Two New Strategies for Pricing Freight Options by Means of a Valuation PDE and by Functional Bounds

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.

Including Jumps in the Stochastic Valuation of Freight Derivatives

“…When the freight rate follows a diffusion process, Gómez-Valle et al [21] provide a partial differential equation and numerical methods could be developed to discretize it. However, if we consider a jump-diffusion process to reflect possible abrupt changes of the spot rate, a valuation equation to obtain freight option prices is not known in the literature.…”

“…In order to fill this gap, in this section, we develop a more general PIDE for pricing freight options assuming that the spot freight rate is given by a jump-diffusion stochastic process. We make a similar reasoning as in [21] for a diffusion process, which is based on pricing standard Asian-style options as in [32], and we extend it to jump-diffusion processes. Moreover, we obtain a partial solution to this PIDE for the GJ model introduced in Section 3.…”

“…Thus, as to provide a framework for pricing freight options, we consider a settlement period [T 1 , T N ] with a fixed time span d = T N − T 1 , which is usually one month. We establish a stochastic process A(t) as an approximation to the average of the spot rate with a continuous version of 1 N ∑ n i=1 S(T i ), as in [21]:…”

“…In the literature, there exist several techniques to obtain the freight option prices with this model, such as [33]. In a similar way to [21], but considering jumps, we calculate a solution to the PIDEs ( 16)-( 17) in Theorem 1 when the average of the spot freight verifies A ≥ dK. Proposition 2.…”

“…We provide a PIDE which allows us to use other different methods to price these derivatives. The original idea of this approach dates back to [21], but they leave unattainable various modeling processes including, for example, price discontinuities which are important in freight markets.…”