Definition 5 If Xt evolves according to dXt =µX(Xt, t)dt+σX(Xt, t)dWt, and if we are dealing with a function f(Xt, t) that is twice differentiable in Xt and once differentiable in t, Itˆo’s lemma gives the first differential of f, df, as
df = ∂f
∂tdt+ ∂f
∂XtdXt+ 1 2
∂2f
∂Xt2(dXt)2
= ∂f
∂t +µX(Xt, t) ∂f
∂Xt +1
2σ2X(Xt, t)∂2f
∂Xt2
dt+σX(Xt, t) ∂f
∂XtdWt,
(4.2.12)
where in the second line, the expression for dXt has been incorporated into the formula. This lemma has some nice properties that will be helpful when solving SDEs1.
Figure 4.2: A simulated GBM process withµ= 0.5 andσ = 0.7
Definition 6 the GBM has the following properties (Miltersen, 2011) 1. The GBM X has positive values for the prices Xt,
2. Price returns over non-overlapping time intervals are independent, 3. X is Markovian, and
4. The distribution of future value returns of X is independent of the historical and current value of X.
A simulated GBM process can be seen in Figure 4.2.
As a concrete example, we will use one that is also considered by Eydeland and Wolyniec (2003), where we assume that the price process is described by the following geometric brownian motion (or SDE)
dSt=µSStdt+σSStdWt, (4.3.2) where S0 = s, Wt is the standard Wiener process, and µS and σS are constants.
Next, we define Zt = lnSt. Furthermore, St is a solution to the SDE, f is the function which is twice differentiable inS and once differentiable in t, we can define a stochastic process Z as
Zt =f(St, t), (4.3.3) and by Itˆo’s lemma we can find the SDE of Z. Writing f as a function of X and t, it becomesf(X, t) = lnX. The relevant derivatives are
∂f
∂t = 0, ∂f
∂X = 1 X, ∂2f
∂X2 =− 1
X2. (4.3.4)
Putting this into the formula for Itˆo’s lemma gives
dZt= (µS−0.5σS2)dt+σSdWt, (4.3.5) where Z0 = lnS0 and Zt ∼ N((µS−0.5σ2S)t, σS2t). Using this expression for the change indZt between times 0 and t, we can express it as
dZt=dlnSt = (µS−0.5σS2)dt+σSdWt
⇐⇒
lnSt= lnS0 + (µS−0.5σS2)dt+σSdWt
⇐⇒
St=S0exp{(µS−0.5σS2)dt+σSdWt},
(4.3.6)
implying thatStis lognormally distributed, such that prices are always non-negative, or equivalently that lnStis normally distributed with an expected value ofS0+(µS− 0.5σS2)t and a variance of σSt. As we defined above, Zt = lnSt, or if reversing this relationship,St= expZt. This means that the spot price is lognormally distributed, and that the conditional expected value and variance are as follows (Eydeland and Wolyniec, 2003)
E[St | F0] =S0eµSt
V ar[St| F0] =S02e2µSt(eσ2St−1).
(4.3.7) The conditional moments will be conditional on the information known at time 0, and this information can be expressed in terms of afiltration F, which basically is a formal way of explaining how information is revealed through time (Lando and Poulsen, 2006).
The famousBlack-Scholes formulais actually based on the assumption that asset prices follow GBMs. So in general, the GBM has been heavily used to model asset prices given its characteristics. However, when modeling commodity prices, other features need to be taken into account. This leads us to introduce the following model.
4.3.2 The 1-factor Model
As is often seen with for example commodity prices, they tend to revert back to a certain long-term level (Dixit and Pindyck, 1994). Instead of modeling the spot prices as geometric Brownian motions, it could be beneficial to model them as
Figure 4.3: A simulated OU process with α= 0.5,σ = 0.7 andκ= 7
mean reverting processes. The simplest of them is the so-calledOrnstein-Uhlenbeck process, abbreviated as OU process, developed by Ornstein and Uhlenbeck (1930) which looks like
dX =κ(α−X)dt+σdW, (4.3.8)
whereκ is the speed of mean reversion and α is the long-term expected level of X.
The larger κ is, the more stable will the process X be around α. For this process, the increments are no longer independent.
Figure 4.3 shows how a simulated OU process might look like. It reverts back to its long-term mean, and has a more stationary distribution than for example the standard Wiener process and the GBM. It is possible to find explicit solutions to the OU process by applying Itˆo’s lemma to specific functions of f.
Following the same line of extensions as above, we can reformulate the OU process as one that describes the return process. This might be more realistic, as the OU process does not in itself guarantee simulation of positive prices or returns
dX =κ(µ−X)Xdt+σXdW, (4.3.9) where proportional changes in X are mean-reverting. Schwartz (1997) wrote an article about the stochastic behavior of commodity prices. The first and most basic model that he investigates is the so-called1-factor model, where the log spot price
is assumed to be mean-reverting
dS =κ(µ−lnS)Sdt+σSdW. (4.3.10) By definingX = lnS, the application of Itˆo’s lemma gives us the same OU-process above forX
dX =κ(α−X)dt+σdW,
where α = µ− σ2κ2, κ > 0 measures the speed of mean reversion to the long-term expected log priceα, and dW is the standard Wiener process or Brownian motion.
What he further does is calculating the conditional first and second moments under the equivalent martingale measure Q. From that, using the properties of the log-normal distribution, he can calculate the futures prices implied by the model under the risk-neutral measure.
4.3.3 Gibson-Schwartz’ 2-factor Model
The next mean reverting model is the2-factor model considered by Schwartz (1997), and a bit earlier developed by Gibson and Schwartz (1990). Instead of letting only the spot price be stochastic, the convenience yield is also considered to be stochastic, and it follows an OU process. The interest rate is assumed to be constant. The two factors are following the joint process
dS = (µ−δ)Sdt+σSSdWS dδ =κ(α−δ)dt+σδdWδ,
(4.3.11) where the mean reversion is incorporated to the model through the stochastic con-venience yield, since the two Wiener processes are positively correlated2 through dWSdWδ = ρdt. If for example there is a positive shock to the spot price process dS, then there probably is a positive shock to the convenience yield process dδ as well given the positive correlation, makingdδtgo up. Then it is likely that the con-venience yieldδt will increase, which in the spot price process will lead to a decrease in dSt. This is, shortly put, how the mean reverting mechanism functions. The process is Markovian, since both processes only depend on current values and not historical ones. DefiningX = lnS, Itˆo’s lemma gives us the process for the log price dX = (µ−δ−0.5σS2)dt+σSdWS. (4.3.12)
2As described in Chapter 3, the correlation is positive given the inventory hypothesis
Again, writing the processes as risk-neutral, they derive the partial differential equation that futures prices must follow and the corresponding solution of futures prices. Schwartz (1997) also considers a 3-factor model, where the interest rate r also is stochastic. The three corresponding processes are correlated through their Wiener increments. In this case, he derives both forward and futures prices since those are not equal whenr is stochastic.
All the three models above are estimated and compared using oil and copper futures data. The estimation technique will be encountered in the next chapter, and is thus not explored further in this chapter. He finds that the increased degree of complexity of the 3-factor model is not outweighed by its performance in general, so that the 2-factor model is quite adequate. The 3-factor model does however perform better in the valuation of long-term futures contracts. Other than that, the 2-factor model performs relatively similar to the 3-factor model, and it is easier to work with. The two models also imply approximately the same futures volatilities, and this is because the volatility of the interest rate process in the 3-factor model is of a smaller order of magnitude than the volatilities of the other variables. Both of them outperform the 1-factor model.