Multi-Factor Energy Price Models
and Exotic Derivatives Pricing
by
Samuel Hikspoors
A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Statistics University of Toronto
c Copyright by Samuel Hikspoors 2008 Multi-Factor Energy Price Models and Exotic Derivatives Pricing
Samuel Hikspoors
Doctor of Philosophy, Department of Statistics University of Toronto, May 2008
Abstract
The high pace at which many of the world’s energy markets have gradually been opened to competition have generated a significant amount of new financial activity. Both academi- cians and practitioners alike recently started to develop the tools of energy derivatives pricing/hedging as a quantitative topic of its own. The energy contract structures as well as their underlying asset properties set the energy risk management industry apart from its more standard equity and fixed income counterparts. This thesis naturaly contributes to these broad market developments in participating to the advances of the mathematical tools aiming at a better theory of energy contingent claim pricing/hedging. We propose many realistic two-factor and three-factor models for spot and forward price processes that generalize some well known and standard modeling assumptions. We develop the associated pricing methodologies and propose stable calibration algorithms that motivate the application of the relevant modeling schemes.
ii A Monique, Jean, Anna et Jean-Paul
Pour tout l’amour et le support...
iii Acknowledgements
This work would hardly have been possible without the support and collaboration of my supervisor and friend, Sebastian Jaimungal. These last few years have been a great learning experience; thanks Seb!
I would also like to thank Kenneth R. Jackson, Luis Seco and Sheldon Lin for having been part of my supervisory committee, and Matt Davison (UWO) for having been my external examinator; Thank you all for your guidance.
Finally, a few friends who very much influenced me over the years: Gu, JP, Tone, Louis, PO, Kam and Lanna; wish you all the best!
iv Contents
1 Introduction 1
2 Background Material 5
2.1 Derivatives Pricing/Hedging ...... 6
2.2 Asymptotic Tools and Terminology ...... 7
3 First Paper: Energy Spot Price Models and Spread Options Pricing 12
3.1 Abstract ...... 12
3.2 Introduction ...... 13
3.3 Real World Dynamics and Pricing ...... 16
3.3.1 Model Specifications ...... 16
3.3.2 Spot Spread Valuation : An Actuarial Approach ...... 19
3.4 Risk-Neutral Dynamics and Pricing ...... 24
3.4.1 Measure Change ...... 25
3.4.2 Forward Prices ...... 27
3.4.3 Spot Spread Valuation ...... 28
3.4.4 Forward Spread Valuation ...... 29
3.5 Spot Prices with Jumps ...... 31
3.5.1 Model Specification ...... 31
3.5.2 Forward Prices ...... 33
v 3.5.3 Spark Spread Valuation ...... 34
3.6 Model Calibration ...... 39
3.6.1 Methodology ...... 40
3.6.2 Some Results: Crude Oil ...... 42
3.7 Conclusions ...... 46
3.8 Acknowledgements ...... 47
4 Second Paper: Asymptotic Pricing of Commodity Derivatives using Stochastic Volatility Spot Models 48
4.1 Abstract ...... 48
4.2 Introduction ...... 49
4.3 Spot Price Models and Main Properties ...... 53
4.3.1 Constant Volatility Models ...... 53
4.3.2 Stochastic Volatility Extensions ...... 56
4.4 Forward Price Approximation ...... 59
4.4.1 One-Factor Model + SV ...... 60
4.4.2 Two-Factor Model + SV ...... 67
4.5 European Single-Name Options ...... 68
4.5.1 Smooth Payoff Function ...... 68
4.5.2 Nonsmooth Payoff: Calls and Puts ...... 73
4.6 European Two-Name Options ...... 74
4.6.1 Smooth Payoff Function ...... 74
4.6.2 Nonsmooth Payoff: Forward Spread ...... 85
4.7 Conclusions and Future Work ...... 86
4.8 Appendix ...... 88
5 Unspanned Stochastic Volatility for Energy Derivatives: A Forward
vi Price Approach 90 5.1 Abstract ...... 90 5.2 Introduction ...... 91 5.3 Model Description ...... 94 5.4 Single-Name Options Valuation ...... 97 5.5 Calendar Spreads Valuation ...... 103
5.5.1 Case I: A Constrained Correlation Structure (ρ12 := 0) ...... 104
5.5.2 Case II: A Constrained Volatility Structure (f := σX ) ...... 107 5.6 Calibration ...... 110 5.7 Conclusions ...... 113 5.8 Appendix I: Sketch of Proof of Theorem 5.4.1 ...... 114 5.9 Appendix II: Expansion Coefficients ...... 116
6 Conclusion 122
vii List of Figures
3.1 The NYMEX Light Sweet Crude Oil spot prices and simulated spot prices based on the calibration in Table 3.1...... 43 3.2 The relative root-mean squared-error of each forward curve based on the calibration in Table 3.1...... 44 3.3 This diagram depicts the evolution of the implied market prices of risk using the calibrated real-world and risk-neutral parameters...... 45
4.1 The annualized running five-day moving volatility of the NYMEX sweet crude oil spot price for the period 10/07/03 to 07/03/06...... 50 4.2 This diagram depicts typical forward curves implied the model for three choices of V . The long-run forward price is set at 61 in the left panel and
59 in the right panel. The spot is 60, β = 0.5 and σX = 0.2...... 67
5.1 Implied Volatility Curves on 4-Jan-2006, for three increasing term. . . . . 111 5.2 Implied Volatility Curves on 10-Jan-2006, for three increasing term. . . . 111
viii List of Tables
3.1 The calibrated real-world and risk-neutral model parameters using the NYMEX Light Sweet Crude Oil spot and futures data for the period 1/10/2003 − 25/07/2006...... 42 3.2 This table shows the evolution of the estimated risk-neutral parameters through time as more recent data is added to the calibration procedure. The average and standard deviation are reported using 176 days onwards. 44
∗ 5.1 The calibrated macro-parameters (volatility expansion method), with Di := √ Di, using the NYMEX Light Sweet Crude Oil implied volatilities on 04/01/2006...... 112
∗ 5.2 The calibrated macro-parameters (volatility expansion method), with Di := √ Di, using the NYMEX Light Sweet Crude Oil implied volatilities on 10/01/2006...... 113
ix Chapter 1
Introduction
The last few years have seen a tremendous increase of activity within the energy risk management industry. The various level of deregulation of many energy markets greatly increased the price volatility. This deregulation process offered the possibility to benefit from market price fluctuations, but it also created the likelihood of large adverse price movements. Most energy companies and major energy consumers have therefore increased the variety of risk management tools and techniques they use. Various financial institu- tions have also developed significant resources to address the hedging and speculative needs of other industry players as well as of their own energy portfolios.
These last financial developments generated a need for new pricing/hedging tech- niques adapted to the special contract structures found within energy markets. Both the fundamentals of the energy price processes and the structures of derivatives contracts significantly differ from their equity and interest rate counterparts. Many practitioners and academics alike have been working on both theoretical and practical issues of this relatively new branch of financial engineering.
The work presented in this thesis can be seen within this financial engineering context. It focuses mainly on the developments of new stochastic models and mathematical tech-
1 1 Introduction 2 niques toward a more complete theory of pricing/hedging of complex energy derivatives. The methods used are rooted in the now classical Martingale theory of contingent claim pricing, with a very particular flavour coming from the type of price processes and con- tract structures found within the energy markets. A distinctive source of difficulty within the present context is the lack of liquidity in spot markets, which explains the fact that most derivative contracts are written on forwards/futures. These futures contracts are typically heavily traded and therefore of a greater interest for risk management purposes. A concept that plays a crucial role in understanding illiquidity issues for spot prices is the cost of carry associated to most physical assets. Commodities are physical assets (in contrast to purely financial assets) that need to be produced, stored and redistributed, making the trading in futures contracts the fundamental price discovery process for market participants. In some case the commodity cannot even be stored (electricity), making its market price behaviour even harder to track and predict. An equally important concept related to illiquidity issues is the convenience yield, a yield which is naturally embedded into the cost of carry and that is meant to measure the benefit of holding the commodities.
The thesis is divided in four core chapters. Chapter 2 contains a concise overview of the minimal background needed in order to make sense of the mathematical and financial developments of the subsequent chapters. This include what the author considers to be a list of essential readings in mathematical finance and energy markets. References to both monographs and journal publications on basic topics are given. This list of readings is far from being complete and is not meant to be anything else than a strong recommendation for whoever would want to learn about the foundations of quantitative finance, with a particular interest in energy derivatives. Each of the subsequent three chapters are based on independent but overall strongly related projects.
Chapter 3 is based on the refereed journal publication Hikspoors-Jaimungal [HJ07]. In this work the authors propose a two-factor generalization of the classical one-factor 1 Introduction 3 exponential-OU model for spot price evolution. They also provide a simple methodology to price both forward contracts and spread options on forward prices. In doing so, the authors give the tools for a more flexible calibration of a mean-reversion spot model to fu- tures price as well as European one-name contracts. An efficient calibration methodology is explained and performed in the last section of the paper.
Chapter 4 is based on the forthcoming refereed journal publication Hikspoors-Jaimungal [HJ08a]. In this paper the authors extend the applications of the asymptotic PDE theory of singular perturbations to the field of commodity derivatives pricing. Based on realistic stochastic volatility (SV) spot price model assumptions, the authors provide a rigorous development of the theory for forward prices as well as single-name European options. The model assumptions include a stochastic volatility improvement of the two-factor spot model proposed in Chapter 3 (or [HJ07]). The authors then proceed to the development of a closed form approximation of the price of any two-name European contracts written on forward prices, a development that significantly extend all previous work on singular perturbation results in finance.
Chapter 5 is based on the paper Hikspoors-Jaimungal [HJ08b]. In this work the authors explore a stochastic volatility forward curve modeling approach in which the unhedgeable ”spot risk” is naturally embedded. The employed methodology generalize the one proposed in [ST08] by including a general stochastic volatility component based on a fast mean-reverting OU-process. This naturally links the concept of unspanned SV to the theory of singular perturbations, as initially applied to commodity derivatives in [HJ08a]. We build a closed form asymptotic approximation to the prices of a large subset of single- name and two-name European contracts on forwards. Interestingly enough, the price of any calendar spread options is completely determined once the model has been calibrated to related liquid single-name contracts.
A review of the relevant literature is provided within the introduction of each of the 1 Introduction 4 three above mentioned chapters. Finally, Chapter 6 concludes with a quick overview of the achieved results and some comments on ongoing and forthcoming work. Chapter 2
Background Material
The mathematical foundations and computational tools of financial engineering have been developed to an extent where most of participants are now specializing in one of the many sub branches of the global field of research. Hundreds of researchers have worked on these developments over the last four decades, making mathematical finance one of the most mature and sophisticated fields of applied mathematics, both from a computational and a theoretical point of view. This body of literature is obviously far too large to be summarized in any useful way. Here we deal only with what we consider to be a solid, yet minimal, background required in order to fully understand the technical developments proposed in this thesis.
We first give a quick overview of some general references where the relevant basic mathematical tools of financial engineering can be found. We do not explicitly review these results as it is assumed that the reader of the present document has already studied the foundations of mathematical finance. We then move to an exposition of what can be seen as an essential primer on both the basic results and terminology necessary for a full appreciation of the theory of singular perturbations in finance.
5 2 Background Material 6
2.1 Derivatives Pricing/Hedging
Many books contain an exposition of the theory and applications of quantitative finance, both from the probabilistic and the PDE approach. The so called bible of financial engineering is John Hull’s practical exposition of the subject [Hul05]. Although there is no need to read Hull’s book in order to understand most of the academic literature, it is nevertheless a must read for anyone who wants to aquire a more complete background than what the purely mathematical structures might suggest. There exist other references that offer a similar background, but we recommend to stick to Hull’s classic introduction, both for the quality of the exposition and its common usage in industry.
For a more mathematically sound foundation, but still very readable, one may consult both of Bjork [Bjo05] and Shreve [Shr04] textbooks. Both of these books treat the basics of Martingale derivatives pricing theory and succeed at succinctly developing the necessary mathematical background. The former is essentially a pedagogically improved version of the monograph [MR05] and is particularly strong in its exposition of interest rates (IR) models and fixed income derivatives pricing. This last reference offers one of the finest exposition of the change of numeraire technique and of its applications to IR derivatives pricing. Shreve’s book has a more complete exposition of the basic mathematical tools of stochastic calculus, and is particularly strong in its exposition of simple equity derivatives pricing. It also contains a clever and original primer on jump processes for financial applications. Oksendal [Oks03] is a more complete reference on stochastic differential equations.
The crucial topic of stochastic volatility models is introduced in Chapter 7 of [MR05] and Chapter 2 of [FPS00a]. Possibly the two most influential papers on the topic are Hull-White [HW87] and Heston [Hes93]; These two references laid the foundations for a very large section of modern mathematical finance.
Just as fundamental as stochastic volatility modeling is the study of non-Gaussian 2 Background Material 7 based models of returns. These often rely on the use of jump-diffusion processes (more generally Levy processes, or even semimartingales) and are extensively studied in numer- ous publications. The first to explicitly tackle the modeling of discontinuous asset price behaviour was Robert Merton in [Mer76]. A large number of researchers have subse- quently contributed to the advances of these techniques in finance, a solid and up to date account of which can be found in the monograph Cont-Tankov [CT04].
Finally and most importantly within the present context, there is an increasing number of books, monographs and articles published on energy derivatives pricing theory and its applications. A simple introduction to the basic modeling and pricing issues can again be found in [Hul05]. We nevertheless recommend a more specialized treatment of the topic such as the one offered in Clewlow-Strickland [CS00] and in the more recent monograph Eydeland-Wolyniec [EW03]. These last references constitute an excellent source of information on energy risk management; there, one can find the essential topics of the field, both from a practical and a theoretical perspective.
2.2 Asymptotic Tools and Terminology
This section presents the minimal background necessary in order to understand the ma- chinery of singular perturbation theory, especially as it is applied to contingent claims pricing/hedging. After having read this section, the work contained in Chapter 4 and Chapter 5 should be fairly self-contained. For a more complete review on the topic, the reader is referred to the monograph [FPS00a].
We start with a quick review of the main properties of the most widely used stochastic process in this thesis; the Ornstein-Uhlenbeck (OU) process, otherwise known as Vasicek’s model for short rates:
dYt = α (φ − Yt) dt + σ dWt , (2.1) 2 Background Material 8
where Wt is a Wiener process (Brownian motion) and all parameters are constants. It is easily seen that this process tends to be dragged back to its long run mean φ at a speed controlled by the parameter α. The magnitude of the random fluctuations is determined by the constant volatility parameter σ.
The solution of the SDE (2.1) with initial condition Y0 := y is well known and is expressed as
Z t −αt −α(t−s) Yt = φ + (y − φ)e + σ e dWs . (2.2) 0
Notice that this process is normally distributed, with mean φ + (y − φ)e−αt and variance
σ2 −αt 2α (1 − e ). A very important distribution related to the solution of a OU process is its long run distribution, that is, its distribution when the time parameter goes to infinity
σ2 (t → ∞). One can directly check that Y∞ ∼ N (φ, 2α ).
The so called infinitesimal generator of the OU process (2.1) is the following dif- ferential operator
∂ σ2 ∂2 L := α (φ − y) + . (2.3) ∂y 2 ∂2y
We make use of this last operator in order to define the invariant distribution of the OU process (2.1) as the distribution of the random variable Y such that
E [Lh(Y )] = 0 , (2.4) for all smooth and bounded functions h(·). For the rest of the present section we will denote the density of this last invariant distribution as Φ(·). Using this notation, we rewrite equation (2.4) as
Z ∞ Φ(y)Lh(y)dy = 0 , (2.5) −∞ 2 Background Material 9 which in turn can be rewritten in terms of the adjoint L∗ of the operator L in (2.4) as
Z ∞ h(y)L∗Φ(y)dy = 0 , (2.6) −∞ with the new operator defined by
∂ 1 ∂2 L∗ := −α ((φ − y) ·) + σ2 . (2.7) ∂y 2 ∂2y
Using the fact that equation (2.6) has to hold for all smooth and positive functions h(y), one can show that Y actually has the same distribution as Y∞, that is, the invariant
2 2 σ2 distribution of the OU process Yt is a normal N (φ, ν ), with ν := 2α .
In what follows as well as in the rest of the thesis, the symbol hgi will denote the aver- aging of the function g(·) with respect to the invariant distribution under consideration. Within the present context, that would be
Z ∞ hgi := E [g(Y∞)] = g(y)Φ(y)dy = 0 . (2.8) −∞
We call this averaging operation centering. As we are about to show, this notation is particularly handy to express concisely some necessary conditions naturally inherited by all solutions of Poisson equations.
We now review a few facts about Poisson equations that will be most useful, espe- cially in the developments of the various proofs contained in Chapter 4 and Chapter 5. In particular, we study some important characteristics of the following Poisson equation
L0X (y) + f(y) = 0 , (2.9)
where the operator L0 is given by
∂ ∂2 L := (φ − y) + ν2 . (2.10) 0 ∂y ∂2y 2 Background Material 10
Notice that this last operator is simply the infinitesimal generator of the OU process
−1 divided by α: L0 = α L. This operator will be at the center of many subsequent investigations. We apply the previously defined centering bracket on both side of equation (2.9) and make use of integration by parts as well as obvious vanishing boundary values to get
hfi = −hL0X i (2.11) Z ∞ = − (L0X (y)) Φ(y) dy (2.12) −∞ Z ∞ ∗ = X (y)(L0Φ(y)) dy (2.13) −∞ = 0 . (2.14)
This last computation clearly shows that solutions to the Poisson equation (2.9) must satisfy the so called centering condition: hfi = 0. This necessary condition will be heavily used in subsequent chapters.
Some growth properties of the solutions of Poisson equations (2.9) will also be of a crucial importance. We simply state the two most relevant such results without proof. First assume that the centering condition hfi = 0 is satisfied and that the function f(·) also has the following growth property
n |f(y)| ≤ C1 (1 + |y| ) , (2.15)
for some n ∈ N and an arbitrary constant C1. If f(y) is bounded (n := 0) then the growth of X (y) is at most logarithmic:
|X (y)| ≤ C2 [1 + log(1 + |y|)] , (2.16)
for some constant C2. On the other hand, if the growth of f(y) is as in (2.15) for n ≥ 1, 2 Background Material 11 then X (y) preserve the same growth structure:
n |X (y)| ≤ C3 (1 + |y| ) , (2.17)
for some constant C3. These last inequalities will provide us with powerful tools for analysing the properties of the various Poisson equations we shall encounter in Chapter 4 and 5. Chapter 3
First Paper: Energy Spot Price Models and Spread Options Pricing
3.1 Abstract
In this article, we construct forward price curves and value a class of two asset exchange options for energy commodities. We model the spot prices using an affine two-factor mean- reverting process with and without jumps. Within this modeling framework, we obtain closed form results for the forward prices in terms of elementary functions. Through measure changes induced by the forward price process, we further obtain closed form pricing equations for spread options on the forward prices. For completeness, we address both an actuarial and a risk-neutral approach to the valuation problem. Finally, we provide a calibration procedure and calibrate our model to the NYMEX Light Sweet Crude Oil spot and futures data, allowing us to extract the implied market prices of risk for this commodity.
12 3 Energy Spot Price Models and Spread Options Pricing 13
3.2 Introduction
Energy commodity markets are fundamentally different from traditional financial security markets in several ways: Firstly, these markets lack the same level of liquidity that the majority of financial markets enjoy. Secondly, the storage costs of most commodities translate into peculiar price behavior; some commodities are extremely difficult to store or cannot be stored at all – electricity being a prime example. Thirdly, partly due to the structural issues surrounding energy price determination, electricity prices are typically exposed to very high volatility and to large shocks. Finally, commodity prices tend to have strong mean reverting trends. These stylized empirical facts are well documented in, for example, [CS00], [CD03], [EW03] and [Hul05].
The world wide energy commodities markets have created a need for a deeper quan- titative understanding of energy derivatives pricing and hedging. We contribute to this program firstly by proposing a two-factor mean-reverting spot price process, both with and without a jump component, and secondly by carrying out the explicit valuation of spread options written on two forward prices. The spot price model is similar in spirit to the two-factor model proposed in [Pil97]; however, in that work the second factor follows a geometric Brownian motion and, therefore, in the long run, the targeted mean blows up. Instead, we chose the mean-reverting level of the first factor to mean-revert to a second long-run mean. Our modeling framework can then be viewed as a perturbation on the standard one-factor mean-reverting approach. This is an appealing approach as the one-factor model has been extensively studied and approximately fits forward price curves. Adding a perturbation on top of this first order model allows us to correct some of the deficiencies of the one-factor model while maintaining tractability. In particular, the second factor perturbation does not change the stationary behavior that the one-factor model enjoys. We delegate the details of our purely diffusive model, and its relation and differences to the classical [Pil97] model, to Section 3.3.1 and of our jump-diffusion model 3 Energy Spot Price Models and Spread Options Pricing 14 to Section 3.5.1.
Much like the financial markets, energy markets are rife with derivative products. However, one product stands out among the many that are traded over the counter: spread options. These options provide the owner with the right to exchange a prespecified quantity of one asset for another, at a fixed cost. An even more popular option is the spread option on forward prices which allows the holder to exchange two forward contracts, possibly with differing maturity dates, rather than the commodity. The holder of such an option receives at maturity T a payoff of
ϕ(F (1) ,F (2) ) := max F (1) − α F (2) − K, 0 . (3.1) T,T1 T,T2 T,T1 T,T2
It is well known that even when the commodity prices are modeled as geometric Brownian motions (GBMs), no closed form solution exists for K 6= 0. As such, we focus on the zero exchange cost case K = 0 which we succeed in valuing in closed form under our two-factor mean-reversion modeling assumptions. Given our closed form solutions, the general case K 6= 0 can be valued using either Monte Carlo or PDE methods with our K = 0 result acting as a control variate.
In a financial markets context, before proceeding to the valuation of derivatives, the objective measure is transformed to an equivalent risk-neutral measure. However, in the context of energy derivatives, due to the illiquidity issue, such a measure change is by no means necessary. Therefore, rather than immediately moving to a risk-neutral valuation procedure, we first present a simple actuarial valuation approach for pricing exchange options in Section 3.3.2. This approach has been adopted by some industry participants and is justified by assuming that the risks associated with the energy prices are non- diversifiable (see for example [Hul05]). [Mar78] first valued exchange options assuming asset prices are GBMs under the risk-neutral measure and by utilizing a measure change induced by treating one of the assets as numeraire. However, since commodities are not 3 Energy Spot Price Models and Spread Options Pricing 15 liquid, their spot prices cannot act as a numeraire. Nonetheless, we show that there is a closely related measure change which renders the valuation in closed form even under the actuarial approach.
Although some industry participants adopt an actuarial valuation procedure, risk- neutral approaches are still very popular. In Section 3.4, we specify a class of equivalent risk-neutral measures which maintains the structure of the real-world dynamics. This allows us to reuse much of the valuation technology developed in Section 3.3.2. Once again, we show that an equivalent measure provides closed form pricing equations for spread options.
Most energy commodities are adequately modeled by diffusive processes; however, electricity prices themselves contain several jumps . Section 3.5 is devoted to a jump- diffusion generalization of our previous results appropriate for spark-spread options – exchange options in which a fuel commodity is exchanged for electricity. Using the affine structure of our two-factor model with jumps, we obtain the forward prices as a solution to a system of coupled ODEs which we explicitly solve. Furthermore, through measure changes and Fourier transform methods, `ala [DPS00] and [CM99], we obtain closed form formula for the price of spark-spread options on forwards.
We complete the paper in Section 3.6 with a calibration procedure that fits the model to spot and forward prices simultaneously. Calibrating to both spot and forward prices allows us to further extract the market prices of risk implied by the data. We apply the calibration procedure to the NYMEX Light Sweet Crude Oil spot and futures data and report on the stability of the implied model parameters as well as on the implied market prices of risk. Interestingly, the real-world mean-reversion rates are found to be higher than the risk-neutral mean-reversion rates. Furthermore, the rate of mean-reversion of the stochastic long-run mean level is lower than the mean-reversion rate of the log-spot price process. We find that these features are reflected in the market prices of risk themselves. 3 Energy Spot Price Models and Spread Options Pricing 16
3.3 Real World Dynamics and Pricing
3.3.1 Model Specifications
A quick glance at historical spot prices for energy markets shows that traditional geometric Brownian motion models, even as a first order model, are inadequate. A successful model must include mean reversion as an essential feature. For early use of such models see the papers by [GS90] and [CS94]. These early one-factor models are a good first order model; however, as energy derivatives will often have maturities extending into months, or even years, such first order models require improvement. They invariably cannot match the term structure of forward prices for example. To this end, [Pil97] first suggested the two-factor mean-reverting model:
(1) dSt = β(θt − St) dt + σS St dWt , (3.2)
(2) dθt = α θtdt + σθ θt dWt , (3.3)
(1) (2) where the two Brownian risk factors are correlated: d[W ,W ]t = ρ dt. In this model,
θt represents the stochastic long-run mean to which spot prices St revert. This additional degree of stochasticity helps to correct some of the biases that a fixed long-run mean pro- duces. In this parametrization, the mean reverting level is a geometric Brownian motion and can therefore grow without bound leading to non-stationary spot price processes. To circumvent this problem, and to assist with obtaining closed form formulae for spread options, we propose to model the log spot-price with an affine two-factor mean-reverting process. Much like Pilipovic’s model, the first factor mean-reverts to stochastic level; however, we ensure that the stochastic mean-reverting level also mean-reverts to a second long-run mean. With such a parametrization, the distribution of spot-prices is stationary, prices do not grow without bound, and the model remains within the Affine modeling 3 Energy Spot Price Models and Spread Options Pricing 17 class.
If the individual assets are driven by a two-factor model, then four driving factors are
(i) (i) required to value spread options – two for each asset. Let {Wt }0≤t≤T and {Zt }0≤t≤T , with i = 1 or 2, denote these four Brownian risk factors and F = {Ft}0≤t≤T denote the natural filtration generated by these processes. The measure P will denote the real-world probability distribution and {Ω, F, P} is used to denote the complete stochastic basis for (i) the probability space. The spot-prices {St }0≤t≤T , with i = 1 or 2, are obtained via an exponentiation of the driving risk-factors. More specifically, the spot-prices are defined as follows:
(i) (i) (i) St := exp gt + Xt , i = 1, 2 . (3.4)
Seasonality is an important feature of some commodity prices; we therefore include the
(i) seasonality term gt using the following popular ans¨atz:
n (i) (i) X (i) (i) gt = A0 t + Ak sin(2π k t) + Bk cos(2π k t) . (3.5) k=1
For calibration stability, n is typically kept small: n = 1 or 2. In our subsequent calcula-
(i) tions we leave gt general, assuming only smoothness and differentiability.
To complete the specification of the two-factor model which drives the spot-prices,
(i) Xt is assumed to satisfy the following coupled SDEs:
(i) (i) (i) (i) (i) dXt = βi Yt − Xt dt + σX dWt , (3.6)
(i) (i) (i) (i) dYt = αi φi − Yt dt + σY dZt . (3.7)
(i) (i) Here, βi controls the speed of mean-reversion of Xt to the stochastic long-run level Yt ; (i) αi controls the speed of mean-reversion of the long-run level Yt to the target long-run (i) (i) mean φi; σX and σY control the size of the fluctuations around these means. 3 Energy Spot Price Models and Spread Options Pricing 18
To reduce the parameter space, the measure P is chosen such that the following simple correlation structure is imposed on the Brownian motions:
(1) (2) d[W ,W ]t = ρ12 dt , (3.8)
(i) (i) d[W ,Z ]t = ρi dt , i = 1, 2 , (3.9) and all other cross correlations are zero. This structure allows the main driving factors
(i) Xt to be correlated to one another and their own idiosyncratic long-run mean-reverting (i) (1) (2) processes Yt , while this structure forces the long-run reverting factors Yt and Yt to have an instantaneous correlation of zero. It is a straightforward, albeit tedious, matter to generalize this correlation structure.
(i) The coupled SDEs (3.6)-(3.7) can be solved by (i) first solving (3.7) for Yt – which is the standard mean-reverting Ornstein-Uhlenbeck process – to obtain
Z t (i) (i) −αi(t−s) (i) −αi(t−u) (i) Yt = φi + Ys − φi e + σY e dZu ; (3.10) s
(i) and then (ii) substituting this result into (3.6) to solve for Xt while accounting for the (i) (i) (i) correlation and feedback of Yt into Xt . After some tedious calculations Xt can be represented as
(i) (i) −βi(t−s) (i) (i) (i) Xt = Gs,t + e Xs + Ms,t Ys Z t Z t (i) −βi(t−u) (i) (i) (i) (i) + σX e dWu + σY Mu,t dZu , (3.11) s s
βi (i) (i) where γi := , and G and M are the deterministic functions αi−βi s,t s,t
(i) −βi(t−s) −αi(t−s) Ms,t := γi e − e , (3.12)
(i) −βi(t−s) (i) Gs,t := φi 1 − e − φi Ms,t . (3.13) 3 Energy Spot Price Models and Spread Options Pricing 19
Armed with our two-factor model and the solutions (3.10)-(3.11), we now focus our at- tention on the valuation of spot spread options and defer the valuation of spreads on forwards and model calibration to sections 3.4.4 and 3.6 respectively.
3.3.2 Spot Spread Valuation : An Actuarial Approach
Much like the financial markets, energy markets are rife with derivative products. How- ever, one product stands out among the many that are traded over the counter: spread options. These options provide the owner with the right to exchange a specified quantity of one asset for another, at a fixed cost. The holder of such an option receives a maturity payoff of
(1) (2) (1) (2) ϕ(ST ,ST ) := max ST − αST − K, 0 . (3.14)
When the cost of exchanging K is set to zero, the option is known as a Margrabe or exchange option [Mar78]. Various approximations for the general (spot) spread option, under simple diffusion processes, have been studied in the literature and the reader is referred to [CD03] for an excellent overview and further references. In the context of electricity markets, this option is known as the spark-spread option and α represents the heat rate of a given plant. The heat rate encapsulates the plant’s profitability by specifying the number of units of the underlying commodity (such as oil or natural gas) which produces one unit of power – this product is studied in Section 3.5. If the exchange is between crude oil and a refined product (such as gasoline) the option is known as a crack-spread option. Many other specific examples of exchange options exist in the energy sector. In all cases, the exchange option can be used to hedge against market movements of spot prices or, alternatively, to speculate on those moves. In either case, a valuation framework is required.
It is difficult and sometimes not viable to store electricity and energy commodities; 3 Energy Spot Price Models and Spread Options Pricing 20 this results in an illiquid spot market. [HP81] demonstrated that the absence of arbitrage is equivalent to the existence of a measure, not necessarily unique, under which the rela- tive price process of tradable assets to the money market account are martingales. Such measures are known as a risk-neutral measures. However, this conclusion has one im- portant assumption – unrestricted and liquid trading of the underlying asset. In illiquid (spot) energy markets, it may be dubious to adopt a risk-neutral pricing framework, and although we ultimately proceed with that program, we first take an actuarial approach. By assuming that the risks associated with the energy prices are non-diversifiable, it is possible to justify an actuarial approach to pricing derivatives [Hul05] which values an option as its discounted real-world expected payoff.
Definition. The actuarial valuation principle assigns the following price to a T -maturity
(1) (2) contingent claim with payoff ϕ(ST ,ST ):
h i P (1) (2) Πt,T := P (t, T ) Et ϕ ST ,ST . (3.15)
The notation Et[A] represents the expectation of A conditional on the filtration Ft.
Throughout the article we assume the (possibly random) interest rates to be independent from other risk factors and write the price at time t of a zero-coupon bond maturing at T as P (t, T ).
In the traditional valuation procedure, expectations are taken under the risk-neutral measure Q; here, however, the relevant measure is the real-world one P. This complicates the problem somewhat. When the asset and the derivative are tradable, it is possible to use a numeraire change to value the Margrabe option; in the present context the asset cannot be used as numeraire and the relevant measure is not the risk-neutral one. Nonetheless, it is possible to adopt a similar strategy; to this end, define the auxiliary 3 Energy Spot Price Models and Spread Options Pricing 21 process
h i (i) P (i) Ht,T := Et ST . (3.16)
(i) If the expectation in (3.16) is computed under a risk-neutral measure, then Ht,T represents (i) the T -maturity forward price, which motivates us to coin Ht,T the T -maturity pseudo- forward price process. At maturity this “price” process coincides with the spot-price
(i) (i) HT,T = ST allowing the actuarial value of the exchange option to be written:
P (1) (2) Πt,T = P (t, T ) Et HT,T − α HT,T . (3.17) +
The pseudo-forward price process has two other notable properties: (i) its expectation
h (i) i (i) P is bounded at all finite times E |Ht,T | = H0,T < +∞ for all t < T , and (ii) it is (i) a P-martingale. These two properties allow a normalized version of Ht,T to assist in transforming the measure P into a particularly convenient measure for pricing. This measure change can, in some sense, be interpreted as being induced by using the pseudo- forward price process as a numeraire asset. The following Theorem contains one of our main tools.
Theorem. 3.3.1 Let {ηt}0≤t≤T denote the Radon-Nikodym process
! (2) de Ht,T η := P := . (3.18) t d (2) P t H0,T
Then, for any A ∈ FT
P Pe(A) = E [I(A) ηT ] , (3.19)
i i and in particular Wft and Zet (i = 1, 2) defined by
Z t h i (2) (2) (2) −β2(T −u) (2) (2) Wft = Wt − σX e + ρ2σY Mu,T du (3.20) 0 3 Energy Spot Price Models and Spread Options Pricing 22
Z t h i (2) (2) (2) (2) (2) −β2(T −u) Zet = Zt − σY Mu,T + ρ2σX e du (3.21) 0 Z t h i (1) (1) (2) −β2(T −u) Wft = Wt − ρ12σX e du (3.22) 0 Z t h i (1) (1) (2) −β2(T −u) Zet = Zt − ρ1ρ12σX e du (3.23) 0 are Pe−Wiener processes with correlation structure
(1) (2) d[Wf , Wf ]t = ρ12 dt , (3.24)
(i) (i) d[Wf , Ze ]t = ρi dt , i = 1, 2 , (3.25) and all other cross correlations zero.
Proof. Given properties (i) and (ii) above and η0 = 1, it is clear that ηt is a Radon- Nikodym derivative process and equation (3.19) immediately follows. To demonstrate
(i) (i) (i) that Wft and Zet are Pe-Wiener processes substitute (3.11) into Ht,T and then compute the expectation explicitly as follows:
h n (i) P (i) (i) −βi(T −t) (i) (i) (i) Ht,T = Et exp gT + Gt,T + e Xt + Mt,T Yt Z T Z T (i) −βi(T −u) (i) (i) (i) (i) +σX e dWu + σY Mu,T dZu t t
(i) (i) (i) −βi(T −t) (i) (i) (i) = exp gT + Gt,T + Rt,T + e Xt + Mt,T Yt (3.26)
Here,
h(t, T ; 2β ) 2 2 R(i) := i σ(i) + γ σ(i) + 2ρ γ σ(i)σ(i) t,T 2 X i Y i i X Y 2 h(t, T ; 2α ) 2 −h(t, T ; α + β ) γ σ(i) + ρ γ σ(i)σ(i) + i γ σ(i) (3.27) i i i Y i i X Y 2 i Y h(t, T ; a) := (1 − e−a(T −t))/a . (3.28) 3 Energy Spot Price Models and Spread Options Pricing 23
(i) The Girsanov kernel can now be read off directly from (3.26) and the solution for Xt (i) and Yt given in (3.10)-(3.11). Through Girsanov’s theorem we find that (3.20)-(3.23) are Pe-Wiener processes. 2
Corollary 3.3.2 The actuarial valuation formula (3.15) can be transformed to
h i P (1) (2) Πt,T := P (t, T ) Et ϕ HT,T ,HT,T (1) (2) ϕ HT,T ,HT,T (2) Pe = P (t, T ) Ht,T Et (2) . (3.29) HT,T
Proof. Use ηt to change the measure. 2
It now remains to compute the expectation appearing in (3.29) under the trans- formed measure Pe. Recall that a process Mt is a Pe-martingale if and only if the process (1) (2) Mt (dPe/dP)t is a P-martingale. Consequently, both Ht,T := Ht,T /Ht,T and (dP/dPe)t are Pe-martingales. This, together with Corollary 3.3.2, reduces the actuarial price of the Margrabe spread option to
(2) Pe Πt,T = P (t, T ) Ht,T Et [(HT,T − α)+] (3.30)
Since Ht,T is a Pe-martingale, its drift under the Pe-measure is zero. Putting this together with equation (3.26), we find that Ht,T satisfies the SDE:
dH t,T (1) −β1(T −t) (1) (2) −β2(T −t) (2) = σX e dWft − σX e dWft Ht,T (1) (1) (1) (2) (2) (2) + σY Mt,T dZet − σY Mt,T dZet (3.31)
This expression clearly shows that Ht,T is a geometric Brownian motion with time de- pendent (but deterministic) volatility; consequently, its terminal value can be expressed in terms of its initial value via HT,T = Ht,T exp (Ut,T ), where Ut,T is a normal random 3 Energy Spot Price Models and Spread Options Pricing 24
1 2 2 variable with mean equal to − 2 (σt,T ) and variance equal to (σt,T ) . Here,
Z T Z T 2 Pe (1) −β1(T −s) (1) (2) −β2(T −s) (2) (σt,T ) = Et σX e dWfs − σX e dWfs t t Z T Z T 2# (1) (1) (1) (2) (2) (2) +σY Ms,T dZes − σY Ms,T dZes t t
(1) (2) (1) (2) = 2 Rt,T + 2 Rt,T − 2 ρ12 σX σX h(t, T ; β1 + β2) . (3.32)
(i) (i) The deterministic functions Ms,T and Rt,T are as in (3.12) and (3.27) respectively. It is now a straightforward matter to recover the final result of this section – a Black-Scholes like expression for the actuarial price of the exchange option.
Proposition 3.3.3 The actuarial value at time t of the T -maturity exchange option is
h (1) (2) i Πt,T = P (t, T ) Ht,T Φ(d + σt,T ) − α · Ht,T Φ(d) (3.33)
with σt,T as in (3.32) and d defined as
log Ht,T − 1 (σ )2 d := α 2 t,T . (3.34) σt,T
3.4 Risk-Neutral Dynamics and Pricing
In complete market settings, there exists a unique equivalent measure which induces the relative price process of tradable assets to be martingales. This measure is known as the risk-neutral measure Q. In the present context, the underlying asset is not tradable in the usual sense due to the illiquidity issue and potentially large storage costs. In the previous section we dealt with this issue by resorting to an actuarial valuation procedure and assigned a price equal to the discounted expectation of the terminal payoff under the real-world measure. However, one can in principle still utilize risk-neutral methodologies 3 Energy Spot Price Models and Spread Options Pricing 25 adjusting for the incompleteness of the market settings. Within such incomplete markets there may exists many equivalent risk-neutral measures; it is the job of the market as a whole, via trading of derivatives, to decide which measure prevails at any one given point in time. In this next section, we provide a class of equivalent martingale measures that maintains the structure of our real-world dynamics for asset prices. These measures are then used to obtain forward prices and value spread options.
3.4.1 Measure Change
In this section, we introduce a class of risk-neutral measure changes which maintains the real-world structure of the asset dynamics. The following Lemma introduces the new measure induced by a four dimensional market price of risk vector.
Lemma 3.4.1 Let {ζt}0≤t≤T denote the Radon-Nikodym process,
Z t dQ (1) (1) (1) (1) (2) (2) (2) (2) ζt := = E λu dWu + ψu dZu + λu dWu + ψu dZu , (3.35) dP t 0 where E(At) is the Dolean-Dade’s exponential of the process At. Then for any A ∈ FT we have,
P Q(A) = E [A ζT ] . (3.36)
In particular the following are Q-Wiener processes:
Z t (1) (1) (1) (2) (1) W t = Wt − λu + ρ12λu + ρ1ψu du (3.37) 0 Z t (1) (1) (1) (1) Zt = Zt − ρ1λu + ψu du (3.38) 0 Z t (2) (2) (1) (2) (2) W t = Wt − ρ12λu + λu + ρ2ψu du (3.39) 0 Z t (2) (2) (2) (2) Zt = Zt − ρ2λu + ψu du (3.40) 0 3 Energy Spot Price Models and Spread Options Pricing 26 with correlation structure,
h (1) (2)i d W , W = ρ12 dt , (3.41) t h (i) (i)i d W , Z = ρi dt , i = 1, 2 , (3.42) t and all other cross-correlations zero.
Proof. Decompose the correlated processes into uncorrelated processes and apply Gir- sanov’s Theorem. 2
Notice that there are no restrictions on the form of the market prices of risk other than the usual integrability ones. In particular, the drifts under the risk-neutral measure Q are not constrained to the risk-free rate. This is precisely the effect of incompleteness in the present context. The following Theorem applies constraints on the market prices of risk such that the risk-neutral dynamics and the real-world one are of the same form.
Theorem. 3.4.2 If the market price of risk processes are chosen as follows:
(i) (i) (i) (i) (i) (i) λt = λ + λX Xt + λY Yt , (3.43)
(i) (i) (i) (i) (i) (i) ψt = ψ + ψX Xt + ψY Yt , (3.44) subject to the constraints ( (i, j) ∈ {(1, 2), (2, 1)} )
(i) (i) ψX = −ρi λX , (3.45)
(i) (j) (i) λ + ρ12λ + ρiψ = 0 , (3.46)
(i) (j) (i) (i) (j) (i) λX + ρ12λX + ρiψX = − λY + ρ12λY + ρiψY , (3.47)
(i) (i) (i) αi = αi − σY ρiλY + ψY , (3.48)
(i) (i) (i) αi φi = αiφi + σY ρiλ + ψ , (3.49)
(i) (i) (j) (i) βi = βi + σX λY + ρ12λY + ρiψY , (3.50) 3 Energy Spot Price Models and Spread Options Pricing 27
(i) (i) then the risk-neutral dynamics of Xt and Yt remain within the same class as (3.6)- (3.7). In particular,
(i) (i) (i) (i) (i) dXt = βi (Yt − Xt ) dt + σX dW t (3.51)
(i) (i) (i) (i) dYt = αi (φi − Yt ) dt + σY dZt . (3.52)
(i) (i) Proof. Insert the expressions for the Q-Wiener processes W t and Zt into (3.6)-(3.7). Collect similar terms and equate coefficients. 2
This ans¨atzmay seem restrictive; however, even though the risk-neutral dynamics remains within the same class as the real-world one, the coefficients driving that dynamics may be significantly different. This flexibility is sufficient for the simultaneous calibration of the risk-neutral and real-world model parameters, while remaining parsimonious.
3.4.2 Forward Prices
Since the risk-neutral dynamics of the driving diffusion processes (3.51)-(3.52) are of the same form as they were under the objective measure (3.6)-(3.7), the forward price curves can easily be extracted from equation (3.26). This is because, within a risk-neutral framework, the forward prices are defined as
h i (i) Q (i) Ft,T := Et ST , (3.53) the precise risk-neutral analog of the pseudo-forward price defined in (3.16). All that remains is to change the P-parameters for the Q-parameters.
Proposition 3.4.3 The forward prices associated to commodity i = 1, 2 are given by
(i) (i) (i) (i) (i) −βi(T −t) (i) (i) F (t, T ) = exp gT + Rt,T + Gt,T + e Xt + M t,T Yt (3.54) 3 Energy Spot Price Models and Spread Options Pricing 28
(i) (i) (i) where the expressions for M t,T , Gt,T and Rt,T are supplied in equations (3.12)-(3.13) and
(3.27) respectively – with the appropriate change of parameters (αi → αi and so on...).
These results can be viewed as an extension of the one-factor model [CF05] study, albeit without jumps. In section 3.5, we address the two factor model with jumps.
3.4.3 Spot Spread Valuation
To value the option under a risk-neutral measure, we follow along the same lines as in Section 3.3.2. In the present context, the measure change is the one induced by using the forward price to drive the measure change. To this end, define a new measure Qe via the Radon-Nikodym derivative process
h i ! Q S(2) (2) dQe Et T Ft,T = h i = (2) . (3.55) dQ Q (2) F t E0 ST 0,T
All steps leading to Proposition 3.3.3 remain valid in this new context, and rather than repeating them, we simply quote our final risk-neutral pricing result.
Proposition 3.4.4 The risk neutral value at time t of the exchange option with maturity T is:
h (1) (2) i Πt,T = P (t, T ) Ft,T Φ(d + σt,T ) − α Ft,T Φ(d) (3.56) where d defined as
2 log Ft,T − (σt,T ) d := α 2 (3.57) σt,T and σt,T as in (3.32) – with all P-parameters replaced by Q-parameters (i.e., αi → αi and so on...). 3 Energy Spot Price Models and Spread Options Pricing 29
(i) It is important to note that the market provides quotes for the forward curve Ft,T , i = 1, 2 for a set of maturities T = {T1,...,Tn}. These curves can be used to calibrate the risk- neutral parameters. Once the model has been calibrated to market data, the resulting pricing rules are just as simple to use as the Black-Scholes formula for a European option
(i) on a single asset. Although the explicit expressions for the forward prices Ft,T and the effective volatility σt,T are somewhat bulky, they involve nothing more complex than exponentiation and are therefore very efficient to calculate.
3.4.4 Forward Spread Valuation
In the previous sections we focused on valuing a spread option based on the future spot prices; however, a more popular derivative product involves the spread between the for- ward prices of the two assets (possibly with differing maturities). Such spreads on forwards pay
(1) (2) (1) (2) ϕ FT,T ,FT,T = FT,T − α FT,T (3.58) 1 2 1 2 +
at the maturity date T where it is implicit that T1 ,T2 ≥ T . We can once again use a measure change to simplify the calculations, this time it is convenient to use the T2- maturity forward price of asset 2, i.e. F (2) , to induce a measure change. In particular we t,T2 define the Radon-Nikodym derivative process
! (2) de Ft,T Q := 2 . (3.59) dQ F (2) t 0,T2
The time t price of the forward spread option is therefore
F Q (1) (2) Πt,T := P (t, T ) Et FT,T − α FT,T 1 2 + = P (t, T ) F (2) Qe (F − α) . (3.60) t,T2 Et T ;T1,T2 + 3 Energy Spot Price Models and Spread Options Pricing 30
Here, F := F (1) /F (2) is the ratio of the two relevant forward prices. In analogy with t;T1,T2 t,T1 t,T2 our earlier calculations, the relative process Ft;T1,T2 is a Qe-martingale and therefore its Qe-dynamics is driftless. Following along the same arguments as in Section 3.3.2, it is easy
∗ ∗ to show that FT ;T1,T2 = Ft;T1,T2 exp{Ut;T,T1,T2 } where Ut;T,T1,T2 is a normal random variable with mean equal to − 1 (σ∗ )2 and variance equal to (σ∗ )2. The explicit form 2 t;T,T1,T2 t;T,T1,T2 for the variance is
2 ∗ 2 (1) (σt;T,T1,T2 ) := γ1σY [h(t, T1; 2α1) − h(T,T1; 2α1)] 2 (2) + γ2σY [h(t, T2; 2α2) − h(T,T2; 2α2)] 2 2 (1) (1) (1) (1) + σX + γ1σY + 2ρ1γ1σX σY h(t, T1; 2β1) − h(T,T1; 2β1)
2 2 (2) (2) (2) (2) + σX + γ2σY + 2ρ2γ2σX σY h(t, T2; 2β2) − h(T,T2; 2β2)
2 (1) (1) (1) − 2 γ1σY + 2ρ1γ1σX σY h(t, T1; α1 + β1) − h(T,T1; α1 + β1)
2 (2) (2) (2) − 2 γ2σY + 2ρ2γ2σX σY h(t, T2; α2 + β2) − h(T,T2; α2 + β2) (1) (2) −2ρ12σX σX exp −β1(T1 − T ) − β2(T2 − T ) h(t, T ; β1 + β2) (3.61) where, h(t, T ; a) is defined in (3.28). The pricing equation (3.60) now reduces to a Black- Scholes like pricing result.
Proposition 3.4.5 The risk neutral value at time t of the forward spread option (3.58) is
h i ΠF = P (t, T ) F (1) Φ(d∗ + σ∗ ) − α F (2) Φ(d∗) (3.62) t,T t,T1 t,T t,T2
∗ 2 ∗ with (σt,T ) as in (3.61) and d defined as
Ft;T1,T2 1 ∗ 2 ∗ log α − 2 (σt,T ) d := ∗ . (3.63) σt,T 3 Energy Spot Price Models and Spread Options Pricing 31
Not surprisingly, the pricing result is very similar to the one in Proposition 3.4.4 and reduces to it when T = T1 = T2.
3.5 Spot Prices with Jumps
The two factor diffusion model captures the main characteristics of most energy spot prices, however, it cannot account for the possibility of sudden jumps in the price data. Such behavior is particularly important for modeling electricity prices and various spreads contingent on electricity and other hedging assets. The most important example of such
(1) (2) option is the so-called spark-spread option which pays FT,T − αFT,T − K at the 1 2 + maturity date T . Here F (1) := QS(1) represents the electricity forward price, F (2) rep- t,T1 Et T1 t,T2 resents the forward price of the commodity used to generate electricity, and α represents the heat rate which encapsulates the number of units of energy that the plant produces per unit of commodity. Notice that the structure of this option allows forward prices of differing maturities to be used as the underlying. As commented earlier on, closed form solutions, even for the purely diffusion case, are not accessible for general strike levels; consequently, we limit ourselves to the exchange option with a strike of zero.
3.5.1 Model Specification
For brevity, we now focus solely on the risk-neutral valuation procedure, and provide model specifications directly under the risk-neutral measure. Typically, when electric- ity prices jump they revert back to normal levels very quickly. A widely used model specification incorporates jumps and diffusions simultaneously as follows:
d ln(St) = α(θ − ln(St−)) dt + σ dWt + dQt . (3.64) 3 Energy Spot Price Models and Spread Options Pricing 32
Regardless of the specification of the jump process Qt, such models suffer from unre- alistically large diffusive volatilities and mean-reversion rates. This occurs because the process must revert very quickly to normal levels after a large jump, implying a high mean-reversion rate α. This in turn induces an artificially large diffusive volatility, since otherwise all diffusive components would revert to the mean extremely quickly and, ex- cluding the jumps, the paths would appear essentially deterministic.
We avoid these problems by splitting the jump component from the diffusion compo- nent and modeling them separately. Specifically, define the (power) spot price by
(1) n (1) (1) o St := exp gt + Xt + Jt , (3.65)
(i) (i) where Xt and Yt satisfy the usual two-factor SDEs (3.51)-(3.52), and the new jump component Jt is defined via
dJt = −κ Jt− dt + dQt , (3.66)
PNt with Qt a compound Poisson process: Qt := 1 li, where Nt is a time inhomogeneous
Poisson process with activity rate λ(s), and {li} the set of i.i.d. jump sizes with distri- bution function Fl(u). Furthermore, Jt− denotes the value of Jt prior to any jump at time t. The jump component Jt mean-reverts to zero with rate κ; typically, κ will be quite large because electricity prices revert back to normal very quickly after a jump. This empirical fact has no direct bearing on the valuation procedure, however, it does attribute to the manner in which we have split the jump component from the diffusion component. We allow the activity rate to vary with time to permit seasonality effects in the rate of jump arrivals; however, we restrict it to be deterministic – it is possible to generalize to stochastic activity rates; however, the additional modeling flexibility renders the calibration process unstable. Finally, it is well known that diffusions and jump pro- 3 Energy Spot Price Models and Spread Options Pricing 33 cesses cannot have any instantaneous correlations, while this does not preclude the jump size from depending on the Brownian risk factors we make the natural assumption that
Nt and {li} are independent of all the Q-Brownian processes.
3.5.2 Forward Prices
(1) Equipped with this jump-diffusion model, we now derive the forward price Ft,T associated (1) with the spot St . As usual, the forward price is the risk-neutral expectation of the asset price at the maturity
h n oi (1) (1) (1) Q (1) Q (1) (1) f(t, Xt ,Yt ,Jt) := Ft,T := Et ST = Et exp gT + XT + JT . (3.67)
Rather than computing this expectation directly, we make use of the affine form of the processes along the lines of [DPS00]. Since f is a Q-martingale, it satisfies the zero-drift (1) (1) condition Af = 0 where A is the generator of the process t, Xt ,Yt ,Jt . The affine ans¨atz:
(1) (1) n (1) (1) o f(t, Xt ,Yt ,Jt) = exp A(t, T ) + B(t, T ) Xt + C(t, T ) Yt + D(t, T ) Jt , (3.68)
(1) with terminal conditions A(T,T ) = gT , B(T,T ) = 1, C(T,T ) = 0 and D(T,T ) = 1, reduces the PDE Af = 0 to the equivalent system of coupled ODEs:
Bt − β1B = 0 , (3.69)
Ct + β1B − α1C = 0 , (3.70)
Dt − κD = 0 , (3.71) (σ(1))2 A + α φ C + X B2 t 1 1 2 (1) 2 Z ∞ (σY ) 2 (1) (1) D·u + C + ρ1σX σY BC = − λ(u) e − 1 dFl(u) (3.72) 2 −∞ 3 Energy Spot Price Models and Spread Options Pricing 34
Although rather tedious, standard methods can be used to solve this system and obtain the forward price.
(1) Proposition 3.5.1 The forward price for the two-factor jump-diffusion spot process St is
n (1) o (1) (1) −β1(T −t) (1) (1) −κ(T −t) Ft,T = exp At,T + e Xt + M t,T Yt + e Jt , (3.73)
(1) where the deterministic function M t,T is provided in equation (3.12),
Z T (1) (1) −κ(T −s) At,T = gT + λ(s) ϕl e − 1 ds t − α1γ1φ1 h(t, T ; α1) − h(t, T ; β1) 1 2 + γ σ(1) h(t, T ; 2α ) + h(t, T ; 2β ) − 2h(t, T ; α + β ) 2 1 Y 1 1 1 1 1 2 + σ(1) h(t, T ; 2β ) 2 X 1 (1) (1) + ρ1γ1 σX σY h(t, T ; 2β1) − h(t, T ; α1 + β1) , (3.74)
and ϕl(u) is the m.g.f. of the individual jump sizes li,
Z ∞ Q u l1 u z ϕl(u) := E e = e dFl(z) . (3.75) −∞
These results can be viewed as an extension of the one-factor model [CF05] study.
3.5.3 Spark Spread Valuation
We now turn to the pricing of the (exchange) spark spread option with T -terminal payoff
(1) (2) (i) FT,T − αFT,T and T ≤ T1,T2 . As usual, the forward prices are expressed as Ft,T := 1 2 + h i n o Q (i) (1) (1) (1) Et ST where St := exp gt + Xt + Jt is the two-factor jump-diffusion spot price (2) n (2) (2)o presented in the previous section and St := exp gt + Xt is the pure diffusion process of Section 3.4. We begin our analysis by rewriting the risk-neutral pricing formula 3 Energy Spot Price Models and Spread Options Pricing 35 in terms of an equivalent measure induced by the forward price process of the purely diffusive asset. In particular,
F Q (1) (2) Πt,T := P (t, T ) Et FT,T − α FT,T 1 2 + = P (t, T ) F (2) Qe [(F − α) ] , (3.76) t,T2 Et T ;T1,T2 + where the measure Qe is induced by h i ! Q (2) (2) Et ST F dQe 2 t,T2 := h i = (2) , (3.77) dQ Q S(2) F t E0 T2 0,T2 and we introduced the ratio process F := F (1) /F (2) . The process F is once t;T1,T2 t,T1 t,T2 t;T1,T2 again a Qe-martingale; however, because of the presence of the jump component, this fact alone does not allow us to extract its distribution. Instead, we make use of transform methods. [CM99] were among the first to illustrate that Fast Fourier transform (FFT) methods can be used to efficiently value European options. The reader is referred to their work for implementation details and other efficiency tricks.
The FFT methods require the m.g.f. of the logarithm of the effective stochastic process
– in our case the process Ft;T1,T2 . To this end, define ZT := ln FT ;T1,T2 so that
(1) (2) (1) (2) (1) (1) Z = A − A + e−β1(T1−T ) X − e−β2(T2−T ) X + M Y T T,T1 T,T2 T T T,T1 T (2) (2) −κ(T1−T ) −M T,T2 YT + e JT , (3.78) and define the corresponding m.g.f. process
ZT Qe u ZT Ψt (u) := Et e . (3.79)
ZT The process Ψt (u) is clearly a Qe-martingale; consequently, it satisfies the zero drift
ZT condition AΨt (u) = 0 (for every u where it is defined) where A is the generator of the 3 Energy Spot Price Models and Spread Options Pricing 36
(1) (2) (1) (2) process t, Xt ,Xt ,Yt ,Yt ,Jt under Qe. Furthermore, since our modeling framework is affine, we employ the ans¨atz
n ZT (1) (1) (2) (2) (1) (1) Ψt (u) := exp A(t, T ) + B (t, T ) Xt + B (t, T ) Xt + C (t, T ) Yt
(2) (2) o + C (t, T ) Yt + D(t, T ) Jt (3.80)
ZT ΨT (u) := exp {u ZT } (3.81)
Here, A(t, T ),B(1)(t, T ),B(2)(t, T ),C(1)(t, T ),C(2)(t, T ), and D(t, T ) are all deterministic functions of time. Note that T1 and T2 have been removed from the arguments for easier readability. Since the boundary condition (3.81) must hold for all terminal values of the
(1) (2) (1) (2) auxiliary processes Xt ,Xt ,Yt ,Yt ,Jt , the deterministic functions must satisfy the induced boundary conditions
(1) (2) A(T,T ) = u A − A ,B(1)(T,T ) = ue−β1(T1−T ), T,T1 T,T2 1 (1) (2) −β2(T2−T ) (1) (3.82) B (T,T2) = −ue ,C (T,T1) = uM T,T1 , (2) (2) −κ(T1−T ) C (T,T2) = −uM T,T2 ,D(T,T1) = ue .
Expanding the PDE Af = 0, rewriting it in terms of an equivalent system of coupled ODEs and solving that system (similar to the analysis in Section 3.5.2) provides the final result.
ZT Qe u ZT Proposition 3.5.2 The transform Ψt (u) := Et e is given by
n ZT −β1(T1−t) (1) −β2(T2−t) (2) Ψt (u) = exp At,T + u e Xt − e Xt
(1) (2) o (1) (2) −κ(T1−t) + M t,T1 Yt − M t,T2 Yt + e Jt (3.83) 3 Energy Spot Price Models and Spread Options Pricing 37
(i) where M t,T is defined in (3.12),
Z T A = u A(1) − A(2) + λ(s) ϕ ue−κ(T1−s) − 1 ds t,T T,T1 T,T2 l t −α1(T1−T ) −α2(T2−T ) +u −α1φ1γ1e h(t, T ; α1) + α2φ2γ2e h(t, T ; α2) 2 (2) −2α2(T2−T ) −β1(T1−T ) − γ2σY e h(t, T ; 2α2) + α1φ1γ1e h(t, T ; β1) 2 2 (2) (2) (2) (2) −2β2(T2−T ) − σX + 2ρ2γ2σX σY + γ2σY e h(t, T ; 2β2)
−β2(T2−T ) − α2φ2γ2e h(t, T ; β2)
(2) (1) −α1(T1−T )−β2(T2−T ) − ρ1ρ12γ1σX σY e h(t, T ; α1 + β2) 2 (2) (2) (2) −(α2+β2)(T2−T ) + 2ρ2γ2σX σY + 2 γ2σY e h(t, T ; α2 + β2) n o i (1) (2) (2) (1) −β1(T1−T )−β2(T2−T ) + ρ12σX σX + ρ1ρ12γ1σX σY e h(t, T ; β1 + β2) 1 2 +u2 γ σ(1) e−2α1(T1−T )h(t, T ; 2α ) 2 1 Y 1 1 2 + γ σ(2) e−2α2(T2−T )h(t, T ; 2α ) 2 2 Y 2 2 2 1 (1) 1 (1) (1) (1) + γ σ + σ + ρ γ σ σ e−2β1(T1−T )h(t, T ; 2β ) 2 1 Y 2 X 1 1 X Y 1 2 2 1 (2) 1 (2) (2) (2) + γ σ + σ + ρ γ σ σ e−2β2(T2−T )h(t, T ; 2β ) 2 2 Y 2 X 2 2 X Y 2 2 (1) (1) (1) −(α1+β1)(T1−T ) − γ1σY + ρ1γ1σX σY e h(t, T ; α1 + β1) 2 (2) (2) (2) −(α2+β2)(T2−T ) − ρ2γ2σX σY + γ2σY e h(t, T ; α2 + β2) i (1) (2) −β1(T1−T )−β2(T2−T ) − ρ12σX σX e h(t, T ; β1 + β2) , (3.84)
ϕl(u) is the MGF of the individual jump sizes (see (3.75)), and the function h(t, T ; ·) is given in equation (3.28).
Now that the transform is explicit, it is possible to use standard Fourier analysis techniques to value the spread option. Under some mild assumptions on the m.g.f. of jump sizes, it is possible to analytically continue the m.g.f. to the entire complex plane. 3 Energy Spot Price Models and Spread Options Pricing 38
For completeness in the exposition, we remind the reader how the pricing equation (3.76) appears in Fourier transformed variables. Firstly,
ΠF = P (t, T ) F (2) Qe [(F − α) ] t,T t,T2 Et T ;T1,T2 +
(2) = P (t, T ) eα F Qe (eZT −α − 1) , (3.85) t,T2 Et +
x where α := log(α). By introducing η(x) := (e − 1)+, the expectation in equation (3.85) reduces to the product of Fourier transforms
1 Z ∞ Qe ZT −α ˜ Et (e − 1)+ = η˜(−p) fZT −α(p) dp , (3.86) 2π −∞
˜ whereη ˜(p) and fZT −α(p) are the Fourier transforms of η(x) and the probability density of ZT − α, respectively. It is well known that
Z ∞ 1 η˜(p) := eipxη(x) dx = (3.87) −∞ p(i − p) whenever =(p) > 1. A simple change of variables reveals that
Z ∞ ˜ ipx Qe ip(ZT −α) −iαp ZT fZT −α(p) := e fZT −α(x) dx = Et e = e Ψt (ip) . (3.88) −∞
Putting these results together leads to our final pricing equation – up to a numerical integration.
Proposition 3.5.3 The price at time t of the exchange option is
Z ∞ e−iαp ΨZT (ip) dp Π = P (t, T ) eα F (2) t , (3.89) t,T t,T2 −∞ −p(p + i) 2π
ZT with Ψt (·) as in Proposition 3.5.2.
Some final remarks are crucial at this point:
1. In our framework, the price process of a spot exchange option is simply given by 3 Energy Spot Price Models and Spread Options Pricing 39
setting T = T1 = T2 in equation (3.89).
2. The integral part of equation (3.89) seems formidable; however, the coefficients are nothing more complicated than exponentials and there exists very efficient numerical methods, such as FFT, for performing the integrals. Therefore, we do not pursue this further, and instead refer the reader to the monograph by [CT04] for further information and references on these topics.
3. The market reveals the entire forward curve and, of course, the risk-free zero coupon bond prices. Before using the valuation formula, the model must be calibrated to these market prices. Once the parameters are calibrated, then the pricing equation (3.89) will provide consistent no-arbitrage prices to the various spread options.
4. It is possible to repeat this analysis when both assets contain jumps. Needless to say, the resulting equations will be bulkier (but not fundamentally more complicated), and although the change of measure will be more subtle, it posses no real problems. However, in real applications, both assets typically do not contain sudden jumps, as one is usually the raw commodity used to produce electricity.
5. Some care must be taken to ensure the integration path in (3.89) remains in the intersection of the regions =(p) > 1 and where the complex continuation of the
ZT function Ψt (z) is analytic in z. However, for typical jump distributions, such as
ZT double exponential and normal, Ψt (z) will be analytic in the region =(p) > 1, and any simple path lying in =(p) > 1 will do.
3.6 Model Calibration
In this section, we finally address the issue of parameter estimation. We perform this last step in two stages. Firstly, in Section 3.6.1 we provide a detailed review of an effi- 3 Energy Spot Price Models and Spread Options Pricing 40 cient method for calibrating the pure diffusion two-factor model to market futures prices, resulting in the risk-neutral model parameters. We also describe how jump parameters can be simultaneously estimated from market spot prices. Secondly, we describe how a method borrowed from interest rate model calibration can be used to estimate the real- world model parameters from a knowledge of spot and future prices. This simultaneous calibration of futures and spot prices to the risk-neutral and real-world measures further allows us to extract the implied market prices of risk. An alternative approach to real- world calibration is to use a well known Kalman Filter approach. Such approaches do not utilize futures prices data and can be quite useful. For more details on the calibration of various two-factor models to spot data and further references on the topic we refer to the work of [BGL04]. Section 3.6.2 concludes with concrete applications of our statistical methodology to the NYMEX Light Sweet Crude Oil data and some further comments.
3.6.1 Methodology
Before proceeding to the calibration process, recall that the log of the forward price
(i) n (i) (i)o associated with the spot St := exp gt + Xt is given by (Section 3.4.2):
(i) (i) (i) (i) (i) −βi(T −t) (i) (i) log Ft,T = gT + Gt,T + Rt,T + e Xt + M t,T Yt (3.90)
(i) (i) (i) (i) −βi(T −t) (i) (i) (i) = gT + Gt,T + Rt,T + e log St − gt + M t,T Yt (3.91)
(i) (i) (i) := U t,T + M t,T Yt . (3.92)
(i) Here, the function U t,T is introduced to simplify notation. Given the spot price data at (i) (i) (i) time t, U t,T is completely determined, while the last term M t,T Yt depends on the hidden long-run mean level Yt. Therefore, a standard nonlinear least-squares optimization cannot be applied directly. Instead, we will express the hidden factor in terms of the remaining model parameters and obtain an optimal fit to the observed futures curve at various time 3 Energy Spot Price Models and Spread Options Pricing 41 points.
(i)∗ Let F p denote the observed futures prices at tp ∈ {t1, ..., tm} with delivery time tp,Tq p p p Tq ∈ {T1 , ..., Tnp } and denote by Θ a point in the (risk-neutral) parameter space Ω of (i) our model. For each given quoted time tp, obtain Ytp (Θ) (as a function of the remaining parameters) such that it minimizes the following sum of squares:
n p 2 X h (i) (i)∗ i Sum(tp, Θ) := log F p − log F p . (3.93) tp,Tq tp,Tq q=1
(i) The optimal Ytp (Θ) is easily found to be
h (i) (i) i Pnp (i)∗ M p log F p − U p #(i) q=1 tp,Tq tp,Tq tp,Tq Yt (Θ) = . (3.94) p h (i) i2 Pnp M p q=1 tp,Tq
Substituting this optimal value into the initial sum of squares (3.93), summing over the range of initial times {tp} and performing a nonlinear least-squares optimization as follows:
m n m 2 ∗ X X h (i) (i) #(i) (i)∗ i Θ := ArgMin U p + M p · Y (Θ) − log F p , (3.95) Θ∈Ω tp,Tq tp,Tq tp tp,Tq p=1 q=1 provides an “optimal fit” of the model to futures prices, therefore obtaining our risk- neutral model parameters (β, α, φ, σX , σY , ρ). An implementation of this methodology naturally requires both futures prices and spot prices at the corresponding futures quote times.
It is worth mentioning that this method does not directly extend to jump-diffusion
(spot) models since the coefficients of the Xt and Jt terms in the forward price (3.73) are unequal. This prevents a simple factorization into functions that are known given
(1) the spot prices and the hidden process Yt . To circumvent that problem a standard alternative methodology is to extract the jump parameters from the spot price data only. Such a calibration can be carried out in two ways: (i) by cutting off all data points 3 Energy Spot Price Models and Spread Options Pricing 42 lying below a given level, so that only spikes remain. From these data points one can then infer the value of the various jump parameters (see for example the discussion in [CS00]); or (ii) by utilizing particle-filter approaches which generalizes the Kalman filter to non-normal innovations (see for example [ABT05]). The standard assumption that the calibrated jump parameters are unchanged when moving to the risk-neutral world is then invoked. Given, the jump parameters it is now possible to repeat the previous futures price calibration process to obtain the risk-neutral diffusive components.
We now turn to the real-world P-parameters (β, α, φ, σX , σY , ρ) estimation problem. Since under any diffusive model for spot prices, a change of measure from the real-world to risk-neutral cannot alter the volatility structure of the model, from equation (3.95) we obtain σX , σY and ρ under P. The remaining set of P-parameters (β, α, φ) are relatively straightforward to obtain. Firstly, we obtain β and φ via linear regression on the spot price data assuming a mean reverting one-factor model for Xt as a proxy to our two-factor model. The one-factor mean-reversion level φ becomes, in our model, the stochastic long- run mean level Yt. Secondly, we perform a similar regression on the estimated hidden
# process Yt which was obtained by minimizing the error on an individual futures curve basis (see equation (3.94)). Equation (3.94) provides a data set which we can use as an input in a regression to find α. We find this procedure to be very stable and, as shown in the next section, leads to reliable parameter estimation.
3.6.2 Some Results: Crude Oil
α β φ α β φ σX σY ρ 0.15 0.31 3.27 0.73 1.07 4.21 33% 63% -0.96
Table 3.1: The calibrated real-world and risk-neutral model parameters using the NYMEX Light Sweet Crude Oil spot and futures data for the period 1/10/2003 − 25/07/2006. 3 Energy Spot Price Models and Spread Options Pricing 43
In this section, we present the calibration results of our two-factor pure diffusion model (Section 3.4) to the NYMEX Light Sweet Crude Oil spot and futures data for the period 1/10/2003 to 25/07/2006. In Table 3.1, we report the calibration results for the real-world and risk-neutral parameters. There are a few notable observations: (i) both real-world mean-reversion rates α and β are significantly larger than the risk-neutral mean-reversion rates α and β, (ii) The real-world long-run mean-reversion level φ is larger than the risk-neutral long-run mean φ, and (iii) in both the real-world and risk-neutral measures, the mean-reversion rates (α and α) of the long-run mean Yt are smaller than the mean- reversion rates (β and β) of the log-spot Xt.
$100
$80
$60 Price $40 Market Spot
$20 Simulation Long Run Mean $0 08/17/03 03/04/04 09/20/04 04/08/05 10/25/05 05/13/06 Date
Figure 3.1: The NYMEX Light Sweet Crude Oil spot prices and simulated spot prices based on the calibration in Table 3.1.
In Figure 3.1, we plot the spot price data together with the stochastic long-run mean
# level Yt implied by the futures prices. For comparison, we also include one simulated sample path based on a simulation of the prices using the real-world model parameters in Table 3.1. Figure 3.2 illustrates the relative root-mean squared-error (RMSE) for each forward curve using the model parameters reported in Table 3.1. The average RMSE per 3 Energy Spot Price Models and Spread Options Pricing 44 curve is 0.7% with only a few dates having relative errors larger than 1%. Recall that the model parameters are fixed over all curves, and are not adjusted on a curve by curve basis. With this in mind, we believe the fit is excellent.
1.8% 1.6% 1.4% 1.2% 1.0% 0.8% 0.6% Relative RMSE 0.4% 0.2% 0.0% 08/17/03 03/04/04 09/20/04 04/08/05 10/25/05 05/13/06 Date
Figure 3.2: The relative root-mean squared-error of each forward curve based on the calibration in Table 3.1.
# Days β α φ σX σY ρ 88 0.38 0.26 3.34 33% 19% -0.97 176 0.52 0.21 3.06 33% 54% -0.79 264 0.62 0.10 2.36 33% 56% -0.73 352 0.61 0.08 1.97 35% 60% -0.64 440 0.52 0.09 2.33 35% 58% -0.71 528 0.43 0.10 2.98 34% 52% -0.95 616 0.34 0.13 3.24 34% 58% -0.96 704 0.31 0.15 3.27 33% 63% -0.96 Average: 0.48 0.12 2.74 34% 57% -0.82 Stdev: 0.13 0.05 0.52 1% 4% 0.14
Table 3.2: This table shows the evolution of the estimated risk-neutral parameters through time as more recent data is added to the calibration procedure. The average and standard deviation are reported using 176 days onwards. 3 Energy Spot Price Models and Spread Options Pricing 45
3
1
-1
-3
-5
-7 Market Price of Risk -9 λ ψ
-11 8/17/03 3/4/04 9/20/04 4/8/05 10/25/05 5/13/06 Date
Figure 3.3: This diagram depicts the evolution of the implied market prices of risk using the calibrated real-world and risk-neutral parameters.
We also investigated the stability of our estimation procedure through time. We calibrated the model to the first T1 calender days and then to the first T2 calender days and so on. The time periods are approximately equally spaced at 88 days from 1/10/2003 to 25/07/2006. We report these calibration results in Table 3.2. The most stable parameters are the volatility σX of the Xt process, the volatility σY of the stochastic long-run mean level Yt, the mean-reversion level α of the stochastic long-run mean Yt and the correlation coefficient ρ. The remaining parameters, although not as unvarying as the previous four, are well behaved. None of the parameters suddenly explode or tend to zero, and always remain realistic.
Finally, since we were successful in extracting the real-world and risk-neutral param- eters, we further extract the implied market prices of risk through Theorem 3.4.2. The evolution of the implied market prices of risk λ and Ψ are displayed in Figure 3.3. Interest- ingly, they are very strongly correlated to one another, becoming almost indistinguishable after one and a half years. This is due to the high correlation coefficient of ρ = −0.96. 3 Energy Spot Price Models and Spread Options Pricing 46
Also, both market prices of risk are negative for essentially the entire time period. This is a reflection of the real-world mean-reversion rates (α and β) and real-world long run mean-reversion level (φ) being higher than the risk-neutral ones ((α, β and φ)). The mar- ket therefore attaches slower reversion rates and lower long run levels than the implied historical levels.
3.7 Conclusions
We introduced a diffusive two-factor mean-reverting process for modeling spot prices of energy commodities. The two-factor diffusive model extends the one-factor mean- reverting model by making the long-run mean a stochastic degree of freedom which itself mean-reverts to a specified level. We also generalized the model to incorporate jumps in the price process such as those observed in electricity prices. To maintain realistic mean- reversion rates and diffusive volatilities we decoupled the jump and diffusive processes. Given our affine modeling framework, we were successful in obtaining expressions for the forward price curves in terms of elementary functions. Through a measure changed induced by the forward price process, our modeling framework allows us to obtain closed form pricing equations for various spread options. We obtained pricing equations under both an actuarial and risk-neutral valuation procedures.
Finally, we provided a method for calibrating both the diffusion and jump-diffusion models to spot and forward prices simultaneously. This simultaneous calibration proce- dure further allowed us to extract the implied market prices of risk. Using the NYMEX light sweet crude oil data set, we demonstrated that the calibration procedure produces realistic and stable implied risk-neutral and real-world model parameters. 3 Energy Spot Price Models and Spread Options Pricing 47
3.8 Acknowledgements
The authors would like to thank Bill Bobey for assistance with acquiring the NYMEX data and Hans Tuenter, OPG Energy Markets, for fruitful discussions on various aspects of the energy markets. Chapter 4
Second Paper: Asymptotic Pricing of Commodity Derivatives using Stochastic Volatility Spot Models
4.1 Abstract
It is well known that stochastic volatility is an essential feature of commodity spot prices. By using methods of singular perturbation theory, we obtain approximate but explicit closed form pricing equations for forward contracts and options on single- and two-name forward prices. The expansion methodology is based on a fast mean-reverting stochastic volatility driving factor and leads to pricing results in terms of constant volatility prices, their Delta’s and their Delta-Gamma’s. Both the standard single factor mean-reverting spot model and a two-factor generalization, in which the long-run mean is itself mean- reverting, are extended to include stochastic volatility and each is analyzed in detail. The stochastic volatility corrections can be used to efficiently calibrate option prices and compute sensitivities.
48 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 49
4.2 Introduction
A quick glance at any commodities price data will reveal the obvious fact that volatility is a stochastic quantity. A now classical and extremely popular model for incorporating this stochasticity of volatility is the [Hes93] model, in which the instantaneous price variance follows a [CIR85] (CIR) like process. [EG98] were among the first to utilize the Heston model in the context of energy derivatives. More recently, [RS06] introduce a stochastic convenience yield model with one underlying stochastic volatility factor in the same spirit of Heston. They make an extensive case study on soybean futures and options data and demonstrate that stochastic volatility is a significant factor. Since Heston inspired stochastic volatility models lead to affine structures, they appear natural; however, the resulting pricing equations are in terms of inverse Fourier transforms rather than explicitly in terms of elementary, or even special, functions. This is not a substantial disadvantage when valuing only a few options; however, in a calibration and trading environment many contracts are involved and consistently calibrating all instruments to market prices would be difficult and time consuming. Furthermore, determining hedge ratios will require computations of the sensitivities of the price to various parameters – the so-called “Greeks” – which, if computed using Fourier methods, may result in further speed reduction. Finally, none of the previous models are tractable enough to provide pricing results, or even approximations, for options on two forward contracts. To circumvent all of these issues, we transport singular perturbation theory techniques, first developed for equity derivatives by [FPS00b] and then for interest rate derivatives in [CFPS04], into the context of commodities and commodities derivatives.
Asymptotic methods have three main advantages over traditional approaches: (i) they naturally lead to efficient calibration across a set of forward contracts; (ii) they lead to approximate, but explicit, closed form pricing equations for a wide class of contingent claims; and (iii) the resulting approximate prices are independent of the specific underlying 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 50
50%
40%
30%
20%
10%
0% 10/07/03 04/24/04 11/10/04 05/29/05 12/15/05 07/03/06 Dates Figure 4.1: The annualized running five-day moving volatility of the NYMEX sweet crude oil spot price for the period 10/07/03 to 07/03/06.
volatility model. Notably, these prices are exact when the mean-reversion rate is large – serendipitously, this is precisely the manner in which the market prices seem to behave. In addition, many of the salient features of option prices – most strikingly the implied volatility smile or smirk – are captured by these methods. [FPS00b] were the first to introduce the use of asymptotic methods in the context of derivative pricing and, together with their collaborators, have written several articles on the application of these techniques to the equity and interest rate markets. To this date, none of these techniques have been applied to the commodities markets where a unique set of challenges arise.
To motivate the validity of asymptotic methods for commodities, we plot the run- ning five-day realized volatility for the NYMEX sweet crude oil spot price for the period 10/07/03 to 07/03/06 in Figure 4.1 which clearly demonstrates the fast mean-reversion of volatility. We therefore model the underlying commodity spot price volatility as a function
σX (Zt) of a fast-mean reverting hidden process Zt. As is well known, commodities, unlike equities, tend to have strong mean-reversion effects in the prices themselves. Secondly, the long-run mean-reversion is not constant through time, rather it is stochastic. These and many other stylized empirical facts are well documented in, for example, [CS00], [EW03] and [Gem05]. Correctly accounting for such behavior together with stochastic volatility and using such models to price derivatives on one and two forward contracts is the main 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 51 contribution of this article.
[HJ07] introduced tractable two-factor mean-reverting models (with and without jumps) and priced forward and spread options on forward contracts. In this article, we successfully determine the asymptotic corrections for forward prices based on stochastic volatility extensions of the one- and two-factor mean-reverting diffusive spot price mod- els. To this end, we quickly review the one- and two-factor spot price models, together with the resulting forward and option prices, in Section 4.3.1. The stochastic volatility extended one- and two-factor mean-reverting models are introduced in Section 4.3.2 and we illustrate that such a model does not provide closed form forward prices. Section 4.4 contains two of our main asymptotic expansion results: the forward prices for the stochas- tic volatility extended one- and two-factor mean-reverting models are shown to be well approximated by adjusted constant volatility results (see equation (4.39) and (4.45)). By calibrating to existing forward prices, the volatility function σX (z) is rendered irrelevant; instead, a new effective pseudo-parameter arises as a smoothed version of the stochas- tic volatility. This pseudo-parameter appears again in the pricing of contingent claims, allowing a consistent calibration between forward and options prices.
It is important to point out that the one-factor commodity model is closely related the IR model examined in [FPS00a]; however, the relevant pricing object is the futures price rather than bond prices. This subtle modification induces a distinct non-vanishing boundary condition into the pricing problem and later on induces modified boundary con- ditions into the option pricing problem. This has the effect of introducing non-commuting operators into the resulting pricing PDEs which we solve using commutator relations (see the solution to equations (4.55) and (4.82)). In addition to the modified boundary condi- tion, we extend the one-factor model to a more realistic two-factor model for commodities. The two-factor model requires its own delicate balancing and analysis to ensure that the asymptotic expansions remain valid. 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 52
Given that the model is calibrated to forward prices, the next task is to determine the price corrections to contingent claims. Since typical single-name contingent claims are written on the forward prices, which we have already approximated, the asymptotic analysis relies on a consistent layering of approximations. In Section 4.5, these asymptotic price corrections to single-name contingent claims are explored. Interestingly, we demon- strate that the corrections depend solely on the Delta’s and Delta-Gamma’s of the option using the constant volatility model (see equations (4.57) and (4.61)). Furthermore, once the free pseudo-parameter arising in the forward price approximation is calibrated to mar- ket prices, the option price corrections are uniquely determined. Section 4.6 contains the extension of these methods to contingent claims written on two forward prices. There are several subtle issues associated with the expansion; nonetheless, we pleasantly find that the resulting price corrections are once again in terms of the Delta’s and Delta-Gamma’s of the constant volatility price (see equations (4.84), (4.85) and (4.99), (4.100)).
Although the techniques employed are quite similar to the single-name option case, the generalization of singular perturbation techniques to multi-name options is quite novel. The analysis requires balancing two fast mean-reverting processes which drive coupled spot price models. The implied forward processes then both factor into the option payoff, requiring a layering of the expansion parameters. It is important to prove that the cor- rection terms are indeed small enough to ignore. Furthermore, due to the non-vanishing boundary condition in the one and two-factor forward prices, commutation relations are once again necessary to solve the resulting PDEs.
We close the paper with conclusions and some comments on ongoing and future work in Section 4.7. 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 53
4.3 Spot Price Models and Main Properties
This section first provides an overview of the standard one- and two-factor constant volatil- ity models for commodity spot price dynamics. For early uses of the one-factor models see [GS90] and [CS94]. The forward prices, call and exchange option prices are also re- viewed. Given these constant volatility models, the stochastic volatility (SV) extensions are then introduced and we briefly demonstrate that the SV extensions lack an affine structure.
We explain why and where asymptotic methods constitute a very useful set of tools in energy markets, as they already have been shown to be for their stocks and interest rate counterparts.
4.3.1 Constant Volatility Models
The One-Factor Model
For completeness, this section provides a quick review of a well known one-factor commod- ity spot price model and its use in derivatives pricing. Let St denote the spot dynamics defined under the risk-neutral measure Q. The standard model assumes
St := exp {gt + Xt} , (4.1)
(1) dXt = β (φ − Xt) dt + σX dWt , (4.2)
(1) where σX is the constant volatility, gt is a deterministic seasonality factor and W is a Q- Wiener process. An important traded commodity instrument is the futures contract with futures price Ft,T . In a no-arbitrage, deterministic interest rate, environment the futures
Q and forward price coincides and the forward price must be given by Ft,T := Et [ST ], where
Q Et [R] represents the expectation of R conditional on the natural filtration Ft generated by the underlying Wiener process(es). The forward price process being a martingale, must 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 54 satisfy the following PDE
AF (t, x) = 0 , (4.3) F (T, x) = egT +x ,
where A is the infinitesimal generator of (t, Xt). Within the present context, a straight- forward calculation provides the following result
σ2 F = exp g + φ 1 − e−β(T −t) + X h(t, T ; 2β) + e−β(T −t) (log(S ) − g ) (4.4) t,T T 2 t t
Here, and in the sequel,
h(t, T ; a) := (1 − e−a(T −t))/a . (4.5)
Turning to the valuation of European contingent claims, let ϕ(FT0,T ) denote the termi- nal payoff at time T0 of a European option written on a forward price. The no-arbitrage price Πt,T0 is the discounted expectation under the risk-neutral measure Q. Specifically,
h R T0 i Q − t rs ds Q Πt,T0 := Et e ϕ (FT0,T ) = P (t, T0) Et [ϕ (FT0,T )] . (4.6)
Here, and in the remainder of this article, interest rates are deterministic, and we denote the T0-maturity zero-coupon bond price contracted at time t by P (t, T0). Following the
martingale techniques employed in [HJ07] Section 3.4, the price Ct;T0,T at time t of a
T0-expiry call option with strike K written on the forward FT0,T can be expressed in the following Black-Scholes like form:
h R T0 i Q − t rs ds Ct;T0,T = Et e (FT0,T − K)+
∗ ∗ ∗ = P (t, T0) Ft,T Φ(d + σt;T0 ) − K Φ(d ) . (4.7)
∗ ∗ Here, d and σt;T0 are functions of the model parameters and time only, and Φ(·) is the 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 55 standard gaussian cdf. A similar result follows for forward exchange option prices:
R T0 (1) (2) F Q − t rs ds Πt;T ,T ,T = Et e FT ,T − αFT ,T 0 1 2 0 1 0 2 + h i = P (t, T ) F (1) Φ(d + σ ) − αF (2) Φ(d) . (4.8) 0 t,T1 t;T0 t,T2
The interested reader is referred to the original article for the precise form of the various coefficients.
The Two-Factor Model : Mean-Reverting Long Run Mean
[HJ07] utilize a two-factor mean-reverting model, in which the long-run mean of the previous one-factor model is itself stochastic and mean-reverts to a second long-run mean. In that work, the authors study the valuation of forward contracts and exchange options and also include jumps into the spot price dynamics. In this article, we focus on the jump-free model; however, much of the results can be extended to the jump case with little additional complication.
In this two-factor model, the Q-dynamics of the spot St is
St = exp {gt + Xt} , (4.9)
(1) dXt = β (Yt − Xt) dt + σX dWt , (4.10)
(2) dYt = α (φ − Yt) dt + σY dWt , (4.11) with correlation structure,
(1) (2) d W ,W t = ρ1 dt . (4.12)
Here, β controls the speed of mean-reversion of Xt to the stochastic long-run level Yt; α controls the speed of mean-reversion of the long-run level Yt to the target long-run mean
φ; σX and σY control the size of the fluctuations around these means. The forward price 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 56 process can be shown to be
−β(T −t) Ft,T = exp gT + Rt,T + Gt,T + e Xt + Mt,T Yt (4.13)
where the expressions for Mt,T , Gt,T and Rt,T are functions of time and the model param-
eters. Even within this more general setting, the call option price Ct;T0,T on a forward as
F well as the exchange option price Πt;T0,T1,T2 on forwards have similar forms to (4.7) and ∗ ∗ (4.8) respectively. More complicated expression for d , σt;T0,T , d and σt;T0,T arise, yet they remain explicit functions only of the model parameters and time. The interested reader is once again referred to [HJ07] for details.
4.3.2 Stochastic Volatility Extensions
The SV Extended One-Factor Model
In this section, the stochastic volatility (SV) extended one-factor model is explored in detail; in particular, the volatility σX is now assumed to be driven by a fast mean- reverting stochastic process. Explicitly, the spot is now modeled under the risk-neutral measure Q as
St = exp {gt + Xt} , (4.14)
(1) dXt = β (φ − Xt) dt + σX (Zt) dWt , (4.15)
(3) dZt = α (m − Zt) dt + σZ dWt , (4.16)
where σX (·) is a strictly positive smooth function bounded above and below by posi- tive constants and with bounded derivatives. We also specify the following correlation structure
(1) (3) d W ,W t = ρ2 dt . (4.17) 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 57
The smoothness and boundedness assumptions on the volatility function σX (·) may ap- pear overly restrictive at first; however, as we later demonstrate, singular perturbation methods remarkably lead to pricing results that are completely independent of its detailed specification.
It is not possible to solve the system of SDEs (4.14)-(4.16) explicitly; nonetheless, we now explore its implications for forward prices. As usual, the forward price is F (t, x, z) =
Q Et,x,z [ST ]. Equivalently, F (t, x, z) can be characterized as the solution of the following PDE: ∂F ∂F ∂F ∂t + β(φ − x) ∂x + α(m − z) ∂z + 1 2 ∂2F 1 2 ∂2F ∂2F (4.18) 2 σX (z) ∂x2 + 2 σZ ∂z2 + ρ2σZ σX (z) ∂x∂z = 0 F (T, x, z) = egT +x
As we now show, a solution to (4.18) can be decomposed into two independent parts; one having a log-affine structure in x and the other being independent of x. First, let Wt be
(1) (3) a Q-Wiener process independent of Wt ,Wt and define the following
−β(T −t) dZet := α(m − Zet) + ρ2σZ σX (Zet)e dt + dWt , (4.19) 1 c(t, z) := σ2 (z)e−2β(T −t) + βφe−β(T −t) , (4.20) 2 X Z T Q M(t, z) := Et,z exp c(s, Zes)ds . (4.21) t
Then, by smoothness and boundedness of c(·, ·) and of the coefficients of dZet, M(t, z) is finite and satisfies the following PDE (see [DPS00])
∂M + α(m − z) + ρ σ σ (z)e−β(T −t) ∂M + 1 σ2 ∂2M + c(t, z) M = 0 ∂t 2 Z X ∂z 2 Z ∂z2 (4.22) M(T, z) = 1 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 58
−β(T −t) By direct, tedious, computations exp gT + e x M(t, z) is seen to satisfy the PDE
−β(T −t) (4.18); consequently, the forward price F (t, x, z) = exp gT + e x M(t, z).
Given the form of M(t, z), the forward prices clearly do not share the natural affine structure that other models often possess (e.g., compare with the constant volatility two- factor model (4.13)). It is also doubtful that an explicit (closed form) solution of the PDE (4.22) exists. Hence, this model appears to suffer from the deficiencies of Heston-like models which require either solving a PDE numerically or resorting to Fourier methods, rendering the models less useful for calibration purposes. Surprisingly, it is possible to partially overcome these difficulties if we accept to limit the range of applicability of our SV model to commodities having fast mean-reverting volatility (α 1). This is indeed the approach we pursue in the rest of this work.
The SV Extended Two-Factor Model
In this section, the stochastic volatility (SV) extended two-factor model is recorded for completeness. Starting with the two-factor model of Section 4.3.1, we make the volatility
σX a function of a fast mean-reverting stochastic process – analogous to the SV extended one-factor model. The spot is now modeled under a Q-measure as
(1) dXt = β (Yt − Xt) dt + σX (Zt) dWt , (4.23)
(2) dYt = αY (φ − Yt) dt + σY dWt , (4.24)
(3) dZt = α (m − Zt) dt + σZ dWt , (4.25) with correlation structure,
(1) (2) (1) (3) (2) (3) d W ,W t = ρ1 dt , d W ,W t = ρ2 dt , d W ,W t = 0 , (4.26)
and restrictions on σX (·) parallel to the previous section. 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 59
Rather than repeating the analysis of the previous subsection, we instead point out that resulting forward prices are not of the affine form. Nevertheless, asymptotic methods will lead to approximate, but explicit, closed form forward and option prices.
4.4 Forward Price Approximation
It is well known that the invariant distribution of the volatility driving factor Zt is Gaus-
2 2 sian with a variance of ν := σZ /2α. The asymptotic expansion revolves around assuming that α 1 and simultaneously holding the variance ν2 of the invariant distribution finite and fixed. As such, our developments are primarily parameterized by the small parameter := α−1. The ultimate goal of this section is to obtain a sound approximation (in a sense to be defined shortly) to the forward price, and in tandem eliminate the dependency of the approximate forward curve on the non-observable Zt.
Such closed form forward price approximations will allow efficient statistical estimation of the model parameters, and lead to tractable pricing of derivatives written on these forward curves. We use the methodology originally applied in [FPS00b] and [CFPS04] for stock and IR options respectively. For detailed discussions on the fundamentals of these asymptotic techniques we refer to the monograph [FPS00a].
Although the work in this section follows the methodology in [FPS00a], there are sev- eral distinctions. Firstly, and most simply, the model is motivated by commodities market behavior. Secondly, the relevant pricing objects are forward prices and not bond prices. This induces distinct boundary conditions which later on play an important distinguishing role and will require commutator relations to solve the resulting pricing PDEs. Finally, we extend the techniques to the more realistic two-factor model. 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 60
4.4.1 One-Factor Model + SV
In this section, we assume that the spot price dynamics is driven by the SV extended one-factor model in section 4.3.2. Recall that
Q F (t, x, z) := Et,x,z [ST ] , (4.27) where the dependence on (:= α−1) is made explicit. Rewriting the PDE (4.18) as
−1 − 1 A F = A + 2 A + A F (t, x, z) = 0 , 0 1 2 (4.28) F (T, x, z) = egT +x , with the three new operators defined as
∂ ∂2 A := (m − z) + ν2 , (4.29) 0 ∂z ∂z2 √ ∂2 A := 2ρ νσ (z) , (4.30) 1 2 X ∂x∂z ∂ ∂ 1 ∂2 A := + β(φ − x) + σ2 (z) , (4.31) 2 ∂t ∂x 2 X ∂x2
highlights the various scales of the individual operators. Note that A0 is the infinitesimal generator of a simple Vasicek (OU) process; A2 is the infinitesimal generator of the process
(t, Xt); while the A1 operator accounts for the correlation between the log spot price Xt and the volatility driver Zt processes. √ Expanding F in powers of
(0) √ (1) (2) 3 (3) F = F + F + F + 2 F + ... (4.32) where we impose the boundary conditions F (0)(T, x, z) := F (T, x, z) := egT +x and F (1)(T, x, z) := 0. We have explicitly assumed that the zeroth order term matches the payoff at maturity, while the first correction term vanishes at maturity. This terminal 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 61 splitting is not required, however it is natural, leading to explicit closed form approxima- tions, and allowing us to prove that the remaining corrections terms are O(). Inserting this last expansion into the PDE (4.28) and collecting terms with like powers √ of gives
1 1 0 = A F (0) + √ A F (0) + A F (1) + A F (0) + A F (1) + A F (2) 0 1 0 2 1 0 √ (1) (2) (3) + A2F + A1F + A0F + ... . (4.33)
√ From this last equation, the coefficients of the various powers of must vanish individu- ally. In the subsequent analysis we investigate these resulting equations and deduce from them the main properties of F (i)(t, x, z) for i = 0, 1, 2 and 3 explicitly.
−1 (0) • –Order Equation : A0F = 0 This holds for all z; therefore F (0) must be independent of z: F (0) = F (0)(t, x).
− 1 (0) (1) • 2 –Order Equation : A1F + A0F = 0
(0) (1) (1) Since F is independent of z, this implies A0F = 0. This further implies F is also independent of z; that is, F (1) = F (1)(t, x).
0 (0) (1) (2) • –Order Equation : A2F + A1F + A0F = 0
(1) (0) (2) Since F is independent of z, this implies the Poisson equation A2F +A0F = 0
(0) and the resulting centering equation hA2F i = 0 is a necessary condition for a solu- tion to exist. Here, and in the remainder of the article, the bracket notation hf(z)i
denotes the expectation of f(Z) where Z ∼ N(m, ν2), the invariant distribution of
(0) the Q-process Zt, as defined in (4.16). Since F is independent of z, the center-
(0) ing equation becomes hA2iF = 0. Remarkably, this is the PDE (4.3) satisfied by the forward price based on the one-factor spot model with constant volatility
p 2 (0) σX := hσX (z)i. Enforcing the boundary condition F (T, x) = exp (gT + x),
(0) implies that F is the one-factor forward price (4.4) with constant volatility σX . 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 62
Up to this order, it is also possible to extract properties of F (2) which will prove
(0) useful in the subsequent analysis. Due to the centering equation hA2iF = 0, notice that
1 A F (0) = (A − hA i) F (0) = σ2 (z) − hσ2 i F (0) , (4.34) 2 2 2 2 X X xx
(0) (2) which allows the zero-order equation A2F + A0F = 0 to be rewritten as
1 1 F (2) = − A−1 σ2 (z) − hσ2 i F (0) = − (ψ(z) + c(t, x)) F (0) , (4.35) 2 0 X X xx 2 xx
where the function ψ is define as the solution of
2 2 A0ψ = σX − hσX i , (4.36)
and c(t, x) is an arbitrary constant of integration. A straightforward calculation also shows that
Z z 0 1 2 2 2 ψ := ∂zψ = 2 2 σX (u) − hσX i Φ(u; m, ν ) du , (4.37) ν Φ(z; m, ν ) −∞
2 2 where Φ(·; m, ν ) is the cdf of N(m, ν ), the invariant distribution of Zt.
1 (1) (2) (3) • 2 –Order Equation : A2F + A1F + A0F = 0 This is a second Poisson equation, but now for F (3). Its centering equation is
(1) (2) (1) (2) hA2F + A1F i = hA2iF + hA1F i = 0 which is easily shown to transform
1 (0) √ 1 (1) − 2 0 (1) (1) 2 0 into hA2iF = 2 ρ2νhσX ψ iFxxx. Define Fe := F and V := ( 2 ) ρ2νhσX ψ i, 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 63
the centering equation is then
hA iF (1)(t, x) = VF (0) , 2 e xxx (4.38) Fe(1)(T, x) = 0 .
Equation (4.38) is the zero boundary version of the usual one-factor forward price √ PDE (4.3) with constant volatility σX and an additional source term of order . Using the previous result that F (0) has the form of the one-factor forward price (4.4),
direct computations show that Fe(1) = −V h(t, T ; 3β) F (0) is a solution to equation (4.38).
Piecing together all of the above partial results, the price approximation based on the first two terms of the expansion (4.32) is succinctly written as
F (t, x, z) ' F (0)(t, x) + Fe(1)(t, x) := ( 1 − V h(t, T ; 3β)) F (0)(t, x) . (4.39)
Intriguingly, the right hand side of (4.39) is independent of the unobservable Zt process. This is an extremely convenient consequence of asymptotic derivative valuation results. It is also worth noting that for calibration purposes, the constant V can, and should, be used as a parameter in its own right. All of the details of the mapping from Zt to the volatility process (σX (Zt)) is averaged out and embedded in the constant V . Rather than specifying the “micro-structure” in the model, it is perfectly valid to specify the “macro-structure” in V as implied from futures price data.
We now state one of our main results on the validity of the approximation (4.39).
2 Theorem. 4.4.1 For any fixed (T, x, z) ∈ R+ × R and all t ∈ [0,T ], we have
(0) (1) F (t, x, z) − F (t, x) + Fe (t, x) = O() , 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 64 where the approximation F (0)(t, x) + Fe(1)(t, x) is defined in (4.39) and F (0)(t, x) as in
p 2 (4.4) with σX replaced by hσX (z)i.
Proof. Define the function Υ(t, x, z) as the error terms of order 2 and higher. Explicitly,
(0) √ (1) (2) 3 (3) Υ := F + F + F + 2 F − F . (4.40)
We first aim at proving that |Υ| = O(). Applying the infinitesimal generator A of
(t, Xt,Zt) on Υ and cancelling vanishing terms, based on our previous analysis of the F (i) functions, we find
−1 − 1 (0) √ (1) (2) 3 (3) A Υ = A0 + 2 A1 + A2 F + F + F + 2 F − F
(2) (3) √ (3) = A2F + A1F + A2F . (4.41)
Now focus on each term from the right hand side of (4.41), paying attention to their growth properties as functions of x, z.
(2) •A2F -Term:
(2) 1 (0) Choosing the constant of integration in (4.35) to be zero, we have F = − 2 ψ(z)Fxx . In addition, since ψ(z) satisfies the Poisson equation (4.36) and since its r.h.s. is bounded and satisfies the centering condition, then ψ(z) grows at most linearly in |z|. Given the form of the forward price (4.4), it is clear that F (0) (and therefore F (2)) is log-linear in x.
(3) (3) •A1F and A2F -Terms:
1 (3) (3) (1) From the 2 -order analysis, F satisfies the Poisson equation A0F + A2F +
(2) (1) (2) A1F = 0 and the centering condition hA2F + A1F i = 0. We then have, 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 65
(1) (2) (1) (1) (2) (2) A2F + A1F = A2F − hA2F i + A1F − hA1F i . Consequently,
√ 1 F (3) = − 2ρ νη(z)F (0) − ζ(z)F (1) , (4.42) 2 xxx 2 xx
0 0 where η(z) and ζ(z) are characterized by solutions of A0η = σX ψ − hσX ψ i and
2 2 A0ζ = σX − hσX i, respectively, with both constants of integration set to zero. Both of these last two Poisson equations satisfy the centering equation and have bounded source terms, implying that η(z), ζ(z) are at most linearly growing in |z| with bounded first derivatives. From these last properties of η(z), ζ(z) and the form
(3) (3) of F in (4.42) as well as the boundedness of σX (z), we conclude that A1F and
(3) A2F are at most linearly growing in |z| and log-linearly growing in x.
(2) (3) The above results allow us to bound the error term Υ . Define N := A2F +A1F +
√ (3) A2F so that equation (4.41) becomes A Υ = N. With this new terminology, the “Feynman-Kac” probabilistic representation of (4.41) can be expressed as (see [KS91], section 5.7):
√ Z T Q (2) (3) Υ (t, x, z) = Et,x,z F (T,XT ,ZT ) + F (T,XT ,ZT ) − N(s, Xs,Zs) ds (4.43) t
We have already demonstrated that N(t, x, z), F (2)(T, x, z) and F (3)(T, x, z) are at most linearly bounded in |z| and log-linearly growing in x. For the N function, this bound is uniform in t ∈ [0,T ]. Furthermore, since σX (·) is bounded, a direct check (or see Lemma
B.1 in [CFPS04]) shows that Xt has finite exponential moments. Similarly for the process
Zt, which implies a bound on its second moment (variance). Therefore, |Υ | = O(), as previously claimed.
We make use of this last partial result and write
(0) (1) (2) 2 (3) (2) √ (3) 3 F − (F + Fe ) = F + F − Υ ≤ |Υ | + F + F , (4.44) 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 66 which, by the properties of F (2) and F (3), completes the proof. 2
We have succeeded in demonstrating that, when the mean-reversion rate is large, the forward prices in the SV extended one-factor model are well approximated by the con- stant volatility price with a small adjustment factor. The correction term is proportional to a parameter V which itself encapsulates the volatility function σX (Zt) information. However, from a calibration and pricing perspective, the detailed specification of this pa- rameter in terms of the underlying volatility function is irrelevant, and instead it should be viewed as a free parameter in and of itself. There is one interesting limit to consider: the limit in which the correlation between the volatility factor Zt and the log spot price process Xt is zero. In this limit, the correction term vanishes identically; however, the market will likely have a non-zero correlation between volatility and spot price returns. In fact, it is well known that for commodities there is an inverse leverage effect which drives volatility higher when spot prices rise.
We would like to make one last comment concerning the SV corrected forward price (4.39): the correction vanishes as T & t while it tends to (1 − V/3β) as T → +∞.
T →+∞ 2 Specifically we have, F (t, x, z) −→ exp{φ + ln(1 − V/3β) + σX /4β}. Consequently, if one fixes the long-end of the log-forward curve and adjusts V , then V will control the mid-term of the forward curves. This is nice feature, because then, V can be viewed as an independent lever affecting the strength of the forward curve hump.
To illustrate this point, in Figure 4.2 we plot sample forward curves with three choices of V . The diagram clearly shows that V affects the strength of the hump. Interestingly, regardless of the sign of V , in this specific example, the forward curve always becomes more humped. 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 67
$61.50 $60.50
$61.25 $60.25
$61.00 $60.00 V = 0 V = 0.3
rd Price $60.75 rd Price $59.75 V = -0.3 a a V = 0 $60.50 $59.50 V = 0.3 Forw Forw $60.25 V = -0.3 $59.25
$60.00 $59.00 02.557.510 02.557.510 Term Term
Figure 4.2: This diagram depicts typical forward curves implied the model for three choices of V . The long-run forward price is set at 61 in the left panel and 59 in the right panel. The spot is 60, β = 0.5 and σX = 0.2.
4.4.2 Two-Factor Model + SV
In this section, we assume that the spot price dynamics is driven by the SV extended two- factor model of section 4.3.2 and look for an approximation to the implied forward prices. We omit the details of the calculations since the formal expansion procedure follows the same steps as in Section 4.4.1, with A2 containing additional terms due to the stochastic long-run mean.
Theorem. 4.4.2 For any fixed (T, x, y, z) ∈ R+ × R3 and all t ∈ [0,T ], we have
Ft,T = {(1 − V1h(t, T ; 3β) β (0) − V2 [h(t, T ; 3β) − h(t, T ; αY + 2β)] Ft,T + O() , (4.45) αY − β
(0) where Ft,T is the two-factor forward curve (4.13) with constant volatility σX replaced by √ p 2 p 0 0 hσX (z)i and the new parameters V1 := 2 ρ2νhσX ψ1i and V2 := 2ρ1ρ2νσY hσX ψ2i.
From a calibration and pricing viewpoint, the detailed composition of V1 and V2 in terms of the initial parametrization is again irrelevant – they should now be considered as parameters in their own right. Furthermore, this approximation is, as in our previous 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 68
forward approximation, independent of Zt. This allows an easy calibration of the two- factor model to futures prices; see [HJ07] and its reference for more details on these topics. Once again, these parameters can be viewed as levers to change strength and now also the shape of the forward-curve hump.
4.5 European Single-Name Options
Forward price determination is only the first stage of the analysis. For a model and method to be of any real use, it must lead to efficient valuation tools for single- and two-name option prices. In this section, we illustrate how the approximate forward prices from the previous section can be utilized to obtain approximate European single-name option prices. In Section 4.6, the issue of two-name contracts is addressed. Both single- and two-name approximations lead to closed form results which depend solely on constant volatility prices, Delta’s and Delta-Gamma’s.
Once again the techniques in [FPS00a] are used to carry out the analysis; however, there are two points which require highlighting: (i) Since the options are written on forward contracts which do not mature to a deterministic value, there is a non-trivial boundary condition in the first order correction term (see equation (4.55)). Nonetheless, we are still able to solve this PDE by exploiting commutator relations. (ii) The asymptotic expansion for forward prices induced by the two-factor spot model requires its own analysis and cannot be extrapolated from the one-factor model results.
4.5.1 Smooth Payoff Function
One-Factor Model + SV
Consider a smooth payoff function ϕ(·) with bounded derivatives and linear growth at infinity. Based on our SV extended one-factor spot price model of Section 4.3.2 we inves- 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 69
tigate the price Π (t, x, z) at time t of the T0-contingent claim ϕ(FT0,T ) on the forward price FT0,T , that is
Q Π (t, x, z) = P (t, T0) Et,x,z ϕ(FT0,T ) . (4.46)
To simplify notation we omit the explicit appearance of T0 and T in the price function.
To obtain an approximation scheme for (4.46), the previous asymptotic result FT0,T = F (0) + F (1) + O() from Theorem 4.4.1 will be used. To this end, consider a power T0,T eT0,T expansion of the option payoff ϕ(F ) around ϕ(F (0) ) – this is valid since we have T0,T T0,T made appropriate smoothness assumptions on ϕ(·)
ϕ F = ϕ F (0) − V h(T ,T ; 3β)F (0) · ϕ0 F (0) + O() . (4.47) T0,T T0,T 0 T0,T T0,T
From (4.46) the price function Π satisfies a similar PDE to the one F satisfies (see (4.28)) with modified terminal conditions. Explicitly,
−1 − 1 ∗ A Π = A + 2 A + A Π (t, x, z) = 0 , 0 1 2 (4.48) Π (T0, x, z) = ϕ (x) ,
∗ where A2 := A2−r(t), r(t) is the short-rate and A0, A1 and A2 are defined in (4.29)-(4.31). √ Expanding Π in powers of , as previously done with F , we have
(0) √ (1) (2) 3 (3) Π = Π + Π + Π + 2 Π + ... , (4.49) and plugging into (4.48) gives
1 1 0 = A Π(0) + √ A Π(0) + A Π(1) + A∗Π(0) + A Π(1) + A Π(2) 0 1 0 2 1 0 √ ∗ (1) (2) (3) + A2Π + A1Π + A0Π + ... . (4.50) 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 70
√ An analysis of the various equations arising from (4.50) order-by-order in – analogous to the study carried out in Section 4.4.1 and specifically for (4.33) – yields
h i Π(0)(t, x) = P (t, T ) Q ϕ F (0) (X ) , (4.51) 0 Et,x T0,T T0 h i Π(1)(t, x) = −V h(t, T ; 3β)P (t, T ) Q F (0) (X ) ϕ0 F (0) (X ) e 0 0 Et,x T0,T T0 T0,T T0
Z T0 Q (0) −V Et,x P (t, u)Πxxx(u, Xu) du , (4.52) t 1 Π(2)(t, x, z) = − ψ(z)Π(0) , (4.53) 2 xx
√ 1 (1) (1) 2 0 where Πe (t, x) := Π , V = 2 ρ2νhσX ψ i is the same parameter which arose in the analysis of the forward price approximation in Section 4.4.1, ψ(z) is defined in (4.36), and the “smoothed” process Xt satisfies the SDE
(1) dXu = β(φ − Xu) du + σX dWu , Xt = Xt . (4.54)
2 2 Here, σX := hσX (z)i. Note that equation (4.52) is, due to its integral part, quite difficult to compute explicitly. It is, however, possible to transform Πe (1) into a much more tractable √ form. From the -order analysis, we find that Πe (1) satisfies the following PDE:
hA∗iΠ(1)(t, x) = V Π(0) , 2 e xxx (4.55) (0) (0) Π(1)(T , x) = −V h(T ,T ; 3β)F ϕ0 F . e 0 0 T0,T T0,T
Using the commutation relation
∗ (0) 3 ∗ 3 ∗ (0) (0) hA2iΠxxx = ∂xhA2i + ∂x; hA2i Π = 3β Πxxx (4.56)
(0) where [A; B] := AB − BA, one can show that G1 := −V h(t, T0; −3β)Πxxx is a solution of
(4.55) with zero boundary condition. Also, a specific solution (say G2) to the homogeneous version of the PDE (4.55) provides a unique solution G1 + G2 to (4.55). Using Feynman- 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 71
Kac with a source to obtain G2, we conclude that
(1) (0) Πe (t, x) = −V h(t, T0; −3β)Πxxx h i −V h(T ,T ; 3β)P (t, T ) Q F (0) (X ) ϕ0 F (0) (X ) . (4.57) 0 0 Et,x T0,T T0 T0,T T0
This last expression is now much simpler to compute for any reasonably well behaved payoff function. It is particularly interesting that the correction terms are dependent only on the zeroth order price, which themselves are determined in terms of the constant volatility model. Furthermore, the first term in the above correction explicitly depends on the Delta-Gamma of the constant vol option price. Contrastingly, the second term can be viewed as the price of a modified payoff assuming constant volatility. For example, if valuing a call option, then the second correction term is the price of an asset-or-nothing option. Finally, the parameter V which controls the impact of stochastic volatility is inherited from the forward price approximation (4.39). We conclude this section by providing the conditions of validity of our price approxi- mation in the following theorem.
2 2 Theorem. 4.5.1 For any fixed (T0, T, x, z) ∈ R+ × R with T0 ≤ T and for all t ∈ [0,T0], we have
(0) (1) Π (t, x, z) − Π (t, x) + Πe (t, x) = O() , where the approximation Π(0)(t, x) + Πe (1)(t, x) is defined in (4.51) and (4.57).
Proof. The proof follows along similar lines to the proof of Theorem 4.4.1. The one main complication is to demonstrate that x-derivatives of Π(0) and Πe (1) have at most exponential growth. This is achieved by appealing to the smoothness properties of ϕ(·) and Lebesgue’s dominated convergence theorem, as similarly done in the more general situation of Section 4.6.1. We provide more details there. 2 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 72
Two-Factor Model + SV
Based on our SV extended two-factor spot price model of Section 4.3.2, we seek an ap-
proximation to the price Π (t, x, y, z) of a T0-contingent claim with payoff ϕ(FT0,T ), i.e.
Q Π (t, x, y, z) = P (t, T0) Et,x,z ϕ(FT0,T ) . (4.58)
The forward approximation F = F (0) +F (1) +O() used in the expansion methodology T0,T T0,T eT0,T is now the one from Theorem 4.4.2. The mathematical developments leading to the next theorem are very similar to those of Section 4.5.1; we therefore concentrate on the precise statement of the main result and omit the proof.
2 3 Theorem. 4.5.2 For any fixed (T0, T, x, y, z) ∈ R+ × R with T0 ≤ T and for all t ∈
[0,T0], we have
(0) (1) Π (t, x, y, z) − Π (t, x, y) + Πe (t, x, y) = O() , where
h i Π(0)(t, x, y) := P (t, T ) Q ϕ F (0) (X ,Y ) , (4.59) 0 Et,x,y T0,T T0 T0 with F (0) as in Theorem 4.4.2, the process X of (4.23) being replaced by its “smoothed T0,T t version”
(1) dXu = β(Yu − Xu) du + σX dWu , Xt = Xt , (4.60)
2 2 σX := hσX (z)i and
(1) (0) (0) Πe (t, x, y) := l1(t, T0)Πxxx + l2(t, T0)Πxxy h i +l(T ,T ) P (t, T ) Q F (0) (X ,Y ) ϕ0 F (0) (X ,Y ) (4.61) 0 0 Et,x,y T0,T T0 T0 T0,T T0 T0 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 73 where,
β V2 l1(t, T0) := − h(t, T0; −2β − αY ) β − αY 2 3 β V2 1 − V1 + 1 − h(t, T0, −3β), (4.62) 2β + αY β − αY
l2(t, T0) := −V2 h(t, T0; −2β − αY ) , (4.63) β l(T0,T ) := −V1h(T0,T ; 3β) − V2 [h(T0,T ; 3β) − h(T0,T ; 2β + αY )] . (4.64) αY − β
Furthermore, V1 and V2 are as in Theorem 4.4.2.
Once again, we find that the SV extended model option prices are written in terms of the constant volatility model prices with a smoothed volatility. The correction terms are again in terms of the various Delta’s and Delta-Gamma’s with coefficient proportional to the parameters Vi which themselves are inherited from the forward price approximation (4.45).
4.5.2 Nonsmooth Payoff: Calls and Puts
When the T0-payoff function ϕ(·) is non-smooth, Theorem 4.5.1 and 4.5.2 can be gen- eralized via a further approximation scheme. The main device is to approximate the non-smooth payoff function by a regularized version – in particular its discounted condi- tional expectation over a very small time – and then prove that the regularized option price well approximates the exact price. The required methodology is, due to the differ- entiability of our one-factor commodity forward call/put option prices (4.7), a simplified version of the one originally developed for stock options in [FPSS03]. We therefore refer to that paper for further mathematical details.
For practical purposes, it suffices to know that the approximate prices developed in Theorems 4.5.1 and 4.5.2 are still valid for non-smooth call/put options as long as they are not used for extremely close to maturity option contracts. In practice, there would be 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 74 no need for such a pricing methodology for small terms since it would be clear whether the option is in or out of the money.
4.6 European Two-Name Options
In this section, we pursue the approximations of options written on two correlated com- modity forwards, where each commodity is driven by an SV extended one- or two-factor mean-reverting model. This is a completely novel extension of the singular perturbation methods. The analysis is more involved than previously; however, the end results inherit similar structures to the single name cases. In particular, the price is found in terms of the constant volatility model price with correction terms depending on the various Delta’s and Delta-Gamma’s. Interestingly, two new parameters arise in this case. These new pa- rameters cannot be calibrated from forward prices, or options on the individual forward prices, instead they should be viewed as a flexibility lever allowing the trader to bias the prices (or equivalently the implied vol skew) upward or downward.
4.6.1 Smooth Payoff Function
Consider a smooth payoff function ϕ(·, ·) having bounded partial derivatives and a linear growth at infinity in each variable. Our main goal is to find a well behaved approximation to the option price Π which as usual is written in terms of the discounted expectation under the risk-neutral measure
Π~(t, ~x,~z) = P (t, T ) Q ϕ F 1 ,F 2 . (4.65) 0 Et,~x,~z T0,T1 T0,T2
Note that we allow the forward contracts to have different maturities, that is, we only require T0 ≤ T1,T2. Most of the important steps in the derivation are explicitly provided for the SV extended one-factor model only, while the main Theorem for the two-factor 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 75 model is simply stated.
One-Factor Model + SV
Here, the joint dynamics of the spot and forward price for the pair of commodities (i = 1, 2) are assumed to satisfy the system of SDEs
(i) n (i) (i)o St = exp gt + Xt , (4.66) h i i Q (i) Ft,T = Et ST , (4.67)
(i) (i) (i) (1i) dXt = βi φi − Xt dt + σXi Zt dWt , (4.68)
(i) (i) (3i) dZt = αi mi − Zt dt + σZi dWt , (4.69)
(11) (12) (1i) (3i) with correlation structure d W ,W t = ρ dt, d W ,W t = ρ2i dt and all others zero. We also assume that the volatility functions σXi(·) are again smooth, strictly positive and bounded functions with bounded derivatives. Also notice that the explicit dependence on the small parameter i := 1/αi has been made. As before (see Section 4.4), the variance
2 2 (i) νi := σZi/2αi of the Zt -invariant distributions are held fixed in the limit of small i. We are now ready to develop an approximation to the price (4.65) which satisfies the PDE
1 1 ~ ~ −1 (1) −1 (2) − 2 (1) − 2 (2) ∗ ~ A Π = 1 A0 + 2 A0 + 1 A1 + 2 A1 + A2 Π = 0 , (4.70) Π~(T , ~x,~z) = ϕ F 1 ,F 2 , 0 T0,T1 T0,T2
~ −1 ~ ~ R t where A is the generator of (t, Mt , Xt, Zt) with Mt := exp{ 0 rs ds} the money market account and the operator 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 76
2 2 ∗ ∂ ∂ ∂ 1 2 ∂ 1 2 ∂ A2 := + β1(φ1 − x1) + β2(φ2 − x2) + σX1(z1) 2 + σX2(z2) 2 ∂t ∂x1 ∂x2 2 ∂x1 2 ∂x2 ∂2 + ρσX1(z1)σX2(z2) − r(t) . (4.71) ∂x1∂x2
~ √ √ Expanding Π in powers of 1 and 2 we have
~ (0) √ (1,1) √ (1,2) (2,1) (2,2) √ (2,3) 3/2 (3,1) Π = Π + 1Π + 2Π + 1Π + 2Π + 12Π + 1 Π
√ (3,2) √ (3,3) 3/2 (3,4) + 1 2Π + 12Π + 2 Π + ... , (4.72)
with T0-terminal condition
∂ϕ 1 2 1(0) 2(0) 1(1) 1(0) 2(0) ϕ FT ,T ,FT ,T = ϕ FT ,T ,FT ,T + FeT ,T FT ,T ,FT ,T 0 1 0 2 0 1 0 2 0 1 ∂F 1 0 1 0 2
2(1) ∂ϕ 2(0) 2(0) 0 + FeT ,T FT ,T ,FT ,T + O( ) (4.73) 0 2 ∂F 2 0 1 0 2
0 i(0) i(1) Here, and in the sequel, := max(1, 2) and Ft,T (Ft,T ) is the first (second, resp.)
i order approximation of the forward price Ft,T , i = 1, 2 (see Section 4.4).
Now, collect terms of the equivalent orders arising from (4.70) on substitution of (4.72)-(4.73), as in the previous section. In the following, we emphasize the new aspects of the present (more general) asymptotic analysis and omit most of details. A study of
−1 −1 −1/2 −1/2 √ √ (0) (1) the 1 ,2 ,1 ,2 , 1/2 and 2/1 - order equations results in Π and Π being
(0) (0) (1) (1) independent of ~z := (z1, z2). Explicitly: Π = Π (t, ~x) and Π = Π (t, ~x).
0 (1) (2,1) (2) (2,2) ∗ (0) • -Order Equation: A0 Π + A0 Π + A2Π = 0 Any solution of the two Poisson equations
1 1 A(1)Π(2,1) + A∗Π(0) = 0 , and A(2)Π(2,2) + A∗Π(0) = 0 . (4.74) 0 2 2 0 2 2 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 77
is a solution of the 0-order PDE. Both Poisson equations have identical centering
∗ (0) ~ ~ conditions hA2Π i = 0, where hf(Z∞)i is defined as the expectation of f(Z∞) with ~ 2 ~ (1) (2) Z∞ ∼ N(~m,~ν ), the invariant distribution of the process Zt = (Zt ,Zt ) defined in
1 ∗ (0) ∗ (0) (4.69) . The centering condition reduces to hA2Π i = hA2iΠ = 0 and enforcing the b.c. (4.73) to zeroth order, implies that Π(0)(t, ~x) is the option price in the
2 1/2 constant volatility one-factor model with σXi := (hσXii) and correlation ρ :=
2 2 1/2 ρhσX1ihσX2i/(hσX1ihσX2i) . The new correlation ρ is in [−1, 1] due to H¨older’s inequality. Explicitly,
h n (1) (2)oi Π(0)(t, ~x) = P (t, T ) Q ϕ F 1(0) X ,F 2(0) X . (4.75) 0 Et,~x T0,T1 T0 T0,T2 T0
(i) Here, the smoothed processes Xt are again defined by
(i) (i) (i) (i) (i) dXu = βi φi − Xu du + σXi dW u , Xt = Xt , (4.76)
(1) (2) with correlation d[W , W ] = ρ.
Using the above solution for Π(0) and following the arguments leading to equation (4.35), but starting with (4.74), we find
1 ρ Π(2,1) = − {ψ (z ) + c (t, ~x,z )} Π(0) + {ψ (~z) + c (t, ~x,z )} Π(0) (4.77) 4 1 1 1 2 x1x1 2 12 12 2 x1x2 1 ρ Π(2,2) = − {ψ (z ) + c (t, ~x,z )} Π(0) + {ψ (~z) + c (t, ~x,z )} Π(0) (4.78) 4 2 2 2 1 x2x2 2 21 21 1 x1x2
1 (1) (2) Since Zt and Zt are independent processes, they also have independent invariant distributions. 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 78
where the ψi’s and ψij’s are defined by
(1) 2 2 (1) A0 ψ1 = σ − hσ i , A0 ψ12 = σX1σX2 − hσX1σX2i , X1 X1 (4.79) (2) 2 2 (2) A0 ψ2 = σX2 − hσX2i , A0 ψ21 = σX1σX2 − hσX1σX2i ,
with the ci’s and cij’s being their respective (arbitrary) constants of integration.
p p (1) (2,3) (2) (2,3) • 1/2 , 2/1 -Order Equations: A0 Π = 0 and A0 Π = 0 These equations imply that Π(2,3) = Π(2,3)(t, ~x) is independent of ~z.
√ ∗ (1,1) (1) (2,1) (1) (3,1) (2) (3,3) • 1-Order Equation : A2Π + A1 Π + A0 Π + A0 Π = 0 Once again decoupling this PDE into two Poisson equations
1 A(1)Π(3,1) + A∗Π(1,1) + A(1)Π(2,1) = 0 , (4.80) 0 2 2 1 1 A(2)Π(3,3) + A∗Π(1,1) + A(1)Π(2,1) = 0 , (4.81) 0 2 2 1
∗ (1,1) (1) (2,1) leads to the centering condition hA2Π + A1 Π i = 0. Inserting the expression for Π(2,1) implies that
√ (0) ∗ (1,1) √1 0 hA2iΠe = ρ21ν1hσX1ψ1iΠx1x1x1 2 2 √ (0) + √ 1 ρρ ν hσ ∂ ψ iΠ , (4.82) 2 21 1 X1 z1 12 x1x1x2 (1) (0) (0) Π(1,1)(T , X~ ) = F 1 ∂ϕ F 1 ,F 2 . e 0 T0 eT0,T1 ∂F 1 T0,T1 T0,T2
(1,1) √ (1,1) where Πe := 1 Π and its boundary condition being induced by (4.73). The
∗ ∗ commutation rules [hA2i; ∂x1x1x1 ] = 3β1∂x1x1x1 and [hA2i; ∂x1x1x2 ] = (2β1+β2)∂x1x1x2 , 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 79
∗ (0) together with the fact that hA2iΠ = 0, allows one to write
∗ (0) (0) (0) hA2i l1(t)Πx1x1x1 + l2(t)Πx1x1x2 = (∂tl1 + 3β1l1)Πx1x1x1
(0) +(∂tl2 + (2β1 + β2)l1)Πx1x1x2 , (4.83)
for l1(t) and l2(t) arbitrary functions of time only. Matching the coefficients of the r.h.s. with coefficients in the r.h.s of the PDE (4.82), and solving the resulting ODEs
for l1,2 (with b.c. l1(T ) = l2(T ) = 0), allows us to solve the PDE (4.82) explicitly
(1,1) V1 (0) (0) Πe (t, ~x) = − h(t, T0; −3β1)Π − V11 h(t, T0; −2β1 − β2)Π 2 x1x1x1 x1x1x2
−V1h(T0,T1; 3β1) P (t, T0)
(1) ∂ϕ n (1) (2)o Q 1(0) 1(0) 2(0) ×E FeT ,T XT FT ,T XT ,FT ,T XT (4.84) t,~x 0 1 0 ∂F 1 0 1 0 0 2 0
p 1 0 p 1 with V1 := 2 ρ21ν1hσX1ψ1i and V11 := 2 ρρ21ν1hσX1∂z1 ψ12i. Equation (4.84) depends solely on the Delta’s and Delta-Gamma’s of the constant volatility price and the constant volatility price of a modified payoff. These individual terms can be computed explicitly in many typical cases – such as Margrabe spread options.
√ ∗ (1,2) (2) (2,2) (1) (3,2) (2) (3,4) • 2-Order Equation : A2Π + A1 Π + A0 Π + A0 Π = 0 Going through similar arguments as above, we find
(1,2) V2 (0) (0) Πe (t, ~x) = − h(t, T0; −3β2)Π − V22h(t, T0; −β1 − 2β2)Π 2 x2x2x2 x1x2x2
−V2h(T0,T2; 3β2) P (t, T0)
(2) ∂ϕ n (1) (2)o Q 2(0) 1(0) 2(0) × E FeT ,T XT FT ,T XT ,FT ,T XT (4.85) t,~x 0 2 0 ∂F 2 0 1 0 0 2 0
√ (1,2) (1,2) p 2 0 p 2 with Πe := 2 Π , V2 := 2 ρ22ν2hσX2ψ2i and V22 := 2 ρρ22ν2hσX2∂z2 ψ21i. 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 80
We now aim at proving the main result of this section, which, according to our general expansion methodology (4.70)-(4.73) and its subsequent analysis, should take the form of
Π~(t, ~x,~z) ' Π(0)(t, ~x) + Πe (1,1)(t, ~x) + Πe (1,2)(t, ~x) , (4.86)
whenever the inverse mean-reversion parameters 1 and 2 are sufficiently small. The precise formulation of this approximation is the subject of our next Theorem.
3 4 Theorem. 4.6.1 For any fixed (T0,T1,T2, ~x,~z) ∈ R+ × R with T0 ≤ T1,T2 and for all t ∈ [0,T0], we have
~ (0) (1,1) (1,2) 0 Π (t, ~x,~z) − Π (t, ~x) + Πe (t, ~x) + Πe (t, ~x) = O( ) , (4.87) where the terms Π(0),Πe (1,1), and Πe (1,2) are defined in (4.75), (4.84), and (4.85). Finally,
0 := max{1, 2}.
Proof. First define the function Υ~(t, ~x,~z) by
3 √ ~ (0) (1,1) (1,2) (2,1) (2,2) 2 (3,1) (3,2) Υ = Π + Πe + Πe + 1Π + 2Π + 1 Π + 1 2Π √ 3 (3,3) 2 (3,4) ~ + 12Π + 2 Π − Π (4.88)
Notice that the Π(2,3)-term has purposefully been included in Υ~ – this is a crucial splitting for the validity of the remaining analysis. The first step toward a proof of
0 Therem 4.6.1 is once again to show that Υ~ = O( ). As similarly executed in Section 4.4, we study the properties of Υ~ via its behavior when acted on by the generator A~. From our previous analysis and the boundary condition (4.73), we have 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 81
A~Υ~ = A∗Π(2,1) + A(1)Π(3,1) + A(2)Π(3,2) 1 2 1 1 ∗ (2,2) (1) (3,2) (2) (3,4) +2 A2Π + A1 Π + A1 Π √ (1) (2) 3/2 + A Π(3,2) + A Π(3,3) + A∗Π(3,1) 1 2 1 1 1 2 √ ∗ (3,2) √ ∗ (3,3) 3/2 ∗ (3,4) +1 2A2Π + 12A2Π + 2 A2Π , (4.89) ~ (2,1) (2,2) 3/2 (3,1) Υ (T0, ~x,~z) = 1Π (T0, ~x,~z) + 2Π (T0, ~x,~z) + Π (T0, ~x,~z) 1 √ √ + Π(3,2)(T , ~x,~z) + Π(3,3)(T , ~x,~z) 1 2 0 1 2 0 3/2 (3,4) 0 +2 Π (T0, ~x,~z) + O( ) .
The detailed growth properties of the first two terms of (4.89) and related boundary conditions will now be examined. All other terms can be shown (with limitless patience!) to have similar bounds.
∗ (2,1) •A2Π -Term:
∗ (2,1) Without loss of generality, we choose c1 = c12 = 0 in (4.77). Then A2Π involves
multiplications of the terms ψ1,ψ12 with up to the fourth order partial derivatives
(0) ∗ of Π and the smooth (linear growth in ~x and bounded in ~z) coefficients of A2. By the boundedness of the source terms in their defining Poisson equations (4.79),
ψ1 and ψ12 are at most linearly growing in their arguments and have bounded first derivative. From the Appendix, the partial derivatives of Π(0) are at most log-
∗ (2,1) linearly bounded in ~x. Aggregating these partial results, A2Π has at most a linear growth in ~z and log-linear in ~x. It also follows from these last arguments that
(2,1) Π (T0, ~x,~z) share equivalent growth bounds.
(1) (3,1) •A1 Π -Term: Since Π(3,1) solves the Poisson equation (4.80) with the corresponding centering 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 82
equations, we can write
1 h i A(1)Π(3,1) = − (A∗ − hA∗i)Π(1,1) + A(1)Π(2,1) − hA(1)Π(2,1)i . (4.90) 0 2 2 2 1 1
Solving this yields,
(3,1) 1 (1,1) ρ (1,1) ρ21ν1 (0) ρρ21ν1 (0) Π = − ψ1Π − ψ12Π + √ ξ1Π + √ η1Π , (4.91) 4 x1x1 2 x1x2 4 2 x1x1x1 2 2 x1x1x2
where ξ1, η1 are solutions of (with constants of integration set to zero)
(1) (1) 0 (1) 0 (1) (1) (1) (4.92) A0 ξ1 = σX ψ1 − hσX ψ1i , A0 η1 = σX ∂z1 ψ12 − hσX ∂z1 ψ12i .
The source terms in (4.92) being bounded, ξ1 and η1 are at most linearly growing and
(1) (3,1) can be chosen with bounded first derivatives. It follows that A1 Π is a linear combination of terms with at most linear growth in ~z multiplied by up to third order ~x-derivatives of Π(1,1) or fifth order ~x-derivatives of Π(0). By the Appendix’s result on log-linear bounds of Π(0) and Π(1,1) under various orders of derivatives, we
(1) (3,1) conclude that A1 Π is at most linearly growing in ~z and log-linearly growing
(3,1) in ~x. It is now straightforward to see that Π (T0, ~x,~z) also shares these growth properties.
We remark that, with the use of similar techniques, the remaining terms from (4.89) can be shown to possess equivalent growth properties.
Letting the functions M(t, ~x,~z) and N(T0, ~x,~z) denote the r.h.s. of the PDE (4.89) and its boundary condition, respectively, a probabilistic representation of the solution is
Z T0 ~ Q ~ ~ ~ ~ Υ (t, ~x,~z) = Et,~x,~z P (t, T0) N(T0, XT0 , ZT0 ) − P (t, u) M(u, Xu, Zu) du . (4.93) t 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 83
(i) (i) From Lemma B.1 in [CFPS04], or by direct computations, the processes Xt and Zt , i ∈ {1, 2} have finite exponential moments, implying the finiteness of the first two moments
(i) of Zt , i ∈ {1, 2}. Applying these considerations to (4.93) finally supplies us with the claimed assertion, that is Υ~ = O(0). We are now ready to conclude our proof:
~ (0) (1,1) (1,2) Π (t, ~x,~z) − Π (t, ~x) + Πe (t, ~x) + Πe (t, ~x) 3 √ √ 3 (2,1) (2,2) 2 (3,1) (3,2) (3,3) 2 (3,4) ~ = 1Π + 2Π + 1 Π + 1 2Π + 12Π + 2 Π − Υ ~ (2,1) √ (3,4) ≤ Υ + 1 Π + ... + 2 2Π (4.94)
~ 0 (2,1) 0 √ (3,4) ≤ Υ + Π + ... + 2Π (4.95)
= O(0) (4.96)
2
It is noteworthy that the two parameters V11 and V22 arise only in the two-name case and are not induced by forward or single-name option prices. These parameters provide the trader with two additional degrees-of-freedom allowing a biasing of a two-name claim upward or downward relative to the single-name case. Equivalently, the two parameters may be used to tweak the implied volatility smile/skew. Furthermore, from the definitions of Vii (see equations (4.84) and (4.85)), if the correlation between the two commodities is zero (ρ = 0) then Vii = 0. Additionally, since each of these coefficients are proportional √ to the product of two correlations and the small parameter i, they should in principle be very small.
Two-Factor Model + SV
This section’s main goal is to find a well behaved approximation to the option price
Π~(t, ~x,~y, ~z) = P (t, T ) Q ϕ F 1 ,F 2 , (4.97) 0 Et,~x,~y,~z T0,T1 T0,T2 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 84 where each of the forward curves F i are based on the SV extended two-factor spot T0,Ti price model of Section 4.3.2. We only state the main result without going through the proof, which follows along similar lines to the previous section.
3 6 Theorem. 4.6.2 For any fixed (T0,T1,T2, ~x,~y, ~z) ∈ R+ × R with T0 ≤ T1,T2 and for all t ∈ [0,T0], we have
~ (0) (1,1) (1,2) 0 Π (t, ~x,~y, ~z) − Π (t, ~x,~y) + Πe (t, ~x,~y) + Πe (t, ~x,~y) = O( ) ,
0 where := Max{1, 2},
h n (1) (2) oi Π(0)(t, ~x,~y) := P (t, T ) Q ϕ F 1(0) X ,Y (1) ,F 2(0) X ,Y (2) , (4.98) 0 Et,~x,~y T0,T1 T0 T0 T0,T2 T0 T0
(i) F i(0) being as in (4.13) with σ replaced by phσ2 (z )i and each X as defined in t,Ti Xi Xi i t (4.76),
(1,1) (1,1) (0) (1,1) (0) (1,1) (0) Πe (t, ~x,~y) := l1 (t, T0)Πx1x1x1 + l2 (t, T0)Πx1x1y1 + l3 (t, T0)Πx1x1x2 h (1) + l(1,1)(T ,T ) P (t, T ) Q F 1(0) X ,Y (1) 0 1 0 Et,~x,~y eT0,T1 T0 T0 ∂ϕ n 1(0) (1) (1) 2(0) (2) (2)o × FT ,T XT ,YT ,FT ,T XT ,YT , (4.99) ∂F 1 0 1 0 0 0 2 0 0 and its symmetric part
(1,2) (1,2) (0) (1,2) (0) (1,2) (0) Πe (t, ~x,~y) := l1 (t, T0)Πx2x2x2 + l2 (t, T0)Πx2x2y2 + l3 (t, T0)Πx2x2x1 h (2) + l(1,2)(T ,T ) P (t, T ) Q F 2(0) X ,Y (2) 0 2 0 Et,~x,~y eT0,T2 T0 T0 ∂ϕ n 1(0) (1) (1) 2(0) (2) (2)o × FT ,T XT ,YT ,FT ,T XT ,YT , (4.100) ∂F 2 0 1 0 0 0 2 0 0
(i) (i) where V1 and V2 are (for each asset i ∈ {1, 2}) given by the parameters value in there respective forward price approximation in Theorem 4.4.2. Furthermore, for each asset 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 85 i ∈ {1, 2}, the various “l-coefficients” are given by
(i) (1,i) βiV2 l1 (t, T0) := − h(t, T0; −2βi − αi) 2(βi − αi) (i) 2 (i) ! V1 3βi V2 1 − + 1 − h(t, T0; −3βi) , (4.101) 2 4βi + 2αi βi − αi V (i) l(1,i)(t, T ) := − 2 h(t, T ; −2β − α ) , (4.102) 2 0 2 0 i i (1,i) l3 (t, T0) := −Viih(t, T0; −2βi − αj) , (i, j) ∈ {(1, 2), (2, 1)} , (4.103)
(1,i) (i) l (T0,Ti) := −V1 h(T0,Ti; 3βi)
(i) βi − V2 [h(T0,Ti; 3βi) − h(T0,Ti; αY i + 2βi)] . (4.104) αY i − βi
Once more, these SV extended two-factor model two-name option prices depend on the constant volatility prices plus corrections depending on the Delta’s and Delta-Gamma’s of the option together with a modified payoff. The V1 and V2 coefficients are inherited from the forward price approximation, while the new parameters V11 and V22 arise as a result of the correlation between the spot prices.
4.6.2 Nonsmooth Payoff: Forward Spread
As similarly argued in Section 4.5.2, it is possible to extend the validity of our pricing re- sults (Theorems 4.6.1 and 4.6.2) to non-smooth T0-payoff functions ϕ(·, ·), such as forward spread option contracts. The arguments behind this assertion follows along the same lines as described in Section 4.5.2. In practice, our stated results apply for dates not extremely close to contract maturity. We again refer to [FPSS03] for further technical explanations within the stock option context. 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 86
4.7 Conclusions and Future Work
This work focused on incorporating stochastic volatility into one- and two-factor mean- reverting commodities spot price models. Although stochastic volatility models have been studied in the context of commodities, none of the previous works produce closed form option prices and also only focus on single-name options. In contrast, we obtained explicit closed form pricing equations for single and two-name options on forward contracts.
By assuming the instantaneous stochastic volatility of the spot price process is driven by a hidden fast mean-reverting OU process, we were successful in obtaining explicit closed form derivative prices. Furthermore, we proved that these explicit approximations are correct up to order . This methodology produced forward and option prices which are independent of the specific mapping between the hidden process and the stochastic volatility. The key consequence is that forward prices can be written in terms of the con- stant volatility model where the effective constant volatility arises due to a smoothening of the hidden process driving the stochastic volatility. Although our results appear some- what daunting at first, they are in fact quite simple in structure and, more importantly, they are explicit. The results can be used to compute the stochastic volatility corrections of many common commodity options such as calls, puts, and spreads. As [FPS00a] sim- ilarly found in equity derivatives, we find, now using commutator methods due to the non-vanishing boundary condition, that these corrections are related to the delta’s and delta-gamma’s of the constant volatility prices and the price of a modified payoff.
Another very important aspect of this asymptotic approach is its ease of calibration. The arbitrary modeling specification is smoothed out and instead we find a new set of free parameters V , Vi and Vii (see equations (4.39) & (4.45) and (4.84) & (4.85) and the arguments leading up to these results). Using a nonlinear least-squares minimization scheme, the first set of these parameters (V or V1-V2) can be calibrated directly from forward price data. The second set of parameters (Vii) arise only in the two-name case 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 87 when the payoff explicitly depends on both names. Consequently, this second class of parameters has two dual and equivalent interpretations: (i) if no market prices for two- name options exists, they are additional model inputs and provide the trader with the flexibility to tweak prices; or (ii) if at least two option prices on two-names exist, they are market determined parameters which can then be used to consistently price other options.
There are many directions left open for future work, the most obvious being applying the model to an extensive data set. It would be instructive to classify the set of com- modities spot price data which are driven by a fast mean-reverting hidden OU process. Then, using the forward price approximations, we extract the market implied parameters and use these implied parameters to price single- and two-name options.
A more mathematically interesting direction to explore, and one we have already begun to, is to apply similar methods in the context of stochastic volatility HJM models for commodities data. [ST08] are the first to introduce a second unspanned stochastic volatility component into the commodities framework using an HJM approach – previous work includes [CS94], [ANP95] and [MS98] who focus solely on the constant volatility cases – however, they resort to an affine Heston-like model and employ transform methods to compute forward and option prices as no closed form results exists. As commented earlier, this is not a huge disadvantage when pricing a few options; however, it becomes very computationally intensive when used as a calibration tool and/or for sensitivity analysis. Our preliminary work gives us confidence that we can indeed find closed form forward and option prices through an SV extension of the standard HJM models using the hidden fast mean-reverting OU process to drive stochastic volatility. 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 88
4.8 Appendix
Here we sketch a proof of some useful bounds on the growth of various partial derivatives of the first three terms in the price expansion (4.72). The main result is stated as follows.
Theorem. 4.8.1 Fix p ∈ N. Then, for any i, n ∈ N s.t. 0 ≤ i ≤ n ≤ p and all t ∈ [0,T0), there exist constants C, C1, C2 such that
∂nΠ(0) ∂nΠ(1,1) ∂nΠ(1,2) , , ≤ C exp (C x + C x ) . (4.105) i n−i i n−i i n−i 1 1 2 2 ∂x1∂x2 ∂x1∂x2 ∂x1∂x2
Proof. We know that for t ≤ u ≤ T0
Z u (i) −βi(u−t) −βi(u−s) (i) Xu = φi + (xi − φi) e + σXi e dWs , (4.106) t
(1) (2) ~ −βi(u−t) with d W ,W t = ρ dt. That is Xu ∼ N(~m, σ¯) with mean mi = φi +(xi − φi) e 2 and covariance matrix given byσ ¯ii = h σXiih(t, u; 2βi) andσ ¯12 =σ ¯21 = ρhσX1σX2ih(t, u; β1+ ¯ 0 0 ~ β2). Let Φt,u be the density of N(~m, σ¯) and Φt,u the one of N(0, σ¯). Also recall from (4.4) that we can write
h σ2 i i(0) (i) −βi(Ti−T0) Xi −βi(Ti−T0) (i) F = exp g + φi 1 − e + h(T0,Ti; 2βi) + e X T0,Ti Ti 2 T0 n o =: C(i) exp D(i)X(i) , (4.107) T0 where we used the symbols C(i) and D(i) to simplify notations. Using these last formulas in (4.75) we can write
Z (1) (2) (0) (1) D (m1+y1) (2) D (m2+y2) 0 Π (t, ~x) = P (t, T0) ϕ C e ,C e Φt,T0 (y1, y2) d~y, (4.108) 2 R where the dependence on the ~x-variable is imbedded in ~m. Smoothness of ϕ and bound- edness of its partial derivatives combined with the dominated convergence theorem (note that r.v. having the density Φ0(·, ·) has finite exponential moments) imply the claimed 4 Asymptotic Pricing of Commodity Derivatives using SV Spot Models 89 bound on the derivatives of Π(0). We also rewrite (4.84) as
Z (1,1) V1 (0) Π (t, ~x) = − h(t, T0; −3β1)Πx1x1x1 (t, m1 + y1, m2 + y2) 2 2 R ! (0) 0 + V11h(t, T0; −2β1 − β2)Πx1x1x2 (t, m1 + y1, m2 + y2) Φt,T0 (~y) d~y
Z (1) (1) D (m1+y1) −V1h(T0,T1; 3β1) P (t, T0) C e 2 R ∂ϕ (1) (2) (1) D (m1+y1) (2) D (m2+y2) 0 × C e ,C e Φt,T (~y) d~y, (4.109) ∂F 1 0 so that the log-linear bound on Π(1,1) follows from the same arguments as in Π(0) case and the previous bound on Π(0). Finally, the log-linear bounds on the partial derivatives of Π(1,2) is found exactly as in
(1,1) the Π case since the arguments are perfectly symmetric with respect to x1 and x2. 2 Chapter 5
Unspanned Stochastic Volatility for Energy Derivatives: A Forward Price Approach
5.1 Abstract
Although very popular, spot price models for commodities cannot fully describe the dy- namics of the forward price curves. On the other hand, forward price models do not allow for the decoupled nature of the very short end of the term structure. [ST08] empirically demonstrate that unspanned volatility exists in commodities prices and develop an HJM inspired stochastic volatility model to account for this behavior. Here, we extend their model to account for a fast mean-reverting driving process for stochastic volatility and, through singular perturbation methods, explicitly compute prices of options on forward contracts and calendar spreads. The expansions are proved mathematically valid and we test the approximations on real data demonstrating its usefulness in calibration and pricing.
90 5 Unspanned SV for Energy Derivatives 91
5.2 Introduction
It is well known that commodities, unlike equities, have mean-reversion trends in their prices. Furthermore, any curiosity check of commodities price data will reveal the obvious fact that volatility is also stochastic. These features together with many other stylized empirical facts are well documented in, for example, [CS00], [EW03] and [Gem05].
To incorporate stochastic volatility (SV) into commodity prices, [EG98] invoked a [Hes93] inspired model. Similarly, [RS06] introduce SV effects into a stochastic conve- nience yield model using a similar modeling approach. Through empirical studies on soybean futures and options data the authors demonstrate that SV is a significant factor. Heston inspired SV models are popular because they are affine and therefore the charac- teristic function of the joint volatility and price process can be obtained in closed form. Option prices can then be written in terms of inverse Fourier transforms involving com- plex functions whose branch cuts and poles must be carefully analyzed to ensure stability of numerical inversions. Such pricing equations cannot be written in terms of elementary functions – or even special functions. Furthermore, these SV models have traditionally been layered on top of constant volatility spot price models which may be inadequate or inappropriate for certain applications.
In this article, the issue of unspanned stochastic volatility is addressed. Quoting from [ST08]:
In order to price, hedge and risk-manage commodity options, it is critical to understand the dynamics of volatility in commodity markets. While volatil- ity is clearly stochastic, it is not clear to what extent volatility risk can be hedged by trading in the commodities themselves or, more generally, their associated futures, forward or swap contracts; in other words, the extent to which volatility is spanned. 5 Unspanned SV for Energy Derivatives 92
The authors then carry out an empirical study on NYMEX light-sweet crude-oil futures and options data from January 1990 to May 2006. They conclusively demonstrate that there is a significant source of unspanned volatility in commodity prices. To model this observed behavior they propose an adaptation of the [HJM92] framework to commodity forward prices and incorporate an unspanned volatility source. Specifically they assume forward prices (in terms of risk-neutral Brownian motions) are modeled as
Z T Ft,T := St exp Yt,s ds (5.1) t where the spot price St and the instantaneous forward cost of carry Yt,T solve the coupled SDEs
St := exp {Xt} , (5.2)
√ dX = δ − 1 σ2 v dt + σ v dW (1) , t t 2 S t S t t √ (2) dYt,T = µt,T dt + σY (t, T ) vt dWt , (5.3) √ (3) dvt = κ (θ − vt) dt + σv vt dWt .
This model is also a Heston inspired model and all pricing results are provided in terms of integrals of the joint characteristic function of the various processes involved. Since the model is affine, the characteristic function is easily obtained; however, it is difficult to extract any intuition on the role that the various parameters play in the final pricing equation and it is difficult to calibrate the model1. To circumvent these issues, we transport singular perturbation theory techniques, first developed for equity derivatives by [FPS00b], then for interest rate derivatives in [CFPS04], and recently for derivatives based on commodity spot models in [HJ08a], to an HJM based forward price model for
1In fact, the authors do not attempt to calibrate their model and instead, somewhat adhocly, use a large multivariate regression based on the [BAW87] formula. 5 Unspanned SV for Energy Derivatives 93 commodities.
Asymptotic methods have three main advantages over traditional approaches: (i) they naturally lead to efficient calibration across a set of forward contracts; (ii) they lead to approximate, but explicit, closed form pricing equations for a wide class of contingent claims; and (iii) the resulting approximate prices are independent of the specific underlying volatility model.
We augment the [ST08] modelling assumption on the spot price and forward cost of carry as follows:
dX = β (φ − X ) dt + σ (Z ) dW (1) , t t X t t (2) dYt,T = µt,T dt + σY (t, T, Zt) dWt , (5.4) (3) dZt = α (m − Zt) dt + σZ dWt .
Here, σX (·) , σY (·, ·, ·) are assumed to be smooth, positive bounded functions, µt,T is Ft- adapted with {Ft}0≤t≤T being the natural filtration generated by all Brownian processes. Further details on our model are provided in section 5.3.
Our framework assumes there exists a hidden mean-reverting process driving volatility of both the spot prices and the forward cost of carry. However, there is no a priori specification on the volatility functions themselves. This modeling framework is more flexible than (5.3) since it allows for very generic volatility relationships between the spot price and the forward cost of carry; this contrasts with (5.3) where the volatilities are all identical up to a constant or deterministic scaling factor.
To approximate option prices, we follow [FPS00a] and [HJ08a] and force the invariant
2 distribution of the volatility driving process Zt to be fixed, i.e. σZ /2α = const., while
1 performing an asymptotic expansion on , α 1. In Section 5.4, we carry out this program for valuing options written on a single underlying forward contract. The main 5 Unspanned SV for Energy Derivatives 94 pricing approximation is provided in Theorem 5.4.1 which shows that the corrections are in terms of the delta, gamma and delta-gamma of the option. As usual in asymptotic expansions, the specific microscopic details of the volatility functions encoded in σX (·) and
σY (·, ·, ·) do not appear in the result, rather their macroscopic averaged out characteristics appear instead.
In Section 5.5, price approximations for calendar spread options are investigated. In- terestingly, the approximation is in terms of the macroscopic parameters already obtained in the single forward contract case. Rather than solving the completely generic case, which we believe contains too many degrees of freedom, we focus instead on two parameter re- duction regimes. The first regime assumes zero instantaneous correlation between the increments of the spot-price and the forward cost of carry processes; however, through the common SV driving factor Zt, these processes inherit a dependence structure. The main result for this case is provided in Theorem 5.5.1. The second regime assume the ratio of instantaneous volatilities of the spot price and forward cost of carry processes is deterministic. This case is the most similar to the [ST08] model, however, the volatility function itself is kept arbitrary.
The paper concludes with an example of parameter estimation in Section 5.6 and concluding remarks in Section 5.7.
5.3 Model Description
Although our model fits the properties of crude oil prices (see [ST08] and [HJ08a]), its wider applicability within energy asset class remains to be statistically investigated. Nonetheless, we believe that the model’s flexibility will draw many applications.
As usual we work on a probability space (Ω, {Ft}0≤t≤T , Q) where {Ft}0≤t≤T is the (1) (2) (3) natural filtration generated by three Brownian process Wt , Wt and Wt with constant 5 Unspanned SV for Energy Derivatives 95
(i) (j) correlation structure d[Wt ,Wt ] = ρi,j dt. The probability measure Q is the risk-neutral measure and we do not address the change of measure from the real-world or statistical measure to the risk-neutral one. The forward price process Ft,T is then defined in terms of the spot price St and an instantaneous forward cost of carry process Yt,T as follows
Z T Ft,T := St exp Yt,s ds , (5.5) t with
St := exp {Xt} , (5.6)
dX = β (φ − X ) dt + σ (Z ) dW (1) , t t X t t (2) dYt,T = µt,T dt + σY (t, T, Zt) dWt , (5.7) (3) dZt = α (m − Zt) dt + σZ dWt ,
where σX (·) and σY (·, ·, ·) are assumed smooth, positive and bounded in all arguments, and µt,T is Ft-adapted for every maturity T . We further impose a decoupling constraint on the volatility parameter of the Yt,T process
−γ(T −t) σY (t, T, z) := f(z)e , (5.8) which is consistent with the empirical observation of increased futures price volatility near contract maturity. Even with this constraint, our volatility and correlation structures are quite general. When addressing calendar spread options, we will impose some further constraints to maintain parsimony in the calibration process.
To value options written on the forward price, we require the SDE for the forward- price process itself. To this end, introduce the integrated forward cost of carry process 5 Unspanned SV for Energy Derivatives 96
R T Yet,T := t Yt,s ds which satisfies the SDE
Z T (2) dYet,T = −Yt,T + µt,s ds dt + σeY (t, T, Zt) dWt , (5.9) t where
Z T σeY (t, T, Zt) := σY (t, s, Zt) ds = h(t, T ; γ)f(Zt) , (5.10) t 1 −γ(T −t) with h(t, T ; γ) = γ 1 − e . (5.11)
Applying Ito’s lemma on the forward price leads to
Z T dF = F µ ds + ρ σ (Z )σ (t, T, Z ) + 1 σ (t, T, Z )σ (t, T, Z ) dt t,T t,T t,s 12 X t eY t 2 eY t eY t t (5.12) (1) (2)o +σX (Zt) dWt + σeY (t, T, Zt) dWt
Since the forward price Ft,T is a Q-martingale, after differentiating the drift with respect to the maturity date T , we obtain the no-arbitrage constraint
µt,T = −ρ12σX (Zt)σY (t, T, Zt) − σY (t, T, Zt)σeY (t, T, Zt) . (5.13)
This is the exact analog of the HJM interest rate drift restriction – in this case forcing the drift of the forward cost of carry to be fully specified by its volatility and the volatility of the spot price process as well. The resulting forward price dynamics is
dFt,T (1) (2) = σX (Zt) dWt + σeY (t, T, Zt) dWt . (5.14) Ft,T
Given this dynamics, we now focus on the valuation of single-name and two-name options on the forward price. 5 Unspanned SV for Energy Derivatives 97
5.4 Single-Name Options Valuation
In this section we consider the asymptotic valuation problem of a general European option
with T0-payoff function ϕ(·) written on the future forward price FT0,T (5.14) (of course
T0 < T ). The volatility driving mean-reversion parameter α in (5.7) is assumed to be very large, justifying the use of singular perturbation theory, as initially applied to equity, IR and commodity derivatives in [FPS00b], [CFPS04] and [HJ08a], respectively. The payoff function ϕ(·) is assumed smooth with linear growth at infinity. These constraints are made to guarantee the convergence of the asymptotic expansion methods and do not introduce any practical limitations as discussed later.
2 σZ 2 By explicitly fixing 2 α =: ν < +∞, and assuming the mean reversion parameter α is very large, we carry out an expansion in terms of the small parameter := 1/α. Let Π(t, x, z) denote the price of the option at time t with current forward price of x and volatility factor of z. Then,
Q Π (t, x, z) = P (t, T0) Et,x,z [ϕ(FT0,T )]
R T0 − t rs ds Q := e E [ϕ(FT0,T )| Ft,T = x, Zt = z] , (5.15)
where P (t, T0) is the T0-maturity zero coupon bond price contracted at time t and the short rate of interest rt is assumed deterministic. To simplify notation we omit the explicit dependence on T0 and T in the price function, and the parameter in the forward price process Ft,T and volatility factor Zt.
The price function Π satisfies a PDE that can be decomposed into operators with √ three scales in powers of . Explicitly,
−1 − 1 A Π = A + 2 A + A Π (t, x, z) = 0 , 0 1 2 (5.16) Π (T0, x, z) = ϕ(x) , 5 Unspanned SV for Energy Derivatives 98
R t − rs ds where the operator A is the infinitesimal generator of the process (e 0 ,Ft,T ) and
A0, A1 and A2 are respectively
2 2 A0 := (m − z)∂z + ν ∂zz , (5.17)
2 2 A1 := A13σX (z)x∂xz + A23σeY (t, T, z)x∂xz , (5.18) 1 A := ∂ + σ2 (z) + σ2 (t, T, z) + 2ρ σ (z)σ (t, T, z) x2∂2 − r . (5.19) 2 t 2 X eY 12 X eY xx t
√ √ Here the constants A13 := 2ρ13ν and A23 := 2ρ23ν have been introduced. Note that in the current model, there is no need for an asymptotic expansion of the forward price process since its explicit dynamics are already given by (5.14). This approach to asymptotic pricing of commodity/energy derivatives is in stark contrast with the model and framework developed in [HJ08a]. In that work, the authors begin with a spot price model, and perform an asymptotic expansion to determine forward prices, and then a second layer of asymptotic analysis to value options on forward prices. There, calibration of the model parameters are required on forward prices themselves. In our present approach, forward prices are exactly matched and do not require calibration; rather, single-name European contracts take over the role as calibration instruments as investigate in Section 5.6. √ We now proceed to the expansion of Π in powers of by writing
(0) √ (1) (2) 3 (3) Π = Π + Π + Π + 2 Π + ... (5.20)
√ Substituting this last expansion in (5.16) and regrouping terms of identical powers of allows a decomposition of the initial PDE into simpler ones, the results of which is here outlined:
−1 − 1 • , 2 –Order Equations These orders, imply that Π(0) and Π(1) are independent of the z-variable, that is: 5 Unspanned SV for Energy Derivatives 99
Π(0) = Π(0)(t, x) and Π(1) = Π(1)(t, x).
• 0–Order Equation The main result of the zeroth-order analysis is that Π(0) is the solution of the classical problem
hA iΠ(0)(t, x) = 0 , 2 (5.21) (0) Π (T0, x) = ϕ(x) .
Here, and in the remainder of the article, the bracket notation hg(z)i denotes the
expectation of g(Z) where Z ∼ N(m, ν2), the Q−invariant distribution of the process
Zt, as defined in (5.7).
The above PDE has the form of an option price in which the underlying forward price follows a Black-like model (GBM with zero drift) with deterministic volatility
2 2 2 σ¯(t, T ) given byσ ¯ := hσX i + hσeY i + 2ρ12hσX σeY i – see equation (5.19).
1 • 2 –Order Equation This order’s equation implies the PDE which Π(1) satisfies
(1) ∗ −γ(T −t) ∗ −2γ(T −t) hA2iΠ = B0 − B1 e + B2 e ∗ −3γ(T −t) 2 (0) 3 (0) +B3 e 2x Πxx + x Πxxx . (5.22) (1) Π (T0, x) = 0 .
Here, the Bi−coefficients are simple deterministic functions of the model’s parame- ters (see the Appendix for the specific form).
n (n) Since the operators hA2i and x ∂x commute (for any positive integer n) as can 5 Unspanned SV for Energy Derivatives 100
h n (n)i directly be checked, i.e. hA2i , x ∂x = 0, it is not difficult to show that
(1) 2 (0) 3 (0) Πa := −(T0 − t)B0 2x Πxx + x Πxxx , (5.23)
∗ ∗ ∗ solves the system (5.22) when B1 := B2 := B3 := 0. A similar argument also implies that
(1) −γ(T −t) −γ(T −T0) −2γ(T −t) −2γ(T −T0) Πb := −B1 e − e + B2 e − e
−3γ(T −t) −3γ(T −T0) 2 (0) 3 (0) +B3 e − e 2x Πxx + x Πxxx , (5.24)
is a solution of (5.22) when B0 := 0 and the form of all Bi-coefficients are reported
∗ in the appendix (note the difference between Bi and Bi ). It then follows that
(1) (1) (1) Π = Πa + Πb is the full solution to the boundary value PDE (5.22).
Collecting the results of the analysis of each order, we can now state our main approx- imation result on single-name European options pricing.
4 Theorem. 5.4.1 For any fixed (t, T0, T, x, z) ∈ R+ × R with t ≤ T0 ≤ T , we have
(0) √ (1) Π (t, x, z) − Π (t, x) + Π (t, x) = O() , (5.25) where
(0) Q ∗ Π (t, x) := P (t, T0) Et,x ϕ(FT0,T ) (5.26)
∗ is the price of the option written on a simpler forward process Ft,T with deterministic volatilities
∗ ∗ 2 2 −2γ(T −t) −γ(T −t) ∗ dFt,T := Ft,T hσX i + hf ie + 2ρ12hσX fie dWt , (5.27) 5 Unspanned SV for Energy Derivatives 101
∗ (1) with Wt an Ft-adapted Q−Wiener process and where the first order correction Π is given by
(1) −γ(T −t) −γ(T −T0) −2γ(T −t) −2γ(T −T0) Π (t, x) = −(T0 − t)B0 − B1 e − e + B2 e − e
−3γ(T −t) −3γ(T −T0) 2 (0) 3 (0) +B3 e − e 2x Πxx + x Πxxx . (5.28)
Proof. The proof follows along similar arguments to those found in Chapter 4 (or [HJ08a]). See the Appendix for a sketch of the main arguments. 2
We now refine this last result to two particular model structures. We introduce two cases of practical importance where calibration against any single-name vanilla option prices will induce the prices of any calendar spread options or single-name European contracts (spreads are treated in Section 5.5).
The first case is when the two driving sources of risk of the forward price process
(1) (2) (5.14), Wt and Wt , are assumed to be independent, i.e. we set ρ12 := 0. In doing so we are not decoupling the spot and forward price processes; however, this assumption leads to a very efficient calibration prescription. The effect of this model constraint on the pricing of single-name European options is provided in Corollary 5.4.2. The second case we investigate is by setting f(z) = σX (z). In this situation, both spot price volatility and the cost of carry volatility are assumed to have the same functional dependence on the
−γ(T −t) driving volatility factor (σY (t, T, z) := σX (z)e ). This again offers very desirable calibration properties as far as the pricing/hedging of calendar spreads and single-name European options are concerned. The effect of this model constraint on the pricing of single-name European options is provided in Corollary 5.4.3.
(1) (1) (1) Corollary 5.4.2 When ρ12 := 0, Theorem 5.4.1 remains in force with Π = Πa + Πb 5 Unspanned SV for Energy Derivatives 102 and
(1) 2 (0) 3 (0) Πa (t, x) := −(T0 − t)C0 2x Πxx + x Πxxx (5.29)
(1) −γ(T −t) −γ(T −T0) Πb (t, x) := − (2C1 + C2 + 3C3) e − e 1 + (C + 3C ) e−2γ(T −t) − e−2γ(T −T0) 2 1 3 1 + C e−3γ(T −t) − e−3γ(T −T0) 2x2Π(0) + x3Π(0) , (5.30) 3 3 xx xxx
where the macroparameters C0, C1, C2, C3 are deterministic functions of the model pa- rameters, as explicitly given in the Appendix. 2
−γ(T −t) Corollary 5.4.3 When σY (t, T, z) := σX (z)e , Theorem 5.4.1 remains in force with
(1) (1) (1) Π = Πa + Πb and
2 (0) 3 (0) Πa := −(T0 − t)D0 2x Πxx + x Πxxx (5.31)
−2 −1 −2 −2 −1 −γ(T −t) −γ(T −T0) Πb := − 2D1γ γ + ρ12 + D2γ 1 + 3γ + 4ρ12γ e − e 1 + D γ−3 + D γ−3 3γ−1 + 2ρ e−2γ(T −t) − e−2γ(T −T0) 2 1 2 12 1 − D γ−4 e−3γ(T −t) − e−3γ(T −T0) 2x2Π(0) + x3Π(0) , (5.32) 3 2 xx xxx
where the macroparameters D0, D1, D2 are deterministic functions of the model parame- ters, as explicitly given in the Appendix. 2
Armed with these last three results, we now move on toward the asymptotic pricing of calendar spreads and explore the connections between these closed-form expressions and those of the previous two Corollaries (for calibration purpose). The practical methodology of the above mentioned calibration to market data will be dealt with in Section 5.6. 5 Unspanned SV for Energy Derivatives 103
5.5 Calendar Spreads Valuation
We now investigate the pricing of European two-name options and, more specifically, calendar spreads. Calendar spreads constitute an important class of risk management tools for various companies having exposures in energy markets and more specifically in the crude oil markets. Although we are primarily targeting the efficient pricing and hedging issues relevant to spreads written on crude oil forwards, it is important to notice that the developments of this section remain valid for calendar spreads on any energy forward price with fast-mean reverting and unspanned SV behavior, as in (5.14). In this section, we specialize to the two parameter structures discussed at the end of the last section. Both structures are realistic in their formulations and will prove versatile for calibration. We will address the calibration issues in Section 5.6.
The asset price is again assumed to follow a fast mean-reverting unspanned SV process with specifications as in Section 5.3. The T0-payoff of a calendar spread is written as
ϕ(FT0,T1 ,FT0,T2 ) := (FT0,T1 − K · FT0,T2 )+ , (5.33)
with K a constant and T0 ≤ T1,T2. The usual (no-arbitrage) pricing argument implies
Q Π (t, ~x,z) = P (t, T0) Et,~x,z [ϕ(FT0,T1 ,FT0,T2 )] (5.34)
where the forward price processes Ft,Ti are defined in (5.14) and its vector of initial condition at time t is expressed as (Ft,T1 ,Ft,T2 ) = ~x := (x1, x2). As before, we assume that α 1 and develop closed-form asymptotic pricing results for calendar spreads under the two parameter structures discussed in the last section. 5 Unspanned SV for Energy Derivatives 104
5.5.1 Case I: A Constrained Correlation Structure (ρ12 := 0)
Here we develop a closed-form asymptotic approximation to the pricing equation (5.34) based on the assumption ρ12 := 0. We will demonstrate that this approximation is provided in terms of the macroparameters defined in Corollary 5.4.2 and is therefore completely determined by the calibration of the model against liquid European single- name contracts.
As before, the price function Π satisfies a PDE which can be decomposed in various power of . Explicitly,
−1 − 1 A Π = A + 2 A + A Π (t, ~x,z) = 0 , 0 1 2 (5.35) Π (T0, ~x,z) = ϕ(~x) ,
−1 where the operator A is the infinitesimal generator of the process (Mt ,Ft,T1 ,Ft,T2 ) with
Mt being the money market account and A0, A1 and A2 being defined by
2 2 A0 := (m − z)∂z + ν ∂zz (5.36)
A := [A σ (z) + A σ (t, T , z)] x ∂2 + [A σ (z) + A σ (t, T , z)] x ∂2 1 13 X 23eY 1 1 x1z 13 X 23eY 2 2 x2z (5.37) 1 1 A := ∂ + σ2 (z) + σ2 (t, T , z) x2∂2 + σ2 (z) + σ2 (t, T , z) x2∂2 2 t X eY 1 1 x1x1 X eY 2 2 x2x2 2 2 (5.38) + σ2 (z) + σ (t, T , z)σ (t, T , z) x x ∂2 − r , X eY 1 eY 2 1 2 x1x2 t
We now proceed (in a very succinct fashion) to the asymptotic analysis of the various √ PDEs arising from the usual expansion in powers of the small parameter
(0) √ (1) (2) 3 (3) Π = Π + Π + Π + 2 Π + ... (5.39)
Once again substituting this expansion in (5.35) and regrouping terms of identical powers 5 Unspanned SV for Energy Derivatives 105
√ of allows a decomposition of the initial PDE into simpler ones, the results of which is here outlined:
−1 − 1 • , 2 –Order Equations To this order, we find that Π(0) and Π(1) are independent of the z-variable, that is; Π(0) = Π(0)(t, ~x) and Π(1) = Π(1)(t, ~x).
• 0–Order Equation The main result of the zeroth-order analysis is that Π(0) is the solution of the classical problem defined by the PDE
hA iΠ(0)(t, ~x) = 0 , 2 (5.40) (0) Π (T0, ~x) = ϕ(~x) .
As in the single-name analysis, the solution to this equation is identical to the price of the option under a Black-like model. In particular, write
∗ dFt (T1) ∗ ∗ = σ(t, T1) dWt , (5.41) Ft (T1)
∗ dFt (T2) ∗ p 2 ∗⊥ ∗ = σ(t, T2) ρ(t)dWt + 1 − ρ (t)dWt , (5.42) Ft (T2)
where
2 2 2 σ (t, T ) = hσX (Z)i + hσeY (t, T, Z)i , (5.43)
hσ2 (Z)i + hσ (t, T ,Z)σ (t, T ,Z)i ρ(t) = X eY 1 eY 2 , (5.44) σ(t, T1)σ(t, T2)
∗ ∗⊥ and Wt and Wt are independent Ft-adapted Q-Wiener processes. Then the solu- 5 Unspanned SV for Energy Derivatives 106
tion to (5.40) can be written as
(0) Q ∗ ∗ Π (T0, ~x) = P (t, T0) E [ϕ(~x)|Ft (T1) = x1,Ft (T2) = x2] . (5.45)
1 • 2 –Order Equation From this order’s equation we first obtain a defining PDE for Π(1)(t, ~x)
hA iΠ(1)(t, ~x) = −hA Π(2)(t, ~x,z)i 2 1 (5.46) (1) Π (T0, ~x) = 0 ,
from which, following the techniques that led to the solution of (5.22) for the single-
1 (1) (1) 2 (1) name analysis, we again find a closed-form solution of Π = Πa + Πb with
(1) 2 (0) 3 (0) 2 (0) (0) Πa := −(T0 − t)E0,t 2x1Πx1x1 + x1Πx1x1x1 + 3x1x2Πx1x1x2 + 4x1x2Πx1x2
2 (0) 2 (0) 3 (0) +3x1x2Πx1x2x2 + 2x2Πx2x2 + x2Πx2x2x2 (5.47)
(1) 2 (0) 3 (0) 2 (0) (0) Πb := E1,t 2x1Πx1x1 + x1Πx1x1x1 + E2,t x1x2Πx1x1x2 + E3,t x1x2Πx1x2
2 (0) 2 (0) 3 (0) +E4,t x1x2Πx1x2x2 + E5,t 2x2Πx2x2 + x2Πx2x2x2 (5.48)
where the deterministic coefficients {Ei,t : i = 1,..., 5} are simple functions of the
same macroparameters Ci and of γ found in the one-name context of Corollary 5.4.2. The detailed expressions of these E-coefficients can be found in the Appendix.
We summarize the previous results in the following Theorem. 5 Unspanned SV for Energy Derivatives 107
6 Theorem. 5.5.1 For any fixed (t, T0,T1,T2, ~x,z) ∈ R+ ×R with t ≤ T0 ≤ T1,T2, we have
(0) √ (1) Π (t, ~x,z) − Π (t, ~x) + Π (t, ~x) = O() ,
(0) Q ∗ ∗ where Π (t, ~x) := P (t, T0) Et,~x ϕ(FT0,T1 ,FT0,T2 ) is the price of the option written on the ∗ ∗ new forward price process Ft,T1 and Ft,T2 satisfying the SDEs (5.41-5.42) and where the first order correction term Π(1) is given by
(1) Π (t, ~x) = Πa(t, ~x) + Πb(t, ~x) . (5.49)
The detailed expressions for Πa and Πb are as in (5.47) and (5.48), respectively, with the deterministic Ei-coefficients being completely determined by the macroparameters Ci and γ of Corollary 5.4.2.
Proof. The proof follows along similar arguments as in [HJ08a], using a version of Feynman-Kac result that can be found in [KS91]. A sketch of the single-name version of the proof is found in the Appendix. 2
5.5.2 Case II: A Constrained Volatility Structure (f := σX)
We again develop a closed form asymptotic approximation to the pricing equation (5.34), this time with the assumption that f(z) := σX (z). This approximation of the calendar spread is now written in terms of the macroparameters defined in Corollary 5.4.3 and is therefore completely determined by the calibration of the model against any liquid European single-name contract. A detailed discussion on the calibration issues is again postponed to Section 5.6.
The price function Π satisfies a PDE that can be decomposed in operators of various 5 Unspanned SV for Energy Derivatives 108
√ power of
−1 − 1 A Π = A + 2 A + A Π (t, ~x,z) = 0 , 0 1 2 (5.50) Π (T0, ~x,z) = ϕ(~x) ,
−1 where the operator A is the infinitesimal generator of the process (Mt ,Ft,T1 ,Ft,T2 ) with
Mt being the money market account and A0, A1 as in (5.36)-(5.37) and A2 now defined by
σ2 (z) A := ∂ + X 1 + h2(t, T ; γ) + 2ρ h(t, T ; γ) x2∂2 2 t 2 1 12 1 1 x1x1 σ2 (z) + X 1 + h2(t, T ; γ) + 2ρ h(t, T ; γ) x2∂2 2 2 12 2 2 x2x2 2 +σX (z) [1 + h(t, T1; γ)h(t, T2; γ)
2 +ρ12 {h(t, T1; γ) + h(t, T2; γ)}] x1x2∂x1x2 − rt , (5.51)
As before, we carry out an asymptotic expansion of the PDE arising from the expansion
(0) √ (1) (2) 3 (3) Π = Π + Π + Π + 2 Π + ... (5.52)
√ into orders of . The analysis is similar to the previous one (see Section 5.5.1) and we omit it for brevity and only state the main result.
6 Theorem. 5.5.2 For any fixed (t, T0,T1,T2, ~x,z) ∈ R+ ×R with t ≤ T0 ≤ T1,T2, we have
(0) √ (1) Π (t, ~x,z) − Π (t, ~x) + Π (t, ~x) = O() ,
(0) Q ∗ ∗ where Π (t, ~x) := P (t, T0) Et,~x ϕ(FT0,T1 ,FT0,T2 ) is the price of the option written on a ∗ ∗ the forward price process Ft,T1 and Ft,T2 satisfying the SDEs 5 Unspanned SV for Energy Derivatives 109
dF ∗ t,T1 := σ(t, T ) dW ∗ , (5.53) F ∗ 1 t t,T1 dF ∗ t,T2 := σ(t, T ) ρ(t) dW ∗ + p1 − ρ2(t) dW ∗⊥ , (5.54) F ∗ 2 t t t,T2 with
2 2 2 σ (t, T ) = hσX (z)i 1 + h (t, T ; γ) + 2ρ12h(t, T ; γ) (5.55) 1 + h(t, T ; γ)h(t, T ; γ) + ρ {h(t, T ; γ) + h(t, T ; γ)} ρ(t) = 1 2 12 1 2 p 2 2 (1 + h (t, T1; γ) + 2ρ12h(t, T1; γ))(1 + h (t, T2; γ) + 2ρ12h(t, T2; γ)) (5.56)
∗ ∗⊥ and Wt and Wt independent Ft-adapted Q-Wiener processes. Furthermore, the first order term Π(1) is given by
(1) 2 (0) 3 (0) 2 (0) Π (t, ~x) = −(T0 − t)G0 2x1Πx1x1 + x1Πx1x1x1 + 3x1x2Πx1x1x2
(0) 2 (0) 2 (0) 3 (0) + 4x1x2Πx x + 3x1x2Πx x x + 2x2Πx x + x2Πx x x 1 2 1 2 2 2 2 2 2 2 (5.57) 2 (0) 3 (0) 2 (0) + G1 2x1Πx1x1 + x1Πx1x1x1 + G2 x1x2Πx1x1x2
(0) 2 (0) 2 (0) 3 (0) + G3 x1x2Πx1x2 + G4 x1x2Πx1x2x2 + G5 2x2Πx2x2 + x2Πx2x2x2
were the deterministic Gi-coefficients being completely determined by the macroparameters
Di and γ of Corollary 5.4.3.
Proof. Similar to what has been done in [HJ08a], using a version of Feynman-Kac result that can be found in [KS91]. 2 5 Unspanned SV for Energy Derivatives 110
5.6 Calibration
In this section we present a calibration procedure of our model to single-name contracts and carry out the analysis on sweet crude oil options and futures data. On comple- tion of this step, other single-name contracts and calendar spread prices are completely determined by the same macroparameters, as previously shown.
Before elaborating on an efficient calibration methodology, we first need a source of sufficiently liquid single-name contracts data. The simplest way to proceed would be to obtain prices for European calls/puts and then to “fit” our model prices to these. How- ever, European commodities contracts are only now becoming more liquid and instead their American counterparts enjoy high liquidity levels. One way to exploit this liquidity is to obtain their implied volatility curves via one of the many available analytical ap- proximation formulas2 for American options on futures. We make use of the well known formula of [BAW87] (also see [GH07] for other approximations) to extract a Black im- plied volatility and calibrate by fitting our European option “Black implied volatilities” to these. In principle it is also possible to directly approximate the American option prices in the fast mean-reverting volatility framework; however, in the interest of space we omit this line of thought.
To illustrate the robustness of the calibration procedure we calibrate the model to option and futures data on the following two dates: 04/01/2006 and 10/01/2006. Figure 5.1 and Figure 5.2 show the implied volatility curves evolution through time (increasing term). As expected, the implied volatility for a fixed term exhibits a skew, while as term increases the smiles flatten out. In addition the curves tend to increase with term. The smoothness of the curves constitute an indirect check on the quality of our data.
It is possible to directly fit the model to implied volatilities rather than resorting to
2Alternatively, one could resort to trees, finite difference or integral equation numerical methods; however, computational time may become an issue. 5 Unspanned SV for Energy Derivatives 111
Figure 5.1: Implied Volatility Curves on 4-Jan-2006, for three increasing term.
Figure 5.2: Implied Volatility Curves on 10-Jan-2006, for three increasing term.
a least-squares fit to prices. For more details on the method in the context of equity derivatives see [FPS00a]. To this end, denote the model Black implied volatility as σ which solves the non-linear equation
(0) √ (1) ΠB(σ ) = Π + Π + ... , (5.58) 5 Unspanned SV for Energy Derivatives 112
(0) where ΠB(·) is Black’s formula for calls/puts on futures (note that ΠB(·) = Π (·)). Write √ the implied volatility as an expansion in as follows
√ σ = σ(0) + σ(1) + ... , (5.59) where σ(0) is the zeroth-order implied volatility
s 2 Z T (0) hσX i −2γ(T −s) −γ(T −s) σ (t, T ) = [(1 + e + 2ρ12e )] ds . (5.60) T − t t
Then expanding the l.h.s. of (5.58) leads to
√ ∂Π(0) √ Π(0)(σ(0)) + σ(1) (σ(0)) + ... = Π(0) + Π(1) + ... . (5.61) ∂σ √ The first terms match exactly by construction, matching the order term provides us with the first order correction to the implied volatility
∂Π(0) −1 σ(1) = Π(1) . (5.62) ∂σ
Given this closed-form expression for the first order approximated implied volatility func-
∗ ∗ ∗ tion, we get the macroparameters (D0,D1,D2) by minimizing the sum of squared distance √ between the implied volatility data and σ(0) + σ(1). We report the values of the optimal parameters at two different times in Table 5.1 and Table 5.2. For the sake of simplicity,
p 2 ∗ ∗ ∗ hσX i ρ12 γ D0 D1 D2 0.313 -0.978 34.115 0.004 -0.103 0.048
Table 5.1: The calibrated macro-parameters (volatility expansion method), with D∗ := √ i Di, using the NYMEX Light Sweet Crude Oil implied volatilities on 04/01/2006.
we here derive a simple expression for σ(1) of equation (5.62). First, one can easily verify the following analytic expressions for the gamma, delta-gamma and vega of the zeroth 5 Unspanned SV for Energy Derivatives 113
p 2 ∗ ∗ ∗ hσX i ρ12 γ D0 D1 D2 0.306 -0.991 38.554 0.033 -0.177 0.084
Table 5.2: The calibrated macro-parameters (volatility expansion method), with D∗ := √ i Di, using the NYMEX Light Sweet Crude Oil implied volatilities on 10/01/2006. order price Π(0):
φ(d ) Π(0) = √1 , (5.63) xx 2 (0) Ft,T σ T − t " # (0) d1 1 Π = −φ(d1) √ + √ , (5.64) xxx (0) 3 2 (0) (Ft,T σ T − t) Ft,T σ T − t √ (0) Πσ = Ft,T T − tφ(d1) , (5.65) where φ(·) is the p.d.f. of a standard normal and
(0) 2 Ft,T (σ ) ·(T −t) log K + 2 d1 := √ . (5.66) σ(0) T − t
From our previous developments we also have that
(1) 2 (0) 3 (0) Π = C · 2Ft,T Πxx + Ft,T Πxxx , (5.67)
with C(·) a function of (D0,D1,D2,T − t) as previously derived. Combining all of the above expressions, it is now straightforward to obtain the following formula
(1) C d1 σ (t, T ) = (0) 1 − (0) 2 . (5.68) σ (t, T ) · (T − t) Ft,T (σ (t, T )) (T − t)
5.7 Conclusions
We introduced a new stochastic volatility extension of the HJM framework which incor- porates an unspanned source of volatility together with mean-reversion of spot prices. By 5 Unspanned SV for Energy Derivatives 114 modeling volatility as a function of a hidden fast mean-reverting process, we are able to provide analytically closed form equations for the price of European options written on the forward contracts as well as options on calendar spreads. Interestingly, the macro- scopic parameters which feed into the prices of calendar spread options are the same as those that appear in the standard options. This is useful for calibration, as it implies that calibrating to single-name forward options is sufficient to consistently price calendar spreads.
5.8 Appendix I: Sketch of Proof of Theorem 5.4.1
This section presents a factual overview of the main steps of the proof of Theorem 3.1. The techniques employed are very similar to those found in [HJ08a]; We refer to this last reference for a more detailed exposition.
We define the function Υ(t, x, z) as the sum of the error terms of order 2 and higher. Explicitly,
(0) √ (1) (2) 3 (3) Υ := Π + Π + Π + 2 Π − Π . (5.69)
We first aim at proving that |Υ| = O(). Applying the infinitesimal generator A of
−1 (Mt ,Ft,T ,Zt) on Υ and canceling vanishing terms, we find
−1 − 1 (0) √ (1) (2) 3 (3) A Υ = A0 + 2 A1 + A2 Π + Π + Π + 2 Π − Π
(2) (3) √ (3) = A2Π + A1Π + A2Π . (5.70)
We now focus on each individual terms on the right hand side of (5.70), paying attention to their growth properties w.r.t. the variables x, z.
(2) •A2Π -Term: 5 Unspanned SV for Energy Derivatives 115
(2) 1 (0) We have that Π = − 2 ψ(z)Πxx , where ψ(z) satisfies a Poisson equation with bounded r.h.s. which satisfies the usual centering condition. Using growth properties of the solutions of Poisson equations, we have that ψ(z) grows at most linearly in |z|. Given this last analysis, it is clear that Π(0) (and therefore Π(2)) is log-linear in x.
(3) (3) •A1Π and A2Π -Terms:
1 (3) (3) (1) From the 2 -order analysis, Π satisfies the Poisson equation A0Π + A2Π +
(2) (1) (2) A1Π = 0 and the centering condition hA2Π + A1Π i = 0. We then have,
(1) (2) (1) (1) (2) (2) A2Π + A1Π = A2Π − hA2Π i + A1Π − hA1Π i . Consequently,
(3) (0) (1) Π = η(z)Πxxx − ζ(z)Πxx , (5.71)
where η(z) and ζ(z) are solutions of Poisson equations satisfying the centering equa- tion and having bounded source terms. These facts imply that η(z), ζ(z) are at most linearly growing in |z| with bounded first derivatives. From these last properties of
(3) (3) (3) η(z), ζ(z) and the form of Π in (5.71), we conclude that A1Π and A2Π are at most linearly growing in |z| and log-linearly growing in x.
(2) (3) The above results allow us to bound the error term Υ . Define N := A2Π +A1Π +
√ (3) A2Π so that equation (5.70) becomes A Υ = N. With this new terminology, the “Feynman-Kac” probabilistic representation of (5.70) can be expressed as (see [KS91], section 5.7):
Q (2) Υ (t, x, z) = Et,x,z Π (T0,FT0,T ,ZT0 ) Z T √ (3) + Π (T0,FT0,T ,ZT0 ) − N(s, Fs,T ,Zs) ds . (5.72) t
We have already demonstrated that N(t, x, z), Π(2)(T, x, z) and Π(3)(T, x, z) are at most 5 Unspanned SV for Energy Derivatives 116 linearly bounded in |z| and log-linearly growing in x. For the N function, this bound is uniform in t ∈ [0,T ]. Furthermore, since σX (·) and σY (·) are bounded, a direct check shows that Ft,T has finite moments. Similarly, for the process Zt has finite exponential moments, which implies a bound on its second moment (variance). Therefore, |Υ| = O(), as previously claimed.
We make use of this last partial result and write
(0) (1) (2) 2 (3) (2) √ (3) 3 Π − (Π + Fe ) = Π + Π − Υ ≤ |Υ | + Π + Π , (5.73) which, by the properties of Π(2) and Π(3), completes the proof.
5.9 Appendix II: Expansion Coefficients
This appendix presents the detailed composition of the various coefficients and macro parameters introduced during the course of the paper. They are all written in terms of the model’s initial parameters.
The Bi-coefficients of Proposition 5.4.1 are given as:
−2 −1 B0 := A13 hσX ∂zφ1i + hσX ∂zφ2iγ + ρ12hσX ∂zφ3iγ
−1 −3 −2 +A23 hf∂zφ1iγ + hf∂zφ2iγ + ρ12hf∂zφ3iγ (5.74)
∗ −1 B1 := B1 γ
−3 −2 := A13 2hσX ∂zφ2iγ + ρ12hσX ∂zφ3iγ
−2 −4 −3 +A23 hf∂zφ1iγ + 3hf∂zφ2iγ + 2ρ12hf∂zφ3iγ (5.75) 5 Unspanned SV for Energy Derivatives 117
∗ −1 B2 := B2 γ /2
−1 −3 −4 −3 := 2 A13hσX ∂zφ2iγ + A23 3hf∂zφ2iγ + ρ12hf∂zφ3iγ (5.76)
∗ −1 B3 := B3 γ /3
−1 −4 := 3 A23hf∂zφ2iγ (5.77)
√ √ with A13 := 2ρ13ν, A23 := 2ρ23ν and the functions φ1 , φ1 , φ3 being defined as the solutions (with zero constant of integration) of the following equations:
−1 2 2 A0 φ1 = 2 σX − hσX i (5.78)
−1 2 2 A0 φ2 = 2 f − hf i (5.79)
A0 φ3 = fσX − hfσX i (5.80) where all dependencies on the variable z has been droped from the notation.
The Ci-coefficients of Corollary 5.4.2 are given as:
−2 C0 := A13hσX ∂zφ1i + A13hσX ∂zφ2iγ
−1 −3 +A23hf∂zφ1iγ + A23hf∂zφ2iγ (5.81)
−3 C1 := A13 hσX ∂zφ2iγ (5.82)
−2 C2 := A23 hf∂zφ1iγ (5.83)
−4 C4 := A23 hf∂zφ2iγ (5.84)
where φ1 and φ2 are again solutions of (5.78) and (5.79), respectively. 5 Unspanned SV for Energy Derivatives 118
The Di-coefficients of Corollary 5.4.3 are given as:
−2 −1 −1 −3 −2 D0 := D1 1 + γ + 2ρ12γ + D2 γ + γ + 2ρ12γ (5.85)
D1 := A13 hσX ∂zφ1i (5.86)
D2 := A23 hσX ∂zφ1i (5.87)
√ √ with A13 := 2ρ13ν, A23 := 2ρ23ν as usual and the function φ1 being again defined as the solutions (with zero constant of integration) of equation (5.78).
The Ei,t-coefficients of Proposition 5.5.1 are given as:
E0,t := C0 (5.88)
−γ(T1−t) −γ(T1−T0) E1,t := − (2C1 + C2 + 3C3) e − e 1 + (C + 3C ) e−2γ(T1−t) − e−2γ(T1−T0) 2 1 3 C3 − e−3γ(T1−t) − e−3γ(T1−T0) (5.89) 3
−γ(T1−t) −γ(T1−T0) E2,t := −2 (2C1 + C2 + 3C3) e − e
−γ(T2−t) −γ(T2−T0) −3 (C1 + C3) e − e 1 + (C + 3C ) e−2γ(T1−t) − e−2γ(T1−T0) 2 1 3
−γ(T1+T2−2t) −γ(T1+T2−2T0) + (C1 + 3C3) e − e
−γ(2T1+T2−3t) −γ(2T1+T2−3T0) −C3 e − e (5.90) 5 Unspanned SV for Energy Derivatives 119
−γ(T1−t) −γ(T1−T0) −γ(T2−t) −γ(T2−T0) E3,t := −2 (2C1 + C2 + 3C3) e − e + e − e
−2γ(T1−t) −2γ(T1−T0) −2γ(T2−t) −2γ(T2−T0) +C3 e − e + e − e 2 − C e−γ(2T1+T2−3t) − e−γ(2T1+T2−3T0) + e−γ(T1+2T2−3t) − e−γ(T1+2T2−3T0) 3 3
−γ(T1+T2−2t) −γ(T1+T2−2T0) +2 (C1 + 2C3) e − e (5.91)
−γ(T1−t) −γ(T1−T0) −γ(T2−t) −γ(T2−T0) E4,t := − (2C1 + C2 + 3C3) e − e + 2 e − e 1 + (C + 3C ) e−2γ(T2−t) − e−2γ(T2−T0) + 2 e−γ(T1+T2−2t) − e−γ(T1+T2−2T0) 2 1 3
−γ(T1+2T2−3t) −γ(T1+2T2−3T0) −C3 e − e (5.92)
−γ(T2−t) −γ(T2−T0) E5,t := − (2C1 + C2 + 3C3) e − e 1 + (C + 3C ) e−2γ(T2−t) − e−2γ(T2−T0) 2 1 3 C3 − e−3γ(T2−t) − e−3γ(T2−T0) (5.93) 3
where the detailed expressions of the Ci-coefficients in terms of the model’s parameters
(with ρ12 := 0) are as in (5.81)-(5.84).
The Gi-coefficients of Proposition 5.5.2 are given as:
G0,t := D0 (5.94)
−3 −2 −2 −4 −3 −γ(T1−t) −γ(T1−T0) G1,t := − 2D1 γ + ρ12γ + D2 γ + 3γ + 4ρ12γ e − e 1 + D γ−3 + D 3γ−4 + 2ρ γ−3 e−2γ(T1−t) − e−2γ(T1−T0) 2 1 2 12 −4 D2γ − e−3γ(T1−t) − e−3γ(T1−T0) (5.95) 3 5 Unspanned SV for Energy Derivatives 120
−3 −2 −2 −4 −3 −γ(T1−t) −γ(T1−T0) G2, t := − 4D1 γ + ρ12γ + 2D2 γ + 3γ + 4ρ12γ e − e
−3 −2 −2 −4 −3 −γ(T2−t) −γ(T2−T0) − 2D1 γ + ρ12γ + D2 γ + 3γ + 4ρ12γ e − e 1 + D γ−3 + D 3γ−4 + 2ρ γ−3 e−2γ(T1−t) − e−2γ(T1−T0) 2 1 2 12
−3 −4 −3 −γ(T1+T2−2t) −γ(T1+T2−2T0) + D1γ + D2 3γ + 2ρ12γ e − e
−4 −γ(2T1+T2−3t) −γ(2T1+T2−3T0) −D2γ e − e (5.96)
−3 −2 −2 −4 −3 −γ(T1−t) −γ(T1−T0) G3,t := − 4D1 γ + ρ12γ + 2D2 γ + 3γ + 4ρ12γ e − e
−3 −2 −2 −4 −3 −γ(T2−t) −γ(T2−T0) − 4D1 γ + ρ12γ + 2D2 γ + 3γ + 4ρ12γ e − e
−4 −3 −2γ(T1−t) −2γ(T1−T0) −2γ(T2−t) −2γ(T2−T0) +D2 γ + ρ12γ e − e + e − e
−3 −4 −3 −γ(T1+T2−2t) −γ(T1+T2−2T0) + 2D1γ + 2D2 2γ + ρ12γ e − e −4 2D2γ − e−γ(2T1+T2−3t) − e−γ(2T1+T2−3T0) 3 −4 2D2γ − e−γ(T1+2T2−3t) − e−γ(T1+2T2−3T0) (5.97) 3
−3 −2 −2 −4 −3 −γ(T2−t) −γ(T2−T0) G4,t := − 4D1 γ + ρ12γ + 2D2 γ + 3γ + 4ρ12γ e − e
−3 −2 −2 −4 −3 −γ(T1−t) −γ(T1−T0) − 2D1 γ + ρ12γ + D2 γ + 3γ + 4ρ12γ e − e 1 + D γ−3 + D 3γ−4 + 2ρ γ−3 e−2γ(T2−t) − e−2γ(T2−T0) 2 1 2 12
−3 −4 −3 −γ(T1+T2−2t) −γ(T1+T2−2T0) + D1γ + D2 3γ + 2ρ12γ e − e
−4 −γ(T1+2T2−3t) −γ(T1+2T2−3T0) −D2γ e − e (5.98)
−3 −2 −2 −4 −3 −γ(T2−t) −γ(T2−T0) G5,t := 2D1 γ + ρ12γ + D2 γ + 3γ + 4ρ12γ e − e 1 + D γ−3 + D 3γ−4 + 2ρ γ−3 e−2γ(T2−t) − e−2γ(T2−T0) 2 1 2 12 −4 D2γ − e−3γ(T2−t) − e−3γ(T2−T0) (5.99) 3 5 Unspanned SV for Energy Derivatives 121
Note that G4 and G5 are equal to G2 and G1 (resp.) after having interchanged T1, T2. Chapter 6
Conclusion
The overall goal of this thesis has been to develop new stochastic models and mathematical methodologies for achieving a better pricing of various energy derivative contracts. By a better pricing is meant a theory that is both sound at its mathematical root and practically tractable from a calibration point of view.
We proposed various realistic two-factor energy spot price models (with and without jump components) in Chapter 3, developing as needed the necessary machinery to use these models for the acurate pricing of a class of two-name contracts, namely spread options. An important point to remember is the particular form of our jump-diffusion model for power and natural gas; there we decoupled the spiky behaviour from its diffusive trend by introducing an independent zero mean-reverting pure jump process. This greatly improves the modeling of spikes in price data as it allow us to reproduce sharp jumps without killing the diffusion’s volatility.
In Chapter 4 we extented the financial applications of singular perturbation to the world of energy/commodity derivatives. We did this by extending most spot models discussed in Chapter 3 with a fast mean-reverting stochastic volatility component. We then proceeded toward a theoretical derivation of closed-form pricing formulas for forward
122 6 Conclusion 123 prices as well as for a very general class of one- and two-name European contracts.
Chapter 5 then explored the theme of commodity derivatives pricing in the presence of some form of unhedgeable risk, that is, in the presence of a risk component that forward price do not span (hence the name unspanned stochastic volatility). We developed this body of work while further assuming a SV component being a function of a fast mean- reverting OU process. Doing so allowed us to construct a pricing structure where the calibration to any liquid one-name contracts completely determines the price of all possible calendar spread options.
Any model and pricing theory that has the ambition to be of direct use in financial applications needs to possess well behaved (stable) calibration properties. We therefore proposed stable yet simple methodologies to fit our model’s parameters to market data. These algorithms have been explicitely discussed and implemented in Chapter 3 and Chapter 5.
A possible future work orientation, and one that we already started to investigate, is to produce a detailed comparative study of the relative speed and stability with which various numerical methods can price swing options. These methods can be broadly di- vided in three essentially different group: least-square Monte Carlo (LSM) method, finite difference scheme for PDEs and multinomial tree approximations, typically binomial and trinomial. Their range of applicability partly depends on the dimensionality of the under- lying stochastic processes to be approximated. Generally speaking, it is well known that trees and finite difference algorithms tend to become very slow when there is more than one source of randomness. This last consideration is the main rationale behind the recent popularity of Monte Carlo methods for pricing/hedging derivatives with early exercise op- tions. In their most general setting, swings are among the most complicated derivatives being traded over the counter. They play a particularly important role within natural gaz and power markets. A careful study of the available numerical schemes would therefore 6 Conclusion 124 be of great interest to most practitioners in the field. Another useful future work avenue would be to carry a vast statistical study, over various enegy asset classes, aiming at testing the presence of an unspanned stochastic volatility component in the asset price data. The presence of such a phenomena has been demonstrated within NYMEX crude oil data, but remains to be investigated for other energy assets. Such a study would determined the extent to which the results of Chapter 5 are likely to find applications outside crude oil markets. Bibliography
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