**6. Discussion and Conclusions**

The freight market is a relatively new market but a very important one nowadays. Therefore, more scientific research is necessary in this area. In the freight market, in order to avoid price manipulations by large participants, setting is against the average value of a freight index. As a consequence, freight derivatives have, in general, average-style payoffs which makes them more difficult to price.

In the freight markets literature, we can find few models and methods to price this kind of derivatives. More precisely, to this end, it is usual to consider very specific parametric models. Here we propose new strategies that open a path to price these freight derivatives with general models, which will facilitate its application in the market by practitioners.

The contribution of this paper is twofold. On the one hand, we prove that the freight option price verifies PDEs with three independent state variables: the spot rate, its delay and the average of the spot rate in the settlement period. This result is notable because it offers a new approach to deal with the freight option valuation problem. Moreover, it opens the door to apply numerical methods for pricing freight options. On the other hand, we find and prove some lower and upper bounds for freight options which allow us to approximate its price. Finally, as an empirical application, we calculate these bounds using the Baltic Dry Index, issued by the Baltic Exchange in London, for freight options with different maturities. In such a case, we observe that the upper bound is close to the option price and then, it could be used as approximation to the price, especially for options in the money.

**Author Contributions:** All authors contributed equally to this article. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded in part by Consejería de Educación, Junta de Castilla y León, grant numbers VA148G18 and VA138G18, and by Ministerio de Economía y Competitividad, grant number MTM2017-85476-C2-P.

**Conflicts of Interest:** The authors declare no conflict of interest.
