[subsection]section *[subsubsection]subsection

# On the Impact of Feeding Cost Risk in Aquaculture Valuation and Decision
Making

Christian Oliver Ewald , Department of Mathematics and Statistics, Umeå
University, Sweden, and Inland University of Applied Sciences, Lillehammer,
Norway, e-mail<EMAIL_ADDRESS>0000-0003-3288-0164 Kevin Kamm ,
Department of Mathematics and Statistics, Umeå University, Sweden, e-mail:
<EMAIL_ADDRESS>0000-0003-2881-0905

We study the effect of stochastic feeding costs on animal-based commodities
with particular focus on aquaculture. More specifically, we use soybean
futures to infer on the stochastic behaviour of salmon feed, which we assume
to follow a Schwartz-2-factor model. We compare the decision of harvesting
salmon using a decision rule assuming either deterministic or stochastic
feeding costs, i.e. including feeding cost risk. We identify cases, where
accounting for stochastic feeding costs leads to significant improvements as
well as cases where deterministic feeding costs are a good enough proxy.
Nevertheless, in all of these cases, the newly derived rules show superior
performance, while the additional computational costs are negligible. From a
methodological point of view, we demonstrate how to use Deep-Neural-Networks
to infer on the decision boundary that determines harvesting or continuation,
improving on more classical regression-based and curve-fitting methods. To
achieve this we use a deep classifier, which not only improves on previous
results but also scales well for higher dimensional problems, and in addition
mitigates effects due to model uncertainty, which we identify in this article.

eep Learning, Optimal Stopping, Real Options, Commodity Futures, Aquaculture

Code availability:  The Matlab/ Python code and data sets to produce the
numerical experiments are available at
https://github.com/kevinkamm/AquacultureStochasticFeeding.

##  1 Introduction

In this paper, we extend the results of [5] for valuation and optimal
decision-making in aquaculture management, taking account of risks attached to
a particular input factor: feed. While feed costs have been clearly identified
as a potential risk factor for aquaculture production, see for example [10]
and [11], feed cost risk has not been taken into account in models for
valuation and decision making of aquaculture businesses. This paper is the
first to explore this topic. Its relevance is increased due to higher recent
volatility in commodity markets.

More specifically, we will discuss whether accounting for the possibility of
feeding cost risk in valuation and decision-making makes a significant impact,
given the current scale of such risk relative to salmon output price risk.
This is a very relevant questions for aquaculture businesses and their
investors. In doing so, we introduce a methodology for comparing stopping
rules from different model frameworks to make a meaningful judgement. First,
we demonstrate a comparison of different stopping times, when conditioned on
single paths of the underlying stochastic process. Second, we train Deep
Classifiers to distinguish between the stopping and continuation region. Both
methods will yield very similar results, demonstrating that the neural
networks learn the decision rule very well. To this end, we will mainly use
the Least-Square-Monte-Carlo regression (LSMC) combined with [9] to perform
the backward induction in the optimal stopping problem for harvesting the
salmons. We also implemented a Deep Learning approach like in [1] and observed
that it performs well for this purpose.

The focus of this article is to understand the impact of feeding costs risk on
the decision-making in harvesting salmon. According to [11, pp. 28 ff. ] a
typical blend of salmon feed consists of soybean meal (30–44 percent), soybean
oil, rapeseed oil, wheat, corn, fish meal, and fish oil. The soy-based
ingredients make up the majority of the blend and for this reason, soybean
futures markets provide a good proxy for future price development of salmon
feed and its risk-characteristics, as well as providing hedging opportunities
for aquaculture businesses.

We will model both commodities, salmon and soy, using two decoupled
Schwartz-2-factor models (cf. [17]). We will investigate the impact of
parameters on the harvesting decision, as well as problems concerning the
estimation of the model parameters. More specifically, we will compare two
different techniques to calibrate the models: The first uses a Kalman filter
as originally suggested by [17], it is being facilitated by a large number of
authors. The second one uses a nested minimization based on [4], and is
facilitated by a somehow smaller number of authors. Both approaches are
conceptually very different. We find that both methods capture the
unobservable state variables well, but lead to very different estimates for
the model parameters. Both of these different parameter sets however result in
a relatively good fit to the futures market data. For pricing futures this
discrepancy may therefore be less of an issue, but, as we demonstrate, it has
a significant impact on the real option valuation and the optimal harvesting
rule discussed in this article. Such model uncertainty seems to be neglected
in the literature and requires more attention in future investigations of real
options.

To assume that the two commodities salmon and soy move independently from each
other may seem crude. In fact from the modeling perspective, this assumption
could be easily dropped, and the optimal stopping problem could still be
solved in the way we propose here. However, for the parameter estimation, it
is a bigger issue to include inter-commodity correlations reliably. We studied
the market data of both commodities and found different results for their
correlation. For data further in the past, we estimated a small (negligible)
correlation, while more recently a more significant correlation can be found.
However, this might be because of a common factor, most likely inflation, and
will not be further pursued here. In this paper, we will focus on stochastic
feeding costs in different scenarios, as described further below.

For an overview of the historical development for salmon commodity pricing and
valuation models, we refer the reader to [5]. Additionally, for a
comprehensive treatment of all the economic factors of fish farming we refer
the reader to [11] and [10]. In this article, we focus on (stochastic) feeding
costs.

[18] discuss salmon feed in detail and consider supply-chain issues, which are
implicitly included in this paper, since in the event of supply shortage, we
would expect the future prices to rise accordingly. Considering supply-chains,
we could also look at storage models such as in [14]. [6] investigate salmon
feed intake and overfeeding, which could be linked to a storage model.

Nevertheless, our article is the first article that looks at the impact of
feed cost risk on the decision-making of aquaculture businesses. There is
nothing alike in the literature.

The remainder of this paper is structured as follows: In Section 2, we
introduce the underlying commodity price model, followed in Section 2.1 by a
description of the salmon farm features considered in this paper. Afterwards,
in Section 2.2, we formulate the optimal decision making problem using both
deterministic and stochastic feeding costs. Numerical results will be
discussed in Section 3. We first describe the market data in Section 3.1,
followed by two calibration algorithms in Section 3.2. Following this, we
explain our methodology for comparing stochastic and deterministic feeding
rules in Section 3.3. For this we involve a deep neural network. Our main
results are presented and discussed in Section 3.4 while the main conclusions
of this paper are summarized in Section 4.

##  2 Mathematical Model and Framework

Henceforth, let $\left(\Omega,\mathcal{F},\mathbb{Q}\right)$ be a probability
space and $\mathbb{Q}$ be a risk-neutral measure. Moreover, let $T>0$ be a
finite time-horizon and $r>0$ a fixed deterministic interest rate.

We will use a multi-commodity framework consisting of two independent
Schwartz-2-factor models directly under $\mathbb{Q}$, i.e.

$\displaystyle dS_{t}^{i}$
$\displaystyle=\left(r-\delta_{t}^{i}\right)S_{t}^{i}dt+\sigma^{i}_{1}S_{t}^{i}dW_{t}^{i,1},\quad
S_{0}^{i}=s_{0}^{i}\in\mathbb{R}_{\geq 0,}$ $\displaystyle d\delta_{t}^{i}$
$\displaystyle=\left(\kappa^{i}\left(\alpha^{i}-\delta^{i}_{t}\right)-\lambda^{i}\right)dt+\sigma_{2}^{i}dW_{t}^{i,2},\quad\delta_{0}^{i}=d_{0}^{i}\in\mathbb{R}$
$\displaystyle d\langle W^{i,1},W^{i,2}\rangle_{t}$
$\displaystyle=\rho^{i}dt,\qquad d\langle W^{i,1},W^{j,2}\rangle_{t}=0,\quad
i\neq j,$

where $W_{t}\coloneqq\left(W_{t}^{i,j}\right)_{i,j=1,2}$ are Brownian motions
generating the $\sigma$-algebra $\mathcal{F}_{t}$ augmented by $\mathbb{Q}$
nullsets satisfying the usual conditions.

The dynamics $dS_{t}^{i}$ describe the i-th commodity’s spot price with
convenience yield described by $d\delta_{t}^{i}$. The parameter
$\sigma_{1}^{i}>0$ is the spot volatility, $\kappa^{i}>0$ the mean reversion
speed of the convenience yield, $\alpha^{i}\in\mathbb{R}$ the long-term mean,
$\lambda^{i}\geq 0$ a risk-premium and $\sigma_{2}^{i}>0$ the volatility of
the convenience yield.

It is straightforward to implement the optimal stopping problem in Section 2.2
so as to include inter-correlations for the Brownian motions or factors for
coupling of the SDEs. However, this is not the focus of this paper and a study
of such effects is deferred to future research.111The coupling of the SDEs
will pose some difficulties in the parameter calibration using standard tools
like Kalman-filtering, since closed form solutions for future prices will not
be available anymore. Inter-correlations might also be difficult to estimate
from the market data and may vary over time or exhibit stochastic behaviour.

Henceforth, $S^{1}_{t}$ will take the role of the salmon spot price and
$S_{t}^{2}$ will take the role of the soybean spot price. Since we assume
independence between $S^{1}_{t}$ and $S^{2}_{t}$, standard tools for
calibration of both commodities can be applied individually and is referred to
Section 3.2. Additionally, since we assume that the commodities are uncoupled,
we can use the results in [17] to obtain an explicit formula of the future/
forward222We assume that our interest rate is deterministic and constant.
Therefore, forward and future prices are the same, cf. [3, p. 456 Proposition
29.6 ]. prices in this model. We have for given spot price $S\in\mathbb{R}$,
convenience yield $\delta\in\mathbb{R}$ and time-to-maturity $\Delta T\geq 0$

$\displaystyle F(S,\delta,\Delta T)$
$\displaystyle=S\exp\left(-\delta\frac{1-e^{-\kappa\Delta T}}{\kappa}+A(\Delta
T)\right)$ $\displaystyle A(\Delta T)$
$\displaystyle=\left(r-\hat{\alpha}+\frac{1}{2}\frac{\sigma_{2}^{2}}{\kappa^{2}}-\frac{\sigma_{1}\sigma_{2}}{\kappa}\right)\Delta
T+\frac{1}{4}\sigma_{2}^{2}\frac{1-e^{-2\kappa\Delta
T}}{\kappa^{3}}+\left(\hat{\alpha}\kappa+\sigma_{1}\sigma_{2}\rho-\frac{\sigma_{2}^{2}}{\kappa}\right)\frac{1-e^{-\kappa\Delta
T}}{\kappa^{2}},$

where $\hat{\alpha}\coloneqq\alpha-\frac{\lambda}{\kappa}$ , omitting
superscripts $i$ referring to the $i$-th commodity for ease of notation, thus
superscripts have to be understood as exponents.

###  2.1 Salmon Farm Parameters

In this subsection, we briefly describe the features of a typical salmon farm,
referring the reader to [11] for a detailed discussion on general economic
issues concerning salmon farms.

We consider a single farm over one harvesting cycle, hence financial values
correspond to the value of a lease, as discussed in [5]. This means that a
fish farmer leases the farm, buys $\mathrm{n}_{0}\in\mathbb{N}$ smolt, young
salmon, and feeds them till they are ready for harvest, and at time of harvest
sells the fish for the market price and returns the farm. The growth of the
fishes over time depends on various factors like water temperature, amount and
quality of feed, health, etc. We simplify this by considering a deterministic
function over time, as so called _Bertalanffy’s growth function_ , which is
given by

$\displaystyle\mathrm{w}(t)$
$\displaystyle\coloneqq\mathrm{w}_{\infty}\left(a-b\,e^{-ct}\right)^{3},$

for parameters $a,b,c$ given in Table 1. This function measures the growth in
kg per fish. We assume a constant mortality rate $\mathrm{m}>0$ and the total
number of fishes $\mathrm{n}(t)$ at time $t$ follows the deterministic
dynamics

$\displaystyle d\mathrm{n}(t)$
$\displaystyle=-\mathrm{m}\,\mathrm{n}(t)dt,\quad\mathrm{n}(0)=\mathrm{n}_{0}.$

Now, we are able to measure the total biomass (in kg) of the fish farm over
time by setting

$\displaystyle\mathrm{B}(t)=\mathrm{n}(t)\mathrm{w}(t).$

We will consider only two varying factors of production costs: harvesting
costs and feeding costs, all other costs including labor, medical treatments,
capital costs, etc., will be treated as constants for simplicity. Harvesting
costs are given by $\mathrm{H}_{0}$ per kg of fish, the total harvesting costs
of the fish farm at time $t>0$ are therefore

$\displaystyle\mathrm{CH}(t)=\mathrm{H}_{0}\,\mathrm{B}(t).$

Feeding costs will be the main focus of this paper. We use a conversion rate
of how much kg of feed will convert to kg of fish

$\displaystyle\mathrm{c}\coloneqq 1.1\frac{\text{\,kg feed}}{\text{\,kg
fish}},$

a reasonable value. Let $F(0)=F_{0}$ be the initial feeding cost for one fish
per year. We will infer the feeding costs from the relative changes of soybean
prices $S_{t}^{2}$, i.e.

$\displaystyle\tilde{S}_{t}^{2}\coloneqq\frac{S_{t}^{2}}{S_{0}^{2}},$

by using

$\displaystyle\mathrm{F}^{\text{stoch}}_{t}=\mathrm{F}_{0}\tilde{S}_{t}^{2},\quad\mathrm{F}^{\text{determ}}_{t}=\mathrm{F}_{0}\mathbb{E}\left[\tilde{S}_{t}^{2}\right],$

and define the discounted cumulative total feeding costs as333The time-
derivative of the growth is denoted by $\partial_{t}\mathrm{w}$ and given by
$\left(\partial_{t}\mathrm{w}\right)(t)=3\mathrm{w}_{\infty}\left(a-b\,e^{-ct}\right)^{2}\,b\,c\,e^{-ct}$.
The term $\left(\partial_{t}\mathrm{w}\right)(s)$ captures the weight increase
of one fish in kg, $\left(\partial_{t}\mathrm{w}\right)(s)\,\mathrm{c}$ the
amount of feed in kg that is required for this growth,
$\mathrm{n}(s)\left(\partial_{t}\mathrm{w}\right)(s)\,\mathrm{c}$ the total
feed and finally
$\mathrm{F}_{s}\,\mathrm{n}(s)\left(\partial_{t}\mathrm{w}\right)(s)\,\mathrm{c}$
the total cost.

$\displaystyle\mathrm{CF}(t)=\int_{0}^{t}{e^{-rt}\left(\mathrm{F}_{s}\,\mathrm{n}(s)\left(\partial_{t}\mathrm{w}\right)(s)\,\mathrm{c}\right)ds}$

for general feeding costs $\mathrm{F}_{s}$, where once more an additional
upper index referring to stochastic vs. deterministic is omitted for ease of
notation.

The parameters in Table 1 are mostly taken from [5] and references therein.
The initial feeding and harvesting costs are estimated from [11, p. 25 Figure
9 ]. We will use the values in Table 1 for the remainder of this paper.

Table 1: Fish farm parameters. Name | Symbol | Value | Unit
---|---|---|---
Bertalanffy growth factor | $a$ | 1.113 | -
Bertalanffy growth factor | $b$ | 1.097 | -
Bertalanffy growth factor | $c$ | 1.43 | -
Asymptotic weight | $\mathrm{w}_{\infty}$ | 6 | kg
Mortality rate | $\mathrm{m}$ | 10 | %
Conversion rate | $\mathrm{c}$ | 1.1 | kg feed/ kg fish
Number of recruits | $\mathrm{n}_{0}$ | 10000 | fish
Time horizon | $T$ | 3 | years
Harvesting possibilities | $N$ | 72 | -
Salmon spot price | $\hat{S}_{0}^{1}$ | see Table 2 | NOK/ kg
Production costs | $\mathrm{PC}$ | $0.5\,\hat{S}_{0}^{1}$ | NOK/ kg
Harvesting costs | $\mathrm{H}_{0}$ | $0.1\,\mathrm{PC}$ | NOK/ kg
Feeding costs | $\mathrm{F}_{0}$ | $0.25\,\mathrm{PC}$ | NOK/ kg year
Salmon initial value | $S_{0}^{1}$ | $\hat{S}_{0}^{1}-\mathrm{PC}+\mathrm{H}_{0}+\mathrm{F}_{0}$ | NOK/ kg
Soy initial value | $S_{0}^{2}$ | 1 | -

###  2.2 Optimal Stopping Problem

Following [5, pp. 8 ff. ], the objective of a fish farmer is to find the
optimal harvesting time for salmon cultivated on the farm, where optimal has
to be understood as maximizing the expected value under a risk-neutral
measure. The latter corresponds to maximizing the financial value of the farm,
when appropriately taking account of relevant risk premia, see [5]. We
consider the current value of the fish at their current weight
$S_{t}^{1}\mathrm{B}(t)$ minus the harvesting costs at $\mathrm{CH}(t)$ and
the cumulative feeding costs up to this point in time $\mathrm{CF}(t)$. Let
$X_{t}^{\text{stoch}}\coloneqq\left(S_{t}^{1},\delta_{t}^{1},S_{t}^{2},\delta_{t}^{2}\right)$
and $X_{t}^{\text{determ}}\coloneqq\left(S_{t}^{1},\delta_{t}^{1}\right)$. The
optimal stopping problem for stochastic and deterministic feeding costs
becomes respectively

$\displaystyle W_{0}^{\text{stoch}}(x)$
$\displaystyle\coloneqq\sup_{\tau^{\text{stoch}}}\mathbb{E}^{\mathbb{Q}}\left[\left.\exp\left(-r\tau\right)\left(S_{\tau}^{1}\mathrm{B}(\tau)-\mathrm{CH}(\tau)\right)-\mathrm{CF}^{\text{stoch}}(\tau)\right|X_{0}^{\text{stoch}}=x\right],$
(1) $\displaystyle W_{0}^{\text{determ}}(x)$
$\displaystyle\coloneqq\sup_{\tau^{\text{determ}}}\mathbb{E}^{\mathbb{Q}}\left[\left.\exp\left(-r\tau\right)\left(S_{\tau}^{1}\mathrm{B}(\tau)-\mathrm{CH}(\tau)\right)-\mathrm{CF}^{\text{determ}}(\tau)\right|X_{0}^{\text{determ}}=x\right].$
(2)

In this paper, we compare the stopping rules obtained from
$W_{0}^{\text{stoch}}(x)$ and $W_{0}^{\text{determ}}(x)$ by evaluating

$\displaystyle
V_{0}^{z}(x)\coloneqq\mathbb{E}^{\mathbb{Q}}\left[\left.\exp\left(-r\tau^{z}\right)\left(S_{\tau^{z}}^{1}\mathrm{B}(\tau^{z})-\mathrm{CH}(\tau^{z})\right)-\mathrm{CF}^{\text{stoch}}(\tau^{z})\right|X_{0}^{\text{stoch}}=x\right]$
(3)

for $z\in\\{\text{stoch},\,\text{determ}\\}$ in an appropriate way as
described in Section 3.3, and investigate the question whether
$V_{0}^{\text{determ}}(x)$ is a good enough approximation of
$W_{0}^{\text{stoch}}(x)$ or alternatively in which cases it is beneficial to
consider the slightly more complicated stopping rule $\tau^{\text{stoch}}$.

We will solve both of the optimal stopping problems numerically by using
least-square Monte Carlo (LSMC), i.e. a backward induction with the [9]
algorithm and refer the reader to [5, pp. 9 ff. ] for the algorithmic details.
We use polynomials up to power two for the regression of the conditional
expectations in our implementation.

######  Remark 2.1.

It is straightforward to use the Deep Optimal Stopping Network by [1] and we
provide the code for using this as well. This allows in principle for high-
dimensional optimal stopping problems, which could become relevant if one
considers further stochastic factors. For example, using soybean futures as a
surrogate for the feeding costs is a more or less crude approximation of
reality, the feed consists of several different ingredients, also including
maize and fish meal (cf. [11, p. 29 Figure 11 ]), all could be modeled
stochastically as well. Introducing stochastic mortality and stochastic growth
of the fish would add to the further dimensions, making a least-square Monte-
Carlo regression less and less appealing, as the computational costs increase,
while the deep neural networks copes well.

##  3 Numerical Experiments

In this section, we will carry out some numerical experiments. In Section 3.1,
we will briefly discuss the historical data used for the calibration process
as described in Section 3.2. After the calibration, we will compare the
different feeding cost scenarios. We introduce two methods for this comparison
in Section 3.3. The first method compares the stopping rules obtained from
LSMC pathwise and the second method relies on classifying the stopping region
with a Deep-Neural-Network (DNN), providing an excellent alternative to the
more classical curve fitting process to identify the optimal exercise
boundary, as employed in [5] for example. Lastly, in Section 3.4, we present
our results and draw our main conclusion: In fact, it is beneficial to
consider stochastic feeding costs, at least under certain scenarios which we
identify.

For the calibration we used Matlab 2023a with the (Global) Optimization
Toolbox and for the DNN (Intel-)Python 3.10 with Tensorflow 2.8.0 running on
Windows 11 Pro, on a machine with the following specifications: processor
Intel(R) Core(TM) i9-13900K CPU @ 3.00 GHz and 2x32 GB (Dual Channel) Kingston
DIMM DDR5 RAM @ 5600 MHz. A GPU did not improve the performance in this case.

For all of our experiments we will fix the interest rate to $r=0.0303$.

###  3.1 Market Data

We use historical future data from 01/01/2006 till 01/01/2023 for both
salmon444The historical salmon future prices are publicly available on
https://fishpool.eu/forward-price-history/ (last accessed 04/07/2023 12:45
CEST). and soybeans. This is illustrated in Figure 1.

Figure 1: Interpolated (nearest neighbour) market future surfaces for salmon
(left) and soy (right) from 01/01/2006 till 01/01/2023. On the $x$-axis are
dates identified by integers, on the $y$-axis the time-to-maturity and on the
$z$-axis the corresponding price. Deep blue associates a low and yellow a high
price.

For the plots we used a nearest neighbour interpolation for fixed date in the
direction of the time-to-maturity. A dark blue color reflects low prices,
while a bright yellow color reflects high prices relative to the individual
commodity. On the left-hand side we can see the evolution of the salmon future
prices over time for different time-to-maturities and on the right-hand side
for the soybean futures. Notably, the volatility of these prices seem to
increase significantly at the end during the Covid-pandemic. For salmon the
prices seem to increase steadily over time with more rapid increase recently.
If we estimate the spot prices from the smallest time-to-maturity in these
pictures, we can also observe that the majority of the behaviour of the future
prices is determined by the spots, since they appear almost constant or with
non-rapid fluctuations on the time-to-maturity axis for a fixed date.

###  3.2 Calibration to Market Data

At the start of this subsection, we would like to remind the reader about two
well-known techniques for the calibration of multi-factor commodity pricing
models with hidden stare variables to historical futures price data. Since our
commodity models are uncoupled we will omit once again the dependence on the
individual commodities throughout this entire section. The first was
introduced by [4] as a nested least square regression the sefcond one is using
a Kalman filter (cf. [17]).

The goal is to find both, the model parameters

$\displaystyle\Pi\coloneqq[\sigma_{1},\ \sigma_{2},\ \kappa,\ \alpha,\
\lambda,\ \rho],$

and the unobservable spots and convenience yields for each date $S\geq 0$,
$\delta\in\mathbb{R}$.

For the calibration procedure we will restrict ourselves to a subset of the
main dataset. In fact we only consider the slice from 01/01/2018 to
01/01/2022, taking only the most recent price developments into account making
a regime-switching model unnecessary.

We will denote in this subsection the number of days, where market future
prices are available by $K\in\mathbb{N}$ and for each date the number of
available maturities by $P_{i}\in\mathbb{N}$. The market future prices at date
$i=1,\dots,K$ with time-to-maturity $\Delta T_{j}$, $j=1,\dots,P_{i}$ will be
denoted by $F^{\text{Market}}_{ij}$ and the model future prices depending on
the parameters $\Pi$ by $F^{\Pi}$.

####  [4].

We would like to minimize the following objective

$\displaystyle\min_{\Pi}\sum_{i=1}^{K}\sum_{j=1}^{P_{i}}\left(\log
F^{\Pi}\left(S_{t_{i}},\delta_{t_{i}},T_{j}-t_{i}\right)-\log
F^{\text{Market}}_{ij}\right)^{2}$

subject to

$\displaystyle(S_{t_{i}},\delta_{t_{i}})\in\operatorname*{argmin}_{S,\delta}\sum_{j=1}^{P_{i}}\left(\log
F^{\Pi}\left(S,\delta,T_{j}-t_{i}\right)-\log
F^{\text{Market}}_{ij}\right)^{2}\text{ for all }i=1,\dots,N.$

We need the constraint since the spot price and convenience yield are not
observable. The optimization for the state variables
$S_{t_{i}},\delta_{t_{i}}$ can be solved efficiently by a linear least-square
regression and for the objective we use Matlab 2023a’s fmincon with the
interior-point method. This is a simplified version of the algorithm presented
in [4] and is only used in order to stress that the model future prices are
mostly determined by the unobservable state variables and can have an equally
good fit to the market future prices for very different parameters $\Pi$.

#### Kalman filtering.

Following [17, pp. 10 ff. ], the idea is to first discretize the log-spot
price $s_{t}\coloneqq\log(S_{t})$ and convenience yield under the historical
measure $\mathbb{P}$, e.g. by using an Euler-Maruyama scheme. They are both
normally distributed. Thus, for a homogeneous time grid
$(t_{i})_{i=1,\dots,K}$ with step-size $\Delta t$, we have

$\displaystyle s_{t_{i+1}}$
$\displaystyle=s_{t_{i}}+(\mu-\delta_{t_{i}}-\frac{1}{2}\sigma_{1}^{2})\Delta
t+\sigma_{1}\Delta W_{t_{i}}^{1},$ $\displaystyle\delta_{t_{i+1}}$
$\displaystyle=\delta_{t_{i}}+\left(\kappa\left(\alpha-\delta_{t_{i}}\right)-\lambda\right)\Delta
t+\sigma_{2}\Delta W_{t_{i}}^{2}.$

Letting $X_{t_{i}}\coloneqq[s_{t_{i}},\delta_{t_{i}}]^{\top}$ we have the
_transition equation_

$\displaystyle X_{t_{i+1}}=c+T\cdot X_{t_{i}}+\eta_{i},$ $\displaystyle
c\coloneqq\left[\begin{array}[]{c}r-\frac{1}{2}\sigma_{1}^{2}\\\
\kappa\alpha\end{array}\right]\Delta t,\quad
T\coloneqq\left[\begin{array}[]{cc}1&-\Delta t\\\ 0&1-\kappa\Delta
t\end{array}\right],\quad
E\coloneqq\left[\begin{array}[]{cc}\sigma_{1}^{2}&\rho\sigma_{1}\sigma_{2}\\\
\rho\sigma_{1}\sigma_{2}&\sigma_{2}^{2}\end{array}\right]\Delta t$

in a so-called state-space formulation. Here $\eta_{i}\sim\mathcal{N}(0,E)$ is
serially uncorrelated. Since the spot prices are non-observable, we need to
_measure_ them. For this we use the log-future prices with different time-to-
maturities $\Delta T_{j}$ to derive the _measurement equation_

$\displaystyle Y_{t_{i}}=\left[d(\Delta
T_{j})\right]_{j=1,\dots,P}+\left[Z(\Delta T_{j})\right]_{j=1,\dots,P}\cdot
X_{t_{i}}+\epsilon_{i},\quad j=1,\dots,P,\ P\in\mathbb{N},$ $\displaystyle
d(\Delta T)\coloneqq A(\Delta T),\quad Z(\Delta
T)\coloneqq\left[\begin{array}[]{cc}1&\frac{-(1-e^{-\kappa\Delta
t}}{\kappa})\end{array}\right],\quad
D\coloneqq\mathrm{diag}\left(d_{1}^{2},\dots,d_{P}^{2}\right),$

where $\epsilon_{i}\sim\mathcal{N}(0,D)$ is serially uncorrelated and
independent of $\eta$. Notice, that $\epsilon$ has to be added as measurement
noise to make the model compatible for Kalman-filtering. The choice of the
covariance-matrix has a significant impact on the estimation. We chose in our
implementation a diagonal covariance matrix $D$. This choice seems to be the
standard for commodity models and has already been used by [17].

Using different panels, i.e., a collection of different time-to-maturities
$\Delta T_{j}$, can yield significantly different parameters. The panels are
assumed to be fixed throughout the entire filtering, which is a disadvantage
compared to [4], since it can use the entire data set without any
interpolation or averaging. In order to compare the Kalman filter to the
method by [4], we use a daily time step for the Kalman filter as well.

The parameters can now be estimated by maximizing the likelihood of
$Y_{t_{i}}$ equalling the market log-future prices and meanwhile estimating
the unobservable spot prices and convenience yields by the Kalman-filter. For
more details we refer to the code and the monographs [7, pp. 100 ff. Chapter 3
] and [2, pp. 635 ff. Chapter 13.3 ]. We use Matlab 2023a’s fmincon with the
interior-point method to minimize the negative log-likelihood.

#### Model uncertainty.

In Figure 2, we compare some calibration results using the soybean data from
01/01/2018 till 01/01/2022 using both a Kalman filter and [4]. We found the
following parameters

$\displaystyle\Pi^{\text{Kalman}}$ $\displaystyle=[0.2044,\ 0.1275,\ 0.1541,\
0.3355,\ 0.0101,\ 0.9065],$ $\displaystyle\Pi^{\text{CS}}$
$\displaystyle=[2.1975,\ 0.4594,\ 0.4381,\ 2.2825,\ 1.3120,\ 0.4893].$

The red color will show all the results using $\Pi^{\text{Kalman}}$ and the
magenta color for $\Pi^{\text{CS}}$. In the upper third of the figure we
compare the model future prices to the market future prices for different
time-to-maturity on the 16/12/2022, which was randomly chosen. We can see that
both models seem to regress the market prices (blue crosses) appropriately and
they are very close to each other. In the middle part of the figure we compare
the inferred spot prices $S_{0}$ for each date from 01/01/2018 till
01/01/2022. We can also see that both calibration methods seem to find very
similar spot prices. A little bit more variation can be seen in the inferred
convenience yields in the lower part of the figure, but still very similar. We
would like to stress at this point, that the model seems to produce very
similar results with extremely different parameters. If one only needs to find
the unobservable spot prices or fit the model to futures prices, then both
methods perform equally well, but if one needs the underlying model for more
complex decision making, this raises a major concern, as in our case. In fact,
for the real option discussed in this paper, the model parameters will have a
significant impact on the harvesting decision. We will address this problem
further in future research.

Figure 2: Close spot and future prices for different model parameters.

We believe that the Kalman filter underestimates the real volatility, while
the method by [4] overestimates it. Since there seems to be a lot of
uncertainty in finding appropriate parameters for our commodity models, we
will use the artificial parameters shown in Table 2 and Table 3 for our
further investigation. These are based on the calibration results, and chosen
such that we can discuss the question of whether stochastic feeding costs
should be considered in aquaculture investment or whether the effect is
negligible. The mean relative price changes of the commodity models using the
different parameters are illustrated in Figure 3. The left subfigure shows the
salmon prices. The red line corresponds to the scenario _down, down_ , the
green line to _down, up_ , and the blue line to _up, up_. Similarly for the
soy spot prices in the right subfigure. The red line corresponds to the
scenario _low volatility_ , the green line corresponds to _medium volatility_
and the blue line corresponds to _high volatility_.

We did a lot of tests and found that the most sensitive parameter for
estimating the benefit of using stochastic feeding costs is $\sigma_{1}^{2}$
for a given salmon scenario. Moreover, the closer the feeding costs are to the
salmon revenue, the more the feeding costs matter, which explains our choices
for selecting the salmon scenarios such that we have a mean decrease, mean
stable and mean increase scenario in prices. This could be achieved by only
adjusting the risk-premium parameter $\lambda^{1}$, since it changes the drift
of the convenience yield and thus the drift of the spot price.

Table 2: Parameters for different scenarios of salmon models. Scenario | $\sigma_{1}^{1}$ | $\sigma_{2}^{1}$ | $\kappa^{1}$ | $\alpha^{1}$ | $\lambda^{1}$ | $\rho^{1}$ | $\delta^{1}_{0}$ | $\hat{S}_{0}^{1}$
---|---|---|---|---|---|---|---|---
down, down | 0.23 | 0.75 | 2.6 | 0.02 | 0.01 | 0.9 | 0.57 | 95
down, up | 0.23 | 0.75 | 2.6 | 0.02 | 0.2 | 0.9 | 0.57 | 95
up, up | 0.23 | 0.75 | 2.6 | 0.02 | 0.6 | 0.9 | 0.57 | 95
Table 3: Parameters for different scenarios of soybean models. Scenario | $\sigma_{1}^{2}$ | $\sigma_{2}^{2}$ | $\kappa^{2}$ | $\alpha^{2}$ | $\lambda^{2}$ | $\rho^{2}$ | $\delta^{2}_{0}$ | $\hat{S}_{0}^{2}$
---|---|---|---|---|---|---|---|---
low volatility | 0.5 | 0.4 | 1.2 | 0.06 | 0.14 | 0.44 | 0 | 1500
medium volatility | 1 | 0.4 | 1.2 | 0.06 | 0.14 | 0.44 | 0 | 1500
high volatility | 2 | 0.4 | 1.2 | 0.06 | 0.14 | 0.44 | 0 | 1500

Figure 3: Mean relative changes of salmon (left) and soy (right) prices for
the different scenarios in Table 2–3.

We have made the code publicly available and encourage the reader to try other
parameter configurations.

###  3.3 Comparison of Exercise Decisions

In order to compare the exercise decisions in the different feeding cost
scenarios, we will employ two different methods. The first one, will be a
pathwise comparison of the the stopping times and the second one will use a
Deep-Neural-Network in order to learn the decision boundary.

#### Pathwise Comparison of Stopping Times.

Let $\left(\Omega,\mathcal{F},\mathcal{F}_{t},\mathbb{P}\right)$ be a filtered
probability space satisfying the usual conditions, where
$\mathcal{F}_{t}\coloneqq\sigma\left(X_{t}\right)$ is generated by a
stochastic process, and $\mathcal{F}^{i}_{t}\subseteq\mathcal{F}_{t}$,
$i=1,2$, be two sub-$\sigma$-algebras. Moreover, let $\tau^{1}$ be an
$\mathcal{F}^{1}_{t}-$stopping time and $\tau^{2}$ an
$\mathcal{F}^{2}_{t}-$stopping time. Now, let us fix a path
$(X_{t}(\omega))_{t\geq 0}$, then conditioned on this path the two stopping
times $\tau^{1}(\omega)\in[0,T]$ and $\tau^{2}(\omega)\in[0,T]$ can be
compared as real-numbers.

Additionally, still conditioned on $(X_{t}(\omega))_{t\geq 0}$, we may compare
the stopped values of a stochastic process $Y_{t}$, i.e.,
$Y_{\tau^{1}(\omega)}$ and $Y_{\tau^{2}(\omega)}$.

#### Deep Decision Classifier.

Given a stopping time $\tau$ achieving (2) we define the continuation and
exercise region at time $t$ by

$\displaystyle\mathcal{E}_{t}^{\tau}\coloneqq\left\\{X_{t}(\omega):\tau(\omega)=t\right\\},\quad\mathcal{C}_{t}^{\tau}\coloneqq\left\\{X_{t}(\omega):\tau(\omega)>t\right\\},$

These two sets are naturally labeled sets allowing for supervised learning.

Now, for each point $t$, we would like to find a function
$f_{t}^{\tau}:\mathbb{R}^{d}\rightarrow[0,1]$, associated to the probability
of value $x\in\mathbb{R}^{d}$ being in $\mathcal{E}_{t}^{\tau}$ and thus with
probability $1-f_{t}(x)$ in $\mathcal{C}_{t}^{\tau}$. For this classification
problem we use at each point in time a Deep-Neural-Network (DNN) for the
function $f_{t}$, whose last activation function is a softmax, using the
categorical crossentropy as a loss function.

For the training of the networks, we sample one million paths of the
commodities, determine the optimal stopping time by e.g. LSMC, and compute the
training sets $\mathcal{E}^{\tau}_{t}$ and $\mathcal{C}^{\tau}_{t}$ at each
point in time $t$. Usually, the number of elements in $\mathcal{C}_{t}^{\tau}$
is much higher than in $\mathcal{E}^{\tau}_{t}$. Hence, we reuse the elements
in $\mathcal{E}^{\tau}_{t}$ for the training of the network, such that for
each batch in the training set we have equal amount of continuations and
exercises. This was crucial for the training of the neural networks. The
specific hyperparameters and model architecture used in this paper are given
in Table 4 and we leave an optimization of these parameters to the reader. The
training for each network takes about $2$ seconds on a CPU in the case $d=2$
and $4$ seconds in the case $d=4$. A GPU did not speed up the training.

Table 4: Hyperparameters and architecture of $f_{t}^{\tau}$ Layer/Activation 1 | Batch normalization/ -
---|---
Layer/Activation 2 | Dense with $d*16$ neurons/ relu
Layer/Activation 3 | Dense with $d*32$ neurons/ relu
Layer/Activation 4 | Dense with $d*16$ neurons/ relu
Layer/Activation 5 | Batch normalization/ -
Layer/Activation 6 | Dense with $2$ neurons/ softmax
Batch size | 128 elements of each $\mathcal{E}_{t}^{\tau}$ and $\mathcal{C}_{t}^{\tau}$
Epochs | Chosen such that the entire training data is used
Optimizer | Adam with constant learning rate $0.001$

Having the trained neural networks for each point $t$, we can solve (2) by
using these stopping rules, i.e. if $f_{t}^{\tau}(x)\geq 0.5$ exercise,
otherwise continue. We will denote the value of the objective using this rule
by $V_{0}(f^{\tau})$. Now, having two different stopping times $\tau^{1}$ and
$\tau^{2}$ we can compare the values $V_{0}(f^{\tau^{1}})$ and
$V_{0}(f^{\tau^{2}})$ with each other. The neural networks have a practical
application, since after training one can ask the neural network in the real
world scenario whether one should exercise or not at each specific point in
time. This is not possible with just using the LSMC.

######  Remark 3.1.

In [5], a simple curve fitting for finding the exercise boundary for the case
of constant feeding costs was employed. However, the choice of the regression
function and the fit led to a lot of uncertainty and large confidence
intervals surrounding the exercise boundary. This is not the case for our DNN.
In higher dimensions, for example $d=4$, as in the case of stochastic feeding
costs, a curve fitting process is very difficult to perform, if not
impossible, since a good intuition for choosing a regression function is hard
to obtain.

As an alternative, one could also use Support-Vector-Machines (SVM) for this
classification problem. In fact, we did this as well and observed in the case
of deterministic feeding costs reasonably good results, i.e., the inferred
stopping rules $f_{t}$ produced values close to the LSMC value of the
objective. However, in the case of stochastic feeding costs the SVMs were not
able to classify the exercise values properly. Therefore, we resorted to DNNs,
which worked very well.

Our DNNs have to be understood as a means to compare different stopping rules
with each other, however we did not optimise these DNNs for performance. Also
they depend on another numerical algorithm to produce the exercise and
continuation sets and therefore cannot be used as an alternative to infer the
stopping region directly.

The reader is referred to [15] for an approach which is using a DNN for
finding the (fuzzy-) stopping region directly.

###  3.4 Results

In this section, we will assume stochastic feeding costs and demonstrate the
differences and improvements when the stopping rules obtained from accounting
for stochastic feeding costs are employed as compared to the case where
deterministic feeding costs are assumed. We compare all the different
scenarios using all the combinations of the parameters given in Table 2–3. For
the validation set and the pathwise comparison of the stopping times, we used
$M=100000$ simulations in the LSMC, which takes roughly $5$ seconds on the CPU
in the case of stochastic feeding costs. We used polynomials up to degree two
as a regression basis.

In Table 5, we show two metrics for the relative improvement of using
stochastic feeding costs compared to deterministic feeding costs. For this we
use the pathwise comparison and the one using Deep Classifiers, denoted by
$\mathrm{RI}^{\tau}\coloneqq\frac{V_{0}^{\tau,\text{stoch}}}{V_{0}^{\tau,\text{determ}}}$
and
$\mathrm{RI}^{f}\coloneqq\frac{V_{0}^{f,\text{stoch}}}{V_{0}^{f,\text{determ}}}$,
respectively.

As expected, we notice in the case of _low vol_ (first row of Table 5) that
the benefit of using a stopping rule accounting for the stochastic nature of
the feed costs is only slightly better than its deterministic counterpart. In
this case deterministic feeding costs seem to be a good approximation. The
benefit of using stochastic feeding costs however increases up to $11.6\,\%$
the higher the volatility, for the scenario of declining salmon prices.

The DNNs $f_{t}$ produce close values compared to the pathwise comparison in
both Table 5 and Table 6, showing that the neural networks are able to
classify the stopping region very well.

In Table 6, we present the corresponding values of the fish farm (3) using the
different stopping rules for all of the scenario combinations. Further, we
stress how much these values change in relation to changes in the risk-premium
$\lambda^{1}$, showing that the decision-making in the real option is very
sensitive to the model parameters.

Table 5: Relative improvements using stochastic feeding costs. Soy Salmon |  down, down |  down, up |  up, up
---|---|---|---
low vol |  $\mathrm{RI}^{\tau}=1.005$
$\mathrm{RI}^{f}=1.004$ |  $\mathrm{RI}^{\tau}=1.002$
$\mathrm{RI}^{f}=1.002$ |  $\mathrm{RI}^{\tau}=1.00019$
$\mathrm{RI}^{f}=1.00007$
medium vol |  $\mathrm{RI}^{\tau}=1.029$
$\mathrm{RI}^{f}=1.029$ |  $\mathrm{RI}^{\tau}=1.019$
$\mathrm{RI}^{f}=1.018$ |  $\mathrm{RI}^{\tau}=1.005$
$\mathrm{RI}^{f}=1.004$
high vol |  $\mathrm{RI}^{\tau}=1.116$
$\mathrm{RI}^{f}=1.116$ |  $\mathrm{RI}^{\tau}=1.088$
$\mathrm{RI}^{f}=1.089$ |  $\mathrm{RI}^{\tau}=1.045$
$\mathrm{RI}^{f}=1.045$
Table 6: Value of (3) using stochastic feeding costs with different stopping rules. Soy Salmon |  down, down |  down, up |  up, up
---|---|---|---
low vol |  $V_{0}^{\tau,\text{determ}}=1920575$
$V_{0}^{\tau,\text{stoch}}=1929700$
$V_{0}^{f,\text{determ}}=1920498$
$V_{0}^{f,\text{stoch}}=1928452$ |  $V_{0}^{\tau,\text{determ}}=2395735$
$V_{0}^{\tau,\text{stoch}}=2401140$
$V_{0}^{f,\text{determ}}=2395467$
$V_{0}^{f,\text{stoch}}=2399075$ |  $V_{0}^{\tau,\text{determ}}=3897624$
$V_{0}^{\tau,\text{stoch}}=3898397$
$V_{0}^{f,\text{determ}}=3896446$
$V_{0}^{f,\text{stoch}}=3896728$
medium vol |  $V_{0}^{\tau,\text{determ}}=1947469$
$V_{0}^{\tau,\text{stoch}}=2005608$
$V_{0}^{f,\text{determ}}=1947200$
$V_{0}^{f,\text{stoch}}=2003756$ |  $V_{0}^{\tau,\text{determ}}=2426319$
$V_{0}^{\tau,\text{stoch}}=2471621$
$V_{0}^{f,\text{determ}}=2425987$
$V_{0}^{f,\text{stoch}}=2469014$ |  $V_{0}^{\tau,\text{determ}}=3931880$
$V_{0}^{\tau,\text{stoch}}=3950537$
$V_{0}^{f,\text{determ}}=3930926$
$V_{0}^{f,\text{stoch}}=3946699$
high vol |  $V_{0}^{\tau,\text{determ}}=2001673$
$V_{0}^{\tau,\text{stoch}}=2232893$
$V_{0}^{f,\text{determ}}=2001876$
$V_{0}^{f,\text{stoch}}=2234064$ |  $V_{0}^{\tau,\text{determ}}=2488125$
$V_{0}^{\tau,\text{stoch}}=2706428$
$V_{0}^{f,\text{determ}}=2482972$
$V_{0}^{f,\text{stoch}}=2704694$ |  $V_{0}^{\tau,\text{determ}}=3997579$
$V_{0}^{\tau,\text{stoch}}=4179171$
$V_{0}^{f,\text{determ}}=3992845$
$V_{0}^{f,\text{stoch}}=4173701$

Let us focus on the first column in Table 6, the _down, down_ scenario. The
values $V_{0}^{\tau,\text{determ}}$ are close too each other regardless of the
volatility scenario for the feeding costs. On the other hand, the values
$V_{0}^{\tau,\text{stoch}}$ differ significantly and increase with higher
volatility. This means if the model is misspecified with e.g. too low
volatility, then the deterministic stopping rules will produce less revenue,
than the stochastic stopping rules, which can benefit from the higher
volatility. To mitigate the model misspecification, the neural networks
$f_{t}$ should be trained for different volatilities, such that a unified
decision rule can be applied for all cases. This improvement as well as using
recurrent neural networks for $f_{t}$ instead of different neural networks for
each $t$ will be left for future research.

If the initial feeding costs would have been $50\,\%$ of the production costs,
then e.g. in the case _down, up_ with _medium vol_ the benefit of using
stochastic feeding costs would increase to roughly $8\,\%$. This is similar
for all the other cases as well and highlights the importance of determining
the correct relationship of salmon revenue and feeding costs. For feeding
costs between $25\,\%$ and $50\,\%$ of the production costs the benefit will
lie in between $1.5\,\%$ and $8\,\%$. Thus, the benefits presented in Table 5
can be understood as a lower bound.

##  4 Conclusion

In this article, we focused on the question of whether accounting for
stochastic feeding costs can significantly improve aquaculture decision making
and increase the financial value of a fish farm as compared to the case when
deterministic feeding costs are assumed. We found a partial answer: If the
volatility of the feed price is high enough, then it is beneficial to use
stochastic feeding costs, otherwise deterministic feeding costs can lead to an
almost equal performance. In none of the cases, however, does the inclusion of
stochastic feeding costs lead to impaired performance, and as the
computational effort only marginally increases when using our proposed
methodology, we highly recommend to adopt relevant models to account for
feeding cost risk. In addition to that, as we demonstrated, a further
advantage is that the inclusion of stochastic feeding costs can make the
optimal stopping rule more robust with respect to model misspecification.

We found that there is a lot of model uncertainty, highlighted by the
significantly different parameters obtained from Kalman filtering and the
technique by [4]. Since this has a major impact on the real option, both value
and exercise, it needs to be addressed in future work. To this end we would
like to add a control to the objective (2) in terms of a model-free decision
rule, based on so-called myopic look ahead and $k$-step ahead rules (cf.
[16]). It might be possible to add such a control also to the Kalman filter
allowing for more robust parameter estimations.

In the line of feeding costs, we have seen that the relative difference of the
salmon prices to the feeding costs is important. If salmon prices are much
higher/ lower than feeding costs and tend to increase/ decrease a lot, then
the stochastic nature of the feeding costs may be negligible/ dominating. To
investigate this further, we could also study a more realistic case, in which
storage for the feed is available, as well as optimal pairs of buying time and
buying quantity combined with an adjustable feeding rate per day. Another
opportunity for future research is to further investigate the correlation
between salmon price and salmon feed and how this should be implemented and
estimated in classical multi-factor commodity models. An application of
reinforcement learning comes to mind to tackle this problem.

Moreover, we are interested in a case where the inter-correlations of the
commodities may be stochastic as well. As this leads to a full correlation
matrix, it can be modeled with a stochastic process taking values in an
appropriate Lie-group like in [13]. For this the techniques developed in [12]
and [8] may be very helpful.

In [11, pp. 40–45 ] it is argued that biological costs due to e.g. salmon lice
treatments are of equal size to the feeding costs. Thus, in future studies,
one should address a similar question for stochastic mortality or at an even
deeper level of detail, look at how host-parasite models and costs for salmon
lice treatments can be implemented with the classical salmon farm model.

## Declarations

The authors have no relevant financial or non-financial interests to disclose.

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