**Lourdes Gómez-Valle 1,†, Miguel Angel López-Marcos 2,† and Julia Martínez-Rodríguez 1,\*,†**


Received: 14 February 2020; Accepted:14 April 2020; Published: 17 April 2020

**Abstract:** Freight derivative prices have been modeled assuming that the spot freight follows a particular stochastic process in order to manage them, like freight futures, forwards and options. However, an explicit formula for pricing freight options is not known, not even for simple spot freight processes. This is partly due to the fact that there is no valuation equation for pricing freight options. In this paper, we deal with this problem from two independent points of view. On the one hand, we provide a novel theoretical framework for pricing these Asian-style options. In this way, we build a partial differential equation whose solution is the freight option price obtained from stochastic delay differential equations. On the other hand, we prove lower and upper bounds for those freight options which enables us to estimate the option price. In this work, we consider that the spot freight rate follows a general stochastic diffusion process without restrictions in the drift and volatility functions. Finally, using recent data from the Baltic Exchange, we compare the described bounds with the freight option prices.

**Keywords:** spot freight rates; freight options; stochastic diffusion process; stochastic delay differential equation; risk-neutral measure; arbitration arguments; partial differential equations
