Recovering Risk-Neutral Densities from Brazilian interest rate options.
| Autor | Haas Ornelas, Jos |
| Cargo | Texto en Portuguese |
-
Introduction
Many techniques have been applied in order to extract market expectations. Building Risk-Neutral Density (RND) from options data is one of them. This information may be useful for financial stability analysis. Supervisory institutions can assess monetary policy impacts on expectations by inferring whether the market is attributing a high probability of a significant change on financial variables, such as interest rate or exchange rate. On the other way, market expectations on financial variables may influence monetary policy decisions. Using option-implied RND, one can calculate, for example, the probability that interest rate will stay inside a specific range of values.
Other areas of research that require densities forecasts may also benefit from Risk-neutral densities. This is the case of strategic asset allocation and market risk models.
Risk-neutral densities may also be used to price illiquid options, where we cannot find a market price. In this way, we would have a price for the illiquid option that would be in a non-arbitrage condition with the other options.
This article aims to evaluate methods for extracting the Risk-Neutral Densities implied in the main Brazilian interest rate option: the IDI (Interbank Deposit Rate Index) option. As far as we know, this is the first paper to recover RND for Brazilian interest rate options. Applications of RND for the Brazilian markets use basically currency options. (1)
The IDI option has some special features that make its pricing different from other fixed income options. In fact, the IDI option is similar to an Asian option on the geometric average of the one-day interbank interest rate (CDI), between the trade date and the expiration of the option. Therefore, option-pricing formulas from traditional models must be adapted to be used with these options.
In order to build RND's, we apply three methods: Shimko (1993) methodology, the fitting of Mixture of two Lognormals, and fitting of a Generalized Beta distribution of second kind (GB2). After the estimation of 1,879 RND's for each method, we have assessed the in-sample goodness-of-fit of each method with option data. The Mixture of Log-Normals provided the best in-sample fitting. We have also calculated the implied Skewness using the Mixture of Log-Normals, and analyzed its behavior in two specific periods of our sample.
The paper is organized as follows: Section 2 revises the Risk Neutral Density literature; Section 3 shows the IDI option main characteristics; on Section 4 our methodology is described; Section 5 gives an overview our dataset; Section 6 presents the results and finally section 7 concludes the paper.
-
Risk-Neutral Density (RND)
Black's (1976) method, assume that the future price's follows a geometric Brownian motion and that the volatility is constant, i.e., options on the same asset should provide the same implied volatility values. However, these are too strong assumptions. In practice, implied volatilities vary along strike prices and expiration dates. One stylized shape of the Strike x Implied Volatility figure is the so-called "smile": options too in-the-money or too out-of-the-money usually results in higher implied volatilities compared to at-the-money options. Depending on the underlying asset, other shapes occur also, such as a "sneer" or "smirk". This constitutes evidence against the Black's method that would produce a flat line shape.
Moreover, for interest rate options, the underlying asset price doesn't follow a geometric Brownian motion. The nominal interest rate can be negative. Therefore, as a consequence of the non-adherence of the Black's method, many other methods and models came out in order to incorporate the underlying asset price dynamics, such as the Vasicek (1977), Heath et al. (1992) and many others. Nevertheless, many practitioners still use the Black's model for interest rate options, given its simplicity. This the case of the Brazilian market of IDI options, subject of this paper.
Once we have a set of option prices for a specific time to maturity, we can recover the risk-neutral probability distribution (Ross, 1976). Breeden and Litzenberger approach (1978) gives an exact formula2 for recover the risk-neutral density:
[[sigma].sup.2]C(K)/[[sigma].sup.2]K[|.sub.S=X] = [e.sup.-rT] pdf(S) (1)
where C(K) is the option price as a function of the strike price K, r is the continuous interest rate of the underlying asset, T is the time to expiration and pdf(S) is the risk-neutral density function as a function of the underlying asset price S.
There are many methods for recovering this risk-neutral density function pdf or the risk-neutral cumulative distribution CDF embedded in option prices. Jackwerth (1999) reviews this literature, and classify these methods into parametric and nonparametric.
Assuming that a CDF is defined by a limited set of parameters, parametric methods just provide ways in order to estimate them. Jackwerth (1999) divides these methods into three groups: i) expansion methods, ii) generalized distribution methods and iii) mixture methods. The expansion methods add a sequence of correction terms in order to obtain a better-fitting distribution. For instance, Jondeau & Rockinger (2001) use Gram-Charlier expansion to extract RND. Generalized distribution methods use distributions with some additional parameters in order to obtain a better fit of the RND. This is the case of the Generalized Beta of Second Kind that we use in our paper which has been used in many articles, as for instance in Liu et al. (2007). Mixture methods create new distributions from combinations of well-known simple distributions like the normal. The most common combination is the Mixture of Two Lognormals, which we use in our paper, and have been used by, for instance, by Coutant et al. (2001).
Non-parametric methods consist of fitting CDF's to observed data by means of more general functions. These methods are divided into three groups: i) kernel methods, ii) maximum-entropy methods and iii) curve fitting methods. Kernel methods use regressions without specifying the parametric form of the function (for example, see Ait-Sahalia & Lo (1998)). Maximum-entropy methods fit the CDF by minimizing some specific form of loss function, as in Buchen & Kelly (1996). Curve-fitting methods try to fit some very flexible curve. This is the case of Shimko (1993) that proposed a curve-fitting method to the smile by fitting a quadratic polynomial.
In our paper, we use three methods for recovering the risk-neutral distribution (RND): i) the approach of Shimko (1993) ii) the fitting of Mixture of two Lognormals (M2N), and iii) fitting of a Generalized Beta distribution of second kind (GB2). We will describe these methods on section 4.
-
IDI Option Characteristics
The IDI option is the main interest rate option traded in Brazil. The average daily value of contracts traded at the Brazilian Exchange BM&F-Bovespa in the period from June 2nd, 2003 to April 23rd, 2009 was US$ 8.86 millions, with 36 trades each day on average.
The IDI option is of European style, and mature in the first business day of the corresponding month of expiration. The underlying asset of this option contract is an index called IDI (interbank deposit index). It is calculated according to the recursive formula:
ID[I.sub.t] = ID[I.sub.t-I](1 + [i.sub.t-1]) (2)
where [i.sub.t-1] is the Average Rate of One-Day Interbank Deposit Certificate (CDI) converted to percentage per day at time t - 1.
...
Para continuar a ler
PEÇA SUA AVALIAÇÃOCOPYRIGHT GALE, Cengage Learning. All rights reserved.
