**4. Lower and Upper Bounds for Freight Options**

For arithmetic standard Asian options, several bounds have been presented in the literature (see for example [14,17]) and they are obtained in terms of European options. Therefore, assuming a specific dynamics of the spot rate in order to know its probability distribution, these bounds can be valuated. For example, in [18] and [7], an optimal lower bound for freight options is provided when the log spot freight price follows a jump-diffusion process with mean reversion.

In this section, we obtain lower and upper bounds for freight options but, unlike what happens in the Asian case, we do not assume a particular expression of the functions in the spot freight stochastic process.

In the following theorem, as in Equation (4), we consider that the freight option is a European option on an FFA.

**Theorem 2.** *Let C*(*t*, *S*; *K*, *T*1, ... , *TN*) *be a freight call option price with settlement period* [*T*1, *TN*] *and strike price K. Then,*

$$e^{-r(T\_N - t)} \left( F(t, S; T\_1, \dots, T\_N) - K \right)^+ \le \mathbb{C}(t, S; K, T\_1, \dots, T\_N) \le \frac{1}{N} \sum\_{i=1}^N e^{-r(T\_N - T\_i)} \mathbb{C}\_{\mathbb{E}}(t, S; K, T\_i), \tag{22}$$

*where F*(*t*, *S*; *T*1, ... , *TN*) *is an FFA with settlement period* [*T*1, *TN*]*, and CE*(*t*, *S*; *K*, *Ti*) *is a European plain vanilla call option with maturity Ti.*

**Proof of Theorem 2.** First of all, note that for a convex function *φ*, *E*[*φ*(*X*)] ≥ *φ*(*E*[*X*]). Therefore, starting with Equation (3) for the freight call option price and taking into account that the maximum function is convex, then

$$\begin{split} &C\left(t,S;K,T\_{1},\ldots,T\_{N}\right) = \mathbf{e}^{-r(T\_{N}-t)}E^{\mathcal{Q}}\left[\left(\frac{1}{N}\sum\_{i=1}^{N}S(T\_{i})-K\right)^{+}|S(t)=S\right] \\ &\geq \mathbf{e}^{-r(T\_{N}-t)}\left(E^{\mathcal{Q}}\left[\frac{1}{N}\sum\_{i=1}^{N}S(T\_{i})-K|S(t)=S\right]\right)^{+} \\ &=\mathbf{e}^{-r(T\_{N}-t)}\left(E^{\mathcal{Q}}\left[\frac{1}{N}\sum\_{i=1}^{N}S(T\_{i})|S(t)=S\right]-K\right)^{+} \\ &=\mathbf{e}^{-r(T\_{N}-t)}\left(F(t,S;T\_{1},\ldots,T\_{N})-K\right)^{+}, \end{split}$$

arriving at the lower bound in Equation (22) which depends on the FFA price *F*(*t*, *S*; *T*1,..., *TN*). In order to deduce the upper bound, we use the following relation

$$\left(\sum\_{i=1}^N a\_i\right)^+ \le \sum\_{i=1}^N (a\_i)^+.$$

which is satisfied for every collection of real numbers {*ai*}*<sup>N</sup> <sup>i</sup>*=1.

If we apply this relation to the option price formula (Equation (3)), we obtain

$$\begin{split} &\mathbb{C}(t,\boldsymbol{S};\boldsymbol{X},\boldsymbol{T}\_{1},\ldots,\boldsymbol{T}\_{N}) = \mathbf{e}^{-r(\boldsymbol{T}\_{N}-t)}\boldsymbol{E}^{\boldsymbol{\Theta}}\left[\left(\frac{1}{N}\sum\_{i=1}^{N}\boldsymbol{S}(\boldsymbol{T}\_{i})-\boldsymbol{K}\right)^{+}|\boldsymbol{S}(t)=\boldsymbol{S}\right] \\ &=\mathbf{e}^{-r(\boldsymbol{T}\_{N}-t)}\frac{1}{N}\boldsymbol{E}^{\boldsymbol{\Theta}}\left[\left(\sum\_{i=1}^{N}\boldsymbol{S}(\boldsymbol{T}\_{i})-\boldsymbol{N}\boldsymbol{K}\right)^{+}|\boldsymbol{S}(t)=\boldsymbol{S}\right] \\ &=\mathbf{e}^{-r(\boldsymbol{T}\_{N}-t)}\frac{1}{N}\boldsymbol{E}^{\boldsymbol{\Theta}}\left[\left(\sum\_{i=1}^{N}(\boldsymbol{S}(\boldsymbol{T}\_{i})-\boldsymbol{K})\right)^{+}|\boldsymbol{S}(t)=\boldsymbol{S}\right] \\ &\leq\mathbf{e}^{-r(\boldsymbol{T}\_{N}-t)}\frac{1}{N}\sum\_{i=1}^{N}\boldsymbol{E}^{\boldsymbol{\Theta}}\left[(\boldsymbol{S}(\boldsymbol{T}\_{i})-\boldsymbol{K})^{+}|\boldsymbol{S}(t)=\boldsymbol{S}\right] \\ &=\frac{1}{N}\sum\_{i=1}^{N}\mathbf{e}^{-r(\boldsymbol{T}\_{N}-t)}\mathbf{e}^{r(\boldsymbol{T}\_{i}-t)}\boldsymbol{C}\_{\boldsymbol{E}}(t,\boldsymbol{S};\boldsymbol{K},\boldsymbol{T}\_{i}) \\ &=\frac{1}{N}\sum\_{i=1}^{N}\mathbf{e}^{-r(\boldsymbol{T}\_{N}-\boldsymbol{T}\_{i})}\boldsymbol{C}\_{\boldsymbol{E}}(t,\boldsymbol{S};\boldsymbol{K},\boldsymbol{T}\_{i}). \end{split}$$

In this case, we obtain the upper bound in Equation (22) which depends on the European plain vanilla call options on the spot freight rate *CE*(*t*, *S*; *K*, *Ti*), with maturities at the different dates of the settlement period, *Ti*, *i* = 1, . . . , *N*.

**Remark 4.** *As we mentioned in the previous section, pricing the freight call option (Equation (3)) is a complex task. However, its lower and upper bounds, presented in Equation (22), are easier to obtain: FFA and European vanilla option prices with several maturities are required. Therefore, the values of these bounds can be used as an estimation of the window where the freight call option price lies.*
