**1. Introduction**

In this global economy, the transport of every kind of goods around the world has become of great importance. In fact, more than 95% of the world trade is carried by marine vessels, see [1].

The freight (transport by vessels) market is usually considered as a part of the commodity market. However, there are important differences between them. Most commodities are real products while a freight is a service and, as a result, it is not storable. Freight rates also present remarkable properties such as high volatility and risk. The cost of sea transport is affected by fleet supply and commodity demand, but also by external factors such as the price of bunker fuel or seasonal pressures, see [2]. As a consequence, freight derivatives were initially provided to protect ship-owners and charterers against risk. Besides, more recently financial institutions also found great opportunities in it.

There are different types of freight derivatives such as futures, forwards or options, but all of them depend on the freight rate in a settlement period before the maturity, see [3] for more detail. Traded freight options are contracts whose payoffs are the difference between the average of freight rates in a settlement period and the strike price. That is, they are arithmetic Asian-style options. This procedure avoids the possible manipulation, by large participants in the market, of the price just at maturity time. Moreover, the transportation of goods usually takes several days and the freight rates change along this time period.

Taking into account that the freight market is very recent, at the moment, not much scientific research has been done yet. When the spot freight follows a geometric process, a framework for its valuation is developed by Koekebakker et al. [4]. Tvedt [5] models the log spot freight rate in shipping by means of a geometric mean reversion process. Prokopczuk [1] considers that the log spot freight follows an Ornstein–Uhlenbeck process and studies the pricing and hedging of freight futures contract. When the log spot follows a jump-diffusion stochastic process, an accurate valuation of freight options is developed by Nomikos et al. [6] and Kyriakou et al. [7].

In general, in order to obtain a freight option price, it is necessary to use the conditional expectation under the risk-neutral measure because there is no valuation partial differential equation (PDE) for pricing this kind of options, unlike what happens with other derivatives (bonds, futures, European options, etc). Therefore, the Monte Carlo method is used to approximate this conditional expectation, see for example [8]. However, this method is very expensive and inaccurate from a computational point of view.

In this paper, we deal with the freight option valuation problem in two ways. On the one hand, we provide a novel partial differential equation whose solution is the freight option price. This PDE depends on three independent state variables: the spot freight rate, its delay and the continuous version of the average of the spot freight rate over a time period. This framework opens a new way to address this valuation problem. For example, this PDE could be used to obtain a partial explicit solution of the freight option in some models. In other cases, the solution could be approximated by using numerical methods for PDE. We obtained lower and upper bounds of the freight option price. These bounds provide valuable estimations to the option prices.

Our contributions need no restrictive conditions on the model: the spot freight follows a general stochastic diffusion process without restrictions in the drift and volatility functions.

The paper is arranged as follows. In Section 2, a one-factor diffusion model to price freight options is introduced. In Section 3, we provide a novel PDE for pricing these kind of options. In particular, for the geometric model, we obtained a partial solution for this price. In Section 4, we provide lower and upper bounds for the freight option prices. In Section 5, we compare these bounds with the freight option prices in a test problem using data from the Baltic Exchange. Finally, Section 6 concludes.

## **2. The Option Pricing Model**

In this section, we consider a general one-factor diffusion model, which we use to price freight derivatives.

Define (Ω, F, {F}*t*≥0,P) as a complete filtered probability space which satisfies the usual conditions and {F}*t*≥<sup>0</sup> is a filtration, see [9,10].

We assume that the spot freight rate follows the diffusion process, under the risk-neutral measure Q,

$$dS(t) = \mu(S(t))dt + \sigma(S(t))dW(t),\tag{1}$$

where *μ*(*S*) and *σ*(*S*) are the drift and volatility of the process, respectively, and *W* is a Wiener process. We suppose that the functions *μ* and *σ* satisfy suitable regularity conditions as follows (see [11]):

**Assumption 1.** *Functions <sup>μ</sup> and <sup>σ</sup> are measurable and there exists a constant C such that, for all x* <sup>∈</sup> <sup>R</sup>*,*

$$|\mu(\mathbf{x})| + |\sigma(\mathbf{x})| \le \mathcal{C} (1 + |\mathbf{x}|)\_{\prime\prime}$$

**Assumption 2.** *There exists a constant D such that, for all x*, *<sup>y</sup>* <sup>∈</sup> <sup>R</sup>*,*

$$|\mu(\mathbf{x}) - \mu(y)| + |\sigma(\mathbf{x}) - \sigma(y)| \le D|\mathbf{x} - y|.$$

The freight call option price at time *t*, with settlement period [*T*1, *TN*], *t* ≤ *TN*, and strike price *K*, can be expressed as *C*(*t*, *S*; *K*, *T*1,..., *TN*), and at maturity it is

$$\mathcal{C}(T\_{\mathcal{N}}, \mathcal{S}; \mathcal{K}, T\_{1}, \dots, T\_{\mathcal{N}}) = \left(\frac{1}{N} \sum\_{i=1}^{N} \mathcal{S}(T\_{i}) - K\right)^{+}. \tag{2}$$

On the other hand, we consider a discount factor *<sup>D</sup>*(*t*) = <sup>e</sup><sup>−</sup> *t* <sup>0</sup> *<sup>r</sup>*(*u*) *du*. If we assume that the riskless interest rate *r* is constant, then *D*(*t*) = e−*rt*. According to the fundamental theorem of asset pricing (see [10]), the price of a freight call option, at time *t*, and strike price *K*, is given by the following conditional expectation

$$\mathbb{C}(t, S; K, T\_1, \dots, T\_N) = \mathbf{e}^{-r(T\_N - t)} \, E^{\underline{\mathcal{Q}}} \left[ \left( \frac{1}{N} \sum\_{i=1}^N S(T\_i) - K \right)^+ \vert S(t) = S \right]. \tag{3}$$

This price can be represented, taking into account that it is a European call option on a forward freight agreement (FFA), by means of the following expectation (see [4])

$$\mathcal{L}(t, \mathcal{S}; \mathcal{K}, T\_1, \dots, T\_N) = \mathbf{e}^{-r(T\_N - t)} \, E^{\underline{\mathcal{Q}}} \left[ \left( F(T\_N, \mathcal{S}; T\_1, \dots, T\_N) - \mathcal{K} \right)^+ \left| \mathcal{S}(t) = \mathcal{S} \right] \right],\tag{4}$$

where *<sup>F</sup>*(*t*, *<sup>S</sup>*; *<sup>T</sup>*1, ... , *TN*) = *<sup>E</sup>*<sup>Q</sup> <sup>1</sup> *<sup>N</sup>* <sup>∑</sup>*<sup>N</sup> <sup>i</sup>*=<sup>1</sup> *S*(*Ti*)|*S*(*t*) = *S* ! is the FFA price with settlement period [*T*1, *TN*]. Finally, note that *<sup>F</sup>*(*TN*, *<sup>S</sup>*; *<sup>T</sup>*1,..., *TN*) = *<sup>E</sup>*<sup>Q</sup> <sup>1</sup> *<sup>N</sup>* <sup>∑</sup>*<sup>N</sup> <sup>i</sup>*=<sup>1</sup> *S*(*Ti*)|*S*(*TN*) = *S* ! = <sup>1</sup> *<sup>N</sup>* <sup>∑</sup>*<sup>N</sup> <sup>i</sup>*=<sup>1</sup> *S*(*Ti*).

## **3. Valuation Partial Differential Equation**

As we have seen in previous sections, freight options are arithmetic Asian-style options, where the average is calculated over a fixed settlement period. Even though in the standard Asian options the settlement period is the total period until maturity, in freight options it is a fixed period close to maturity.

With respect to the standard Asian options, geometric ones usually have an exact pricing formula, however, for arithmetic Asian options such a price does not exist. In the literature, this fact has led to use different methodologies for acceptable and tractable valuation: Monte Carlo simulation approach (see [12,13]) and numerical methods for the PDE provided in [14], as in [15,16], where the spot freight rate follows a geometric process. Moreover, when the arithmetic average is calculated on a fixed period lower than in the standard Asian options, there is not a valuation equation for pricing these freight options.

Therefore, Equation (3) is, nowadays, the main available method to price this kind of derivatives. Unfortunately, in general, it is not an easily manageable form for the empirical application. In order to provide a new framework that allows us to price the freight options in a different way, here we develop a PDE for pricing freight options when the spot freight follows a general diffusion stochastic process. To this end, we will make a similar reasoning for pricing standard Asian options, as in [14], but we need to incorporate a new variable, the delayed spot freight rate. Moreover, when the spot freight rate follows a geometric process, we obtain a partial solution to this PDE.

First, we consider a settlement period [*T*1, *TN*] such that *d* = *TN* − *T*<sup>1</sup> is a fixed time span, for example, one month. Then, we introduce a continuous version of the average of the spot price, for *t* ≤ *TN*, as the process *A*(*t*):

$$A(t) = \begin{cases} \int\_0^t S(z) \, dz, & \text{if} \quad 0 \le t \le d, \\\\ \int\_{t-d}^t S(z) \, dz, & \text{if} \quad \quad t > d. \end{cases} \tag{5}$$

We write Equation (5) in differential form and obtain the following stochastic delay differential equation

$$dA(t) = \begin{cases} \begin{array}{ll} S(t) \, dt, & \text{if} \quad 0 \le t \le d, \\\\ \left( S(t) - S(t-d) \right) dt, & \text{if} \quad \quad t > d, \end{cases} \end{cases} \tag{6}$$

In order to obtain the equation that verifies the freight option price, we introduce a new variable which is a delay of the spot freight rate along a time period *d*. We denote this delayed spot freight rate as the new variable

$$X(t) = \begin{cases} \ S(0), & \text{if} \quad 0 \le t \le d\_r \\\\ S(t-d), & \text{if} \quad -t > d. \end{cases}$$

Therefore, we can rewrite Equation (6) as

$$dA(t) = \begin{cases} S(t) \, dt, & \text{if} \quad 0 \le t \le d, \\\\ \left( S(t) - X(t) \right) \, dt, & \text{if} \quad \quad t > d. \end{cases}$$

Then, the process *A*(*t*) depends on the spot freight rate and on its delay value *X* as a new variable. In this case, we can approximate the average value of the spot freight rate in the discrete Equation (2), by means of Equation (5), in a continuous way as

$$\mathbb{C}(T\_{\mathcal{N}}, \mathbb{S}, X, A; K, T\_{\mathcal{I}}, \dots, T\_{\mathcal{N}}) = \left(\frac{1}{d} A(T\_{\mathcal{N}}) - K\right)^{+},\tag{7}$$

and the expectation in Equation (3) as

$$\mathbb{C}\left(t, \mathbb{S}, X, A; \mathbb{K}, T\_1, \dots, T\_N\right) = \mathbf{e}^{-r(T\_N - t)} \, \mathbb{E}^{\mathbb{Q}} \left[ \left(\frac{1}{d} A(T\_N) - \mathcal{K}\right)^+ \mid \mathcal{S}(t) = \mathcal{S}, X(t) = X, A(t) = A\right]. \tag{8}$$

The following theorem provides a PDE satisfied by the freight call option price.

**Theorem 1.** *The freight call option price function C*(*t*, *S*, *X*, *A*; *K*, *T*1, ... , *TN*) *in Equation (8) satisfies, when d* < *t* < *TN, the following PDE*

$$\begin{aligned} \mathbf{C}\_t + \mu(\mathbf{S})\mathbf{C}\_S + \mu(\mathbf{X})\mathbf{C}\_X + (\mathbf{S} - \mathbf{X})\mathbf{C}\_A + \frac{1}{2}\sigma^2(\mathbf{S})\mathbf{C}\_{SS} + \frac{1}{2}\sigma^2(\mathbf{X})\mathbf{C}\_{XX} - r\mathbf{C} &= \mathbf{0}, \\ \mathbf{S} > 0, \quad \mathbf{X} > 0, \quad A > 0. \end{aligned} \tag{9}$$

*However, when* 0 < *t* < *d, the function C in Equation (8) verifies the PDE*

$$\begin{aligned} \mathbf{C}\_t + \mu(\mathbf{S})\mathbf{C}\_S + \mathbf{S}\mathbf{C}\_A + \frac{1}{2}\sigma^2(\mathbf{S})\mathbf{C}\_{SS} - r\mathbf{C} &= \mathbf{0}, \\ \mathbf{S} > \mathbf{0}, \quad X > \mathbf{0}, \quad A > \mathbf{0}. \end{aligned} \tag{10}$$

**Proof of Theorem 1.** Applying arbitrage arguments in the market, the discounted freight option price is a martingale under the risk-neutral measure Q, see [10]. That is,

$$\begin{aligned} &E^{\triangle} \left[ D(T\_N) \mathcal{C}(T\_{N\prime}, \mathcal{S}, X, A; K, T\_1, \dots, T\_N) \middle| \mathcal{S}(t) = \mathcal{S}, X(t) = X, A(t) = A \right] \\ &= D(t) \mathcal{C}(t, \mathcal{S}, X, A; K, T\_1, \dots, T\_N). \end{aligned}$$

Then, in the development of *d*(*D*(*t*)*C*(*t*, *S*, *X*, *A*; *K*, *T*1,..., *TN*)), the *dt* term must be zero.

Note that *dSdS* = *σ*2(*S*)*dt* and

$$dXdX = \begin{cases} \ 0, & \text{if} \quad 0 < t < d\_{\tau} \\\\ \sigma^2(X) \, dt, & \text{if} \quad \quad t > d. \end{cases}$$

Moreover, *dSdX* = 0, because *dW*(*t*)*dW*(*t* − *d*) = 0, and *dAdA* = *dSdA* = *dXdA* = 0. Therefore, by means of Ito Lemma, for *d* < *t* < *TN*, we obtain

$$\begin{cases} d(\mathbf{e}^{-rt}\mathbb{C}) = \mathbf{e}^{-rt} \left( -r\mathbb{C} + \mathbb{C}\_{l} + \mu(\mathcal{S})\mathbb{C}\_{S} + \mu(X)\mathbb{C}\_{X} + (\mathbb{S} - X)\mathbb{C}\_{A} + \frac{1}{2}\sigma^{2}(\mathcal{S})\mathbb{C}\_{\mathcal{S}S} + \frac{1}{2}\sigma^{2}(X)\mathbb{C}\_{XX} \right) dt \\ \qquad + \mathbf{e}^{-rt} \left( \mathbb{C}\_{S}\sigma(\mathcal{S})dW(t) + \mathbb{C}\_{X}\sigma(X)dW(t-d) \right), \end{cases} \tag{11}$$

and for 0 < *t* < *d*,

$$d(\mathbf{e}^{-rt}\mathbf{C}) = \mathbf{e}^{-rt}\left(-r\mathbf{C} + \mathbf{C}\_l + \mu(\mathbf{S})\mathbf{C}\_S + \mathbf{S}\mathbf{C}\_A + \frac{1}{2}\sigma^2(\mathbf{S})\mathbf{C}\_{\mathcal{SS}}\right)dt + \mathbf{e}^{-rt}\mathbf{C}\_S\sigma(\mathbf{S})d\mathcal{W}(t). \tag{12}$$

Finally, the vanishing of the *dt* terms in Equations (11) and (12) leads to Equations (9) and (10), respectively.

**Remark 1.** *This result allows us to address the valuation problem of freight options in a new way: We obtain a pure final value problem associated to a PDE whose solution gives the freight option price. However, it is very difficult to solve this problem, except in some particular cases. Next, we will consider one of these situations.*

In the freight options literature, some stochastic processes are commonly used to describe the dynamic of the spot freight rate. In particular, it is usual to consider a geometric process where the functions in Equation (1) are *μ*(*S*) = *μS* and *σ*(*S*) = *σS*, with constants *μ* and *σ*. In such a case, in the literature there exist some techniques to approximate the freight option prices although none of them are exact solutions. However, in a similar way to [14], here we value the option on the FFA when the average of the spot freight verifies *A* ≥ *dK*, by solving the PDEs Equations (9) and (10) in Theorem 1.

**Proposition 1.** *Let μ*(*S*) = *μS and σ*(*S*) = *σS be the drift and volatility of the process (Equation (1)), respectively, with μ and σ constants. Then, the following function is solution to the PDEs, seen in Equations (9) and (10) and verifies the final condition of Equation (7) when A* ≥ *dK:*

$$C(t, S, X, A; K, T\_1, \dots, T\_N) = 0$$

$$\begin{cases} \left( \frac{1}{d} A - K \right) e^{-r(T\_N - t)} + \frac{\varepsilon^{-r(T\_N - t)}}{d\mu} \left( S(e^{\mu(T\_N - t)} - 1) - X(e^{\mu(T\_N - d)} - 1) \right), & 0 \le t \le d, \\\\ \left( \frac{1}{d} A - K \right) e^{-r(T\_N - t)} + \frac{\varepsilon^{-r(T\_N - t)}}{d\mu} (S - X)(e^{\mu(T\_N - t)} - 1), & d \le t \le T\_N. \end{cases} \tag{13}$$

**Proof of Proposition 1.** First of all, we change the time variable by considering *τ* = *TN* − *t*. Then, from Equations (7) and (9) we have the initial value problem

$$\mathbf{C}\_{\tau} = \mu \mathbf{S} \mathbf{C}\_{S} + \mu \mathbf{X} \mathbf{C}\_{X} + (\mathbf{S} - \mathbf{X}) \mathbf{C}\_{A} + \frac{1}{2} \sigma^{2} S^{2} \mathbf{C}\_{SS} + \frac{1}{2} \sigma^{2} X^{2} \mathbf{C}\_{XX} - r \mathbf{C}, \quad 0 < \tau < T\_{N} - d,\tag{14}$$

$$\mathbb{C}(0, S, X, A; K) = \left(\frac{1}{d}A - K\right)^{+}.\tag{15}$$

For *A* ≥ *dK*, as in [14], we look for a linear solution to this problem as:

$$\mathcal{L}(\tau, S, X, A; K) = \left(\frac{1}{d}A - K\right)B\_1(\tau) + (S - X)B\_2(\tau), \quad 0 \le \tau \le T\_N - d,\tag{16}$$

where *B*<sup>1</sup> and *B*<sup>2</sup> are functions depending only of time. Replacing Equation (16) in the PDE Equation (14), we obtain that the functions *B*<sup>1</sup> and *B*<sup>2</sup> must verify the following system of ordinary differential equations

$$\begin{aligned} B\_1'(\tau) &= -r B\_1(\tau), \\ B\_2'(\tau) &= (\mu - r) B\_2(\tau) + \frac{1}{d} B\_1(\tau). \end{aligned}$$

From Equation (15), we get the initial conditions *B*1(0) = 1 and *B*2(0) = 0. Solving this system, we obtain the solution to the problem Equations (14) and (15)

$$\mathbf{C}(\tau, S, X, A; K) = \left(\frac{1}{d}A - K\right)\mathbf{e}^{-r\tau} + (S - X)\frac{\mathbf{e}^{-r\tau}}{d\mu}(\mathbf{e}^{\mu\tau} - 1), \quad 0 \le \tau \le T\_N - d. \tag{17}$$

Now, the same change of variable *τ* in Equation (10), and the value of Equation (17) in *TN* − *d*, provide the initial value problem

$$\text{C}\_{\text{7}} = \mu \text{SC}\_{\text{S}} + \text{SC}\_{A} + \frac{1}{2} \sigma^{2} \text{S}^{2} - r \text{C}, \quad T\_{\text{N}} - d < \text{\textdegree} < T\_{\text{N}}.\tag{18}$$

$$\mathbf{C}(T\_N - d, \mathbf{S}, X, A; K) = \left(\frac{1}{d}A - K\right)\mathbf{e}^{-r(T\_N - d)} + (\mathbf{S} - X)\frac{\mathbf{e}^{-r(T\_N - d)}}{d\mu}(\mathbf{e}^{\mu(T\_N - d)} - 1). \tag{19}$$

Again, we look for a linear solution as

$$\mathcal{C}(\mathbf{r}, \mathbf{S}, X, A; K) = \left(\frac{1}{d}A - K\right)A\_1(\mathbf{r}) + SA\_2(\mathbf{r}) + XA\_3(\mathbf{r}), \quad T\_N - d \le \mathbf{r} \le T\_{N\prime} \tag{20}$$

where *A*1, *A*<sup>2</sup> and *A*<sup>3</sup> are functions of time.

If we replace Equation (20) into the PDE Equation (18) we obtain that *A*1, *A*<sup>2</sup> and *A*<sup>3</sup> verify the system of ordinary differential equations:

$$\begin{aligned} A\_1'(\tau) &= -r A\_1(\tau), \\ A\_2'(\tau) &= (\mu - r) A\_2(\tau) + \frac{1}{d} A\_1(\tau), \\ A\_3'(\tau) &= -r A\_3(\tau), \end{aligned} \tag{21}$$

and from Equation (19) we derive the initial conditions

$$\begin{aligned} A\_1(T\_N - d) &= \mathbf{e}^{-r(T\_N - d)}, \\ A\_2(T\_N - d) &= \frac{\mathbf{e}^{-r(T\_N - d)}}{d\mu} (\mathbf{e}^{\mu(T\_N - d)} - 1), \\ A\_3(T\_N - d) &= -\frac{\mathbf{e}^{-r(T\_N - d)}}{d\mu} (\mathbf{e}^{\mu(T\_N - d)} - 1). \end{aligned}$$

Solving the system Equation (21) with the previous initial conditions, we obtain the solution

$$\mathbb{C}(\tau, S, X, A; K) = \left(\frac{1}{d}A - K\right) \mathbf{e}^{-r\tau} + S \frac{\mathbf{e}^{-r\tau}}{d\mu} (\mathbf{e}^{\mu\tau} - 1) - X \frac{\mathbf{e}^{-r\tau}}{d\mu} (\mathbf{e}^{\mu(T\_N - d)} - 1), \quad T\_N - d \le \tau \le T\_N.$$

Finally, if we return to the original time variable *t*, we obtain the expression in Equation (13) for *C*: which provides the call freight option price when *A* ≥ *dK*.

**Remark 2.** *Note that the solution that provides Equation (13) is only valid for A* ≥ *dK. Unfortunately, for other values of the continuous average of the spot rate we do not have an explicit expression for the freight call option price. Therefore, even in this simple case, the partial solution to the PDE that we get is not sufficiently to price the freight option. However, it could be useful for the numerical solution of the problem, as we remark in a later section.*

**Remark 3.** *Although knowing the PDE problem previously described is not sufficient, in general, to get the exact price of the option, we could use numerical methods in order to approximate its solution. However, this is a very hard problem. On the one hand, the PDE involves four independents variables: the time, the spot rate, its delay and the average of the spot rate in the settlement period. Then, it is necessary to design suitable specific numerical methods for this expensive multidimensional problem. On the other hand, the application of numerical methods for a pure final problem requires appropriate boundary conditions. In this sense, for the specific stochastic processes considered in Proposition 1, we can use Equation (13) to obtain such boundary conditions (in a similar way to [14] for Asian options). In any case, the numerical approach of this problem is beyond the scope of this work.*
