**5. Empirical Application**

In this section, we analyze the accuracy of the bounds obtained in Section 4. To this end, we consider that the spot freight rate follows a geometric process, which is widely used in the literature (as, for example, in [4]). That is, we assume that the spot freight rate follows a geometric stochastic process

$$dS(t) = \mu S(t)dt + \sigma S(t)dW(t). \tag{23}$$

We estimated the parameters in Equation (23) using Baltic Dry Index data from 2013 to 2019. This index, daily issued by the London-based Baltic Exchange, is mostly used in the freight market. As the spot freight rate follows a geometric Brownian process, we use maximum likelihood obtaining the values *μ* = 0.0041 and *σ* = 0.3738.

Assuming that the market price of risk is *λ* = 0, then, the drift under the risk-neutral measure is equal to the drift under physical measure

$$
\mu S(t) - \lambda = \mu S(t).
$$

The bounds in Theorem 2 are obtained in the following way. The lower bound is computed using the FFA price obtained by [4] for a geometric Brownian motion. As far as the upper bound is concerned, the prices of the European plain vanilla call options on the spot freight rate are obtained in a similar way that in [19], but considering that *λ* = 0.

In order to compare the bounds in Equation (22) with the freight option price, we approximate the latter using the Monte Carlo simulation technique, which has been proved to be a flexible and handy method to price options (see, for example, [13]). We approximate the expectation in Equation (3) using a daily time step (Δ*t* = <sup>1</sup> <sup>252</sup> ) and the previously established parameters. We generated 100,000 paths and consider that the settlement period is 1 month and the interest rate is 0.5%. We assumed that the spot freight rate is *S*<sup>0</sup> = 1034.6, which is the average of the Baltic Dry Index from January 2013 to January 2019, and different strike prices from 70% to 130% of this spot freight rate. In order to increase the precision of this technique, we used the antithetic variable method as a variance reduction technique, see [13].

Tables 1 and 2 show several option prices and their corresponding bounds for different maturities (1 and 3 months, respectively) and strike prices (as percentages of the spot price). Both tables confirm the validity of the bounds in Equation (22).

We conclude that the window defined by the bounds, when the maturity is 1 month, is narrower than the one obtained with a maturity of 3 months. In both cases the maximum width of the window is for options at the money (30.56 monetary units for 1 month and 70.05 for 3 months). Moreover, around the spot price the upper bound is closer to the option price than the lower bound. This fact can be observed clearly in Figure 1 that plots the option prices (solid line) and their corresponding lower and upper bounds (dotted and dashed lines, respectively) for several strike prices with maturities of 1, 3, 6 and 12 months. Note that, the higher the maturity the wider the window but, in all cases the behavior of the upper bound fits the option price better than the lower bound.

Finally, Table 3 shows the differences, absolute and relative, between each bound and the freight option price for different values of the strike price, and with a maturity of 3 months. The last row provides the mean of the differences presented in the previous rows. As we can see, when the option is out of the money, the differences in both bounds are higher than when the option is in the money. In fact, these differences reach a maximum when the option is at the money. If we compare the mean of the differences, we observe that the upper bound is much more accurate than the lower bound. Therefore, in this case the upper bound is a good estimation of the option price.


**Table 1.** European freight call option prices with a maturity of 1 month, for several strikes and their corresponding lower and upper bounds.

corresponding lower and upper bounds.


**Table 2.** European freight call option prices with a maturity of 3 months, for several strikes and their

 6WULNHSULFH **0DWXULW\PRQWK** /% 8% 3ULFH 6WULNHSULFH **0DWXULW\PRQWKV** /% 8% 3ULFH **0DWXULW\PRQWKV** /% 8% 3ULFH **0DWXULW\PRQWKV** /% 8% 3ULFH

**Figure 1.** The lower and upper bounds and the option prices according to the strike prices. Maturities: 1, 3, 6 and 12 months.

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**Table 3.** Absolute and relative differences between the freight call option prices with a maturity of 3 months and their lower and upper bounds, for several strike prices.
