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José Renato Haas Ornelas
Aquiles Rocha de Farias
Building Risk-Neutral Densities (RND) from options data can provide
variable. This paper uses the Liu et all (2007) approach to estimate the
option-implied risk-neutral densities from the Brazilian Real/US Dollar
exchange rate distribution. We then compare the RND with actual exchange
rates, on a monthly basis, in order to estimate the relative risk-aversion of
investors and also obtain a real-world density for the exchange rate. We are
the first to calculate relative risk-aversion and the option-implied real world
Density for an emerging market currency. Our empirical application uses a
sample of exchange-traded Brazilian Real currency options from 1999 to
2011. The RND is estimated using a Mixture of Two Log-Normals
distribution and then the real-world density is obtained by means of the Liu
et al. (2007) parametric risk-transformations. Our estimated value of the
relative risk aversion parameter is around 2.7, which is in line with other
articles that have estimated this parameter for the Brazilian Economy. Our
out-of-sample evaluation results showed that the RND has some ability to
forecast the Brazilian Real exchange rate. However, when we incorporate
the risk aversion into RND in order to obtain a Real-world density, the out-
of-sample performance improves substantially. Therefore, we would suggest
not using the “pure” RND, but rather taking into account risk aversion in
order to forecast the Brazilian Real exchange rate.
Banco Central do Brasil, Gerência-Executiva de Riscos Corporativos e Referências Operacionais; Ibmec
Fundação Getúlio Vargas, Rio de Janeiro, EBAPE. E-mail:
Extracting market expectations is one of the most important tasks in economics
policy decisions. It can be also useful for corporate and financial institutions decision
making. Many techniques have been applied in order to extract market expectations,
among them building Risk-Neutral Density (RND) from options prices is one of the
most used. In this sense the papers of Shimko (1993), Rubinstein (1994) and Jackwerth
and Rubistein (1996) were the first to empirically obtain RND. Using option-implied
RND, one can calculate, for example, the probability that exchange rate will stay inside
a specific range of values. Any empirical application in finance that that requires
densities forecasts may also take advantage of Risk-neutral densities.
On the other hand, many papers had focused its attention on the estimation of the
subjective density, if these densities are not equal, the risk aversion adjustment indicates
the investors’ preferences for risk. The first to recover empirically RRA was Jackwerth
(2000). He used the historical density as the subjective density. There are other ways to
obtain the RRA, as for example the approach introduced by Bliss and Panigirtzoglou
Most of the works that have studied RRA estimation have used options on
stocks. But, as pointed out by Micu (2005) and Bakshi, Carr and Wu (2008), it is
important to address the same estimation using currency option data in order to obtain a
global risk premium.
In this paper we estimate RND from the Brazilian Real/US Dollar (USD/BRL)
the relative risk-aversion of investors and also obtain a real-world density for the
exchange rate distribution. This is done for a sample of USD/BRL options traded at
BM&F-Bovespa from 1999 to 2011. The RND is estimated using a Mixture of Two
Log-Normals distribution and then the real-world density is obtained by means of the
Liu et al. (2007) parametric risk-transformations. The relative risk aversion is calculated
for the full sample, and is in line with previous studies of the Brazilian economy using
stock and consumption data. An out-of-sample goodness-of-fit evaluation is carried out
to evaluate the performance of the risk-neutral and real world densities.
Summing up our contributions are: We are the first to calculate RRA parameter
for the Brazilian Real Exchange rate. Second, we evaluate the RND and RWD density
forecasts for the USD/BRL and obtain a very good out-of-sample fit for the Real World
Density, with mixed results for the RND.
The paper is organized as follows: in Section 2 we give an overview of the RND
Section 4 we present our estimation algorithm. In Section 5 we describe our sample
data. In Section 6 we present our results and Section 7 concludes.
Once we have a set of option prices for a specific time to maturity, we can
for recovering this RND function implied in option prices. Jackwerth (1999) reviews
this literature, and classify them into parametric and non-parametric methods.
Parametric methods assume that the risk-neutral distribution can be defined by a
to estimate the set of parameters. For instance we can use the Generalized Beta of
Second Kind or the Mixture of two log-Normals in order to obtain the RND. Abe et all
(2007) was the only paper so far that analyzed the forecast ability of RND for the
Brazilian Real, and used the Generalized Beta of Second Kind.
Non-parametric methods consist of fitting CDF’s to observed data by means of
more general functions. Among the non-parametric methods are the kernel methods and
the maximum-entropy methods. Kernel methods use regressions without specifying the
parametric form of the function (for example, see Ait-Sahalia and Lo, 1998).
Maximum-entropy methods fit the distribution by minimizing some specific loss
function, as we can see in Buchen and Kelly (1996).
In our paper, we use the Mixture of two Lognormals (M2N) method for
recovering the risk-neutral distribution (RND). We will describe this method on section
Once we have a RND of an asset, we may use it to forecast its behavior.
in the equity market is known as Equity Risk Premium.
For short-term forecasts, this premium is usually small if compared with the
the size of this premium may be relevant. In this way, if we are trying to forecast over a
longer time period, it would be important to use a distribution which includes the risk
premium, and this is usually called “real-world” distribution.
Transformations from a risk-neutral density g to a real-world density h can be
representative agent with a power utility function and constant relative risk aversion
(RRA) denoted by c. The marginal utility is proportional to x
and the real-world
on next section.
We use a Mixture of Log-Normals to model the Risk-Neutral Densities. More
lognormals densities g:
ሻ ൌ ݓ כ ݂݀
ሻ ሺ1 െ ݓሻ כ ݂݀
ሺݔ|ܨ, ߪሻ ൌ ൫ݔߪ√2ߨܶ൯
parameters of the distribution. We do that by making the expectation of the distribution
equal to Dollar Future Contract price:
ሺ1 െ ݓሻܨ
Therefore, we have a total of five parameters, but only four free parameters. This
are the expectation of the two distributions of the mixture, while the sigmas
The price of an European call option is the weighted average of two Black
(1976) call option formulas C
(F, T, K, r, T):
, ݓ, ܭ, ݎ, ܶሻ ൌ ݓܥ
, ݓ, ܭ, ݎ, ܶሻ ሺ1 െ ݓሻܥ
, ݓ, ܭ, ݎ, ܶሻ (4.4)
The parameters estimation of the M2N was done using an adaptation of the
for the Brazilian Real/U.S. Dollar Exchange rate
squared errors of the theoretical and actual option prices.
Once we have the RND, we calculate the RRA parameter following the Liu et al.
world density h defined by (3.1) when there is a representative agent who has constant
RRA equal to c. If g is a single lognormal density then so is h. The volatility parameters
for functions g and h are then equal but their expected values are respectively F and F
also a mixture of two lognormals. For a M2N g (x|
) given by (4.1), it is
shown by Liu et al. (2007) that the real-world density h is also a Mixture of Lognormals
with the following density:
, ܿሻ ൌ ݄ሺݔ|ݓԢ, ܨ
Where the new set of transformed parameter is:
The original algorithm of Jondeau and Rockinger is available at the website:
. Among the changes we have done in the algorithm, we
use formula (4.3) to reduce the number of parameters.
൰ ൌ 1 ൬
1 െ ݓ
The real-world density has a closed-form representation because the cumulative
probabilities for the standard normal distribution. However, the calibration of this
transformation requires the estimation of the RRA parameter c, which ideally should be
calculated over a long time series of data.
Our dataset consists of put and call option prices traded at BM&F Exchange
from March 1999 to February 2011. In order to avoid overlap of data, we took only
options with about one month (20 business days
) before the expiration date, and this
the first day of the month. The dataset has 1,460 daily average option prices, with 938
calls and 522 puts. Therefore, we have built RND with 10.2 options on average.
Besides the USD/BRL Options data, we have collected also data from the future
contract of the USD/BRL exchange rate (DOL Futures) and futures contract of Average
Rate of One-Day Interbank Deposit (DI Futures), both with expiration at the same date
as of the respective option. Finally, for each expiration date we collected the USD/BRL
spot exchange rate, called PTAX
, which is the underlying asset of both options and
Brazilian Reais per U.S. Dollar, which means that an appreciation (depreciation) of the
Brazilian Real decreases (increases) the exchange rate.
We may have problems with the lack of synchronism between the traded time of the
option and the DOL and DI Futures, since we are using the average price of the day.
This may include some noise in our risk-neutral densities.
The period of the sample starts just after the end of the almost-fixed exchange rate
regime in Brazil. There were various upward shocks in the exchange rate (i.e.
devaluation of the Brazilian Real) during the period, including the period of the
When we had less than 5 options traded 20 business days before expiration, we used the business day
The PTAX is the daily average spot exchange rate, calculated by the Central Bank of Brazil. The time
there is a downward trend in the exchange rate after the overshooting that followed the
free-float in 1999, which means appreciation of the Brazilian Real against U.S. Dollar.
expiration cycles, which are all non-overlapping. For estimation, we minimized the
squared errors of the actual option price and the theoretical option price of the Risk-
Neutral Distribution. The mean squared error divided by the future exchange rate in our
estimation was 0.21% and the median 0.0251%.
6.2 Relative Risk Aversion Estimation
We have calculated a Relative Risk Aversion (RRA) for the full sample using
parameters estimated last section and then maximize the log-likelihood function with
the RRA being the only free parameter. This is done for the 143 expiration cycles,
which are all non-overlapping, as seen before. Therefore we have 143 RND g
ෞൟ and aim to maximize the following
The estimated RRA parameter c using equation 6.1 is 2.6959
and the p-value of
the null hypothesis of this parameter being equal to zero is 7.45%, so that there is
evidence of some risk premium for the Brazilian Real.
This is in line with previous papers that have performed RRA estimation for the
2.202 (median 1.70) using quarterly data with seasonal dummies and values between
2.64 e 6.82 (median 4.89) using annual data for the period 1975 to 1994. Nakane and
In fact, the RRA parameters calculated here are negative, since all our quotes are Brazilian Reais per
have the RRA for our Risky asset, the Brazilian Real, we just change the signal.
also GMM estimation. Catalão and Yoshino (2006), using quarterly data, obtain GMM
estimates of 0.8845 and 2.119 for the period 1991 to mid 1994 (Pre Real Plan) and mid
1994 to 2003 (Post Real Plan), respectively. Also, Araújo (2005) using GMM
estimation found similar ranges with a quarterly data for the period 1974 to 1999. For
constant relative risk aversion, he found a mean of 2.17.
In order to assess estimation robustness we made some tests. If you take out the
15.85%. When you take out the last 12 months it goes to 2.5963 with a p-value of
9.34%. This shows that there is some robustness on estimated data regarding sample
Another robustness exercise made was estimating 100 months rolling windows.
Table 1: Descriptive Statistics
Once we have the Relative Risk Aversion parameter and the Risk-Neutral
Risk transformation as described on section 4. Graph 1 below shows typical
distributions for our estimated RRA parameter (2.7). This is the densities on July 2006
for options expiring on August 2006. Note that the Real World Density appears on the
left of the RND, and this sounds counter intuitive, since the inclusion of risk-aversion
usually shifts the distribution to the right. The explanation is that we are using the
exchange rate quoted as Brazilian Real per US Dollar, i.e., we are not quoting the “risk”
currency, but the other currency.
which shows the difference between the Risk-Neutral and Real-World densities. We see
that the RND has more mass to the right, as well as a fatter right tail.
Graph 1 – Risk-Neutral and Real-World Densities for July, 2006
Risk-Neutral and Real-World Densities
Risk-Neutral minus Real-World Densities
and Real-World densities. For the real world densities, we need to choose a RRA
parameter in order to use the risk-transformation. Although Liu et al. (2007) use their
own estimates for the RRA, we consider that using the in-sample estimates for the RRA
would make the evaluation not truly out-of-sample, since at least one parameter is
However, in fact our estimates for the RRA are pretty much in line with articles
we have decided to use an RRA varying from 0 (the Risk-Neutral) to 4.
Our density forecast evaluation is based on Berkowitz (2001) and Crnkovic and
the following way:
ܷ ൌ ሼܷ
ሽ ൌ ൛݃
If the forecast density models are good, this series U must be a Uniform
series using the inverse of the standard normal distribution, generating a Z series:
ܼ ൌ ሼܼ
ሽ ൌ ሼΦ
If the forecast density models are good, this series Z should follow Standard
Normal distribution. Thus, we may apply usual normality tests like the Kolmogorov-
Smirnov in this series Z in order to assess the quality of the density forecast. Berkowitz
(2001) proposes a test that besides testing standard normality, also tests for first order
autocorrelation in the Z series. The Berlowitz Likelihood Ratio statistics must follow a
Results are on Table 2:
p-value Berkowitz LR p-value
Table 2: Goodness of fit statistics for selected RRAs
respectively seasonally adjusted quaterly data, quarterly
data with seasonal dummies and annual data.
AR stands for Araujo (2003) and IS stands for In-sample
Relative Risk Aversion
Graph 3- Density Forecats Results
Kolmogorov distance, while the RWD would perform well, including the RRA=2.17
estimated by Araujo(2003), which we believe is a true out-of-sample estimation for the
RRA parameter, since his time period finishes near the beginning of our time period. A
RRA around 3 would bring the best out-of-sample results using Kolmogorov as we can
see on Graph 3.
Regarding the Berkowitz LR test, both RRA and RWD performed well and
approach as we can see on Graph 3.
Overall, there is evidence that the addition of a risk premium in the RND using a
We have estimated the USD/BRL option-implied Risk-Neutral Densities using
Aversion and the Real-World density, and performed an out-of-sample evaluation of the
density forecast ability. This paper is the first to calculate the RRA parameter implied in
option prices for an emerging market currency. Our estimated value of the RRA
parameter is around 2.7, which is in line with other articles that have estimated this
parameter for the Brazilian Economy, such as Araújo (2005) and Issler and Piqueira
Our out-of-sample evaluation results showed that the RND has some ability to
the out-of-sample analysis of the RND forecast ability for exchange rate options.
However, when we incorporate the risk aversion into RND in order to obtain a Real-
world density, the out-of-sample performance improves substantially, with satisfactory
results in both Kolmogorov and Berkowitz tests. Therefore, we would suggest not using
the “pure” RND, but rather taking into account risk aversion in order to forecast the
Brazilian Real exchange rate.
Given this good performance in the out-of-sample assessment, a suggestion for
article for calculations of market risk and portfolio optimization. We would also suggest
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235 Revisiting bank pricing policies in Brazil: Evidence from loan and
250 Recolhimentos Compulsórios e o Crédito Bancário Brasileiro
Paulo Evandro Dawid e Tony Takeda