Estimating risk‐neutral freight rate dynamics: A nonparametric approach

We present a new method for estimating the unobservable drift of the risk-neutral spot freight rate process from Forward Freight Agreement (FFA) prices in the absence of a closed-form solution and demonstrate robustness via numerical simulations. Moreover, we conduct empirical experiments involving estimation of standard parametric models and a nonparametric model using Baltic Exchange data. We find that our nonparametric approach yields the lowest FFA pricing errors across maturities. Finally, we estimate the market price of risk, analyze its behaviour in-sample and out-of-sample and observe that, when estimated using our nonparametric approach, it evolves consistently with the indices under study.

empirical results, whereas Section 5 presents the procedure for extracting the market price of risk and its analysis. Section 6 concludes the paper.

| MODEL SETTING
In this section, we adopt a diffusion process for the dynamics of the freight rates and show some models for pricing FFA contracts. Then, we derive a new result which allows us to estimate the drift of the freight rate process under the risk-neutral measure.
Let (Ω,  ,  ≥ { } t 0 , ) be a complete filtered probability space, which satisfies the usual conditions, where  ≥ { } t 0 is a filtration (e.g., see Protter, 2005;Shreve, 2004). Therefore, there exists a risk-neutral measure  equivalent to . We assume that, under the measure , the spot freight rate evolves according to the general model where  μ S ( ) and σ S ( ) are, respectively, the drift and volatility of the process, and  W is the standard Brownian motion. We assume that the functions  μ and σ satisfy suitable regularity conditions as follows (see Oksendal, 2003): 1. Functions  μ and σ are measurable and there exists a constant C such that, for all (1 + ).

There exists a constant D such that, for all
The FFA is a cash-settled financial contract which gives its owner the average spot price over the settlement period T T [ , ] N 1 , with ⋯ T T T < < < N 1 2 , where T 1 and T N are, respectively, the first and last trading days in a month. We denote by F t S T T ( , ; , …, ) (2) where the terms in the sum represent the conditional expectations under the risk-neutral measure on the information at time t. If ≤ T t T < k k +1 , for ∈ k N {1, …, − 1}, the right-hand side of (2) equals In the literature, there are some very well-known models for pricing FFA contracts that we will use as test problems in Section 3. Koekebakker et al. (2007) assume that the spot freight rate follows a geometric Brownian motion: GÓMEZ-VALLE ET AL.

| 3
Then, under the  measure, where  α α ϕσ = − and ϕ is the market price of risk coefficient. This fact yields a closed-form solution for the FFA price given by (see Koekebakker et al., 2007 for more details); we dub this Model 1. This model is basic and provides a very simple and manageable expression for the FFA price. Nevertheless, the freight market is a nontradable transportation service hence its dynamics looks quite different from a geometric Brownian motion. For instance, Tvedt (1997) accounts for mean reversion in the spot freight rate process using and, under the  measure, where  m m ϕσ = − ; we call this Model 2. This model has been used widely in commodity, energy, and freight markets (e.g., see Schwartz, 1997;Tvedt, 1997). The futures price at time t with maturity time ≥ T t is given by which, from Schwartz (1997), for the spot process (7) is equal to . Then, the FFA price with settlement period T T , is given by the arithmetic average of the futures prices with maturities T where ∼ F is given by (8). In both models, the market price of risk coefficient is assumed to be constant resulting in a closed-form solution for the FFA price. However, in the literature, there is neither empirical evidence nor consensus about which stochastic process and market price of risk function are the best to model the spot freight rates and this can easily lead to model misspecification. Moreover, there has been a recent tendency to use nonparametric methods to identify and estimate the functions of the stochastic processes (see Stanton, 1997 in bond pricing;Gómez-Valle & Martínez-Rodríguez, 2016 in commodity markets;or Adland, 2003;Adland & Cullinane, 2006 in freight markets). Nonparametric techniques do not impose a specific parametric structure and, as such, reduce potential misspecification errors.
We describe briefly how the nonparametric estimation works. Suppose a data set consisting of n pairs of ob- . We consider a model of the type Y m S ε i n = ( ) + , = 1, …, , where m S ( ) is an unknown function and ε represents the random error in the observations or the variability from sources not included in S. The error terms ε { } i i n =1 are assumed to be independent and identically distributed with mean equal to zero and finite variance. For example, a candidate function for m S ( ) is the Nadaraya-Watson (NW) estimator (see Härdle, 1990) given by , K is the Gaussian kernel and h the bandwidth parameter. To apply this nonparametric estimation technique to the FFA price model, we first estimate the volatility of the stochastic process (1). As the volatility is not affected by the change of measure, we can use the spot freight rate process under the physical measure, and spot freight rate market data. We approximate the volatility in (12) based on the Euler discretization 2 and by considering the NW estimator, gives (for more details, see Fusai & Roncoroni, 2008;Stanton, 1997). Finally, we need to estimate the risk-neutral drift. Usually, in the literature, this is inferred from the FFA closedform solution. An alternative method is to estimate the drift under the  measure using the spot freight rate observations and, then, the market price of risk from the FFA closed-form solution. However, this is not always possible, especially in nonparametric models, where an explicit price is not known. We solve this problem by proving a result which links the risk-neutral drift with the directional derivative of the FFA price with respect to the time points in the settlement period.
Theorem 1. Let S be the spot freight rate evolving according to process (1) and F t S T T ( , ; , …, ) N 1 the FFA price given by (2) where ≤ t T N . Then, and dividing by N , we obtain We calculate the conditional expectation under the  measure of the above equality and from (2) and for the Itô integral being a martingale, we get GÓMEZ-VALLE ET AL.
where the second approximation is due to (15) and (16).
We use as the observations Y i in (10) the right-hand side of (17), with the shortest equidistant maturities, and the spot freight rates as S i . Then, any technique can be applied to estimate the risk-neutral drift, such as the nonparametric NW estimator (11). An important merit of this result is that it can be applied to estimate the risk-neutral drift directly from FFA prices in the market, independently of any assumptions one makes about the underlying model or the estimation technique used. We will call dRNP this approach based on the directional derivative of the FFA price and the risk-neutral process. This allows us to evaluate derivatives without estimating either the market price of risk or the physical drift, hence reduces potential model misspecification.
In the rest of this paper, this approach, implemented with nonparametric estimation, will be referred to as the Nonpar model. Its main advantage over the previous parametric models (Models 1 and 2) is that it uses fully functional procedures to identify and estimate the functions of the stochastic processes appearing in the most recent literature (see, among others, Adland, 2003;Adland & Cullinane, 2006;Bandy & Phillips, 2006). Therefore, it does not impose any arbitrary restrictions to either the functions of the spot freight rate processes or the market price of risk.

| NUMERICAL SIMULATIONS
To illustrate the precision of our proposed approach, in this section we carry out numerical experiments with test problems to study the robustness and efficiency of our estimation method in Theorem 1. We test the approximation in problems with exact solutions to calculate and analyze the errors. As we are not testing here the model realism, but rather an approximate against an exact solution, we do not use real market data (except only for the parameter estimates mentioned next from previous research).
We investigate our method for Models 1 and 2 in Section 2 as test problems. We conduct Monte Carlo simulations of the spot rate based on (3) and (6), respectively. We calculate FFA prices for each model from (5) and (9), respectively, for different values of the market price of risk coefficient. We then analyze the error from misspecifying the risk-neutral drift, in particular, by setting this equal to the drift under the  measure; we refer to this as the risky process (RP) assumption.
First, we use the parameter values obtained by Gómez-Valle et al. (2020) for (3), that is, α = 0.0041 and σ = 0.3738; and Gómez-Valle and Martínez-Rodríguez (2021) for (6), that is, k = 0.4041, m = 6.8805, and σ = 0.3740. Both estimations were made using Baltic Dry Index (BDI) data (Baltic Exchange, 2020) from January 2013 to January 2019 as a proxy for the spot freight rate. The BDI is a composite daily freight index that records freight rates for transporting dry-bulk commodities, like, iron ore, coal, grain, and other agricultural commodities, across 20 different shipping routes. We simulate 5000 spot freight rate daily trajectories for these parameter values.
Second, to estimate the risk-neutral drift, we need FFA prices for three short equidistant maturities, such as 1-3 months. We compute these using the previous parameters, the closed-form solutions (5) and (9), and various values of the market price of risk coefficient ∈ ϕ {0, ±0.02, ±0.05, ±0.2}. We then estimate the risk-neutral drifts for both parametric models by means of Theorem 1, the second-order approximation (17) and nonlinear least squares, using, respectively, for Models 1 and 2 where F t S ( , ) j i i is the price of the FFA with maturity j months for observed spot freight rate S i , i n = 1, …, . The volatility parameter is estimated, as it is not affected by the change of measure, using the simulated spot rate process under the measure .
We consider the mean absolute percentage error (MAPE) and the root mean square percentage error (RMSPE) for a given maturity τ, , are, respectively, the estimated and simulated FFA prices for maturity τ. In particular, we consider, here, τ = 1, 2, 3, 6, and 12 months. We also denote by MAPE and RMSPE the corresponding errors based on all maturities: In Tables 1 and 2, we report the errors from comparing the FFA prices obtained from the simulations of each of Models 1 and 2 with those calculated based on the RP misspecified risk-neutral drift and the dRNP risk-neutral drift of Section 2. We find that pricing FFA contracts using the dRNP approach yields very small errors. More specifically, these do not exceed 10 −3 in the case of Model 2, whereas in the case of Model 1 these appear to be as low as 10 −8 with some | 7 discrepancies observed depending on the value of the market price of risk coefficient. Expectedly, the errors increase considerably when the FFA contracts are priced under the RP drift, in particular, by a factor of 10 2 up to 10 3 , or even 10 4 in some cases, compared with our dRNP approach. Tests based on alternative parameter values yielded similar results. It is worth noting that, even when the market price of risk is assumed to be equal to zero (i.e., ϕ = 0), the errors from the RP drift remain high. In this case, the parameters of the stochastic process under the  measure are estimated using the spot freight rate observations in the market which can lead to possible misspecification in the estimation of the drift (see Adland, 2003;Stanton, 1997). Our results confirm the importance of an accurate estimation of the risk-neutral drift for an accurate pricing of the freight derivatives, therefore, in what follows, we study further the dRNP drift approach.

| DATA AND EMPIRICAL APPLICATION
In this section, we estimate the functions of the risk-neutral spot freight rate process and price FFA contracts with different maturities for the dry-bulk freight rates. We focus on the Capesize, Panamax, and Supramax dry-bulk carriers which are the most popular contract categories and jointly account for more than 95% of the total derivatives volume in the dry-bulk FFA market. In addition, this is the oldest market for freight derivatives with continuous price data going back to at least 2006. At the larger end of the scale, Capesize carriers have a carrying capacity of 180,000 dead-weight (dwt) tons and are the largest vessel category in the dry-bulk sector. Capesize vessels are primarily employed for the transportation of iron ore from South America and Australia to the Far East (primarily China) and coal from North America, Australia, and South Africa to the Far East and North Europe. The name is attributed to the fact that this type of vessel is too large to transit the Panama canal, hence it has to navigate around Cape Horn or Cape of Good Hope. Panamax carriers have a capacity of about 74,000 mt dwt and, as the name implies, are the largest-sized vessels that can transit the Panama Canal. Finally, Supramax vessels have a cargo-carrying capacity of 58,000 mt dwt and are more versatile than the other vessel categories in terms of commodities they carry and the routes they serve. They are also equipped with cranes and, as such, can operate in ports where there is not sufficient infrastructure. According to Clarksons Shipping Intelligence Network (SIN), as of October 2020 the Capesize, Panamax, and Supramax sectors accounted, respectively, for 40%, 25%, and 23% of the total cargo-carrying capacity of the dry-bulk fleet. In addition, these vessels are almost exclusively used for the transportation of the so-called "major bulks," including commodities, such as iron ore, grain, coal, bauxite, and alumina and phosphate, and jointly account for about 25% of the overall global volume of trade. Therefore, these vessel types are representative of the overall bulk fleet and their dynamics.
Our key variables are the spot and forward freight rates for the Capesize, Panamax, and Supramax sectors known, respectively, as the Baltic Capesize Average of five key time-charter routes (BCI 5TC), the Baltic Panamax Average of four key time-charter (BPI 4TC), and the Baltic Supramax Average of five time-charter routes (BSI 5TC), which are published by the Baltic Exchange (2020) on a daily basis. 1 The sample period for our data is from January 2006 to December 2019. We also consider the FFA rates that price the forward value of the index. These contracts are available at any given point in time for various maturities up to 7 years ahead. A particular issue when using forward prices is that individual contracts expire, complicating the empirical analysis. As such, we create synthetic forward contracts with fixed maturities. For example, we create a 30-day-ahead synthetic contract using contracts that expire in M and L days, where M L < 30 < : We use Equation (18) to construct daily forward contracts with maturities of up to 2 years ahead, which are then used in our empirical application.
In Figure 1, we present the daily time plot of the BCI, BPI, and BSI from January 2006 to December 2019. It is important to note that the three indices have a very similar behavior, although the BCI has, in general, the highest values, especially during the period 2006-2010. We note a persistent rally in freight rates during the period 2006-2008, which culminates with the indices reaching their peaks in July 2008. For example, following the financial crisis, the BCI dropped by more than 95% from a peak of 230,000 $/day in July 2008 to a low value of 5000 $/day by December of the same year. Since then, the market exhibits very high volatility with large swings, which is typical of the behavior of dry-bulk freight rates.
We estimate all the functions of the spot risk-neutral process with data from January 2006 to December 2017 and keep data from January 2018 to December 2019 to analyze the results for robustness, that is, we divide our data in the in-sample period (January 2006-December 2017) and out-of-sample period (January 2018-December 2019).
First, using the in-sample data, we estimate the drift and volatility of the process under the  measure in (1) as established in Section 2. When applying (17) to estimate the risk-neutral drift, we require three FFA prices with the shortest equidistant maturities; here, we use prices with maturities of 1-3 months. As the BCI and BPI reached very high values, especially in the period 2007-2008 (see Figure 1), we choose to work on the log-scale, that is, use as observations S ln i instead of S i in (11) to achieve more accurate nonparametric estimations of  μ S ( ) in (17). For the BSI, we use the original spot freight rate S i observations as these are lower than BCI and BPI's. Also, to improve the estimation of the risk-neutral drift of the BSI, we use the Local Linear Regression (LLR), instead of the NW estimator, to reduce the bias at the boundaries (see Loader, 1999 for more details); however, this increases considerably the computational cost and does not always provide better results than NW. Figure 2 exhibits the scatter plots of the observed BSI and the logarithm of BCI and BPI over the in-sample period against the approximation (17) and the nonparametrically estimated risk-neutral drift. We observe that there are only a few very low and high spot rate observations. Moreover, the high values are very scattered, especially for the BCI.
For comparison, we also estimate the risk-neutral drift and volatility of the parametric spot freight benchmark Models 1 and 2. We apply the usual methodology to estimate the parameters via their corresponding closed-form solutions (5) and (9) and nonlinear least squares. The volatility of Model 1 is estimated from the discretization of the spot freight rate process under the  measure as in Section 3. and BSI. Figure 3 shows for each index the estimated risk-neutral drifts  μ S ( ) of the two parametric models and the nonparametric model together with its associated confidence bands obtained by bootstrap (see . In all the cases, we observe a decreasing trend, but the behavior of the three models varies. In Models 1 and 2, the shape is determined by the functions imposed to the risk-neutral drifts, whereas in the Nonpar model this is determined by the behavior of the BCI, BPI, and BSI observations in the sample period. The nonparametric risk-neutral drifts of the BCI and BPI are smoother than the BSI's. (The smoothness of the nonparametric estimation depends on the bandwidth h as well as the magnitude of the independent variables S ln i or S i .) Finally, we price FFA contracts under all the models and for maturities from 1 to 24 months. For Models 1 and 2, we use their corresponding closed-form solutions (5) and (9). For the Nonpar model, we use, by definition, with ∼ F satisfying the partial differential equation for the futures price  for i N = 1, …, ; we solve (19) using the Crank-Nicolson finite difference method (e.g., see Duffy, 2006). To obtain the FFA prices in the out-of-sample period, we use spot freight rates in the out-ofsample period and parameters and nonparametric functions estimated using the in-sample data. Table 4 reports the in-sample and out-of-sample FFA pricing errors for all the different cases for the Capesize sector. First, we compute the MAPE τ and RMSPE τ for maturities τ = 1, 3, 6, 12, and 24 months. (Additional maturities between 1 and 24 months led to similar conclusions. This is also valid for all three indices studied in the paper.) We find that Model 1 is the worst-performing and that Nonpar yields the lowest errors. As a measure of the aggregate performance of each model, we report also the MAPE and RMSPE computed across all maturities (last column). The superiority of Nonpar is also maintained both in-sample and out-of-sample. Table 5 reports the relevant FFA pricing errors for the Panamax sector. We observe that in the in-sample period and for maturities up to 3 months, Nonpar results in the lowest errors, but they are slightly higher for longer maturities from 6 to 24 months; it also yields the lowest out-of-sample errors. The relationship between the errors of Models 1 and 2 is similar to that in the Capesize sector. Finally, Table 6 presents the FFA pricing errors for the Supramax sector. Nonpar takes the lead once more both in-sample and out-of-sample, whilst the relationship between the errors of Models 1 and 2 remains unaffected.

| MARKET PRICE OF RISK
In this section, we study the market price of risk, which is essential for pricing derivatives based on no-arbitrage principles. The market price of risk is not observed, rendering its estimation quite nontrivial. Derivative pricing models with known closed-form solutions are usually used to infer the market price of risk from market quotes of traded derivatives. However, model misspecification can lead to inaccurately estimated market price of risk. A potential way about this problem is to implement, instead, a nonparametric model approach; the main challenge there is the absence of an explicit solution for any derivative price that could be used to this end, therefore T A B L E 4 Mean absolute percentage errors (MAPEs) and root mean square percentage errors (RMSPEs) for Models 1, 2, and Nonpar for the Capesize sector prohibiting the estimation of the market price of risk. Our proposed approach is able to remove this major block, as we explain next. Following Weron (2008), the market price of risk of the spot freight rate is given by which, therefore, requires estimation of both drifts under the  and  measures.  μ S ( ) can be estimated using the closed-form model solution for any traded freight derivative or, if this is unknown as in the case of a nonparametric model, via Theorem 1. Here, we follow the same approach and use the same data as in Section 4; refer to Figure 3 for the outcome of this estimation. Under the  measure, μ S ( ) can be inferred from the observed spot freight rate series. In particular, we adopt a similar nonparametric approach to the volatility (recall Equations 13 and 14) based now on the first moment from an Euler discretization of (12), that is, and, from the NW estimator (11), we get Finally, for Model 1 we use maximum likelihood estimation as in Gómez-Valle et al. (2020), whereas for Model 2 we transform first (6) to an Ornstein-Uhlenbeck process and then use least squares as in Gómez-Valle and . Table 7 reports for Models 1-2 and each index the resulting parameter estimates under the  measure.
On the basis of the estimation procedure delineated above, Figure 4 shows the estimated market price of risk λ S ( ) for each model and for the observed values S of each index. We observe varying patterns for the different models. More specifically, Model 1 exhibits a linear and increasing market price of risk, whereas for Model 2 this is slightly nonlinear and increasing for BPI and BSI, but decreasing for BCI. These patterns reflect the parametric structures imposed by Models 1 and 2 (see Equations 3-4 and 6-7). On the other hand, the nonparametric market price of risk is nonlinear in all cases. We also observe in Figure 4 that the changes of the nonparametric market price of risk are quite perceptible especially in regions of very low and high observations. This behavior can be attributed to boundary bias (see Loader, 1999) which is very common in nonparametric estimation, especially when there is scarcity of very high or low observations as in our case (see Härdle et al., 2004). Figure 5 shows the market price of risk from 2009 to 2017. (We do not present results for the period 2006-2008 which includes the global financial crisis; the high volatility and values of freight rates in that period resulted in not accurate estimates for the nonparametric model.) Models 1 and 2 return always positive market prices of risk; as these models assume a constant market price of risk coefficient, the sign of the market price of risk will depend on this and remain unchanged over time. Without such a restriction, the nonparametric market price of risk has instead both positive and negative values and this is frequently observed also in other financial markets, such as in the electricity market (e.g., see Weron, 2008) and the fixed-income market (see Ahmad & Wilmott, 2007). Furthermore, this result supports the assertion that the risk premium in the freight market is time-varying and depends on the prevailing state of the freight market (see Adland & Cullinane, 2005). Indeed, by superimposing on the plots in Figure 5 the observed BCI, BPI, and BSI values, we see that, in general, the nonparametric market price of risk follows the movement pattern of the corresponding index more closely than the parametric models. Finally, Figure 6 shows the estimated market price of risk in the out-of-sample period (2018-2019). As before, the nonparametric market price of risk follows very closely the movements of the observed index and takes both negative and positive values. On the contrary, for Models 1 and 2 this barely changes and is always positive.
In summary, in the in-sample as well as the out-of-sample periods, we observe that the nonparametric market price of risk is negative when the spot freight rate drops, that is, when investors need extra return for taking risk (e.g., see Wilmott, 1998). This is consistent with the view that when freight rates are low, long hedgers are F I G U R E 4 Market price of risk for Models 1, 2, and Nonpar as a function of the BCI (top), BPI (middle), and BSI (bottom). BCI, Baltic Capesize Average; BPI, Baltic Panamax Average; BSI, Baltic Supramax Average [Color figure can be viewed at wileyonlinelibrary.com] willing to pay a premium to lock in favorably low freight rates; similarly, short hedgers are willing to accept a discount to lock in high freight rates. Overall, its qualitative behavior is similar to that of the corresponding index.

| CONCLUSIONS
In this study, we propose a novel nonparametric approach for estimating the drift of risk-neutral freight rate dynamics. We prove a result which links the risk-neutral drift of the spot freight rate with the directional derivative of FFA prices. This development, jointly with some approximations, allows us to estimate the risk-neutral drift and even the market price of risk, without knowledge of a closed-form solution of any freight derivative, using only readily available market data, such as spot freight rates and FFA prices. Moreover, the estimated risk-neutral drift and market price of risk can F I G U R E 5 BCI (top), BPI (middle), and BPI (bottom) and the corresponding estimated market prices of risk for Models 1, 2, and Nonpar over the in-sample period (2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016)(2017) be used to price any other freight derivative as they determine the equivalent martingale measure that is necessary for pricing under no-arbitrage arguments.
We illustrate the efficiency and robustness of our approach via numerical experiments using two popular parametric models in the literature as test problems. Furthermore, we consider several Baltic indices to calibrate the riskneutral drift with our approach and price FFA contracts. As a result, we show the superiority of our methodology, implemented by means of a nonparametric technique, for the in-sample and out-of-sample periods. Finally, we obtain the market price of risk and show that this is time-varying and its behavior is closely related to the corresponding Baltic index.
The implications of this study are of potential interest to both researchers and market participants. On the one hand, the precise estimation of the risk-neutral drift of the spot freight rate is necessary to price accurately any freight derivative. On the other hand, the market price of risk, obtained from the drift under the risk-neutral and physical measures, helps us understand the behavior and risk attitudes of market participants. Moreover, it allows us to deduce in the short term the expected risk that market participants must assume. F I G U R E 6 BCI (top), BPI (middle), and BPI (bottom) and the corresponding estimated market prices of risk for Models 1, 2, and Nonpar over the out-of-sample period (2018)(2019)