4.4 Relationship between Spot and Futures Prices
4.4.2 Pricing of Futures Contracts
Before we start elaborating on the spot price model we have chosen to work with, we here briefly explain the relationship between the spot price of commodities and futures contracts. From Chapter 3, we know that the best estimator of the future spot price is the current futures price
F(t, T) =E[ST | Ft],
where Ft is the filtration at time t. In order to avoid arbitrage opportunities, this relation must hold. This is not always the case, as the risk aversion of investors might cause the futures price to be both higher or lower than the expected future spot price given the nature of the current markets (Geman, 2005). By working in the risk-neutral word with theQ-probability, we can circumvent this inequality since we do not have to take into account any individual risk premium.
Let St be a stochastic process representing the spot price dynamics of an asset.
Assume a constant risk-free interest rate,r >0, so that the futures price and forward
price are equal because of the constant discounting. The explanation below is thus equally relevant for pricing of forwards as it is for pricing of futures contracts. Given that F(t, T) is the futures price at time t≤T, the payoff from the futures contract maturing in timeT, entered at time t, is
S(T)−F(t, T).
As we know that the discounted value of the futures contract at time t under the risk-neutral measureQis a martingale, it must be equal to the discounted expected value of the futures contract at timeT . And given that a there is no cost to enter a futures, we have
e−r(T−t)EtQ[S(T)−F(t, T)] = 0,
where EtQ represents the expectation with respect to Q. With this we have the formula for the futures price
F(t, T) = EtQ[S(T)]. (4.4.2) The futures price is therefore the expected spot price under the risk-neutral measure Q.
As a concrete example, we will below explain how the futures price is esti-mated assuming a two-factor model according to Gibson and Schwartz (Gibson and Schwartz, 1990; Schwartz, 1997). The spot price, assuming it has a lognormal distribution, is modeled under Q as
dS = (r−δ)Sdt+σSSdWS∗ dδ= [κ(α−δ)−λ]dt+σδdWδ∗,
wheredWS∗dWδ∗ =ρdtandλis the market price of convenience yield risk (Schwartz, 1997). Accordingly, futures prices must satisfy the following PDE
1
2σ2SS2FSS +σSσδρSFSδ+1 2σ2δFδδ
+ (r−δ)SFS+ (κ(α−δ)−λ)Fδ−FT = 0
(4.4.3)
subject to the boundary conditionF(S, δ,0) = S. Since the spot price is assumed to follow a lognormal distribution, and since the spot price and the convenience yield are jointly normally distributed, parameters for the transition density can be found (Erb, L¨uthi, and Otziger, 2010) under P. To find the equivalent parameters under
Q, we need to set µ = r and α = ˜α = α−λ/κ. The futures price under Q in t maturing at timeT, which is the solution to the PDE above, is given by3
F(S, δ, T) =EtQ(ST) =Stexp
A(T)− 1−e−κT κ δt
, (4.4.4)
with
A(T) =
r−α˜+1 2
σ2δ
κ2 − σSσδρ κ
T + 1
4σδ21−e2κT κ3 +
κα˜+σSσδρ−σδ2 κ
1−eκT κ2 .
(4.4.5)
Based on this procedure, we can move on to our spot price model and how we consequently estimated the futures prices in this model. As we will see, the futures price is rather similar to the one we just wrote from the two-factor model.
3For a detailed demonstration, please see Erb, L¨uthi, and Otziger (2010)
The 3-factor Spot Price Model
Having looked at different spot price models in the previous chapter, we decided on which model to choose for this analysis. Since we are dealing with a proxy hedging problem, some further considerations are needed in order to obtain as good results as possible. This chapter will first begin with a description of our data. Based on the characteristics of the data, we will introduce the 3-factor model that we have chosen as to match our data.
This chapter is organized as follows. The first section introduces the concrete problem we are facing, and what important characteristics should be taken into account to appropriately model our strategy. The second section gives an overview of our data. The third section mathematically describes the model. The fourth section goes into detail about how we estimate the model parameters, and the methods we have used to do so. Section five states the results of the estimations, and in the last and sixth section these results will be discussed.
5.1 Rationale
How should firms hedge when there is no perfect hedging instrument at hand? This question can be answered in several ways, and for this thesis we have chosen to focus on two rather specific studies. The first article by Ankirchner, Dimitroff, Heyne, and Pigorsch (2012), which we focus on is rather used as input to our model, and demonstrates how to optimally cross hedge when the spread between the futures used for hedging and the risky commodity is stationary. This spread, or so-called basis, becomes a source of risk since the underlying asset is not perfectly correlated
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to the futures contract. It is therefore of the essence to choose a futures contract that is highly correlated to the risky commodity. Given their findings of a strong positive long-term correlation between the two, they assume that the prices are cointegrated.
This is also supported by empirical tests. Instead of imposing a cointegrated vector, they assume that the spread between the log prices is stationary. The long-term relationship is therefore an essential part of their analysis, which allows for inclusion of the basis-risk which varies in a stationary way over time. This is modeled as an OU-process. While their futures price is modeled as a GBM, we chose another path.
The second, and mainly contributing, article we focus on is one by Bertus, God-bey, and Hilliard (2009). The main motivation of this article is also how to best strategize a cross hedge. They derive minimum variance hedges by assuming that the commodity price moves according to the two-factor model (Gibson and Schwartz, 1990), and that the log spread moves as a mean reverting OU-process. It is therefore a jointly estimated 3-factor model. They specifically focus on the airline industry, and cross hedge jet fuel risk with crude oil futures given their high positive correla-tion.
Our concrete problem is a large energy corporation that wants to hedge away risk in German gasoil. There is no liquid German futures market for gasoil, hence the need to hedge with another commodity. For this purpose, we have chosen two potential hedges: ARA gasoil and Brent crude oil futures. We will for both of these look at different hedging strategies, and finally compare their effectiveness.