As carried out in
Hambly et al. (
2009), from now on, we assume that the model is specified in the risk-neutral measure and that the spot price is driven by the following one-factor process:
where
$h\left(t\right)$ is a deterministic function,
$F(0,t)$ is the forward curve derived from quoted products and
$X\left(t\right)$ is an OU-BCTS process. This market can easily be turned into a multifactor one, for instance by adding a second CTS process obtaining a tempered stable version of the two factor Gaussian model of
Schwartz and Smith (
2000). We nevertheless focus on the model (
31) to better highlight the results obtained for the OU-BCTS and OU-CGMY processes.
5.1. Call Options
We consider a daily strip of
M call options with maturity
T and strike
K, namely a contract with payoff
Such a contract is commonly used for hedging purposes or for the parameters’ calibration. It normally encompasses monthly, quarterly and yearly maturities but is not very liquid and is generally offered by brokers.
We assume that the market model (
31) is driven by a full seven-parameter OU-BCTS process with infinite activity and finite variation. The literature is rich in efficient numerical methods that one could employ to price European options under this market assumptions other than the Monte Carlo method. Among others, for instance, the CONV approach in
Lord et al. (
2007) consists of reformulating the risk-neutral valuation as a convolution, which is then accomplished numerically via FFT. Alternatively, one adopts wavelet-based pricing methods such as those presented in
Ortiz-Gracia and Oosterlee (
2013,
2016), which are applicable to a wide range of contracts.
We price the strip of calls using the FFT-based technique of
Carr and Madan (
1999) given the
chf $\varphi (u,t)$ of the of
$\mathrm{log}\phantom{\rule{0.166667em}{0ex}}S\left(t\right)=\mathrm{log}F(0,t)+h\left(t\right)+X\left(t\right)$
where
$h\left(t\right)$ is given by (
33) and
${\psi}_{Z}(u,t)$ by (
21). Briefly, each call option price
${c}_{m}(K,{t}_{m})$ can be written as (we set the interest rate to zero and drop
m for simplicity)
where
$k=\mathrm{log}K$ and
$q(s,t)$ is the risk-neutral density of the log-prices.
$c(k,t)$ is not square-integrable and its Fourier transform cannot be computed in contrast to the function
$\tilde{c}(k,t)={e}^{\alpha \phantom{\rule{0.166667em}{0ex}}k}\phantom{\rule{0.166667em}{0ex}}c(k,t)$,
$\alpha >0$, which is now square-integrable. Consider now the Fourier transform of
$\tilde{c}(k,t)$
the initial problem can be recast as
Carr and Madan (
1999) found an analytical formula for
$\psi (u,t)$, namely
Accordingly, call values are determined by combining the equations above and performing the required integration with FFT. We refer the reader to
Carr and Madan (
1999) for the details on the method and in particular to the choice of
$\alpha $.
The calibration and the parameters estimation is not the focus of this numerical experiment, and we rather take parameters sets available in the literature. In this example, we take those of
Poirot and Tankov (
2006) (plus
b and
${c}_{n}$) and let
${\alpha}_{p}$ and
${\alpha}_{n}$ vary:
$(b,{\beta}_{p},{\beta}_{n},{c}_{p},{c}_{n})=(0.1,2.5,3.5,0.5,1)$; for simplicity, we consider a flat forward curve with
$F(0,t)=20,t>0$.
Table 5 shows the values relative to a strip of
$M=30$ daily at-the-money call options with maturity
$T=1/12$ with different pairs of
${\alpha}_{p},{\alpha}_{n}$. We observe that fixing one of
${\alpha}_{p}$ or
${\alpha}_{n}$, the value of the option is increasing when the other one increases. Moreover,
Figure 1a illustrates the variability of the option price with respect to the strike price
K, where the dotted lines represent the values obtained with
$N={10}^{5}$ MC simulations plus and minus three times the estimation error (the root-mean squared error divided by
$\sqrt{N}$). In addition,
Figure 1b compares the price of at-the-money options
${C}_{m}={c}_{m}(K,{t}_{m}),m=1,\cdots ,M$,
$K=20$ obtained with the FFT method with those estimated once again with
$N={10}^{5}$ MC simulations. In these last two examples, we selected
${\alpha}_{p}={\alpha}_{n}=0.5$.
As far as the MC method is concerned, the simulation of the skeleton of the process is accomplished running the procedure explained in
Sabino and Cufaro Petroni (
2022) based on Proposition
4 two times because of the bilateral OU-BCTS; the acceptance rejection step to draw from the law of
V in (
15) assumes a piecewise approximation of the dominating functions into
$L=1000$ terms. The results calculated with the FFT method and with the MC method are totally consistent. On the other hand, it is well-known that the FFT approach is faster. Indeed, they can price vectors of strike prices, thus accelerating the process a lot. Nevertheless, a side product of the MC approach are percentiles or other statistics, which are widely used by practitioners for risk-management purposes.
5.2. Asian Options
As a second financial application, we consider the pricing of Asian options. In contrast to the previous example, we assume that the market dynamic is driven by an OU-CGMY process with infinite activity and finite variation with
$C={c}_{p}={c}_{n}$,
$G={\beta}_{n}$,
$M={\beta}_{p}$ and
$Y={\alpha}_{p}={\alpha}_{n}$ that we estimate from real data. We consider historical day-ahead prices of the Italian gas market PSV in the four-year time window from 1 January 2016 to 31 December 2019 for a total of 1461 values, see
Figure 2.
PSV is the main meeting point between supply and demand of the gas market in Italy. Here, the wholesale gas price is defined, and on the basis of this value, the gas suppliers evaluate the price of the gas raw material to be applied to end customers. There are two main types of contracts between market operators: take or pay and spot, and we focus on the latter type. Spot contracts are signed in the hubs, that is the hubs between the natural gas pipelines, usually located on the border between two states. The gas prices in Europe are strongly interdependent; therefore, the PSV price is affected by the evolution of the main gas prices in Europe, e.g., the Dutch TTF and the German THE and, of course, their values are also influenced by LNG and oil prices as well as from gas imports, e.g., from Algeria and Russia.
As mentioned, in this paper we rely on the
reduced-form process approach which consists of specifying a stochastic process for the price evolution from a parameterized family of mean-reverting processes (see
Eydeland and Wolyniek 2002). Similar examples including different commodity types can be traced back to
Schwartz (
1997), who studied oil prices, and after to
Benth and Šaltyté Benth (
2004), with applications to UK spot gas and Brent crude oil spot prices, or to
Geman (
2007) for natural gas and oil prices. Alternatively, one could adopt the different point of view of fundamental and hybrid models with a stronger focus on price forecasting rather that on risk-neutral pricing and hedging. To this end, the recent study by
Berrisch and Ziel (
2002) gives a rigorous data analysis of European gas prices, including the impact of autocorrelation, seasonality, risk premia, temperature, storage levels and the price of European Emission Allowances and the related fuel prices of oil, coal and electricity.
Coming back to the calibration problem, firstly, we check if the time- eries of the logarithm of the day-head prices is nonstationary using the augmented Dickey–Fuller test. On the selected dataset, it turns out that the ADF Statistic is $-1.791886$ with p-value equal to $0.384497$ and with critical values $-3.435$, $-2.864$ and $-2.568$ relative to $1\%,5\%$ and $10\%$, respectively. Hence, we have to accept the null hypothesis that the time series is nonstationary and does have a time-dependent structure.
Consequently, we remove the seasonality component from the logarithm of the day-ahead time series using a decomposition with a double cosine plus a linear trend, see
Figure 3. Secondly, we assume that the residuals of the deseasonalized time series, denoted
${s}_{k}$,
$k=1,\cdots ,1461$, follow an autoregressive model AR(1)
where
$\Delta t=1/365$ (one day) and
${\u03f5}_{k},k=1,\cdots $ is distributed according to the law of
$Z(\Delta t)={Z}_{p}(\Delta t)-{Z}_{n}(\Delta t)$ in (
4) of an OU-CGMY process. The mean reversion rate
b can be estimated using the standard least-squares method, however, according to Proposition 1, the
pdf of
$Z(\Delta t)$ is not known in closed form; therefore, one has to resort to MC-based techniques or to a generalized method of moments. On the other hand, before performing the parameter estimation, we Taylor expand
${\mathrm{\Lambda}}_{a}$ in Proposition 1
and observe that for
$\Delta t$ small,
${X}_{2}$ in Proposition 1 can be neglected (accordingly for the bilateral case), which is reasonable for
$\Delta t=1/365$. It turns our that under this approximation, labeled
Approximation 1,
$Z(\Delta \phantom{\rule{0.166667em}{0ex}}t)\sim \mathcal{BCTS}\phantom{\rule{-0.166667em}{0ex}}\left(Y,Y,\frac{M}{a},\frac{G}{a},C\phantom{\rule{0.166667em}{0ex}}\frac{1-{a}^{Y}}{Y\phantom{\rule{0.166667em}{0ex}}b},C\phantom{\rule{0.166667em}{0ex}}\frac{1-{a}^{Y}}{Y\phantom{\rule{0.166667em}{0ex}}b}\right)$, and we estimate the model parameters adapting the generalized method of moments to match the first four cumulants in (
28) and (29), namely, we perform the constrained optimization
Table 6 shows the estimated parameters, whereas
Table 7 reports the sample cumulants
$\widehat{{c}_{i}},i=1,\cdots ,4$ and the fitted cumulants
${\overline{c}}_{i}={c}_{i}(\widehat{C},\widehat{G},\widehat{M},\widehat{Y}),i=1,\cdots ,4$. The fit is in agreement with the sample cumulants, taking into account that the first and the fourth ones are very much close to zero.
We now come back to the initial problem of the pricing of Asian options with MC simulation. MC methods are known to be much slower than FFT techniques that can also be tailored to the pricing of Asian options (see
Zhang and Oosterlee 2013b). Nevertheless, the former approach provides a view on the distribution of the potential cash flows of derivative contracts, giving precious information to risk managers or to trading units.
We recall that the payoff at maturity
T of an Asian option with European style and strike price
K is
In addition to this simulation procedure, we consider here two approximations: the first one is
Approximation 1 defined in the parameter estimation; the second one—as also assumed in
Benth et al. (
2018) dealing with the
normal inverse Gaussian-driven OU processes—takes advantage of the approximation of the law of
$Z(\Delta t)$ with that of
$a\phantom{\rule{0.166667em}{0ex}}L(\Delta t)\sim \mathcal{BCTS}\left(Y,Y,\frac{M}{a},\frac{G}{a},C\phantom{\rule{0.166667em}{0ex}}\Delta \phantom{\rule{0.166667em}{0ex}}t,C\phantom{\rule{0.166667em}{0ex}}\Delta \phantom{\rule{0.166667em}{0ex}}t\right)$,
$a={e}^{-b\phantom{\rule{0.166667em}{0ex}}\Delta \phantom{\rule{0.166667em}{0ex}}t}$ (
Approximation 2).
In order to highlight the differences between the estimations returned by the exact method and those with the two alternatives, we consider two Asian options, both of them with
$I=90$ daily settlements. The second option, however, is a forward start contract whose first settlement date occurs after 30 days. The MC option values, their relative errors and computational times in seconds are reported in
Table 8 and
Table 9 with different
Y’s and number of simulations
N.
Irrespective to the combination of
Y and
N, for the option that starts settling after one day, the exact solution and
Approximation 1 return very close values, whereas
Approximation 2 is slightly biased. In contrast, for the forward start contract, although the time steps for
$m>1$ coincide and are very small, for the simple fact that the first time step is relatively high, the estimated prices returned by the two non-xact simulation schemes are quite biased and do not offer an acceptable alternative any longer. More importantly, the bias cannot be controlled by increasing the number of simulations, as shown in
Table 9. The cause of this difference it that for forward start options the compound Poisson term cannot be ignored, be it only for one simulation time point, and it is not computationally convenient to simulate 30 extra points with the two approximate methods.
These observations suggest a mixed strategy for parameters estimation and exact simulation of the OU-BCTS processes: Approximation 1 or Approximation 2 for the former task and use the exact and unbiased strategy for the path simulation to avoid being forced to always simulate the process on an unnecessary fine time grid.
5.3. Swing Options
A swing option is a type of contract used by investors in energy markets that lets the option holder buy a predetermined quantity of energy at a predetermined price (strike), while retaining a certain degree of flexibility in both the amount purchased and the price paid.
Let the maturity date
T be fixed and the payoff at time
$t<T$ be given by
${(S\left(t\right)-K)}^{+}$, where
K denotes the strike price. In addition, we assume that only one unit of the underlying can be exercised at any time period. Let
$V(n,s,t)$ denote the price of such a swing option at time
t given the spot price
s which has
n out of
N exercise rights left. For
$m=1,\cdots ,{M}_{S}$, the dynamic programming principle allows us to write (see
Bertsekas 2005)
and
$V(n,s,T)={(S\left(T\right)-K)}^{+}$,
$0<n\le N$ and
$V(0,s,t)=0$. In order to solve the recursion equation, we rely on the modified version of the LSMC, introduced in
Longstaff and Schwartz (
2001) and detailed in
Boogert and de Jong (
2008,
2011), where the continuation value is approximated with a linear regression with
$m=1,\cdots ,{M}_{S}$
In our experiments, we used simple power polynomials with
$B=3$, but the regression may be performed on a different set of basis functions as well (see
Boogert and de Jong 2011 for a comparison with other basis functions).
Several other approaches have been proposed: for instance one may solve the recursion by adapting the method of
Ben-Ameur et al. (
2007) or one might use the quantization technique of
Bardou et al. (
2009). Alternatively, one can also use the tree method of
Jaillet et al. (
2004) or the Fourier cosine expansion in
Zhang and Oosterlee (
2013a) taking advantage of the explicit form of the
chf of the OU-BCTS process. On the other hand, a swing contract can be seen as a degenerate case of gas storage, namely with no injection and withdrawal cost equal to the strike (see
Boogert and de Jong 2008), and one then could even adopt the recent approach of
Boonstra and Oosterlee (
2021) based on the COS method. In this last example, we assume an OU-CGMY-driven market model with
$Y<0$, namely a combination of mean-reverting compound Poisson processes with positive and negative jumps. We consider a different set of parameters compared with the cases illustrated so far, namely, we take
$(b,C,G,M)=(25,80,10.5,15.5)$ and let
Y vary. The parameters are very different than the other two examples and are chosen to mimic a realistic price path as shown in
Figure 5. We also remark that, due to the fact that energy markets are very seasonal and spikes occur in clusters due to for instance, cold spells, one could assume that the intensity of the compound Poisson processes is a seasonal time-dependent function. The results in
Section 3.2 and the simulation algorithms in
Section 4 can be easily adapted taking a step-wise approximation of the intensity function.
Table 10 shows the values and MC errors relative to the pricing of a
$120-120$ swing option with maturity
$T=1$ and strike price
$K=20$, namely, the holder has
$N=120$ rights and can exercise all of them. We also take a flat forward curve
$F(0,t)=20,t>0$. The computational time is now dominated by the stochastic optimization, which is independent of the specific price dynamics. On the other hand, the time needed by the path generation is similar to that shown in
Table 4. We observe that the LSMC combined with Algorithm 1 produces unbiased results for all selected
Y’s, and apparently
$2\times {10}^{4}$ simulations are required to attain an acceptable convergence. In contrast to the Asian option case, it does not make sense to adopt the approximation of the law of
$Z\left(t\right)$ in (
3) with that of
${e}^{-k\phantom{\rule{0.166667em}{0ex}}t}L\left(t\right)$ (
Approximation 2 in
Section 5.2) because this approach returns another compound Poisson process and therefore does not provide a computational advantage.