

have several limitations because it assumes that the price of the underlying asset evolves

according to the geometric Brownian Motion (GBM) with a constant expected return and a

constant volatility. The volatility is constant until maturity and also across all quoted strikes,

Andr´

e Santos is at BPI, SA, Lisboa, Portugal and Jo˜

ao Guerra is at CEMAPRE and ISEG-Technical Uni-

versity of Lisbon, Lisboa, Portugal. Wewould like to thank the editor and an anonymous referee for helpful

suggestions. Jo˜

ao Guerra was financially supported by FCT—Fundac¸ ˜

ao para a Ciˆ

encia e Tecnologia through

the strategic project PEst-OE/EGE/UI0491/2011 and by the European Union in the FP7-PEOPLE 2012

ITN Programme under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN

STRIKE - Novel Methods in Computational Finance).

JEL Classification: G13, C13, C15

*Correspondence author,ISEG (Mathematics Department), Rua do Quelhas, n. 6, 1200-781 Lisboa, Portugal. Tel:

+351-213925849, Fax: +351-213922781, e-mail: jguerra@iseg.utl.pt

Received April 2013; Accepted February 2014

The Journal of Futures Markets, Vol. 35, No.7, 655–678 (2015)

©2014 Wiley Periodicals, Inc.

Published online 1 April 2014 in Wiley Online Library (wileyonlinelibrary.com).

DOI: 10.1002/fut.21668

656 Santos and Guerra

which ignores phenomena such as volatility smile and as such distorts probabilities for ex-

treme scenarios. In fact, higher volatilities for strike prices deep out-of-the-money make it

more likely that future prices will be very different from current market values. This in turn

increases the probability of these option prices being in-the-money in the future and leads

to more expensive prices for deep out-of-the-money options, when compared to prices cal-

culated through the Black and Scholes model. This results in fatter tails of the true RND

when compared with a lognormal RND.

To tackle these problems, various methods have been suggested to extract Risk-Neutral

Density Functions (RNDs) from option prices and several studies have been carried out to

examine the robustness of these estimates and their information power.

In this work, we compare four methods of extracting RNDs from European type ex-

change rate options for the US dollar and Brazilian real exchange rate (USD/BRL). These

methods are the Mixture of Lognormal distributions (MLN), the Smoothed Implied Volatil-

ity Smile (SML), the Density Functional Based on the Confluent Hypergeometric function

(DFCH), and the Edgeworth expansions (EE). We test the stability of the estimated RNDs

and their robustness as regards small errors by randomly perturbing option prices by half

of the quotation of the tick size as in Bliss and Panigirtzoglou (2002). The “true” RND was

estimated using the method developed in Cooper (1999), who generated pseudo options

prices from Heston’s stochastic volatility model.

In Cooper (1999), the authors compared the MLN model with the SML method in terms

of accuracy and stability using the summary statistics approach and in Bu and Hadri (2007)

the DFCH method was compared with the SML method using the root mean integrated

squared error measure (RMISE).

The analysis presented here extends the analysis of Cooper (1999) and of Bu and Hadri

(2007). We added the DFCH and EE methods to the Cooper analysis and the MLN and EE

methods were added to the Bu and Hadri analysis.

We conclude from our analysis that in the majority of the cases the DFCH and MLN

outperformed the SML and the EE methods in capturing the “true” implied skewness and

kurtosis. However, using the RMISE criterion (which is less sensitive to the tails of the

distribution), the DFCH outperformed the other methods as the best estimator of the “true”

RND.

The remainder of this work is organized into seven sections.

Section 2 describes the five models used in this work (MLN, SML, DFCH, EE, and

Heston).

In Section 3, we present the measures used to evaluate the performance of the four

models tested (MLN, SML, DFCH, and EE) in terms of accuracy and stability.

The results of the Monte Carlo simulation experiments and the comparisons of the

models tested are presented and discussed in Sections 4 and 5. In Section 4, we analyze

the accuracy and stability performance using the “true” RNDs generated by the Heston

parameters proposed in Cooper (1999). In Section 5, the Heston parameters were calibrated

taking into account the observed quotes for the USD/BRL European options between June

2006 and February 2010.

In Section 6, we analyze the historical RND summary statistics obtained using the

MLN, SML, DFCH, and EE methods for the USD/BRL in the time period described above.

Finally, in Section 7 some concluding remarks are presented.

2. OPTION PRICING AND EXTRACTION OF RND

In this Section, we give an overview of the different methods used to estimate the RND.