Real Options in Petroleum: Geometric Brownian Motion and Mean-Reversion with Jumps
Mikael PELET New College Oxford University
A dissertation submitted in partial fulfillment of the requirements for the degree of MSc. in Mathematical Modelling and Scientific Computing September 12, 2003
Acknowledgements Special thanks to my supervisor, Dr William Shaw and to Dr Jeff Dewynne, for playing a major role in directing me through this dissertation. I would like to extend my gratitude to Dr Andy Wathen, my College advisor, for his sound advice and to Dr Hilary Ockendon for her continuous support. And a very special thanks to the EPSRC for allowing me to come and study this year at Oxford University.
I, Mikael Pelet, hereby declare that the content of this dissertation is entirely my own work (except where otherwise indicated), that it has not been submitted for a degree of any other university, and that all the assistance I have received has been fully acknowledged. Mikael Pelet New College September 12, 2003
1
Abstract The traditional Real Options model, developed by Paddock, Siegel & Smith, assumes the underlying stochastic variable to follow a Geometric Brownian process. In the case of a commodity, however, basic microeconomics and politics dictates that the price of the commodity ought to be related to its long-run marginal production cost, which is not the case with a Geometric Brownian process. This process, moreover, does not take into consideration the possible arrival of abnormal information, likely to generate discrete stochastic shocks. By not modelling accurately the underlying’s price in the case of a commodity, the Geometric Brownian Motion may lead to great errors in estimating the value of an undeveloped petroleum reserve and in finding the optimal investment strategy in the reserve. In this dissertation, I first developed a numerical scheme for the Geometric Brownian Motion model. Then I built an underlying model that presented a better economic logic for petroleum prices, i.e. a model using Mean-Reversion with Jumps. I afterwards implemented a numerical scheme for this model and did a sensitivity analysis of the model parameters. I finally performed a comparison between the two models.
Contents 1 Introduction
5
1.1
Real Options Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2
Modelling Oil Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3
Outline of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2 The Real Options Approach of Investment under Uncertainty 2.1
2.2
Generalities on Solving Real Option Problems . . . . . . . . . . . . . . .
9 9
2.1.1
Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . .
10
2.1.2
Contingent Claims Analysis . . . . . . . . . . . . . . . . . . . . .
10
The Value of a Developed Reserve . . . . . . . . . . . . . . . . . . . . .
10
3 Model Using Geometric Brownian Motion
13
3.1
The Value of an Undeveloped Reserve and the Optimal Development Rule 13
3.2
Solving the Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.2.1
Finite Difference Method . . . . . . . . . . . . . . . . . . . . . .
16
3.2.2
At the Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.2.3
Projected Gauss-Seidel Solution Scheme . . . . . . . . . . . . . .
18
3.3
Precision of the Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.4
Influence of the Numerical Scheme Parameters . . . . . . . . . . . . . .
20
4 Model Using Mean Reversion with Jumps
22
4.1
Stochastic Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4.2
Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
4.3
Explicit Finite Difference Method . . . . . . . . . . . . . . . . . . . . . .
27
4.4
Discretization of the Jump Term . . . . . . . . . . . . . . . . . . . . . .
28
4.5
Stability of the Explicit FDM, Computation and Comments . . . . . . .
29
4.6
Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
1
4.7
Influence of the Numerical Scheme Parameters . . . . . . . . . . . . . .
5 Models Results and Comparison 5.1
5.2
5.3
33 35
Traditional Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
5.1.1
Influence of the GBM Model Parameters . . . . . . . . . . . . . .
35
5.1.2
Comparison with other Results . . . . . . . . . . . . . . . . . . .
37
5.1.3
How to Use the Results . . . . . . . . . . . . . . . . . . . . . . .
38
Mean-Reversion with Jumps Model . . . . . . . . . . . . . . . . . . . . .
39
5.2.1
General Shape of the Solution . . . . . . . . . . . . . . . . . . . .
39
5.2.2
Influence of the MRJ Model Parameters . . . . . . . . . . . . . .
39
5.2.3
Evolution of the Option Value over Time . . . . . . . . . . . . .
44
Comparison between the Two Models . . . . . . . . . . . . . . . . . . .
44
6 Conclusion
46
6.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
6.2
Possible Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
A Calculation details
49
A.1 Autoregressive Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . B MATLAB Code
49 51
B.1 American Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . .
51
B.2 Mean-Reversion with Jumps . . . . . . . . . . . . . . . . . . . . . . . . .
54
2
List of Figures 1.1
Yearly oil price history from 1900 to 1957 . . . . . . . . . . . . . . . . .
6
1.2
Monthly oil price history from 1957 to 2003 . . . . . . . . . . . . . . . .
7
3.1
Grid disposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.2
Option value and corresponding ∆ . . . . . . . . . . . . . . . . . . . . .
19
3.3
Comparison with Black-Scholes answers for the option value . . . . . . .
20
3.4
Comparison with Black-Scholes answers for ∆ . . . . . . . . . . . . . . .
20
3.5
What if Pm changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.6
What if the grid gets finer . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.7
A very precise result . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
4.1
Real world jumps distribution . . . . . . . . . . . . . . . . . . . . . . . .
23
4.2
Random jumps distribution . . . . . . . . . . . . . . . . . . . . . . . . .
25
4.3
Discretization of the jump . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4.4
Influence of Pm and dΦ on the MRJ model . . . . . . . . . . . . . . . .
34
4.5
Influence of the grid size on the MRJ model . . . . . . . . . . . . . . . .
34
5.1
Option price and threshold evolution when r changes . . . . . . . . . . .
36
5.2
Option price and threshold evolution when δ changes . . . . . . . . . . .
36
5.3
Option price and threshold evolution when σ changes
. . . . . . . . . .
36
5.4
Option price and threshold evolution when q changes . . . . . . . . . . .
37
5.5
Critical value for development of oil reserves . . . . . . . . . . . . . . . .
37
5.6
MRJ Option Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
5.7
Influence of the volatility on the MRJ model . . . . . . . . . . . . . . .
41
5.8
Effect of volatility on the MRJ threshold . . . . . . . . . . . . . . . . . .
41
5.9
Influence of the mean-reversion speed on the MRJ model . . . . . . . .
41
5.10 Influence of the jump arrival frequency on the MRJ model . . . . . . . .
42
5.11 Threshold function of the jump arrival frequency . . . . . . . . . . . . .
42
3
5.12 Influence of the long-run average oil price on the MRJ model . . . . . .
42
5.13 Influence of the exogenous discount rate on the MRJ model . . . . . . .
43
5.14 Influence of the jump-up average and standard deviation on the MRJ model 43 5.15 Influence of the economic quality of the reserve on the MRJ model . . .
43
5.16 Threshold comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
6.1
47
Probability density function of Φ . . . . . . . . . . . . . . . . . . . . . .
4
Chapter 1
Introduction 1.1
Real Options Theory
Financial derivatives have known a great success in the 1980s. This lead to the development of the theory of Real Options (or theory of irreversible investment under uncertainty). Indeed, the main idea underlying Real Options is that financial derivatives, which are traded on many markets at extremely high volumes, can also be used to model micro-economic decisions. A simple analogy can illustrate this point: take for instance a company that wishes to invest in a project. Imagine that this company possesses the option to wait for better conditions to implement this investment. This option is in fact very much like a financial derivative, and its underlying consists of the economic variables that will condition the future value of the project (such as the market share, the value of the products sold or bought or the intensity of demand). This option is a so-called “Real Option” and its valuation is similar to that of financial derivatives. The Real Options approach appears to be much better than the traditional method of Net Present Value, thereafter denoted NPV, in order to decide whether to invest or not. In the NPV approach, expected future cash flows to leaseholders are determined, discounted to the present and summed to yield the lease value. To determine expected cash flows and proper discount rates, it is first necessary to specify a statistical distribution (not necessarily independent) for exploration costs, quantities of hydrocarbon reserves, development costs, hydrocarbon prices and operating costs. But by boiling down all the possibilities for the future into a single scenario, NPV does not account for the ability of executives to react to new circumstances - for instance, spend a little up front, see how things develop, then either cancel or go full speed ahead.
1.2
Modelling Oil Prices
The most common Real Options model was developed by Paddock, Siegel and Smith [18] between 1983 and 1988. It models the underlying’s price as a Geometric Brownian
5
Motion. If one considers commodities, however, this model may not be accurate enough since the natural choice for modelling commodities seems to be mean-reversion processes. Indeed, basic microeconomics theory tells us that the price of a commodity ought to be tied to its long-run marginal production cost, which varies largely across the countries mainly because of the geologic features. Generally, marginal interaction between production and demand, between depletion and new reserve discoveries leads to smooth price changes. In the case of oil, as it is a cartelized commodity, the situation is a little bit more complicated. Indeed, the oil price is also tied to the long-run profit-maximizing price sought by cartel managers. The role of OPEC furthermore remains very important in the production game of the petroleum industry as most of the lower cost countries belong or are influenced by the OPEC cartel. For instance, the large rise in oil prices in February-April 1999 (Figure 1.2) is mainly due to the articulation power of OPEC in reducing the production. In Figure 1.1 and 1.2, one can see the evolution of oil prices over the last century. The prices for these figures as well as throughout this dissertation are real prices, they have been compounded with inflation and are expressed in terms of 04/2003 US$. For the purpose of this dissertation, Figure 1.2 is more relevant, as the time scale is better adapted to the Real Options problem.
Figure 1.1: Yearly Prices of Brent and Similar Oils from 1900 to 1957
6
Figure 1.2: Monthly prices of Brent and Similar Oils from January 1957 to April 2003. Source: The IMF’s International Financial Statistics site: http://imfStatistics.org
Eventually, although the oil prices have sensible short-term oscillations, they tend to revert back to a “normal” long-term equilibrium level. Moreover, as expected, there is much evidence that oil prices are mean-reverting. For instance, Pindyck & Rubinfeld [22], using a Dickey-Fuller unit root test, rejected the random walk hypothesis for very long time series (more than 100 years). They do point out, though, that the oil price reversion to a long-run equilibrium level is likely to be slow. Other tests have been performed on the futures markets showing evidence of mean reversion. For more references, see Dias et al [8]. Hence the arrival of routine information over an infinitesimal time interval can be considered as generating only marginal (small) price adjustments, possibly modelled by a continuous diffusion process. If, however, one takes into consideration the arrival of abnormal information, like very important news, one will expect such arrivals to generate a discrete stochastic shock, a jump. This feature can be modelled with Poisson processes. Dixit and Pindyck [11] have presented a model that combines Geometric Brownian Motion with Poisson Jumps, but a model that presents a better economic logic is of course a model which uses Mean-Reversion with Poisson Jumps to model petroleum prices. 7
1.3
Outline of the Dissertation
In the second chapter, I shortly introduce the context we place ourselves in, as well as the two procedures for solving Real Options Problems: Dynamic Programming and Contingent Claims Analysis. I then present the approach developed by Paddock, Siegel & Smith [18] to appraise developed reserves. In the third chapter, I develop the traditional Geometric Brownian Motion model, thereafter denoted by GBM. I show its underlying theory and its numerical implementation. I also discuss the precision of the latter. In the fourth chapter, I develop the more complex Mean-Reversion with Jumps model, thereafter denoted by MRJ. I derive a new differential equation related to this new approach of modelling the underlying and I estimate the parameters of this model. In the fifth chapter, I first discuss the influence of the various model parameters on the option price and threshold of the two models. I then compare the two models. The sixth chapter presents a summary of the results as well as some possible ways in which the MRJ model could be improved. In the Appendixes, details about some calculations are to be found, along with the Matlab code that was used to numerically implement the two models.
8
Chapter 2
The Real Options Approach of Investment under Uncertainty The valuation of offshore leases is an important issue in itself. Firms perform valuations as inputs to their bidding decisions. The government uses valuation to establish presale reservation prices and to study the effect of policy changes on revenues it expects to receive from lease sales. Because the bidding process involves billions of dollars, it is important to obtain accurate valuations. Interestingly, government valuations have tended to underestimate industry bids. Embedded in any approach to valuing petroleum leases is a rule specifying when and if a firm should explore and develop a particular leased property (i.e. exercise its option). The valuation and exploitation of an offshore oil tract can be viewed as part of a multistage investment problem. The first stage involves exploration - seismic tests and drilling to find out how much oil is present and the cost of extracting it. The second stage (which would only occur if the exploration results are favorable) involves development - the installation of the platforms and production wells that are needed to extract the oil. The last stage involves the extraction of the oil over a period of years. The development expenditures convert undeveloped reserves into developed reserves. It is also important to bear in mind that the government subjects the leaseholder to relinquishment requirements that dictate how long a company can wait before beginning exploration and development.
Consider an oilfield discovered in the concession area, and suppose that the oilfield is delineated (small geological uncertainty), so that is not optimal to continue the appraisal phase. The focus of the Real Option models is the development decision. The model identifies the optimal investment strategy and the oilfield value.
2.1
Generalities on Solving Real Option Problems
I now place myself in the case of the investment decision of a firm in a stochastic environment. At any time t the firm can invest in a project yielding an operating
9
profit that depends on a decision variable P (here the market price of oil) ruled by a particular stochastic process. In the literature, real options models are developed through two techniques closely related to each other: dynamic programming and contingent claims analysis. These techniques essentially differ about the assumptions they involve concerning investors, the financial markets and the discount rates used by investors. Due to these differences, the results they yield are similar although not identical. One common characteristic of these models is the implicit assumption that actions are taken instantaneously: the project is started as soon as the investor has decided to invest. Using the stationary property and the Markovian property of the cash flows generated by the project, traditional real options models consider that the investment starts as soon as the decision variable P hits some constant optimal level (see [11]).
2.1.1
Dynamic Programming
The first valuation method used in the real options literature amounts to finding the optimization program of an investor through dynamic programming arguments. This investor is generally risk neutral and has rational expectations. He maximizes the present value of the cash flow generated by the investment through an appropriate timing of the investment decision.
2.1.2
Contingent Claims Analysis
The other valuation method relies on an analogy between real and financial investment decisions. The firm has an option to invest in a project and the value of this option can be found by the usual contingent claim valuation framework. The main assumption of this framework is that capital markets must be complete: there must exist an asset or a dynamic portfolio of assets spanning the stochastic changes in the project value function F (P, t).
2.2
The Value of a Developed Reserve
I denote here the quantity of barrels of oil in the ground (the volume of the petroleum reserve) by B and the market value of one barrel from the reserve by V . I make the common assumption that this value is proportional to oil prices. Of course, this is a strong simplification and one could improve the model by taking a non-constant coefficient of proportionality between V and P . I consider that the operating project value W (P ), that is, the project value after the investment, can be conveniently given by the following equation: W (P ) = B · V (P ) = B · q · P.
(2.1)
In my case V follows the same stochastic process as P . The proportion factor q describes the economic quality of a developed reserve; a higher q implies higher market value for a barrel of oil in the ground (higher expected operational profit in present value from this underlying asset). 10
Now let R be the return over an instant of time to the owner of the developed reserve. The return will have two components: the flow of profit from production and the capital gain on the remaining oil. Due to the fall in pressure in the oilfield resulting from exploitation, the flow of production from a developed reserve is best modelled as an exponential decline and a fraction ω of the oil is produced each year: dB = −ωBdt.
(2.2)
Then, if I take Π as being the after-tax profit from producing and selling a barrel of oil, the return R can be expressed as Rdt = ωBΠdt + d(BV ) = ωBΠdt + BdV − ωV Bdt.
(2.3)
I now have to assume that the rate of return on the developed reserve follows a given stochastic process. I will place myself in the case of the GBM model and assume that the rate of return follows a Brownian motion process: Rdt = µdt + σdz, BV
(2.4)
where the continuous time uncertainty is represented by the volatility σ and the Wiener increment dz, and µ is the risk-adjusted expected rate of return required by a competitive capital market. Combining equations (2.3) and (2.4) gives the following equation for the dynamics of V , the unit value of a developed reserve: dV = (µ − δ)V dt + σV dz,
(2.5)
where δ represents the payout rate from a unit of producing developed reserve: δ=ω
Π−V . V
(2.6)
The economic quality of a developed reserve, linking the per-barrel value of a developed reserve and the market price of oil is usually taken as being equal to one third. For the after-tax profit Π, I have to subtract the per-barrel costs, which are about 30 percent of P , and the corporate tax rate, net of depreciation allowances, which is about 34 percent. Hence, the after-tax profit on a barrel of oil is about 46 percent of the price. Finally, ω is usually around 12%, which leads to: δ = 0.12
0.46P − 0.33P ≈ 0.05. 0.33P
Thus holding a developed reserve is like holding a stock that has dividend yield of about 5 percent. This is illustrated by the comparison presented in Table 2.1 between an American Call Option and an Undeveloped Oil Reserve.
11
Call Option
Undeveloped Reserve
Stock price
Value of developed reserve
Exercise price
Cost of development
Time to expiration
Relinquishment requirement
Volatility of stock price
Volatility of value of developed reserve
Dividend on stock
Net production revenue from developed reserve less depletion
Table 2.1: Comparison between an American Call Option and an Undeveloped Oil Reserve
12
Chapter 3
Model Using Geometric Brownian Motion 3.1
The Value of an Undeveloped Reserve and the Optimal Development Rule
Given equation (2.5) for the value of a developed reserve, I can now determine the value of an undeveloped reserve as well as the optimal timing rule for its development. Since there are a variety of financial instruments that can be used to replicate fluctuations in the price of oil (for example, futures contracts, forward contracts, and the shares of oil companies), spanning clearly holds and contingent claims methods can be used to value an undeveloped reserve. I am going to work with unitary values or per-barrel values. Of course, it is also possible to work with total values. N P V is used to express net present value per barrel: N P V = V (P ) − D = q · P − D,
(3.1)
where D is the per-barrel cost of developing the reserve (that is, the “exercise price” of the option). Since I have chosen V (P ) = q · P where q is a constant, V and P follow the same stochastic process. So similarly to (2.5), dP = (µ − δ)P dt + σP dz.
(3.2)
Now let F (P, t) denote the value of a one-barrel unit of undeveloped reserve. Using equation (2.5), I will construct a risk-free portfolio, determine its expected rate of return and equate that expected rate of return to the risk-free rate of interest. I consider the following portfolio: hold the option to invest (worth F (P, t)), and go short n units of the project. One important issue is whether you can actually hedge the risk in any project related to the oil industry, and if so, how? That is, how in practice do I get short my n units of the project.
In general by investing in the petroleum industry, what I have to fear is that after investing, the oil price decreases too much for my investment to be profitable. Say for 13
instance that I want to buy a futures contract on oil with a 10 years expiration date. Indeed, if I sell in 10 years future time, and that the price goes down, I have hedged my risk. I still face two problems however: • 10 years futures do not exist; • If I have a problem in production, I am selling oil that I do not have. To solve the first problem, what I do is use swaps: I get long in the short term and short in the long term. I buy tomorrow and sell in three months; in three months less one day, I repeat the operation. And with the rule of sums, (xi − xi+1 ) = first − last term, at last, I have bought tomorrow and sold in ten years (of course, there is a discount factor e−rt intervening). This what I wanted: if the price of the barrel goes down, my note compensates for the loss.
Now going back to the portfolio, its value is φ = F (P, t) − nV . This portfolio is dynamic in the sense that if P changes, n may change from one short interval of time to the next. Hence, the composition of the portfolio will be changed. However, over each short interval of length dt, I hold n fixed. An investor holding a long position in the project will demand the risk-adjusted rate of return µ · V , which equals the capital gain α · V (α = µ − δ) plus the dividend stream δ · V . Since the short position includes n units of the project, it will require that n · δ · V dollars are paid out per time period; otherwise no rational investor will enter into the long side of the transaction. Taking this payment into account, the total return from holding the portfolio over a short time interval dt is dF (P, t) − n with
dV dP − δnV dt, dP
dV dP = qdP. dP
Since n is held fixed over this short interval, I do not have any terms involving
(3.3)
(3.4) dn dP .
Now, Itˆo’s lemma applied to F (P, t) with P satisfying (3.2) gives 1 dF (P, t) = Ft (P, t)dt + FP (P, t)dP + FP P (P, t)σ 2 P 2 dt, 2
(3.5)
as (dP )2 = σ 2 P 2 dt, and the total return on the portfolio is 1 FP P (P, t)σ 2 P 2 dt + FP (P, t)dP + Ft (P, t)dt − nqdP − δnqP dt. 2
(3.6)
In order to eliminate the drift in the real world, I take 1 n = FP (P, t), q
14
(3.7)
and the total return on the portfolio becomes 1 2 2 σ P FP P (P, t)dt + Ft (P, t)dt − δP FP (P, t)dt. 2
(3.8)
If the return is risk-free, to avoid arbitrage possibilities, it must equal rφdt = r[F (P, t) − FP (P, t)P ]dt,
(3.9)
with r being the risk-free rate of return. Hence, 1 2 2 σ P FP P (P, t)dt + Ft (P, t)dt − δP FP (P, t)dt = r[F (P, t) − FP (P, t)P ]dt. 2
(3.10)
Dividing by dt and rearranging gives the following differential equation that F (P, t) must satisfy: 1 2 2 σ P FP P + (r − δ)P FP − rF = −Ft . (3.11) 2 In addition, F (P, t) must satisfy the following boundary conditions that are typical for American Call Options: F (0, t) = 0, F (P, T ) = max[V (P ) − D, 0], ∗
∗
F (P , t) = V (P ) − D, ∗
∗
FP (P , t) = VP (P ) = q.
Absorbing barrier at P = 0
(3.12a)
Expiration optimal condition
(3.12b)
Value matching at
P∗
Smooth pasting condition
(3.12c) (3.12d)
Equation (3.12a) is standard for call options and arises from the observation that if P goes to zero, it will stay at zero (this is an implication of the stochastic process (3.2) for P ). Therefore the option to invest will be of no value if V = 0. Let the instant t = T be the expiration of the concession option. At this time the owner has two alternatives: to develop the field immediately or to give up the concession, returning the tract to the National Agency. As the firm will choose the maximum between NPV and zero, condition (3.12b) just says that at expiration, the option to develop will be exercised if V (P ) > D. Conditions (3.12c) and (3.12d) address the early exercise feature of American options and come from consideration of optimal investment. The first one is the value-matching condition: upon investing, the firm receives a net payoff V (P ∗ ) − D, P ∗ being the price at which it is optimal to invest. The last equation (3.12d), known as “smooth pasting condition” (or “high-contact”), is equivalent to the optimum exercise condition, so alternatively the earlier exercise test can be performed (the maximum between the lived option and the payoff V − D). If F (P, t) was not continuous and smooth at the critical exercise point V (P ∗ ), one would do better by exercising at a different point.
15
3.2 3.2.1
Solving the Equation Finite Difference Method
Equation 3.11 can be solved numerically by using a Crank-Nicholson 1 Finite Difference Method (thereafter denoted FDM). The FDM consists in transforming the continuous domain of the P and t state variables by a network or mesh of discrete points (grid). The Partial Differential Equation (thereafter denoted PDE) is converted into a set of finite difference equations, which can be solved by using the appropriate boundary conditions. The solution is reached by proceeding backward through small intervals (∆P )s until finding the optimal path P ∗ (t) to every t. The Crank-Nicholson form of this method corresponds to a specific choice of finite differences for this substitution. I assume the following discretization: F (P, t) ≡ F (i∆P, j∆t) ≡ Fi,j , where 0 ≤ i ≤ m and 0 ≤ j ≤ n with n =
(3.13)
T ∆t .
Now make the substitutions: FP P ≈
1 [Fi+1,j+1 − 2Fi,j+1 + Fi−1,j+1 ] 1 [Fi+1,j − 2Fi,j + Fi−1,j ] + , 2 (∆P )2 2 (∆P )2 FP ≈
1 [Fi+1,j+1 − Fi−1,j+1 ] 1 [Fi+1,j − Fi−1,j ] + , 2 2∆P 2 2∆P [Fi,j+1 − Fi,j ] Ft ≈ , ∆t F = Fi,j .
(3.14a) (3.14b) (3.14c) (3.14d)
I use the “forward-difference” for the t variable. Applying these approximations to the PDE and the boundary conditions, I get the following difference equation (assuming σ2 that |r−δ| = O(1), which is consistent with the case I study. Otherwise I would have used an upwind scheme): h1 F 1 2 1 Fi+1,j − 2Fi,j + Fi−1,j i i+1,j+1 − 2Fi,j+1 + Fi−1,j+1 σ (i∆P )2 + 2 2 (∆P )2 2 (∆P )2 h1 F 1 Fi+1,j − Fi−1,j i i+1,j+1 − Fi−1,j+1 + + (r − δ)i∆P 2 2∆P 2 2∆P Fi,j+1 − Fi,j − rFi,j + = 0, ∆t 1
This is also spelt Nicolson. Apologies if I wrote it the wrong way round.
16
i 1 2 2h σ i (Fi+1,j+1 − 2Fi,j+1 + Fi−1,j+1 ) + (Fi+1,j − 2Fi,j + Fi−1,j ) 4 h i 1 + (r − δ)i (Fi+1,j+1 − Fi−1,j+1 ) + (Fi+1,j − Fi−1,j ) 4 Fi,j+1 − Fi,j − rFi,j + = 0, ∆t and finally, Ai Fi−1,j + Bi Fi,j + Ci Fi+1,j = ai Fi−1,j+1 + bi Fi,j+1 + ci Fi+1,j+1 , where
¢ − σ 2 i2 + (r − δ)i ∆t, ¡ 22 ¢ Bi = 1 + σ 2i + r ∆t, ¡ ¢ Ci = − 14 σ 2 i2 + (r − δ)i ∆t, ¡ ¢ ai = 14 σ 2 i2 − (r − δ)i ∆t,
Ai =
1 4
¡
2 2
bi = 1 − σ 2i ∆t, ¡ ¢ ci = 41 σ 2 i2 + (r − δ)i ∆t.
3.2.2
(3.15) (3.16a) (3.16b) (3.16c) (3.16d) (3.16e) (3.16f)
At the Limits
We now study the boundary conditions of the FDM. Firstly, there is no problem for i = 0, as A0 = a0 = 0. Secondly, the upper space limit is approximated by Pm , such P → ∞. Hence, if FP P = 0, Fm+1 = 2Fm − Fm−1 , that Pm = m∆P and FP P → 0 as D which allows us to eliminate Fm+1 from equation 3.15. The new equation at the limit is thus: Am Fm−1,j + Bm Fm,j + Cm (2Fm,j − Fm−1,j−1 ) = am Fm−1,j+1 + bm Fm,j+1 + cm (2Fm,j+1 − Fm−1,j+1 ), or Am Fm−1,j + B m Fm,j = am Fm−1,j+1 + bm Fm,j+1 ,
(3.17)
with Am = Am − Cm B m = Bm + 2Cm am = am − cm bm = bm + 2cm . An alternate method would be to consider the new differential equation when S → ∞, with FP P ∼ 0: ½ Ft + (r − δ)P FP − rF = 0, F (P, T ) = qP − D, 17
which gives F ∼ qP e−δ(T −t) − De−r(T −t) , and Fm,j = qPm e−δ(T −j∆t) − De−r(T −j∆t) . For the strike price, I choose to place D/q between two oil price steps, so that the oil price derivatives of F remain bounded on the grid.
Figure 3.1: Grid disposition
3.2.3
Projected Gauss-Seidel Solution Scheme
Let us now define: Zi = ai Fi−1,j+1 + bi Fi,j+1 + ci Fi+1,j+1 , so that equation 3.15 becomes: Ai Fi−1,j + Bi Fi,j + Ci Fi+1,j = Zi .
(3.18)
Similarly to the method presented by Wilmott, Dewynne and Howison [25], I will use the projected Gauss-Seidel method to solve the scheme. The algorithm for finding the solution is iterative. At each time step, I start with an initial guess U(0) = c, c being the final value obtained at the previous time step, and the payoff function at the initial step. At each iteration, I create the vector (k+1)
, U1,j
(k)
(k)
U(k+1) = (U0,j from the current vector
(k+1)
(k+1)
, . . . , Um,j ), (k)
U(k) = (U0,j , U1,j , . . . , Um,j ),
(3.19) (3.20)
using the following process: for each i = 0, 1, . . . , m, I sequentially calculate: ¡ ¢ ˜ (k+1) = 1 Zi − Ai U (k+1) − Ci U (k) , U i,j i−1,j i+1,j Bi
(3.21)
and compare it with the payoff w: (k+1)
Ui,j
(k+1)
˜ = max(wi , U i,j
).
(3.22)
This procedure implies the use of a loop. The stopping condition for this loop are either ||U(k+1) − U(k) || < ²1 ||Z − MU(k) || < ²2 18
the change, or the residual.
The residual, for an american option, is X ¡ (k) (k) (k) ¢2 Zi − Ai Ui−1,j − Bi Ui,j − Ci Ui+1,j . Un >payoff (k)
I then take U(k+1) as the solution, Ui,j = Fi,j+1 , and reset the initial guess c = Fj+1 .
3.3
Precision of the Scheme
For Matlab Code, see Appendix B.1. I used the following procedure to evaluate the precision of the numerical scheme trough a graphical method. If there is no dividend (δ here) for the option I want to price, the price of an American Call is the same as the price of an European Call. Hence, I can determine the precision of my scheme by comparing the values I obtain with the exact values given by Black-Scholes. I will perform that verification on the option price, but also on the first of the “Greeks”, ∆ = ∂F ∂P . The shapes of the functions are as shown in Figure 3.2. The parameters for this figure are T = 10 years, r = 0.05, δ = 0, σ = 0.22, D = 5.25, Pm = 30, dP = 0.5, dt = 0.005, q=1/3 and the authorized variation in residual is 10−6 . All the calculations presented here have been preformed with the same parameters, unless otherwise stated.
0.35
6
0.3
5
0.25
4
0.2 Delta
Value of the concession (Option)(US$/bbl)
7
3
0.15
2
0.1
1
0.05
0 10
0 10
8
8
30 25
6 10 0
15 10
2
5 0
20
4
15 2
Time (years)
30 25
6
20
4
Time (years)
Oil Prices (US$/bbl)
(a) Option value over 10 years
5 0
0
Oil Prices (US$/bbl)
(b) Value of ∆ over 10 years
Figure 3.2: Option value and corresponding ∆ obtained with the previous method One can see in Figure 3.3 that the error over the whole price range is of order 10−3 , i.e. 0.1%. The error in ∆ is of same order (Figure 3.4). The error tends to get bigger ∆t at the limits of the mesh. This is normal if one considers the relevant α = 21 σ 2 P 2 (∆P . )2 This α is not constant over the grid and tends to get big at the upper P boundary. Moreover, the error tends to fluctuate around the strike price. This is particularly visible when the exercise price is at a grid-point. This situation should be avoided as much as possible, as it leads to discontinuities in the derivatives of the option value. The best position for the strike price is right in the middle of two oil price grid-points, as I took it. 19
−3
x 10 0.5 Relative difference with Black−Scholes
Difference with Black−Scholes
0.005 0 −0.005 −0.01 −0.015 −0.02 −0.025
0 −0.5 −1 −1.5 −2 −2.5 −3 −3.5 −4 10
−0.03 10 8
8
30
30
25
6
25
6
20
4 10
2 0
15 10
2
5 0
Time (years)
20
4
15
(a) Difference between the Black-Scholes values and the calculated values
5 0
Time (years)
Oil Prices (US$/bbl)
0
Oil Prices (US$/bbl)
(b) Relative difference between the BlackScholes values and the calculated values
Figure 3.3: Comparison between the Black-Scholes values and the option values obtained with the previous method
−3
x 10
−3
x 10
Relative difference with Black−Scholes Delta
Difference with Black−Scholes Delta
0.5 0 −0.5 −1 −1.5 −2 −2.5 10 8
−2
−4
−6
−8 10
25
30
10 0
15 10
2
5 0
20
4
15 2
25
6
20
4
Time (years)
0
8
30 6
2
Time (years)
Oil Prices (US$/bbl)
(a) Difference between the Black-Scholes ∆ values and the calculated values
5 0
0
Oil Prices (US$/bbl)
(b) Relative difference between the BlackScholes ∆ values and the calculated values
Figure 3.4: Comparison between the Black-Scholes ∆ values and the ∆ values obtained with the previous method
3.4
Influence of the Numerical Scheme Parameters
The next question is: what happens to the error when the grid size, the oil price axis truncation value, or the allowed error in the residual change? In each of the following figures, all the parameters dP , Pm , dt, T , D and the authorized variation in the residual remain the same, apart from the parameter mentioned in the caption of the corresponding figure. While increasing Pm can prove a good idea as it increases precision (Figure 3.5), one can notice in Figure 3.6 that a finer grid does not necessarily lead to a better result. This is due to the presence of the term in σ, the second oil price derivative. The iterative solver does not converge as well when one doubles the number of points, as the iteration multiplies the error. The solution is to decrease the authorized variation in the residual. Eventually, good precision can be obtained by taking the parameters of Figure 3.7.
20
−4
x 10
−3
x 10
3 Relative difference with Black−Scholes
1
Difference with Black−Scholes
0.5 0 −0.5 −1 −1.5 −2 −2.5
2 1 0 −1 −2 −3 10
−3 10 8
8
50 6
50 6
40
40
30
4
30
4
20
2
2
10 0
Time (years)
0
20
(a) Evolution when Pm = 45
10 0
Time (years)
Oil Prices (US$/bbl)
0
Oil Prices (US$/bbl)
(b) Relative evolution when Pm = 45
Figure 3.5: Evolution when Pm change
−3
x 10 0.5 Relative difference with Black−Scholes
Difference with Black−Scholes
0.005 0 −0.005 −0.01 −0.015 −0.02 −0.025
0 −0.5 −1 −1.5 −2 −2.5 −3 −3.5 −4 10
−0.03 10 8
8
30
30
25
6
25
6
20
4 10
2 0
15 10
2
5 0
Time (years)
20
4
15
(a) Evolution when dP =0.25 and dt=0.001
5 0
Time (years)
Oil Prices (US$/bbl)
0
Oil Prices (US$/bbl)
(b) Relative evolution when dP =0.25 and dt=0.001
Figure 3.6: Evolution when dP and dt change
−5
−4
x 10
x 10
Relative difference with Black−Scholes
Difference with Black−Scholes
5
0
−5
−10
−15
−20 10
4 2 0 −2 −4 −6 −8 10
8
8
60 50
6 20 0
30 20
2
10 0
40
4
30 2
Time (years)
60 50
6
40
4
Time (years)
Oil Prices (US$/bbl)
(a) Variation with Black-Scholes
10 0
0
Oil Prices (US$/bbl)
(b) Percentage variation
Figure 3.7: Difference with Black-Scholes when T = 10 years, D = 5.125, dP = 0.25, dt = 0.001, Pm = 60 and the authorized variation in residual is 10−20
21
Chapter 4
Model Using Mean Reversion with Jumps 4.1
Stochastic Process
Again, let P be the spot price of a barrel of oil. As I said before, the prices most of the time change continuously as a mean-reverting process, but sometimes they change discretely by jumps. In this way, the oil prices follow the stochastic differential equation (SDE): £ ¤ dP = η(P − P ) − λk dt + σdz + dq, P ½ 0, with probability 1 − λdt, dq = (4.1) Φ − 1, with probability λdt, k = E(Φ − 1), where E(Φ − 1) is the expected value of Φ − 1 with respect to the density of Φ. Equation (4.1), which describes the rate of variation of oil prices (dP/P ), has three terms on the right side. • The first term is the mean-reverting drift: the petroleum price has a tendency to go back to the long-run equilibrium mean P with a reversion speed η. It is compensated by λk for the Poisson jump expected value. • The second term presents the continuous time uncertainty which is represented by the volatility σ and the Wiener increment dz. • The third term is the jump term, with the Poisson arrival parameter λ (there is a probability λdt that a discrete jump will occur) and jump size Φ. The “minus one” appears as a matter of convention, as I define the probability density function (pdf) of Φ only over positive values and then translate the jump size.
One can determine the frequency and size of the jumps in the real world, if one sorts out the data on the oil price. Figure 4.1 shows the distribution of the jumps. 22
Figure 4.1: Real world jumps distribution: how many jumps occurred in 46 years and what was their size
Here is the code used to sort out the data: jump=zeros(1,34); for i=1:length(price)-1 if price(i+1)/price(i)=0.1 & price(i+1)/price(i)=0.2 & price(i+1)/price(i)=3.2 & price(i+1)/price(i)=3.3 jump(34)=jump(34)+1; end end
I chose to consider any price change of over 20% in a month as a jump. As one can see in Figure 4.1, in 46 years (I have taken only the period over which I had monthly data, i.e. between 1957 and 2003), 4 monthly jumps down and 6 jumps up have occurred. In 23
reality, however, some of these jumps were consecutive, therefore they are concatenated and we are left with 3 jumps down and 4 jumps up. Of course, as the market tends to absorb any jump in a period going from one to three month, I could have defined a jump as taking place over a period of more than one month. Besides, as I only consider large jumps here, this model is not adapted for trading in commodities markets. It is clearly designed for projects of exploration and production in petroleum. The jumps-up quoted here have taken place during the Yom Kippur war and Arabian oil embargo in 1973/1974, during the Iran revolution and Iran/Iraq war in 1979/1980, during the Kuwait invasion by Iraq in 1990, and in 1999 as the OPEC and its allies created a supply shock. Jumps-down happened in 1957, in 1986 due to Saudi-Arabia price war and in 1991 after the Iraq defeat. One notices that the jumps have random size. Hence I chose to model Φ as having a particular probability distribution with mean k+1, represented by two truncated-normal distributions: one normal distribution for the jump-up and one for the jump-down (see Figure 4.2 below). In case of jump, this abnormal movement has the same chances of being up or down. There is not enough data available in order to determine the exact parameters of both jumps. Hence, for reasons of simplicity, I defined the probability density function of Φ as: pdf(Φ) =
Φ−m1 2 Φ−m2 2 1 1 1 1 −1( ) −1( ) √ e 2 σ1 √ e 2 σ2 , + 2 σ1 2π 2 σ2 2π
(4.2)
with m1 = 21 , m2 = 2, σ1 = 18 , σ2 = 27 . Hence, E(Φ) = V (Φ) = and k =
m1 +m2 2
m1 + m2 , 2 σ12 + σ22 (m1 + m2 )2 + , 2 4
(4.3) (4.4)
− 1.
Figure 4.2 shows that the exact size of each jump is uncertain. The same figure indicates that, in case of jump-up, the price is expected to double, whereas in case of jump-down the price is expected to drop by half. I am interested in large jumps (as shown in the figure) but with low frequency (rare events). This probability density function presents the drawback that the average jump size is bigger than 1. Thus over a long enough period, jumps will tend to increase the oil price, although the probability of jump up is the same as the probability of jump down. Hence, one could improve the model by defining a different jump distribution which eliminates this feature. In addition, it is clear that Φ is not only positive, as the tails of the pdf are non-zero at infinity. But as a simplification, the pdf in considered as being bounded at 0.
24
Figure 4.2: Random jumps distribution: probability density function of Φ
4.2
Optimization Problem
Again, the instant t = T is the expiration of the concession option. As with the GBM model, it is necessary to derive both the value of the concession (the value of the option to invest) F (P, t), and the optimal decision rule (the threshold) P ∗ (t ≤ T ). The decisions are to develop, or to wait, or even to give up. The solution procedure can be viewed as a maximization problem under uncertainty. I use the Bellman-dynamic programming framework (see Dixit & Pindyck, [11], chapter 4) to solve the stochastic optimal control problem. I want to maximize the value of the concession, the option F (P, t), seeking the instant when the price reaches a level P ∗ (the threshold) in which one type of action is optimal. Bellman’s Principle of Optimality states that an optimal policy has the property that, whatever the initial action, the remaining choices constitute an optimal policy with respect to the subproblem starting at the state that results from the initial actions. In short, that if at each step one chooses the optimal path, it leads to a globally optimal path. This, of course, is not always true as shown by the Fermat principle. Hence here, the Bellman equation, also called fundamental equation of optimality, is: ¤ ½ £ ¾ V (P ) − D, E[F (P + dP, t + dt)e−ρdt ] , for all t < T F (P, t) = max , (4.5) [V(P) - D, 0], for t = T P ∗ (t) where ρ is an exogenous risk-adjusted discount rate and not necessarily a Capital Asset Pricing Model (CAPM) risk-adjusted discount rate for the underlying asset which would require a complete market. 25
Indeed, CAPM looks at risk and rates of return and compares them to the overall market. If I use CAPM I have to assume that most investors want to avoid risk (risk averse), and those who do take risks, expect to be rewarded. µ = r + (rmarket − r)βxm ,
(4.6)
where µ is the risk-adjusted discount rate, r is the risk-free discount rate, rmarket is the market discount rate and βxm is the coefficient of correlation between returns on the particular asset x and the whole market portfolio m. In case of incomplete markets, the risk-adjusted discount rate can be determined: 1. by “market-estimate”; 2. by choosing a risk-premium (hence getting the discount rate), specifying a utility function for the investor; 3. by simply choosing an arbitrary exogenous value.
If I consider here the jump-risk as being systematic (correlated with the market portfolio), it is not possible to build a riskless portfolio. The market is not complete for this model with non-diversified jump risk, and one simply chooses an exogenous discount rate. Using the Bellman equation and Itˆo’s Lemma, it is possible to build a partial differentialdifference equation (PDE). I know:
and
1 ∂2 ∂F ∂F dP + dt, dF = dP 2 + ∂P 2 ∂P 2 ∂t ¡ ¢ 2 dP 2 = η(P − P ) − λk P 2 dt2 + σ 2 P 2 dz 2 + P 2 dq 2 + . . . dt · dz + . . . dt · dq + . . . dz · dq.
One knows that according to the definition of a Wiener process, E(dz) = 0 and E(dz 2 ) = dt. One can also neglect all the terms whose expected value will be smaller than dt, i.e. the terms in dq 2 , dt · dz, dt · dq, dz · dq. Now from 4.5 is a term in E[P FP dq] remaining. One knows that: ½ 0 with probability 1 − λdt, P FP dq = (Φ − 1)P FP with probability λdt.
(4.7)
Hence, E[P FP dq] =λdtE[FP (P, t)(P Φ − P )] =λdtE[F (P Φ, t) − F (P, t)], 26
(4.8) (4.9)
and I get the following PDE: © ª £ ¤ 1 2 2 σ P FP P + η(P¯ − P ) − λE[Φ − 1] P FP + Ft + λE F (P Φ, t) − F (P, t) = ρF, (4.10) 2 with the same boundary conditions as for the GBM model. Equation (4.10) is a PDE of parabolic type and is solved using the numerical method of finite differences in the explicit form (see Section (4.3)). The parameters estimation is discussed in Section (4.6).
4.3
Explicit Finite Difference Method
To solve (4.10) I use the finite difference method in the explicit form. Assuming the same discretization as previously, the partial derivatives are here approximated by the differences: FP P ≈ [Fi+1,j − 2Fi,j + Fi−1,j ]/(∆P )2 ,
(4.11a)
FP ≈ [Fi+1,j − Fi−1,j ]/2∆P,
(4.11b)
Ft ≈ [Fi,j − Fi,j−1 ]/∆t.
(4.11c)
I use the “central-difference” approximation for the P variable and the “backwarddifference” for the t variable. Applying these approximations to the PDE and the boundary conditions, I get the following difference equation:
2 Fi+1,j 1 2 2 σ (i∆P )
− 2Fi,j + Fi−1,j (∆P )2 ½ ¾ ¡ ¢ Fi+1,j − Fi−1,j ¯ + η P − (i∆P ) − λk (i∆P ) 2∆P £ ¤ Fi,j − Fi,j−1 + + λE FiΦ,j − Fi,j = ρFi,j−1 , ∆t
£ ¤ £ ¤ and since E FiΦ,j − Fi,j = E FiΦ,j − Fi,j , I get: £ ¤ Fi,j−1 = p+ Fi+1,j + p0 Fi,j + p− Fi−1,j + pjump E Fi.Φ,j , ˜
27
(4.12)
where
" # 2 i2 2 .η.∆P ∆t σ i.(η.P ) i i.λ.k p+ = + − − , ∆t.ρ + 1 2 2 2 2 " # ∆t 1 2 2 0 −σ i −λ , p = ∆t.ρ + 1 ∆t " # ∆t σ 2 i2 i.(η.P ) i2 .η.∆P i.λ.k − p = − + + , ∆t.ρ + 1 2 2 2 2 ∆t λ, ∆t.ρ + 1 £ ¤ k =E Φ−1 .
pjump =
(4.13a) (4.13b) (4.13c) (4.13d) (4.13e)
The boundary conditions are the same as for (3.12):
4.4
F0,j = 0,
(4.14a)
Fi,n = max[qi∆P − D, 0],
(4.14b)
∗
Fi∗ ,j = qi ∆P − D,
(4.14c)
Fi∗ +1,j − Fi∗ −1,j = 2q∆P.
(4.14d)
Discretization of the Jump Term
The last term in equation (4.12) represents the jump term contribution to the real option value. Jumps can occur with probability pjump , and the expectation term is equivalent to a numerical integration of the options value over the new oil prices after the jump occurrence (i.∆P.Φ). To solve the numerical integration I perform a discretization over the jump size distribution (Φ). Besides, one can test that the probability of a jump tends toward one, by testing the jump with this code: dPhi=1e-4; L=3/dPhi
% Step size for discretisation of PDF of Phi % Number of steps
A_Phi=0; for l = 1:L Phil = (l-1/2)*dPhi; prob = dPhi*pdf_phi(Phil); A_Phi=prob+A_Phi; end A_Phi
28
Figure 4.3: The jump is discretized: the Φ axis is divided in L small partitions, with LdΦ = 3. The central value Φl−1/2 of each partition is associated to the corresponding probability from the respective partition, i.e. p(Φl ) = dΦf (Φl−1/2 )
For dΦ=0.5, I obtain AΦ =0.71435, but for dΦ=0.1, AΦ =0.99988, as expected. Now E[FiΦ,j ˜ ]=
L X
p(Φl )F (iΦl ).
(4.15)
l=1
For any i in the interval 0 to m, iΦl is a real number. Hence, approximation or interpolation methods have to be used to better approximate the new oil price after a jump (i.∆P.Φ) to some point (i.∆P ) inside the grid, while taking into account the truncation of the grid at Pm . Essentially, the higher the number of the partitions the better the accuracy of the result. This numerical procedure can handle any kind of jump size distribution.
4.5
Stability of the Explicit FDM, Computation and Comments
In order to obtain stability, the discrete steps ∆P and ∆t must be chosen so that all the coefficients (p+ , p0 , p− ) from equation (4.12) are positive for any value inside the grid. Therefore the stability of the explicit FDM determines the choice of ∆P and ∆t. This condition is very demanding. The coefficient that usually poses problem is p0 . 29
Small enough time steps have to be taken in comparison to the number of oil price steps so that the first term in p0 is bigger than the second term. This implies the need for a fine time grid and a not so fine oil price grid, which would result in a very long computation time and a poor precision in oil prices, especially penalizing for the threshold value. It is furthermore not possible to calculate the option value for small volatility σ with this numerical scheme. Indeed, for the scheme to be consistent, one needs it to be diffusion dominated (i.e. 21 σ 2 À |η(P¯ − P ) − λE[Φ − 1]|) so that the drift term in FP does not take over the term in FP P . As for the GBM case, this feature is true here and if it was not, one could implement an upwind scheme, giving better stability but loosing precision. Finally, one could ask why I did not use the same numerical scheme as for the GBM model, i.e. Crank-Nicholson. This is because it would raise the issue of whether or not to work out the expectation term as an average over the two times and then working with non tri-diagonal matrices.
4.6
Parameter Estimation
Consider the following format for my model: dP = η(P − P )dt + σdz + (Φ − 1)dq 0 − λkdt P
(4.16)
where dq 0 = 1 with probability λdt and zero otherwise. The last two terms are the compensated-Poisson jump components (k = E[Φ − 1]). Now, using Itˆo’s Lemma for the jump-diffusion case and the function x = log P (so that ∂x/∂t = 0; ∂x/∂P = 1/P ; ∂ 2 x/∂P 2 = −1/P 2 ) I get after some simplification:
dx = d(log P ) (4.17) " Ã ! # P log P σ2 = η P− − log P dt + σdz + (Φ − 1)dq 0 − kλdt. log P P 2η Comparing it with the simple equation dx = η ∗ (x − x)dt + σ ∗ dz + jump-terms,
(4.18)
I find the parameters: P , log P Ã ! log P σ2 P− , P 2η
η∗ = η x =
σ ∗ = σ.
(4.19) (4.20) (4.21)
30
From these equations one gets the values of η, P and σ.
log P , P σ = σ∗, P σ2 P = x+ . log P 2η η = η∗
(4.22) (4.23) (4.24)
One could see here that there is a drawback with this method, as η and P¯ depend on the oil price. Thus we must determine a representative oil price. I chose to use the mean of the price from 1957 to 2003. Nonetheless, I am now looking for the mean-reverting parameters (x, η ∗ , σ ∗ ). For the estimation job, I put out the sample data referring to jumps, in order to estimate only the mean-reverting parameters. I take the discrete time version of the equation 4.18. As pointed out in Dixit & Pindyck [11], it is a first order autoregressive process. Specifically, equation (4.18) is the limiting case as ∆t → 0 of the following Autoregressive Process (see Appendix A.1): ∗
∗
xt − xt−1 = x(1 − e−η ) + (e−η − 1)xt−1 + ²t ,
(4.25)
where ²t is normally distributed with mean zero and standard deviation σ² , and σ²2 =
σ∗2 ∗ (1 − e−2η ). ∗ 2η
(4.26)
Hence one can estimate the parameters by running the regression xt − xt−1 = a + bxt−1 + ²t
(4.27)
and then calculating a x = − , b η ∗ = − log(1 + b) and s 2 log(1 + b) σ ∗ = σ² . (1 + b)2 − 1 where σ² is the standard error of the regression.
31
(4.28) (4.29) (4.30)
These operations are performed very easily in Matlab: price; % price history excluding the jumps x=log(price); for i=1:length(price)-1 delta_x(i)=x(i+1)-x(i); x_short(i)=x(i); end p = polyfit(x_short, delta_x, 1);% pop = polyval(p, x_short);% res = delta_x - pop;% a = p(2) b = p(1) disp(’mean of residual, should be zero =’);% disp(mean(res)) sigma_eps = var(res)% eta_star = -log(1+b)% x_bar = -a/b% sigma_star = sigma_eps*sqrt(2*log(1+b)/((1+b)^2-1)) %%%%%%% eta = 12*eta_star * log(mean(price))/mean(price)% sigma = 12*sigma_star% P_bar = mean(price)*x_bar/log(mean(price)) + sigma^2/(2*eta)
I finally obtain the parameters I was looking for: a (monthly)
1+b (monthly)
σ (% p.a.)
(($/bbl)−1 .year−1
P¯ ($/bbl)
0.033587
0.989109
18.11
0.031735
20.1189
η
Table 4.1: Estimated parameters for the Mean-Reverting model
I am now ready to implement the scheme into Matlab. For the calculations, I took into consideration the values generally adopted in the literature. For the volatility, Paddock, Siegel and Smith [18] mentioned that estimates based on data over the past 30 years would put σ at about 0.15, but that industry forecasts might be somewhat higher. Dias and Rocha [8] have made several more complex tests in order to estimate this parameter. 32
They obtained values going from 14.6 to 32.2. I decided to follow their lead and took σ = 22% p.a. For the long-run equilibrium price P¯ , Dias and Rocha [8] point out that it can be bounded by two values: • the OPEC long-run price goal of about 22 US$/bbl; • the long-run marginal cost from non-OPEC countries, 19 US$/bbl. Taking into account that most of the values usually used in the various models are between 18 and 22, I chose to set P¯ = 20 here. λ was calculated accordingly to the data: three jumps down and four jumps up occurred in 46 years (from 1957 to 2003). Hence, λ=0.15 p.a. Now, the exogenous discount rate ρ is probably the most difficult parameter to estimate. As stressed earlier, the systematic jump risk assumed here does not allow for hedging all the risk (see section 4.2). Hence, I used the official discount rate that reports the present value of proven oil reserves to stock market investors, ρ = 10 p.a. Besides, there is one practical “market-way” of relating ρ to the net convenience yield δ. I take δ as being similar to a dividend, like for the GBM model. In this case, it can be calculated as being the difference between the discount rate (total required return) and the expected capital gain (E(dP/P )). Hence here, δ(t) = ρ(t) − η(P¯ − P (t)).
(4.31)
The value for the investment D, per-barrel cost of developing the reserve, was taken to be a representative value for offshore oilfields, D=5 US$/bbl. Finally, the values adopted for the base case are the following: • Time to expiration T = 5 years; • Exogenous discount rate ρ = 10% p.a.; • Long-run average oil price P¯ = 20 US$/bbl; • Annual frequency of jumps λ = 0.15 p.a.; • Volatility of the diffusion process σ = 22% p.a.; • Reversion speed η = 0.03 (US$/bbl)−1 .year−1 ; • Investment D=5 US$/bbl.
4.7
Influence of the Numerical Scheme Parameters
The Matlab Code can be found in Appendix B.2. The first point to notice in Figure 4.4 is that the truncation value of the oil price grid 33
Pm does not have as much effect on the option price as for the GBM model. This only holds as long as Pm remains above a certain value, which was found to be Pm = 40 in this case. The same remark can be made about the discretization of the jump. The step size dΦ does not influence the result much, as long as it remains above a certain value. The limit in this case was found to be dΦ = 0.25. In this section, the parameters have been taken as being dP = 0.5, dt = 5 · 10−4 , Pm = 45, T = 5 years, D = 5.25, dΦ = 0.05 and q = 1/3 unless otherwise stated. 3.5
22
3
21
2.5
20 Threshold (US$/bbl)
Value of the concession (Option)(US$/bbl)
dΦ=0.1
2
1.5
1
5
18
16
Payoff
0
19
17
Reference Option Value
0.5
0
dΦ=0.05
10 15 Oil Prices (US$/bbl)
20
15
25
0
(a) Option price value when Pm = 90
0.5
1
1.5
2
2.5 Time (years)
3
3.5
4
4.5
5
(b) Threshold evolution when dΦ=0.1
Figure 4.4: Influence of Pm and dΦ on the MRJ model Though Pm and dΦ did not matter much, the grid size did, as shown in Figure 4.5, where one can observe the effect of a change in dP over the option price and the threshold. 22
10
dP=0.5 9
21 dP=0.25
20
7
Threshold (US$/bbl)
Value of the concession (Option)(US$/bbl)
8
6
5
4
3
19
18
17 dP=0.25
2 dP=0.5
16
1
0
0
5
10
15
20 25 Oil Prices (US$/bbl)
30
35
40
15
45
(a) Option price evolution when dP = 0.25
0
0.5
1
1.5
2
2.5 Time (years)
3
3.5
4
4.5
(b) Threshold evolution when dP =0.25
Figure 4.5: Influence of the grid size on the MRJ model
34
5
Chapter 5
Models Results and Comparison 5.1 5.1.1
Traditional Model Influence of the GBM Model Parameters
What happens to the option price and the threshold when parameters such as r, δ, σ or q vary. The parameters for the numerical scheme were the same as for Figure 3.7. The base case to which all the values are compared is r = 5%, δ = 5%, σ = 22%, q = 1/3. It is to be seen in Figure 5.1 that the interest rate increases both the option value and the threshold. In order to understand why, imagine that I leave the money to pay an eventual option exercise in the bank. If the interest rate increases, I have less incentive to exercise the option using that money; or in terms of present value, exercising the option late instead of earlier is more attractive because the present value of the exercise price (the investment cost) is lower for the delay strategy. The dividend (convenience) yield δ has an opposite effect compared with the interest rate, as shown on Figure 5.2. Dividend yield is like an opportunity cost of holding the underlying asset. So, only the one who owns the underlying asset earns the flow of benefits associated with it (dividend yield). Higher δ means a lower value on waiting to invest, therefore the threshold is lower and the live option has lower value. Figure 5.3 shows that the volatility increases the option value and the threshold. As uncertainty increases, delaying the investment proves to be a better strategy.S Figure 5.4 shows that a greater economic quality of the reserve leads to greater option value but decreases the threshold. All other parameters remaining identical, the project has greater value and it is more interesting to invest if q is bigger. Finally, the option price increases with r, σ and q and decreases with δ. Similarly, the threshold increases with r, σ but decreases this time with q and δ. Moreover, as expected, higher time to expiration means higher option value. But it is well known that there is an upper bound for this case. The perpetual American option (with infinite time to expiration, as an option to develop a land) has a known analytic solution that is the upper bound for the option with the expiration time. See Dixit & Pindyck [11].
35
15
45
r=10%
10
35 Threshold (US$/bbl)
Value of the concession (Option)(US$/bbl)
40
5
30
r=5%
25
r=10% 20
r=5% 0
0
10
20
30 Oil Prices (US$/bbl)
40
50
15
60
(a) Option price evolution when r increases
0
1
2
3
4
5 Time (years)
6
7
8
9
10
(b) Threshold evolution when r increases
Figure 5.1: Option price and threshold evolution when r changes
30
5
δ=5% 4.5
25 Threshold (US$/bbl)
Value of the concession (Option)(US$/bbl)
4
3.5
3
2.5
2
δ=10%
20 1.5 δ=5%
1
δ=10%
0.5
Payoff 0
0
5
10
15 Oil Prices (US$/bbl)
20
25
15
30
(a) Option price evolution when δ increases
0
1
2
3
4
5 Time (years)
6
7
8
9
10
(b) Threshold evolution when δ increases
Figure 5.2: Option price and threshold evolution when δ changes
15
45
σ=35%
10
35 Threshold (US$/bbl)
Value of the concession (Option)(US$/bbl)
40
5
30
σ=22%
25 σ=35% 20 σ=22%
0
0
10
20
30 Oil Prices (US$/bbl)
40
50
15
60
(a) Option price evolution when σ increases
0
1
2
3
4
5 Time (years)
6
7
8
9
(b) Threshold evolution when σ increases
Figure 5.3: Option price and threshold evolution when σ changes
36
10
15
30
q=1/3
10 Threshold (US$/bbl)
Value of the concession (Option)(US$/bbl)
25
Payoff when q=2/3
20
q=2/3
15
5
10
q=2/3 q=1/3 Payoff when q=1/3 0
0
5
10
15 Oil Prices (US$/bbl)
20
25
5
30
(a) Option price evolution when q increases
0
1
2
3
4
5 Time (years)
6
7
8
9
10
(b) Threshold evolution when q increases
Figure 5.4: Option price and threshold evolution when q changes
5.1.2
Comparison with other Results
As I said before, the GBM model was first introduced by Paddock, Siegel and Smith [18] in 1987. As a verification, I made the very same calculations. My results are presented on Figure 5.5 and Table 5.1. They are very similar (until the fourth digit) to those obtained at that time. 2
1.9 σ=0.25
1.8
Hitting Boundary
1.7
1.6
1.5
1.4
σ=0.142
1.3
1.2
1.1
1
0
5
10
15 20 Time (years)
25
30
35
Figure 5.5: Critical value for development of oil reserves (Shows V (P ∗ )/D for δ = 0.04 and r = 0.0125) Figure 5.5 shows the ratio V (P ∗ )/D (the threshold) as a function of the number of years to expiration for two values of σ. It is interesting to note that at expiration, V (P ∗ )/D = 1 so that the standard NPV rule applies. When the time to expiration increases, this ratio grows and reaches an asymptote that can be greater than two. Table 5.1 shows the value of the development option (value of an undeveloped reserve, i.e. F (V, t)/D) per dollar of development cost. When V (P )/D is less than one, the value of the reserve is less than the development cost. Hence, even a NPV approach would tell us not to develop the reserve. But if V (P )/D is greater than one, the NPV standard criterion tells us to develop the reserve, while Figure 5.5 shows us that a ratio much bigger than one should be expected in order for that decision to be optimal.
37
V (P )/D 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15
T =5 0.0181903 0.0277936 0.0405532 0.0569072 0.0772516 0.1019398 0.1312860 0.1655732
σ = 0.142 T = 10 0.0286743 0.0396675 0.0533608 0.0700809 0.0901562 0.1139179 0.1417014 0.1738481
T = 15 0.0338309 0.0452272 0.0591480 0.0758961 0.0957849 0.1191380 0.1462903 0.1775881
σ = 0.25 T =5 T = 10 0.0750321 0.1057525 0.0929930 0.1250862 0.1131118 0.1461212 0.1353737 0.1688461 0.1597523 0.1932472 0.1862127 0.2193099 0.2147147 0.2470190 0.2452145 0.2763592
Table 5.1: Some option values for various parameters, calculated with the GBM model. Precision of 10−4 .
5.1.3
How to Use the Results
It is now time to ask how does one use the results I have obtained, the value of the Real Option, to price an undeveloped reserve. For that purpose, I will use an example introduced by paddock, Siegel and Smith [18] and used by Dixit and Pindyck [11]. Consider an undeveloped reserve which, if developed, is expected to yield 100 million barrels of oil and has a ten year relinquishment requirement. Let some assumptions be made: • the value of the developed reserve is $12 per barrel; • the payout rate is 5% (just as a reminder, the payout rate is the net production revenues less depletion as a fraction of the reserve value); • development takes three years; • the present value of development cost is $11.79 per barrel. Now the undeveloped reserve could be valued as follows: • First I calculate the present value of the developed reserve, because of the three years development time. The correct discount rate is the difference between the risk-adjusted rate µ and the expected rate of growth of the reserve value (µ − δ), i.e. the payout rate δ. Hence, the present value of the developed reserve is V 0 = e−3∗0.05 ($12) = $10.32. • Now the critical ratio (developed reserve value to the present value of the development cost) is V 0 /D = $10.32/$11.79 ≈ 0.90. Being less than one, the development option is out-of-the-money. • It is time to use Table 5.1 to calculate the value of the undeveloped reserve. Assuming that σ is 0.142, the option value per dollar of development cost is 0.05336 and the total development cost is ($11.79)(100 million)=$1179 million. Hence the total value of the undeveloped reserve is (0.05336)($1179 million) = $62.91 million. 38
Thus, the undeveloped reserve has a value of $63 million due to its option value, although it would not be profitable yet to develop it because of the oil prices. Imagine though that the conditions on the oil market change in a way such that σ was 0.25. This new perceived value of oil prices would increase the value of the undeveloped reserve to (0.14612)($1179 million) = $172.28 million!
5.2 5.2.1
Mean-Reversion with Jumps Model General Shape of the Solution
Figure 5.6 represents the option value over time, for a relinquishment parameter of T = 5 years and D = 5.125. The parameters for this Figure are dP = 0.25, dt = 5 · 10−4 , dΦ = 0.05, Pm = 45, λ = 0, σ = 22%, η = 0.03.
10
Value of the concession (Option)(US$/bbl)
9 8 7 6 5 4 3 2 1 0 6 4 2 0 Time (years)
0
5
10
15
20
25
30
35
40
45
Oil Prices (US$/bbl)
Figure 5.6: MRJ Option Value
5.2.2
Influence of the MRJ Model Parameters
Let us now see what influence the various model parameters have on both the option value and the threshold. For this, I chose numerical scheme parameters so that the computation did not take too long. Of course, the precision was not that good, especially for the threshold where one sees that the step size used for the grid did not allow me to obtain a smooth function. Nonetheless, it is sufficient in order to see the option price and threshold evolve with the various parameters. Following the remarks of section 4.7, the important point is to keep the same grid size for each computation leading to a comparison of model parameters.
39
First, one can notice on Figures 5.7 and 5.8 that an increase in σ increases the option value and the threshold. This result is similar to the one observed with the GBM model. Figure 5.9 shows the influence of the mean-reversion speed η. An increase in η increases the option value as well as the threshold. Indeed, as the reversion speed is bigger, the oil price is more likely to remain around its long-run average price. And as the strike price is below the long-run average oil price, the option value increases. Besides, when η is zero, the threshold is below the long-run average oil price, while it is above the long-run average oil price for any other positive value of η. Figures 5.10 and 5.11 show that the jump arrival factor λ tends to diminish both the option value and the threshold. For higher jump frequency, the option price and the threshold for immediate investment are lower. The economic explanation is that the probability of a jump makes the investor more willing to invest. This is due to the bias in the probability density function I chose for the jump size, as the average jump size is greater than one. Figure 5.12 shows the influence of the long-run average oil price P¯ over the option price and immediate investment threshold. One can notice that a lower P¯ decreases the option value as well as the threshold. Indeed, when the long-run average oil price is closer to the strike price, the expected profit decreases. Furthermore, as the expected oil price is lower, the threshold decreases as well. The influence of the exogenous discount rate ρ is shown in Figure 5.13 and tends to reduce both the threshold and the option price. Indeed, given a fixed drift, the dividend or convenience yield δ has to adjust to ρ due to the relation 4.31. By increasing the convenience yield, the value of waiting decreases as do the option value and the threshold. In order to visualize the influence of the average jump-up size m2 (Figure 5.14), the jump distribution was modified so that the general shape remained the same. Only the jump-up was modified with an average at 3 instead of 2 and a standard deviation of 2/3 instead of 2/7. I noticed then that the modification of the jump-up had a negative influence over the option price and threshold. The explanation is the same as for the influence of the jump arrival rate. As the jump-up has a greater mean, the oil value is expected to be bigger, hence decreasing the opportunity value of waiting. The influence of the economic quality of the reserve shown on Figure 5.15 is the same as for the GBM model. Finally, as one can see on Table 5.3, the option value increases for higher volatility, for higher mean-reversion speed, for lower jump arrival rate, for higher long-run average oil price, for lower exogenous discount rate, for lower average jump-up and for higher economic quality of the reserve. The outcome is the same for the threshold, except for the economic quality of the reserve, as the threshold decreases if q increases.
40
1.8
20 σ=15%
19.5
1.4
1.2 Threshold (US$/bbl)
Value of the concession (Option)(US$/bbl)
1.6
1
0.8
19 σ=10%
18.5
0.6 σ=15% 0.4
18
σ=10%
0.2
0
0
2
4
6
8 10 12 Oil Prices (US$/bbl)
14
16
18
17.5
20
(a) Option price evolution when σ changes
0
0.5
1
1.5
2
2.5 Time (years)
3
3.5
4
4.5
5
(b) Threshold evolution when σ changes
Figure 5.7: Influence of the volatility on the MRJ model
Figure 5.8: Effect of volatility on the MRJ threshold
21
2.5
η=0.03
20
19 Threshold (US$/bbl)
Value of the concession (Option)(US$/bbl)
2
1.5
1
η=0
18
17 η=0.03 0.5
16 η=0
0
0
2
4
6
8
10 12 Oil Prices (US$/bbl)
14
16
18
20
15
22
(a) Option price evolution when η changes
0
0.5
1
1.5
2
2.5 Time (years)
3
3.5
4
4.5
(b) Threshold evolution when η changes
Figure 5.9: Influence of the mean-reversion speed on the MRJ model
41
5
3
24
λ=0%
23
λ=10%
22
2 Threshold (US$/bbl)
Value of the concession (Option)(US$/bbl)
2.5
1.5
λ=0%
21
20 λ=30% 19
1 λ=10% 18
λ=30%
0.5
17
0
0
5
10 15 Oil Prices (US$/bbl)
16
20
(a) Option price evolution when λ changes
0
0.5
1
1.5
2
2.5 Time (years)
3
3.5
4
4.5
5
(b) Threshold evolution when λ changes
Figure 5.10: Influence of the jump arrival frequency on the MRJ model
Figure 5.11: Threshold function of the jump arrival frequency for the base case
22
1.5
21
1
Threshold (US$/bbl)
Value of the concession (Option)(US$/bbl)
Mean−Reverting Oil Price is 20 20
Mean−Reverting Oil Price is 20
19
Mean−Reverting Oil Price is 15
18
0.5 Mean−Reverting Oil Price is 15
17
16
0
0
2
4
6
8 10 12 Oil Prices (US$/bbl)
14
16
18
15
20
(a) Option price evolution when P¯ changes
0
0.5
1
1.5
2
2.5 Time (years)
3
3.5
4
4.5
(b) Threshold evolution when P¯ changes
Figure 5.12: Influence of the long-run average oil price on the MRJ model
42
5
22
1.5
ρ=10%
ρ=15%
20 1
Threshold (US$/bbl)
Value of the concession (Option)(US$/bbl)
21
ρ=10%
19
18
0.5
17 ρ=15%
16
0
0
2
4
6
8 10 12 Oil Prices (US$/bbl)
14
16
18
15
20
(a) Option price evolution when ρ changes
0
0.5
1
1.5
2
2.5 Time (years)
3
3.5
4
4.5
5
(b) Threshold evolution when ρ changes
Figure 5.13: Influence of the exogenous discount rate on the MRJ model 22
1.5
Average Jump− Up Size is 2
20 1
Threshold (US$/bbl)
Value of the concession (Option)(US$/bbl)
21
Average Jump− Up Size is 2
Average Jump− Up Size is 3
19
18
0.5
17
Average Jump− Up Size is 3
0
0
2
4
6
8 10 12 Oil Prices (US$/bbl)
14
16
16
18
15
20
(a) Option price evolution when m2 and σ2 change
0
0.5
1
1.5
2
2.5 Time (years)
3
3.5
4
4.5
5
(b) Threshold evolution when m2 and σ2 change
Figure 5.14: Influence of the jump-up average and standard deviation on the MRJ model 12
22 q=1/3 20
18 q=2/3
8 Threshold (US$/bbl)
Value of the concession (Option)(US$/bbl)
10
Payoff for q=2/3
6
16
14
12
4 Option value for q=2/3
Payoff for q=1/3
10
2 8
Option value for q=1/3
0
0
5
10 Oil Prices (US$/bbl)
15
6
20
(a) Option price evolution when q change
0
0.5
1
1.5
2
2.5 Time (years)
3
3.5
4
4.5
(b) Threshold evolution when q change
Figure 5.15: Influence of the economic quality of the reserve on the MRJ model
43
5
5.2.3
Evolution of the Option Value over Time
On Table 5.2, one can see the evolution of price with respect to the expiration date. The percentages shown are the variation between the value corresponding to one expiration date to the next. One can notice that these variation are much more important out-ofthe-money than in-the-money, and that the variations in the option price are much more important than the variations in threshold. Indeed, if the expiration date goes from 2 to 5 years, the option value out-of-the-money (P = 15) increases by 21%, the option value in-the-money increases by 7.6%, whilst the threshold only increases by 2.4%. Hence, taken that the bid bonus is proportional to the option value, a later expiration date implies greater bid bonus without influencing the threshold value too much. T (years) 1 2 3 4 5
F (P = 15) 0.5114 0.6856 0.7798 0.8346 0.8673
% in F 34.1% 13.7% 7.0% 3.9%
F (P = 18) 1.0681 1.1522 1.2006 1.2294 1.2472
% in F 7.9% 4.2% 2.4% 1.4%
P ∗ (0) 20 20.5 20.75 21 21
% in P ∗ 2.5% 1.2% 1.2% 0.0%
Table 5.2: Sensitivity of the option value and threshold with time to expiration for the base case (Prices in $/bbl)
5.3
Comparison between the Two Models
One important issue for comparing the two models is to unify the variables. Let us take r = δ = 5% for the GBM model. The convenience yield δ is not a direct input parameter for the MRJ model (it is implicit, endogenous of the model) and depends on the price level. Hence, in order to compare the two models we need to choose the right price. It has been seen that δ = ρ − η(P¯ − P ). Hence, by replacing by the latter values, I get P = 18.3. Mean-Reversion + Jump Process: F (P = 18.3$/bbl, t = 0) Base No σ = 5% σ = 10% No σ = 23%, Case Jump Reversion λ=0 (λ = 0) η=0 and η = 0 1.3005 1.7743 1.0500 1.0804 1.0760 1.3425 Geometric Brownian Motion: F (P = 18.3$/bbl, t = 0) Base r = 10% r = 10% Case and and r = δ = 5% δ = 5% δ = 10% 1.4667 1.9084 1.3412
Table 5.3: Option value at P = 18.3 US $/bbl
44
As one can observe on Table 5.3, it is hard to make a comparison between the two models. Only for η = 0, implying ρ = δ, can one compare the case of GBM with r = δ and MRJ with no reversion, no jump and the same volatility σ = 23%. As expected, the values are very close, as are the graphical solutions. Jump-Reversion GBM
F (P = 12) 0.5487 0.3689
F (P = D/q) 0.8427 0.8002
F (P = 18) 1.2472 1.3959
P ∗ (t = 0) 21 26.8
P ∗ (t = 5) 17.75 15.6
Table 5.4: Comparing the two Real Options methods in the base case ($/bbl) 28 Threshold for GBM 26
Threshold (US$/bbl)
24
22 Threshold for Jump−Diffusion 20
18
16
14
0
0.5
1
1.5
2
2.5 Time (years)
3
3.5
4
4.5
5
Figure 5.16: Threshold comparison: Jump Diffusion and Geometric Brownian Motion Figure 5.16 and Table 5.4 show that in the base case, the MRJ threshold is smaller than for GBM when the time to expiration is important. But the situation is reversed closer to expiration. Furthermore, the option price is higher in the MRJ case than in the GBM case for an option out-of-the-money, but the situation is reversed for an option inthe-money, while the prices are similar at-the-money. Hence by using the GBM model, one would invest later, and in projects that may have been rejected if using MRJ closer to the deadline, and one may underestimate or overestimate the value of the project depending on the in-the-money/out-of-the-money situation.
45
Chapter 6
Conclusion 6.1
Summary
The advantage of jump-reversion process is that it better describes the reality of both economics (microeconomic logic) and statistical time-series (explaining the skewness, fatter tails, abnormal movements of oil prices). But there is a cost. The two problems with jump-diffusion processes are the impossibility of building a riskless portfolio and the difficulty of parameters estimation. Nonetheless, the MRJ model is definitely more suitable to Real Options related to commodities like oil. There can be huge differences between the option values yielded by this model and the GBM model. It is of course difficult to compare the values of the two models, as they do not require the same assumptions nor use the same parameters. One could however still observe that the behaviour of both models relative to their parameters was as similar as it possibly could be. Indeed, both the GBM and MRJ option values increase with the volatility and the economic quality of the reserve, and their threshold increases with the volatility and decreases with the economic quality of the reserve. Besides, by noting the relation between the payout rate and the exogenous discount rate in the MRJ model, one notices that an increase in these parameters leads to lower option values and threshold in both cases. Some parameters in the MRJ model do not have equivalents in the GBM model and vice versa. For instance, the fact that the GBM option value and threshold increase with the risk-free interest rate. Or the fact that the MRJ option value and threshold increase for higher mean-reversion speed, for lower jump arrival rate, for higher long-run average oil price and for lower average jump-up. But considering the case where the risk-free interest rate is equal to the payout rate and to the MRJ exogenous discount rate, and where there are no jumps and the mean-reversion speed is zero, the same solution is obtained. In the base case I chose to study, using the GBM model would lead to later investing, and in projects that may have been rejected if using the MRJ model. Furthermore, the value of the project may be either underestimated or overestimated depending on the in-the-money/out-of-the-money situation. Finally, the only drawback to be noticed by using the MRJ model with the scheme 46
that I implemented here is that the computation time is much longer than that for the GBM model with the corresponding numerical scheme.
6.2
Possible Improvements
The MRJ model presented here could be extended in several ways. • One could implement a numerical scheme based on projected Gauss-Seidel, as used with the GBM model scheme. This was not done because of time constraints. As a Crank-Nicholson Scheme has a faster convergence rate, this may decrease the computation time significantly. • One could try a more complex probability density function for the jump size, although the available data for the oil price did not allow us to come up with a precise statistical distribution. One could also try to use two uncorrelated distributions for jumps-up and jumps-down. • Furthermore, I later discovered that another Mean-Reversion with Jumps SDE, could be used to better equilibrate the Jumps Process: d(logP ) = η(log(P¯ ) − log(P ))dt + σdz + dq, ½
0, with probability 1 − λdt, Φ − 1, with probability λdt, k = E(Φ − 1).
dq =
The Jump distribution is shown in Figure 6.1. It is the sum of two normal distributions. With this probability density function, the jump size and direction are perfectly random. In case of Jump-up, the prices are expected to double and E(Φ)up = log(2) = 0.6931. In case of Jump-down, the prices are expected to halve and E(Φ)down = log(1/2) = −0.6931.
Figure 6.1: Probability density function of Φ
47
• One could allow the equilibrium price to be stochastic. • One could use the models introduced by Schwartz [23] and model uncertainty through two or three stochastic variables like the oil price, its convenience yield and interest rates and combine them with my model. • As a complement to the latter point, one could improve Schwartz’ assumption that the operational costs are deterministic and independent of the commodity price P . Indeed, this latter assumption does not hold as the correlation between operating costs and oil price are very high (see Adelman [2]). My model, by taking the economic quality of a reserve (q) as constant, assumes the opposite: that there is a perfect correlation between these two values. The reality being in between, a more realistic model could allow for stochastic costs with a positive correlation with the price process.
48
Appendix A
Calculation details A.1
Autoregressive Process
I have: dx = η ∗ (¯ x − x)dt + σ ∗ dz Let us define u as:
(A.1)
∗
u = eη t x
(A.2)
Hence, ∗
∗
du = η ∗ eη t xdt + eη t dx ∗ η∗ t
=η e
= η∗e
η∗ t
∗ η∗ t
xdt + η e
x ¯dt − η e
η∗ t
x ¯dt + σ ∗ e
(A.3) ∗ η∗ t
∗ η∗ t
xdt + σ e
dz
(A.4)
dz
(A.5)
Then, Z e
η∗ t
η∗ t
xt − e
η∗ t
xt−1 = ut − ut−1 = x ¯(e
−e
η ∗ (t−1)
)+σ Z
∗
∗
xt = e−η xt−1 + x ¯(1 − e−η ) + e−η t σ ∗ I finally obtain
t
t−1
and ∗
∗
∗
t
t−1
∗
eη s dzs ,
∗
eη s dzs .
(A.6)
(A.7)
∗
xt − xt−1 = x ¯(1 − e−η ) + (e−η − 1)xt−1 + ²t , (A.8) R R t η∗ s ∗s ∗t t η ∗ −η dzs . And since t−1 e dzs is the integral of a stochastic with ²t = σ e t−1 e process of mean zero and variance σ ∗ , ²t is a stochastic process of mean zero and variance σ² . h³ Z t ´2 i ∗ 2 ∗ 2 −2η ∗ t σ² = σ e E eη s dzs (A.9) Rt
t−1
η ∗ s dz s t−1 e
As one knows, the term have two interesting properties:
is an Ito integral. Ito integrals such as
h³ Z E
t−1 f (s, zs )dzs
´i
t
t−1
Rt
f (s, zs )dzs 49
=0
(A.10)
and h³ Z E
´³ Z
t
t−1
f (s, zs )dzs
Z
´i
t
t−1
g(s, zs )dzs
t
= t−1
E[f (s, zs )g(s, zs )]ds
A consequence of the second property is called the Ito isometry: h³ Z t ´2 i Z t E f (s, zs )dzs E[f (s, zs )2 ]ds = t−1
(A.11)
(A.12)
t−1
Hence, here, according to the Ito isometry, Z σ²2 = σ ∗ 2 e−2η =e =
−2η ∗ t σ
∗t
∗2
2η ∗
t
t−1
·
2η ∗ s
(A.13)
¸t
e
σ∗2 ∗ (1 − e−2η ) ∗ 2η
50
∗
e2η s ds
(A.14) t−1
(A.15)
Appendix B
MATLAB Code B.1
American Option Pricing
In order to launch the computations on several remote computers, this program has been constructed as a function in which one can enter the desired parameters. Besides, each time the program is launched, it asks for the output file name and a progression bar shows the evolution of the calculation. function y=GBM(r,delta,sigma,D,Pm,dP,dt,change_limit,q) % Mikael Pelet % MSc Mathematical Modelling % OCIAM, Oxford University
% Summer 2003 & Scientific Computing
% Real Options, Geometric Brownian Motion % Solution of the Finite Difference Problem format compact filename = input(’Enter test number:
’, ’s’);
%%%%%%%%%%%% % Real Option Parameters: %r = 0.05; %sigma = 0.23; %delta = 0.05; T % q
% Real Risk Free Rate of Interest % Volatility % Payout rate
= 10; = 1/3;
% Lease period (Expiry) (in years) % Economic quality of the reserve
% Finite Difference Grid: % % % %
Pm dP dt D
= = = =
20; 0.05; 5e-4; 5+dP/2;
% % % %
Truncation of the space grid Space Grid interval Time Grid interval Development cost in dollars
% Projected Gauss-Seidel parameters %change_limit = 1e-10; residual_limit = change_limit; TooMany = 500;
% maximum variation in change % maximum variation in residual % maximum number of iterations
t_stor=100;
51
m = Pm/dP n = T/dt
% Number of space steps, better be an integer % Number of time steps, better be an integer
%%%%%%%%%%%% % Initialisation for i = 0:m P(i+1) = dP*i; % Prices scale payoff(i+1) = max( q*P(i+1) - D , 0); % Payoff constraint A(i+1) = 1/4 * ( -sigma^2*i^2 + (r-delta)*i );% FDM coefficients B(i+1) = 1/dt + 1/2*sigma^2*i^2 + r; C(i+1) = -1/4 * ( sigma^2*i^2 + (r-delta)*i ); a(i+1) = 1/4 * ( sigma^2*i^2 - (r-delta)*i ); b(i+1) = 1/dt - 1/2*sigma^2*i^2; c(i+1) = 1/4 * ( sigma^2*i^2 + (r-delta)*i ); end vold = payoff; time(n+1)=n*dt; Stor(n/t_stor+1,:) = vold; thres(n+1)=D/q; % For comparison with Black-Scholes, calculation of Delta %Delta(n+1,1)= 0; % for i=1:m-1 % Delta(n+1,i+1) = (vold(i+2) - vold(i))/(2*dP); % end %Delta(n+1,m+1)= Delta(n+1,m); %%%%%%%%%%%% % Time loop h = waitbar(0,’Please wait...’); for t = n-1:-1:0 time(t+1)=t*dt; %%%%%%%%%%%% % Gauss Seidel % set up rhs of Crank Nicholson equations rhs(1) = b(1)*vold(1) + c(1)*vold(2); rhs(m+1) = (a(m+1)-c(m+1))*vold(m) + (b(m+1)+2*c(m+1))*vold(m+1); for i=1:m-1 rhs(i+1) = a(i+1)*vold(i)+b(i+1)*vold(i+1)+c(i+1)*vold(i+2); end % copy old value as initial guess guess = vold; % Projected Gauss-Seidel iteration change =1; residual = 1; count = 0; while (change > change_limit) & (residual > residual_limit)... & (count < TooMany ) temp = guess(1); guess(1) = (rhs(1) - C(1)*guess(2))/B(1);
52
if guess(1) < payoff(1) guess(1) = payoff(1); end change = (guess(1)-temp)^2; for i=1:m-1 temp = guess(i+1); guess(i+1) = (rhs(i+1)-A(i+1)*guess(i)-C(i+1)*... guess(i+2))/B(i+1); if guess(i+1) < payoff(i+1) guess(i+1) = payoff(i+1); end change = change + (guess(i+1)-temp)^2; end temp = guess(m+1); guess(m+1) = (rhs(m+1) - (A(m+1)-C(m+1))*guess(m))... /(B(m+1)+2*C(m+1)); if guess(m+1) < payoff(m+1) guess(m+1) = payoff(m+1); end change = change + (guess(m+1)-temp)^2; % As an alternative, just apply asymptotic... % European boundary conditions % Computation of the residual if guess(1) > payoff(1) diff = rhs(1) - B(1)*guess(1) - C(1)*guess(2); residual = diff^2; end if guess(m+1) > payoff(m+1) diff = rhs(m+1) - (A(m+1)-C(m+1))*guess(m) -... (B(m+1)+2*C(m+1))*guess(m+1); residual = residual + diff^2; end for i=1:m-1 if guess(i+1) > payoff(i+1) diff = rhs(i+1) - A(i+1)*guess(i) -... B(i+1)*guess(i+1) - C(i+1)*guess(i+2); residual = residual + diff^2; end end count=count+1; if count==TooMany disp(’Loop did not end properly’) end end %%%%%%%%%% % Threshold for i=1:m
53
if (guess(i+1)==payoff(i+1)) & (guess(i)~=payoff(i)) thres(t+1)=i*dP; end end vold = guess;
% New initial guess
% Reduction of vectors sizes if t/t_stor==floor(t/t_stor) tt=t/t_stor; Stor(tt+1,:) = vold; % Storage %Delta(tt+1,1)= 0; %for i=1:m-1 % Delta(tt+1,i+1) = (guess(i+2) - guess(i))/(2*dP); %end %Delta(tt+1,m+1)= Delta(tt+1,m); end waitbar((n+1-t)/(n+1),h) end close(h) F_init=guess;
% Initial value of the option
save(filename)
B.2
Mean-Reversion with Jumps
function y=MRJ(dP,dt,lambda,sigma,eta,Pm,dPhi) % Mikael Pelet % MSc Mathematical Modelling % OCIAM, Oxford University
% Summer 2003 & Scientific Computing
% Real Options, Mean-Reversion with Jumps % Solution of the Finite Difference Problem format compact filename = input(’Enter test number:
’, ’s’);
%%%%%%%%%%%% % Real Option Parameters : rho = 0.10; % % lambda = 0.15; % sigma = 0.22; % eta = 0.03; Pbar D T q
= = = =
Exogeneous discount rate % Annual frequency of jumps % Volatility % Reversion speed
20; % Average oil price (US$/bbl) 5+dP/2; % Development cost (US$) 5; % Lease period (Expiry) (in years) 1/3; % Economic quality of the reserve
% Finite Difference Grid : % Pm = 45; % dP = 0.1; % dt = 1e-4;
% Truncation of the space grid % Space Grid interval % Time Grid interval
54
% Jump characteristics % Reference to function pdf_phi m1 = 1/2; m2 = 2; % dPhi=5e-2; % Step size for discretisation of PDF of Phi % Number of steps : L=3/dPhi % Grid Parameters : m = Pm/dP n = T/dt L=5/dPhi
% Number of space steps, better be an integer % Number of time steps, better be an integer % Has to be an integer
if L ~= round(L) disp(’The number of steps for PDF of Phi is no integer!’) return end
%%%%%%%%%%%% % Initialisation t_stor=100; k = (m1+m2)/2 -1; A = dt/(rho*dt+1); for i = 0:m P(i+1) = dP*i; % Prices scale payoff(i+1)= max( q*P(i+1) - D , 0); % Payoff constraint pplus(i+1) = A*1/2*(sigma^2*i^2+i*eta*Pbar-i^2*eta*dP-i*lambda*k); pzero(i+1) = A*(1/dt-sigma^2*i^2 -lambda); pminus(i+1)= A*1/2*(sigma^2*i^2-i*eta*Pbar+i^2*eta*dP+i*lambda*k); pjump(i+1) = A*lambda; if
pplus(i+1)
