MEAN REVERTING MODELS FOR ENERGY OPTION PRICING 1 ...

MEAN REVERTING MODELS FOR ENERGY OPTION PRICING ALI LARI-LAVASSANI, ALI A. SADEGHI, AND ANTONY WARE Abstract. Various one to three factor mean reverting processes are investigated in the context of energy markets. Results of Natural gas and crude oil market data calibrations are presented. Numerical implementations of the multi-factor models are discussed via binomial trees, finite difference method and Monte Carlo simulation.

1. Introduction Commodity markets are different from financial markets, and energy markets are among the most complex commodity markets. This is principally due to physical constraints such as the local nature of the markets, mild to severe inelasticity in supply and demand, and volume constraints on transfer and storage. These strict physical constraints impact the entire dynamics of energy markets and give rise to very specific features. The processes modeling the spot price of energy commodities are not lognormal. They tend to exhibit time variable and proportional volatility, strong mean reversion and seasonality on several time scales. Therefore, standard formulas used in financial practice for option pricing, risk evaluation and management, and modeling of the forward price curves are not directly applicable to these markets. Volumetric risk and constraints are specific features of energy markets and must be properly modeled and valued. The modeling of energy price data and processes is extremely complex. Indeed, the lack of sufficient data poses serious challenges in testing the accuracy of any real model. And even when data are available, since the markets are illiquid and constrained at various degrees, the statistical measurements performed on these data do not unfold the complete dynamics of the market, and can be in addition highly unstable. In this paper we limit ourselves to considering some mean reverting price processes of the diffusion type relevant for energy prices. In particular we do not consider the added complexity Date: October 10, 2001. Key words and phrases. Mean Reverting Models, Multifactor Models, Energy Markets, Binomial Trees, Finite Difference Methods, Monte Carlo Simulation, Numerical Methods, Option Pricing, Energy Derivatives. A. Lari-Lavassani is partially supported by a grant from the National Science and Engineering Research Council of Canada. A. Sadeghi is supported by a Postdoctoral Fellowship from the Faculty of Management of the University of Calgary, and the Canadian Network of Centres of Excellence, MITACS. The authors are grateful to Brad Tifenbach for his contributions to this work. 1

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ALI LARI-LAVASSANI, ALI A. SADEGHI, AND ANTONY WARE

introduced by jump processes. Our goal is to present a range of approaches to the numerical implementation for the processes we discuss. 2. Mean Reverting Processes for Modeling Energy Prices We introduce here several one to three factor models while motivating the ideas behind the notion of mean reversion. 2.1. One Factor Models. A general one factor stochastic differential equation (SDE) typically assumes the form (2.1)

dSt = µ(St , t)dt + σ(St , t)dZt ,

where µ and σ satisfy the usual conditions for the existence of solutions of the above SDE, and dZt is a standard Brownian motion, with E(dZt ) = 0, and E((dZt )2 ) = dt, see [KP, 1999]. The classical lognormal model is a particular case: (2.2)

dSt = µSt dt + σSt dZt .

A one factor mean reverting SDE takes then the form (2.3)

dSt = α(L − St ) dt + σStγ dZt , γ = 0 or 1,

where α > 0, σ > 0 and L are constants. Note that when γ = 0 the noise is additive and when γ = 1 the noise is proportional. A simple application of the Ito Lemma proves that, qualitatively, the case with additive noise is the linearization (about L) of the case with proportional noise. On the other hand, the ordinary differential equation followed by the mean (denoted by an upper bar) of these SDEs is: dS = α(L − S) dt. Then S = L is the equilibrium of this ordinary linear differential equation, whose solutions are given by S(t) = L + c e−αt for a constant c. Therefore, as t −→ ∞, S(t) −→ L. In other words, the mean S(t) reverts on the long run to L, hence the terminology mean reverting. In a finite time horizon, L plays the role of an attractor, in the sense that when St > L, the drift term α(L − St ) < 0, hence St decreases, and when St > L a similar arguments shows that St increases. In short the trend is that St hovers around L without getting too far from it. This simple case captures the intuition behind all mean reverting processes. This model is proposed for the spot price of energy commodity in [P, 1997]. Generalizations of it to the case where L is made time dependent to capture seasonality are discussed in [T, 2000].

MEAN REVERTING ENERGY PRICE PROCESSES

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2.2. Two Factor Models. To go to the more complex level of two factor models, keeping the same mean, the SDE (2.3) can be generalized by either allowing the long run mean L or the volatility σ to be governed by an SDE. This leads to two distinct two factor models, with different dynamics. The first model assumes a stochastic long run mean: (2.4)

dSt = α(Lt − St ) dt + σ Stγ dZt dLt = µ Lt dt + ξ Lδt dWt , where γ, δ are either 0 or 1,

and where α, µ, σ and ξ are constants, and dZt and dWt are uncorrelated Brownian motions. This model which is introduced in [P, 1997] enables the attractor or market equilibrium L to be modeled more realistically as a random process. A more detailed analysis of the dynamic and numerical implementation of this model can be found in [LSW, 2001], and [T, 2000] which studies in addition, the more interesting case where the parameter µ is allowed to be a function of time in order to capture seasonal effects. As in the one-factor case, the case of additive noise (δ, γ = 0) is similar to the linearization of the case with proportional noise (δ, γ = 1). The second generalization is the two factor model where volatility is allowed to be stochastic: (2.5)

dSt = α(L − St ) dt +

√

σt Stγ dZt

dσt = µ (σ0 − σt ) dt + ξ σtγ dWt ,

γ = 0 or 1,

where α, L, µ, ξ and σ0 are constants. We mention, without going into further details that from a dynamic point of view, one common feature of these two models is that their deterministic parts, giving the mean of the systems, are linear hyperbolic systems, i.e. the real parts of their eigenvalues are non-zero. Once this point is understood, then it is relatively easy to move to three factor models: the deterministic part of the system should be made of a three dimensional linear hyperbolic system. 2.3. Three Factor Models. A canonical way to proceed would be to take three by three hyperbolic matrices for the drift of the system. A complete classification of these systems is beyond the scope of this presentation. Let us just state that some normal forms in this classification consist in upper triangular matrices with zero entries below the diagonal and non-zero entries on the diagonal. These entries then become the eigenvalues of the system and since they are non-zero the system is hyperbolic, see [R, 1995] for full details. Hyperbolicity is crucial if the systems are be structurally stable and hence proper for numerically stable discretization as they are used in computations and simulations. We note that this is exactly what was done in the case of the two two-factor models above, for a more detailed analysis we refer to [LSW, 2001].

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We finally mention that the number of positive and negative eigenvalues gives respectively the dimensions of the unstable and stable manifolds of the mean or expectation of the SDE, and hence governs the dynamics of the resulting SDE, on the average. Simple numerical investigations in terms of stability of these systems reveal the following intuitive fact. Systems with two negative eigenvalues, that is a two dimensional stable manifold attracting the dynamics of the mean to the origin and hence preventing it from growing exponentially, and one positive eigenvalue, leading to a one dimensional unstable manifold along which exponential growth takes place, and which we dedicate to the shocks induced by the long run mean, result in systems that are more robust to calibrate and work with. We now exhibit two different types of three factor mean reverting systems explained above. The first is a three factor mean reverting model with an intermediate level of stochastic mean Lt , governing the growth or decay of St , and a second level of stochastic mean reversion Mt , inducing higher local shocks to St via Lt . This system is a candidate for modeling the spikes of electricity prices.

dSt = α(Lt − St ) dt + σStγ dZt (2.6)

dLt = µ (Mt − Lt ) dt + ξ Lγt dWt dMt = β Mt dt + λ Mtγ dVt , γ = 0 or 1,

where α, σ, µ, ξ, β and λ are constants. Finally, a different three factor mean reverting model with both stochastic mean and volatility is:

dSt = α(Lt − St ) dt + (2.7)

√

σStγ dZt

dLt = µ Lt dt + ξ Lγt dWt dσt = β (σ0 − σt ) dt + λ σtγ dVt , γ = 0 or 1,

where α, σ0 , µ, ξ, β and λ are constants. Local shocks to St are triggered by combined nonlinear effects of Lt and σt . 2.4. Modeling capabilities of the two and three-factor mean reverting models. Modeling electricity price processes is currently an active area of academic and industrial research. The main difficulty here is to capture the spikes occurring in the price, where for a short period of time, prices rise up to twenty times higher than the average figures. A fundamental question is to identify different classes of dynamics capable of driving these price processes. One model


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used in the industry is the two factor mean reverting system (2.4). The question is whether it is able to produce such big spikes over short periods of time. In fact the answer to this question depends highly on the relative values of the parameters involved in the system. While for some combinations it is hard to see any spiky effect, there are other values which are clearly able to produce significant spikes. In Figure 1, spiky behavior is simulated with this two factor model. The parameters used to run the simulation are quoted in the figure.

Figure 1. Spikes generated by the 2-factor model (2.4).

The additional flexibility provided by the 3-factor model (2.6) might be preferred as a means of reproducing spikes. As shown in Figure 2 this model is capable of doing so. This sample path was simulated using an initial price of the asset St , S0 = $40, with an initial ‘excitable’ equilibrium level L0 = $100 and an initial value of the ’shock’ market level M0 = $400. The remaining parameters used to run this simulation are indicated on the figure. The sample path displayed could represent the hourly price of power over nearly 250 days of a trading year. Note that the two simulated sample paths above were obtained with constant coefficients. If the coefficients are allowed to vary in time, seasonal or quasi-periodic effects can then be created in the dynamics, resulting in more realistic realizations.

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Figure 2. Spikes generated by the 3-factor model (2.6). 3. Market Data Analysis One major problem encountered in the practical use of the above multifactor models is that direct market data only exists for the spot price St of the commodity, whereas these models have additional “hidden” variables such as the long run mean Lt or the stochastic volatility σt . Additional relevant information is however available in the form of forward curves, and here one is faced with two basic alternatives. One option is to model the forward curve directly, and possibly build an implied model of the underlying spot price as a second step. The other (which we will use in this paper, and develop more fully below) is to start with a model for the spot price, and to deduce from this a model for the forward curve. In the absence of forward data, the hidden variables must be filtered to unfold or reconstruct the phase space of the dynamics. Once the hidden variables are recovered, maximum likelihood techniques can be used to estimate the parameters of the system. In general these tasks constitute the most difficult part of any stochastic modeling and its implementations. In our multi-factor models, when noise is additive, the system is linear and, at least theoretically, Kalman filters can be used to uncover the hidden variables. However, practical implementations are often subject to numerical instabilities and care must be exercised. On the other hand, when noise is proportional, the filtering problem becomes nonlinear and hence a lot more difficult to perform. For the case of the two factor model (2.4) with proportional noise, [T, 2000]


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had developed a numerically robust methodology for filtering and estimating its parameters, including the more interesting and difficult case where the drift µ of Lt is allowed to be time dependent to capture seasonal effects. 3.1. Forward curve models. We shall describe in detail a procedure for building a model for the forward curve in the case of one-dimensional models with affine drift terms. For multifactor models the procedure is analogous, and we restrict ourselves to quoting some of the resulting formulae. Consider St governed by (2.1). As described in [W,1998] for example, the value of a derivative contract V (S, t; T ) (with expiry time T ) satisfies a partial differential equation (3.1)

Vt + R(S, t)VS +

σ 2 (S, t) VSS = rV, 2

where R(S, t) is a risk-adjusted drift, given by µ(S, t) − σ(S, t)λ(S, t) for some λ1. We model R as an affine function of S independent of t: R(S, t) = AS + B. Note that R does not depend on the expiry time T (or any other parameters pertaining to the contract), but only on S and (but not in our model) the market time t. If we consider the case when V is a forward contract to take delivery of a unit of the commodity at a price E at time T , then at expiry (t = T ), V (S, T ; T ) = S − E, and is linear in S 2. We can look for a solution of (3.1) that remains linear in S: V (S, t; T ) = α(t; T )S + β(t; T ). We find that αt S + βt + (AS + B)α = r(αS + B), so that, equating the coefficients of each power of S: αt = (r − A)α βt = rβ − Bα, 1λ 2If

can be thought of as a market price of risk term. the constract is for delivery over a significant length of time, say from T1 to T2 , then the final condition is

not quite S − E: the appropriate value can be found as an integral of values of contracts for delivery at times between T1 and T2 .

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and α(T ; T ) = 1, β(T ; T ) = −E. This is a system of ordinary differential equations with solution α(t; T ) = e(r−A)(t−T ) B −A(t−T ) r(t−T ) β(t; T ) = e (e − 1) − E , A so that our forward contract has value V (S, t; T ) = e

r(t−T )

Se

−A(t−T )

B −A(t−T ) − E − (1 − e ) . A

At the market date, arbitrage considerations dictate that V = 0, i.e. the price E at which the contract is struck, E should be zero. Thus E = E(St , t; T ) =

B St + A

e−A(t−T ) −

B . A

The forward curve at time t predicted by the model is a collection of values E(St , t; Ti ) for times Ti > t. Values for A and B may be obtained by matching this with observed forward curves. If a similar analysis is carried out for the model (2.4) for example, the forward curve is given by E(St , Lt , t; T ) = St e−A(t−T ) +

αLt e−B(t−T ) − e−A(t−T ) , B−A

where the risk-adjusted drifts of St and Lt are ASt + αLt and BLt respectively. Notice that the unobservable Lt appears in this formula, and may then be estimated at each time t by matching the observed forward curve at time t. The approach we adopt is to obtain αLt (for each t), and A and B by this method, and then to use the time series for St and Lt to estimate α, µ, σ and ξ. 3.2. Calibrations. In Figures 3 and 4, we illustrate historical price processes for Henry Hub Natural Gas and Light Sweet Crude Oil, as quoted on NYMEX in the period January 1 1997 December 31 1999. Each commodity has been calibrated for the two-factor model (2.4), using methods described above. In each case we plot the values of Lt providing the optimal fit with the forward data, and show the resulting model parameters (all parameters shown are annualized). We finally mention the important issue of how well a proposed model fits actual market data. To illustrate our point we consider the spot price of NYMEX Henry Hub Natural gas over the year 1999. We generate the distribution of daily returns (∆St /St ), on the price of natural gas over 1999 in two different ways. On one hand, we use historical data to generate this distribution and plot the result in Figure 6. On the other hand, we run one sample path simulation of the calibrated two factor model (2.4), with the parameters given in Figure 3 and plot the distribution in Figure 5.


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Note the qualitative similarities between the distribution plots in Figures 6 and 5, and the very good numerical agreement of the first four moments of the historical and simulated distributions as they are quoted in these figures. We first conclude that the nontrivial values for the Skewness and Kurtosis in the return distribution coming from the real historical data is an indication of the deviation of this return from lognormality. This is a simple confirmation of our earlier statement about energy market price dynamics not being lognormal. We next note that different simulations of the above two factor model, with the provided calibration, yield different distributions of daily returns with values for the first four moments, near the ones quoted in Figure 5. This illustrates the fact that this calibrated two factor model has the capability of generating states of the world similar to the actual market dynamic from which the historical spot data came from, and is therefore a good model for these markets.

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4. Numerical Methods Numerical methods in finance aim at achieving several goals. Consider a price process St and a time horizon, which one can assume without loss of generality to be [0, T ]. The first aim is to find E(f (ST )|S0 ), the expectation of a given functional f of St at the time T . This leads to a methodology which, under certain assumptions on risk neutrality, enables one to price options on St . A second goal is to analyze risk on portfolios of assets and liabilities. One approach we adopt to solve these two problems is to discretize the SDEs giving St and to use simulations to reproduce the distribution of returns at the given time T . The distribution of returns at each instance of time T embodies the whole information needed for both option pricing and risk analysis purposes, such that E(f (ST )|S0 ), value at risk, cash flow at risk and other risk measurements can then be easily estimated. We also present a finite difference method for option pricing, which is based on a discretization of a partial differential equation (PDE) (derived from the mean-reverting SDEs in much the same way as the Black-Scholes equation is derived from a log-normal model for the underlying asset). The third approach we discuss is the binomial tree, which can be thought of either as a finite difference discretization of the option-pricing PDE or as a discretization of the underlying SDE. Very little has been published about numerical implementations of mean reverting underlying price processes. For this reason we present in this paper a range of numerical techniques for them. To simplify the presentation we fully develop numerical implementation schemes for the two factor model (2.4) in the more complicated case of proportional noise. Our goal is to provide the numerical algorithms necessary to implement this model. We therefore implicitly assume that all processes are risk neutralized so that the expectations thus calculated reflect the fair market price of options. We mention the fact that binomial and finite difference algorithms are model dependent and will have to be redesigned for systems other than (2.4), while Monte Carlo routines would require minor readaptations. 4.1. Binomial Trees. Since the construction of trees is quite well known we shall be brief. Note that the formulas available in the literature are for lognormal processes. We give here formulas for a binomial tree appropriate for (2.4) and refer to [LSW, 2001] for a detailed and complete presentation. Several trinomial trees are developed for this model in [T, 2000] and [LT, 2001]. We first partition the interval [0, T ] into N + 1 time steps i∆t with ∆t = T /N , and next discretize the stochastic process St with a recombining binomial tree with nodes Sji involving j = 1, ..., i + 1 possible values of St at the time step i, i = 0, ..., N . We define the up jump size at the node (i, j) by uij so that, over the time interval indexed form i to i + 1, Sji goes


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i+1 either up to Sj+1 = Sji uij with probability pij or down to Sji+1 = Sji dij with probability (1 − pij )

with uij = 1/ dij . A similar construction is carried out for the process Lt in terms of a binomial i i i discretization Lik with an up jump given by wki to Li+1 k+1 = Lk wk with probability qk or a down

jump to Li+1 = Lik hik with probability (1 − qki ) with wki = 1/hik . It is established in [LSW, k 2001] that the following values, which are approximations to the order ∆t, result in individual recombining binomial trees: √ 1 µ − ξ2 √ + ∆t ; wki = eξ ∆t ; dik = 1/ wki 2 2ξ √ Lik /Sji − 1 σ √ 1 i pj = + (α − ) ∆t ; uij = eσ ∆t ; dij = 1/hij . 2 2σ 4

qki =

Then, since in (2.4) the two Brownian motions are uncorrelated up to the order ∆t, the following direct product tree is a recombining binomial tree for (2.4), where, at the time step i, a typical i node on this tree denoted by Tj,k = (Sji , Lik ) branches into four nodes given by:

Nodes i+1 Tj+1,k+1 = (Sji uij , Lik wki )

Probabilities pij qki

i+1 Tj,k+1 = (Sji dij , Lik wki )

(1 − pij )qki

i+1 Tj+1,k = (Sji uij , Lik hik )

pij (1 − qki )

i+1 = (Sji dij , Lik hik ) Tj,k

(1 − pij )(1 − qki ).

i Once the tree Tj,k is populated according to the above formulae, option pricing follows standard

routes, see [W, 1998]. 4.2. Finite Difference Methods. As described by Wilmott [W, 1998], the partial differential equation governing the price of an option V (S, L, t) when the underlying asset is governed by a 2-factor SDE such as (2.4) is (4.1)

∂V ∂V ∂V σ2 ∂ 2V ξ2 ∂2V + [α(L − S) + σλS ] + [µL + ξλL ] + S 2 2 + L2 2 = rV, ∂t ∂S ∂L 2 ∂S 2 ∂L

where λS and λL are market price of risk functions associated with the asset price S and the long-run mean L respectively. In the computational testing reported in this paper, these terms have been set to zero (corresponding to an assumption that the price processes are already risk neutralized). Incorporating non-zero values for λS and λL introduces no additional numerical complexity. This equation runs backwards in time, from the expiry time T , when the option value is given by the payoff function (typically a function of S alone) to the present, when t = 0.

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Our first step in obtaining a solution is to discretize in time. Discretization in the other variables will follow as a second step. In order to discretize in time effectively and stably, we use a semi-Lagrangian approach. The interval [0, T ] is partitioned into N + 1 time steps: 0 = t0 < t1 < · · · < tN = T. We shall use equally-spaced time steps, although nothing in our formulation makes this necessary. The semi-Lagrangian method involves writing the equation in terms of Lagrangian coordinates within each time step. Thus, for each time step, we define particle paths S and L satisfying dS = α(L − S), dt dL = µL, dt

(4.2) (4.3) n

S(tn−1 ) = S, L(tn−1 ) = L.

n

We define S = S(tn ) and L = L(tn ). Then n

S = Se−α∆tn +

α L eµ∆tn − e−α∆tn , α+µ

where ∆tn = tn − tn−1 . Using Euler’s method to discretize the Lagrangian form of the equations in time now results in the semi-discrete equation n

σ 2 2 ∂ 2 V n−1 S (S, L) 2 ∂S 2 ξ 2 2 ∂ 2 V n−1 + ∆tn L (S, L) = r∆tn V n−1 (S, L). 2 2 ∂L

n

V n (S , L ) − V n−1 (S, L) + ∆tn (4.4)

n

n

Note that as we are solving backwards in time, the values V n (S , L ) are assumed to be known from previous calculations, and (4.4) is an elliptic equation that must be solved to find V n−1 (S, L). In order to be able to solve the equations numerically, we restrict the domain to the bounded interval (S, L) ∈ Ω, with Ω = [Smin , Smax ] × [Lmin , Lmax ]. Thus we restrict S and L by only considering values satisfying 0 < Smin < S < Smax < ∞ and 0 < Lmin < L < Lmax < ∞. We assume that (4.4) holds in the interior of Ω, while on the boundaries we impose (1 + r∆tn )V n−1 (S, L) − ∆tn

ξ 2 2 ∂ 2 V n−1 n n L (S, L) = V n (S , L ) 2 2 ∂L

when S = Smin , Lmin < L < Lmax , (1 + r∆tn )V n−1 (S, L) − ∆tn

σ 2 2 ∂ 2 V n−1 n n S (S, L) = V n (S , L ) 2 2 ∂S

when L = Lmin , Smin < S < Smax , and n

n

(1 + r∆tn )V n−1 (S, L) = V n (S , L ) when S = Smax or L = Lmax or S = Smin and L = Lmin .


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Thus we have two one-dimensional problems to solve along the S = Smin and L = Lmin boundaries. Once these are solved, the values obtained provide Dirichlet boundary conditions for (4.4). To solve (4.4) we use a finite difference discretization on a grid of points (Sj , Lk ), with j = 0, . . . , J and k = 0, . . . , K. The points are given by Sj = Smin

Smax Smin

j/J

,

and Lk = Lmin

Lmax Lmin

k/K

.

We further define ∆+ Sj = Sj+1 − Sj

∆+ Lk = Lk+1 − Lk

∆− Sj = Sj − Sj−1

∆− Lk = Lk − Lk−1

∆0 Sj = (Sj+1 − Sj−1 )/2

∆0 Lk = (Lk+1 − Lk−1 )/2.

Then, if we let Wj,k denote our approximate value for V n−1 (Sj , Lk ) (4.4) is replaced by the discrete equation, for j = 1, . . . , J − 1 and k = 1, . . . , K − 1, σ 2 ∆tn Sj2 (1 + r∆tn )Wj,k + 2∆+ Sj ∆− Sj

∆+ Sj ∆− Sj − 0 Wj−1,k + 2Wj,k − 0 Wj+1,k ∆ Sj ∆ Sj 2 2 + ξ ∆tn Lk ∆ Lk ∆− Lk + − 0 Wj,k−1 + 2Wj,k − 0 Wj,k+1 2∆+ Lk ∆− Lk ∆ Lk ∆ Lk n

n

= V (S j , Lj ). n

n

The value of V (S j , Lj ) is found by linear interpolation between the neighbouring grid points. The boundary problems are discretized in an entirely analogous manner. It remains to solve these discrete equations, and this is done using the standard Successive Over Relaxation (SOR) algorithm. The relaxation parameter was dynamically adjusted at each time step to minimize the number of iterations performed. In the results quoted below, the tolerance (relative l2 norm) was set to 10−6 . 4.3. Monte Carlo Simulation. We first review the definitions of convergence modes which are of interest from a theoretical and numerical view point. Any numerical scheme for the approximation of an SDE modeling St begins with the discretization of that SDE and leads to Stochastic Difference Equations. The main parameter which characterizes any discretized scheme is the maximum step size ∆t. The interval [0, T ] is usually divided into equal subintervals ∆t = Tn , although in general the step sizes do not need be equal.

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We consider below a multidimensional version of the mean reverting model (2.4) with proportional noise. That is a multi-asset setting with d assets Stk given by: dStk

= α

k

(Lkt

−

Stk )dt

+

d X

σ kj (St , t)dWtkj

j=1

dLkt = β k Lkt dt + τ k Lkt dZtk , where, St = (St1 , · · · , Std ) and Wtkj and Ztk are independent N (0, 1) Gaussian random variables, k = 1, . . . , d. 4.3.1. Euler Discretization. The stochastic Euler discretization is a direct extension of the classical Euler method used in the treatment of ordinary differential equations, where

dx dt

is approx-

imated by its first order Taylor expansion. In the stochastic context, we also need an approximation for the Ito term dWt , which, in fact refers to Wt+∆t − Wt . Since a Brownian motion has independent normally distributed increments, Wt+∆t − Wt has the same distribution as W∆t √ which equals ∆tZ, where Z is a standard Gaussian N (0, 1) random variable. This leads to the √ expression dWt = ∆tZ. As a result, the Euler scheme for equation (2.4) can be written as k St+∆t

=

Stk

+α

k

(Lkt

−

Stk )∆t √ k

dLkt = β k Lkt ∆t + τ k Lt

+

d X

√ σ j (St , t) ∆twkj

j=1

∆tz k ,

where (wkj )k,j=1,...,d is a d-dimensional matrix consisting of standard N (0, 1) random variables. Although in the deterministic context the Euler scheme is a first order approximation, its strong order of convergence in the stochastic context is 0.5. This is because the Euler scheme ignores some first order terms in the stochastic Taylor expansion of St+∆t around St . However, under certain conditions, it retains its first order approximation property in the weak sense. This issue is addressed by the Milstein scheme below. For full details and proofs we refer to [KP, 1999]. 4.3.2. The Milstein scheme. To gain efficiency (accuracy with less variance) in Monte-Carlo simulations, one can increase either the number of time steps or the number of simple paths. These imply a higher computational cost. If one uses higher order approximation schemes, then the desired degree of accuracy can be achieved by using fewer time steps. The main idea of the Milstein scheme is to incorporate the entire first order term of the stochastic Taylor expansion. This results in a more complicated recursive formula, which has the advantage of having a strong order of convergence equal to 1. The formula, for the general


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multi-factor mean reverting systems above, becomes k St+∆t = Stk + (Lt − St )∆t +

d X

√ σ kj (St , t) ∆tZj

j=1

+

d X d X d X

σ lj1 (St , t)

j1 =1 j2 =1 l=1

∂σ lj2 (St , t) Ij1 j2 ∂Stl

where Ij1 j2 =

Z∆tZs t

dWτj1 dWsj2 .

t

The main difficulty here is to compute the above double Ito integral terms. In some special cases, see [KP, 1999] this integral reduces to easy forms. For instance if the volatility matrix is only a function of time, i.e. σ(St , t) = σ(t) (which is the case in the our models with additive noise), the derivatives in the Milstein correction vanish. The Milstein scheme then reduces to the Euler scheme which then acquires a strong order of convergence equal to one. Another special case is when the underlying processes Stk are uncorrelated, such that the matrix is diagonal and the Milstein term includes only Ij1 j1 , which using the Ito identity Ij1 j1 = 0.5((∆W j1 )2 − ∆t) becomes easy to simulate. Finally for the model (2.4), this integral reduces to the simple form, Ij1 j2 = ∆W j1 ∆W j1 = ∆t Z 1 Z 2 , where Z 1 and Z 2 are independent N (0, 1), and hence are easy to simulate in practice. For a more complete and detailed expose of Monte Carlo simulation as applied to option pricing we refer to [LSW, 2000] and [H, 2000] and the references therein. 4.4. Numerical Option Pricing Experiments. 4.4.1. Option pricing with a 1-factor model. Consider the one-factor model (2.3) with γ = 0. The parameters of the model can be calibrated from historical prices by considering the Euler discretisation of (2.3): √ Si+1 − Si = α∆t(L − Si ) + σ ∆tWi ,

(4.5)

where Si stands for Si∆t , and the {Wi } are i.i.d. N (0, 1) random variables. √ In order to simplify our notation we write α = α∆t and σ = σ ∆t. We also will use the notations N −1 1 X S= Si , N i=0

S 12

N −1 1 X = Si Si+1 , N i=0

N −1 1 X 2 S2 = S , N i=0 i

δ=

S N − S0 N

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and N −1 1 X ∆= (Si − Si+1 )2 . N i=0

Note that these are all observable quantities. First, summing (4.5) from i = 0 to i = N − 1 and dividing by N gives N −1 1 X δ = αL − αS + σ Wi ≈ αL − αS. N i=0

(4.6)

Secondly, multiplying (4.5) by Si and summing gives (4.7)

S 12 = S 2 + αLS − αS 2 + σ

N −1 1 X Si Wi ≈ S 2 + αLS − αS 2 . N i=0

Thirdly, squaring (4.5) and then summing gives (4.8)

N −1 1 X 2 ∆ = α L − 2α LS + α S 2 + σ W ≈ α2 L2 − 2α2 LS + α2 S 2 + σ 2 . N i=0 i 2

2

2

2

2

If we assume that the approximations made in (4.6–4.8) are in fact exact we have three equations for three unknowns which may be solved to give α=

S 12 − Sδ − S 2 2

S − S2

SS 12 − S 2 δ − SS 2 S 12 − Sδ − S 2 q σ = ∆ − (α2 L2 − 2α2 LS + α2 S 2 ).

L=

We employ these formulae, using the 1998 natural gas and oil prices, as shown in Figures 3 and 4 (and with ∆t = 1/250, representing one trading day as a fraction of the total number of trading days in a year), and we obtain the parameters (shown to 6 d.p.) Gas

Oil

σ

2.133753

6.438117

α

9.454015

8.651607

L

2.485906 13.752057

With these values for the parameters we price options using both binomial trees and MonteCarlo simulations. We take for the initial values S0 = $2.31 for gas, and S0 = $17.62 for oil, set strike prices of $2.40 for gas and $15 for oil. The risk-free interest rate is taken to be 5%, there is no dividend payment, and we price call options with expiry times of 3 months, 6 months and 1 year.


Gas B

17

Oil

M-C

True

B

M-C

True

3 months

0.00982 0.01000 0.00983 0.17888 0.17811 0.17884

6 months

0.00970 0.00972 0.00971 0.17930 0.17942 0.17942

1 year 0.00959 0.00949 0.00946 0.17496 0.17569 0.17510 Table 1. Call option prices (in dollars) for gas and oil commodities with various expiry times calculated using a binomial tree with 2560 time steps, and MonteCarlo simulation with 100000 samples (and using antithetic variable variance reduction).

The binomial tree calculations were performed with nodes Sji and probabilities pij given by √ Sji = S0 + (2j − i)σ ∆t 1 1 − e−α∆t i i √ 1 + (L − Sj ) pj = , 2 σ ∆t with the probabilities replaced by max(0, min(1, pij )) to ensure that they remained between zero and one. These choices of nodes and probabilities mean that the tree can price futures contracts exactly (in the absence of probability truncation). The Monte-Carlo simulations were carried out using the fact that, since (2.3) is an OrnsteinUhlenbeck process, it has an explicit solution (4.9)

St = L + (S0 − L)e−αt + σ

r

1 − e−2αt Zt , 2α

where for each t > 0, Zt is an N (0, 1) random variable (although for different t’s the variables are not uncorrelated). Thus the use of Euler discretisation is not necessary in order to obtain option values via Monte-Carlo simulation: one may use (4.9) directly to provide samples of the random variable St . The Monte-Carlo results in Table 1 were obtained in this manner using 100000 samples with antithetic variable variance reduction. The binomial results in Table 1 were obtained using 2560 time steps. We show the relative errors calculated via relative error =

calculated value − true value , true value

where in each case the ‘true value’ was provided by a binomial tree calculation using 10240 time steps. In Figures 7 and 8 we illustrate that the two methods are converging exactly as √ expected: for the Monte-Carlo simulations, the theoretical error estimate is std(v)/ N , where v is a collection of N simulated option values, while for the binomial trees the error bound is inversely proportional to the number of time steps.

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4.4.2. An example of option valuation with a 2-factor model for the underlying asset. We compare here the performance of the above numerical algorithms on a test case. We would like to price a call option on Henry Hub natural gas, denoted by St and modeled by (2.4) with proportional noise, with the parameters calibrated from the 1998 historical prices as shown in Figure 3. The option is struck at $2.20, has an expiry time of 1 month, and the risk-free interest rate is taken to be 5%. We take for initial values L0 = S0 = $2.20. We price this option using the three different methods introduced above, namely, the binomial tree, the finite difference method and Monte Carlo simulation. The binomial tree option pricing calculations were performed with a number of time steps N , resulting in a total of (N + 1)(N + 2)/2 nodes on the tree. The error estimate derived in [LSW, 1999] indicates that the errors are proportional to 1/N . An analysis of the actual error estimates at each time step of our computations, represented in a log-log plot in Figure 9, shows that the results are entirely consistent with the theory. The accuracy of the Monte-Carlo simulation depends on two factors: the number of time-steps used, and the number of simulations performed. If the error incurred as a result of too few time steps is large, then the effect of taking a large number of simulations will simply be to reduce the randomness in the value obtained, but not make it any more accurate. On the other hand, if too few simulations are performed, then increasing the number of timesteps will have the effect of replacing one random variable with another, but neither being necessarily accurate. In practice, it is desirable to achieve a balance in the contributions to the error from these two sources. In Figure 10, errors are shown for Monte-Carlo estimates using various numbers of time steps and simulations. For purposes of comparison, an estimate obtained using 108 simulations and 320 time steps was used as a proxy for the exact value. Finite difference methods were also applied to this problem. Instead of giving a single option value, they actually generate a host of values, corresponding to a range of current asset prices S and values of the long run mean L. In Figure 11 we illustrate these values as a surface plotted as a function of the initial values of S and L. The option value of value $0.1332 is obtained using N = 1280 and J = K = 400, and with Smin = Lmin = 0.0001, and Smax = Lmax = 100.


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4.4.3. Valuing a spread option between natural gas and power. Energy options are often based on multi-assets. Spread options are quite popular in practice. However beyond log normal processes for the underlying factors, no general numerical scheme is available. We consider here a spread option between two energy commodities such that the dynamics of one is governed by a the two factor mean reverting process (2.4) and we view it as 1998 Henry Hub Natural gas with the parameters calibrated in Figure 5. The other commodity is power and its dynamics is governed by the three factor model (2.6), where the parameters and initial values are chosen to be those used to generate Figure 2. Viewing this as a spark-spread, we use a heat ratio of 10, and hence assume that the initial price of gas is S0 = $25, instead of $2.5 and that the initial value of its the long run mean is also L0 = $25. We set the strike price at $50, the time horizon at 30 days and the interest rate at 5%. We note that due the high dimensionality of this problem binomial trees and finite difference methods are no longer efficient. We resort instead to Monte Carlo simulation and obtain, for this option, the price of $226.72 after 100000 simulation runs. The efficiency of this option pricing via Monte Carlo simulations is again measured by the variance of the final results. Figure 12 plots the variance versus the number of simulations. One should notice the fast rate of convergence. This is principally due to the fact that mean reversion prevents the spread of price distributions and forces fast convergence. In short, Monte Carlo simulations yield very good results for high dimensional mean reverting processes.

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REFERENCES [KP, 1999] Kloeden Peter E. and Platen Eckhard. 1999. Numerical Solution of Stochastic Differential Equations. Springer-Verlag. [K, 1998] Kwok Yue-Kuen. 1998. Mathematical Models of Financial Derivatives. Springer-Verlag. [LSW, 2000] Lari-Lavassani, Ali, Sadeghi, Ali A. and Wong, Hilda. Monte-Carlo Simulation for HighDimensional Option Pricing. Preprint, 2000. [LSW, 2001] Lari-Lavassani, Ali; Simchi, Mohammedreza and Ware, Antony. A Discrete Valuation of Swing Options. 1999. Preprint 28 pages. [LT, 2001] Lari-Lavassani, Ali and Tifenbach, Bradley, 2001. A general framework for trinomial trees. To appear in the proceedings of the 2001 International Conference on Computational Science (ICCS 2001). Springer-Verlag Lecture Notes in Computer Science. [P, 1997] Pilipovic Dragana. 1997. Valuing and Managing Energy Derivatives. McGraw-Hill. [R, 1995] Robinson Clark. 1995. Dynamical Systems. CRC Press. [T, 2000] Tifenbach, Bradley. 2000. Numerical Methods for Modeling Energy Spot Prices. Master’s Thesis, University of Calgary. [W, 1998] Wilmott, Paul. 1998. Derivatives: The Theory and Practice of Financial Engineering. John Wiley & Sons. [W, 2000] Wong, Hilda. 2000. Advanced Monte-Carlo Simulations in Option Pricing. Master’s Thesis, University of Calgary.


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Dr. Ali Lari-Lavassani is the Director of the Mathematical and Computational Finance Laboratory in the Department of Mathematics and Statistics of the University of Calgary (http://finance.math.ucalgary.ca/). He is also the founder and CEO of QuantRisk.com (www.quantrisk.com). Dr. Ali A. Sadeghi is a Research Fellow in the Faculty of Management and the Mathematical and Computational Finance Laboratory of the University of Calgary. Dr. Tony Ware is an Assistant Professor in the Department of Mathematics and Statistics of the University of Calgary and a member of the Mathematical and Computational Finance Laboratory. All three authors can be reached at: The Mathematical and Computational Finance Laboratory. Dept of Mathematics and Statistics. University of Calgary. Calgary, Alberta T2N 1N4, Canada E-mail address: [email protected] [email protected] [email protected]

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Figure 3. Natural gas in 1997-1999: shown are the spot price St and the values of the long run mean Lt .

Figure 4. Light sweet crude oil in 1997-1999: shown are the spot price St and the values of the long run mean Lt .


Figure 5. Distribution of natural gas daily returns in 1999, obtained via simulation of the model 2.4.

Figure 6. Distribution of natural gas daily returns in 1999, obtained from historical data.

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24


Figure 7. Convergence history for binomial tree calculations of call option prices for gas and oil commodities with various expiry times, with the number of time steps ranging from 20 to 5120. The relative errors were calculated by comparison with the result from a binomial tree calculation with 10240 time steps. The theoretical error estimates are in fact estimates of the variance of the error.


Figure 8. Convergence history for Monte-Carlo calculations of call option prices for gas and oil commodities with various expiry times, with the number of samples ranging from 100000 to 100000000. The relative errors were calculated by comparison with the result from a binomial tree calculation with 10240 time steps.

25

26


Figure 9. Plot of errors in option value computation using a binomial tree

Figure 10. Errors in option values calculated by Monte-Carlo simulation.


Figure 11. Option values calculated using the finite difference method for a range of initial values for S and L.

Figure 12. Variance of a spread option calculations using Monte Carlo simulations.

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