Subjective Probability Density Functions from FX Option Prices: Predictive Power and Performance on a Carry Trade Strategy

• Date: 01 June 2018
• Published date: 01 June 2018
• Author: Tiago Neves,André Santos,João Guerra
• DOI: http://doi.org/10.1111/irfi.12146

Subjective Probability Density

Functions from FX Option Prices:

Predictive Power and Performance

on a Carry Trade Strategy*

ANDRÉ SANTOS

†

,JOÃO GUERRA

‡

AND TIAGO NEVES

§

†

Quantitative Advisory, EY, Lisbon, Portugal,

‡

Departamento de Matemática and CEMAPRE, ISEG, Universidade de Lisboa, Lisbon,

Portugal and

§

Enterprise Risk Services, EY, London, UK

ABSTRACT

In this article, we extracted the risk-neutral densities (RNDs) and subjective

probability density functions of the US Dollar/Brazilian Real (USD/BRL) ex-

change rate and evaluated its performance in predicting the future realizations

of the USD/BRL exchange rate. The RNDs were estimated using two structural

models and three nonstructural models. In the first category, we included the

Variance Gamma-OU model and the CGMY Gamma-OU model. In the second

category, we included the density functional based on confluent

hypergeometric function model, the mixture of lognormal distributions

model, and the smoothed implied volatility smile. The density functional

based on confluent hypergeometric function and the CGMY Gamma-OU pro-

duced 1-month term densities (RND and subjective probability density func-

tion) with the highest forecasting power of the 1-month USD/BRL exchange

rate. Finally, we applied the CGMY Gamma-OU model to extract a sample of

subjective cumulative probabilities of 1-month USD/BRL movements, and

used them as explanatory variables in predictive time series models, whose de-

pendent variable was the 1-month carry trade return. Its predictive power was

then tested and confirmed in three trading strategies that over performed the

standard carry trade strategy in terms of annualized cumulative returns.

JEL Codes: G13; C13; C15; F31

Accepted: 22 June 2017

An option is a contingent claim whose payoff depends on the future state of the

underlying asset, and as result, option prices are influenced by the expectations

about the future underlying asset prices. For a particular asset and maturity, the

* We would like to thank the editor and an anonymous referee for valuable comments that improved

the paper. The opinions expressed in this paper are those of the authors and are not necessarily en-

dorsed by the authors’employers. João Guerra was partially supported by the Project CEMAPRE –

UID/MULTI/00491/2013 financed by FCT/MCTES through national funds.

© 2017 International Review of Finance Ltd. 2017 1

International Review of Finance, 2017

DOI: 10.1111/irfi.12146

International Review of Finance, 18:2, 2018: pp. 253–286

DOI:10.1111/irfi .12146

© 2017 International Review of Finance Ltd. 2017

availability of option prices for various exercise prices allows the estimation of

risk-neutral densities (RNDs).

In financial markets, there is only a range of discrete option prices; there-

fore, it is necessary to estimate the RNDs through a smoothing function. The

RND will vary depending on the model applied on its estimation, and so, the

choice of a reliable model is very important. The use of a standard option

pricing model such as Black and Scholes is not recommended because of

its limitations. Black and Scholes assumes that the asset price is modeled

by a geometric Brownian motion with a constant expected return and vola-

tility. The constant volatility assumption contradicts the empirical evidence

where asset prices exhibit different implied volatilities across maturities and

strikes; therefore, a realistic model shall assume stochastic volatility through

time.

For the period between June 2006 and September 2013, we estimated RNDs

using option prices with the US Dollar/Brazilian Real (USD/BRL) exchange rate

as underlying asset. The RNDs were generated using two categories of models:

structural and nonstructural. Structural models assume specific dynamics for

the price or volatility process. Nonstructural models allow the estimation of an

RND without describing any stochastic process for the price or volatility of the

underlying asset.

In the latter, we used two parametric models, the density functional based on

confluent hypergeometric function (DFCH) and the mixture of lognormal distri-

butions (MLN), and one nonparametric model, the smoothed implied volatility

smile (SML). In the former, we applied models based on Lévy processes: the

Variance Gamma-OU (VG Gamma-OU) and the CGMY Gamma-OU. The Lévy

models chosen are rich and sophisticated stochastic volatility models that gener-

ate jumps for asset returns and volatility. Consequently, they are able to generate

RNDs with statistical properties similar to the ones observed empirically, that is,

with leptokurtic distributions and fat tails.

The option prices incorporate investors’risk preferences; hence, the RND

is not a direct measure of the subjective probabilities that investors attribute

to future underlying asset prices. The subjective probability density function

(SPDF) can be extracted from the RND as long as we know investors’risk

preferences, and based on that, Bliss and Panigirtzoglou (2004) suggested

the estimation of the risk aversion assuming that if investors are rational,

their SPDF is equal to the distribution of observed realizations, called objec-

tive distribution.

Consequently, we expect that the SPDF provides accurate forecasts, with the

difference between RND and objective distribution arising from the rational

investor risk aversion. Various authors estimated the SPDF assuming stationary

over long periods (Ait-Sahalia and Lo 2000; Jackwerth 2000). We believe that

Bliss and Panigirtzoglou (2004) approach is more suitable to extract the SPDF,

because it estimates probabilities that are more reactive to market changes;

therefore, we decided to use Bliss and Panigirtzoglou method to convert RNDs

into SPDFs.

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254 © 2017 International Review of Finance Ltd. 2017

After testing the predictive power of the 1-month SPDFs, we had the ambition

to understand the practical utility of it. With the goal of predicting the direction

of the 1-month carry trade return on the USD/BRL exchange rate, we extracted

sample points of the SPDFs (subjective cumulative probabilities of certain move-

ments of the exchange rate USD/BRL) and use them as explanatory variables in

time series models. Based on the predictions of the carry trade return, we simu-

lated a few trading strategies using futures on the BRL/USD (the inverse of the

pair USD/BRL means price of Brazilian Real in terms of dollars –hereafter called

BRL) and compared its profitability against a benchmark carry trade strategy

where we were always long on a BRL future. All trading strategies based on the

SPDFs exhibited much higher annualized and cumulative returns than the stan-

dard carry trade strategy, hence showing the good predictive power of SPDFs.

Our work goes beyond previous studies as it includes not only the analysis of

the market sentiment but also the forecasting power of the RNDs and SPDFs.

Moreover, it shows how to use the information embedded in the SPDFs to enter

in profitable trading strategies.

This paper is divided into six sections. Section i describes the nonstructural

and structural models used to estimate the RNDs, and Section ii defines the meth-

odology used to transform the RNDs into SPDFs. The data used in this work are

presented in Section iii. In Section iv, we assess the predictive power of the

RND and SPDF through the comparison between the estimated densities and

the realized outcomes of the USD/BRL exchange rate. The profitability of trading

strategies based on these densities is analyzed in Section v, and finally, the con-

clusions are presented in Section vi.

To help the readers understand the economic environment between June

2006 and September 2013, in Appendix A, we analyzed the probabilities pro-

vided by the RNDs and SPDFs and described the main market and political events

that occurred in that time.

I. RISK-NEUTRAL DENSITY ESTIMATION

A. Option prices and the risk-neutral density

The call option value is given by the discounted value of its expected payoff at

the expiration date T.

CX;TðÞ¼erT∫∞

XfS

TXðÞdS T(1)

where Xis the exercise price, S

T

is the price of the underlying asset at T, and ris the

risk-free interest rate. The expectation is taken under the risk-neutral measure.

Breeden and Litzenberger (1978) deduced the relationship between the option

prices and the RND by taking the second derivative of equation (1) with respect

to X, that is,

d2CX;TðÞ

dX 2¼erTfS

T

ðÞ:(2)

Predictive Power of FX Option Prices

© 2017 International Review of Finance Ltd. 2017 3

Predictive Power of FX Option Prices

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