A developed four-factor model based on the Schwartz model system

Transcription

1 A developed four-factor model based on the Schwartz model system Christian O. Ewald and Zhe Zong March 2, 2016 Abstract: In this paper, the three-factor mode, which was proposed by Schwartz in 1997, will be run with the extended Kalman filter to catch the implied stochastic volatility of the spot price of the underlying asset (in this case, WTI crude oil). Then, a new, expanded four-factor model will be constructed and be compared with the three-factor model. In the four-factor model, the stochastic volatility of the spot price of the underlying asset will be considered to be a stochastic process following the Brownian motion. Key words: Futures contract of WTI crude oil, state space model, extended Kalman Filter, Factor model system, three-factor model, four-factor model 1 Introduction At the beginning of this paper, it is essential to consider briefly the development of the oil commodity markets and their derivative markets throughout the past several decades. The dramatic change in the status of financial derivatives has been noticed since 1973, when Black and Scholes published their famous paper and the first options exchange opened in Chicago (Fan, 2008). As for the energy commodity market, crude oil provides energy for almost every aspect of human activity, such as heating oil, aviation fuel, fuel oil, primary crude oil distillates and gasoline. As such, crude oil has unquestionably been thought of as one of the most important physical commodities for our day-to-day lives and for scientific research. Today, there are two price benchmarks of crude oil in the world, the Brent and the WTI (West Texas Intermediate). In this paper, we will focus on the WTI, which is also known as Texas light sweet and has a relatively low-density and low sulphur content. To be more specific, its APT (American Petroleum Institution) gravity is around 39.6, and its specific gravity is about Also, the the corresponding author ( z.zong.1@research.gla.ac.uk ) 1

2 WTI contains about 0.24% sulphur, which is sweeter than Brent s 0.37%. The WTI is mainly traded in North America, and its futures contracts are mainly traded on the New York Mercantile Exchange (the CME group). All futures of WTI traded on the CME are physical settlement, which means that all futures are related to real delivery when they mature, and the place of settlement of the WTI is Cushing, Oklahoma. Since the oil crisis in 1973, the traditional way of trading long-term oil contracts has changed (Alizadeh, Lin and Nomikos, 2008). In 1978 and 1983, heating oil and crude oil were introduced to the New York Mercantile Exchange (NYMEX). Then, oil has been traded as the biggest commodity in the world commodity markets since the 1980s. According to Geman (2003), since then, trading oil has been seen not only as a primary physical activity, but also as a worldwide financial activity. Therefore, crude oil and oil related products have become more attractive to financial investors. In the meantime, the era of crude oil price stability ended, and a new era of tremendous price fluctuation of price has begun. To be more specific, the price of crude oil fluctuated between a relatively narrow range of 10 dollars and 20 dollars per barrel between 1874 and 1974 (the price was measured by US dollar in 1997) (Rühl, 2008). As a result of the increase in its fluctuation, the price of crude oil keeps moving up, especially after At the end of 1998, the price of crude oil bottomed out at 12 U.S. dollars per barrel. However, after 10 years, the price peaked at 145 dollars per barrel in 2008, which was a new record in the history of trading in crude oil. Even though the price of crude oil rapidly decreased after the historical peak, it has rebounded and keeps fluctuating around the historically high level, until As has been explained, crude oil and oil-related commodities have played increasingly crucial roles not only in our daily lives but also in the financial markets. Even though an increasing number of scientists have paid more attention to this domain, there are lots of problems in the derivative markets. For example, one of the most important problems that needs to be solved is the lack of a practical and unified point of view for pricing a futures contract (not only for a crude oil commodity, but also for precious metals, agricultural products, and so forth). In most financial institutions, investors still price a futures contract based on a simple relationship between the prices of futures contracts and the interest rate, because there are difficulties and too many factors must be considered in pricing a futures contract. More than that, some of those momentous factors cannot even be observed in the markets. The easiest case can be described as follows: The basic way to value a futures contract is F T (r δ)(t t) (t) = S(t)e, where S(t) is the spot price of the underlying asset at time point t; δ is the convenience yield of the underlying asset, which is defined as the yield if an investor holds the commodity instead of the futures contract of the commodity; r is the interest rate; and T is the maturity of the futures contract. In trading crude oil, since most crude oil is traded by futures contracts, the trading price of spot crude oil might not be a suitable measurement of the spot price of crude 2

3 oil. In addition, the convenience yield of crude oil is absolutely unobservable in markets. However, with the development of the pricing a particular futures contract, the convenience yield δ plays an extremely important role and has drawn more attention. Furthermore, even if r can be considered an observable constant, it will be helpful to provide a more accurate result when it is considered to be instantaneous. However, the stochastic interest rate is also unobservable. Most previous studies focused on estimating the unobservable convenience yield of the underlying asset. Apart from the Schwartz factor model system, which will be introduced in the following paragraphs, Casassus and Collin-Dufresne (2005) suggested that the convenience yield of an underlying asset might be obtained by the inverse mathematical function of the futures pricing model, which means that the convenience yield could be expressed as follows: δ(t) = f 1 (F T (t), S(t), r, T, t). In addition, Lewis (2005) suggested that the convenience yield is an indication of an uncertain market and is positively correlated with volatility across various commodity markets. Moreover, the convenience yield was considered to be an exogenous random variable by Gibson and Schwartz (1990). On the other side, a number of other studies considered the changes in the convenience yield to be an endogenous result of the interactions among supply, demand, and storage decisions (Brennan (1958), Deaton and Laroque (1992), Pindyck (2001), Routledge, Seppi and Spatt (2000), Working (1949)). Furthermore, Hong (2001) combined the idess of exogenous and endogenous variables, and pointed out a strongly seasonal variation of the international oil convenience yield due to an imbalance between supply and demand by studying the price spread. In order to calculate the unobservable convenience yield, many studies have tried doing it from different perspectives. For example, Carmona and Ludkovski (2004) suggested a variant of the Schwartz model with time-dependent parameters to calculate the hidden state variables; Geman (2003) pointed out that the convenience yield was essential a difference between the benefit of holding a physical commodity and the cost of storage, and the convenience yield should be strongly related with the level of the inventory. She also proposed that there is a negative relationship between the volatility of a commodity and the world inventory level, and, hence, the price of any particular commodity and its volatility were positively correlated; Cassassus and Collin-Dufresne (2005) successfully obtained a stochastic convenience yield which was implied from assumed observable prices of commodity futures and interest rates based on a system similar to the Schwartz model. The system of price a particular futures contract, which this paper will discuss, was constructed by Schwartz (Schwartz, 1997). In 1990, Gibson and Schwartz proposed a PDE(partial differential equation) for two factors (spot price and convenience yield), which the pricing formula of a futures contract should follow. However, they did not propose a closed form of the solution of the PDE. 3

4 Later, Bjerksunk (1991) solved a PDE which was proposed by Brennan and Schwartz (1979), and proposed a close form of the PDE. Then, Schwartz (1997) adjusted the solution for pricing a futures contract and estimated the results in one-, two- and three-factor model by using the Kalman filter. The traditional three factors are the spot price of the underlying asset, the convenience yield of the underlying asset and the instaneous interest rate, respectively. In 2008, Yan and Li constructed a four-factor model with the fourth-factor exchange rate, and estimated the result by using the least squares method. This paper aims to construct and test the effectiveness of a four-factor model with the consideration of the stochastic volatility of the spot price of the underlying asset. Then the four-factor model will be compared with the traditional three-factor model with an implied stochastic parameter of the volatility of the spot price of the underlying asset. In this paper, the original three-factor model (Schwartz, 1997) and the proposed four-factor model will be introduced in the second section in detail. In the third section, the methodology of the implementation of both models will be introduced. In the fourth part, the assumptions of this paper and the collected data will be described. Then, the empirical results and the conclusions will be discussed in detail. 2 Mathematical Model 2.1 The Schwartz Three-Factor Model Before 1997, people normally thought of only two stochastic processes the spot price and the convenience yield of the underlying asset when they tried to price a forward price. In 1997, Schwartz developed a new three-factor model, in which the stochastic interest rate was considered to be the third unobservable factor. Since then, the instantaneous interest rate was introduced in the model system. Similarly, with the two-factor model, the three stochastic processes can be described as follows: ds = (µ δ)dt + σ 1 Sdz 1 dδ = κ( α δ)dt + σ 2 dz 2 dr = α(m r)dt + σ 3 dz3 with dz1dz 2 = ρ 12 dt dz1dz 3 = ρ 13 dt 4

5 dz 2dz 2 = ρ 23 dt where µ, α and m representing the long-term return on investing in oil, the long-term convenience yield and the interest rate, respectively; κ and α are the coefficient of reverting in the stochastic processes, respectively; similarly, σ 1, σ 2 and σ 3 are the volatilities of spot price, convenience yield and the interest rate, respectively; dz 1, dz 2 and dz 3 are increments of Brownian motions and following normal distribution N(0, dt 1/2 ) with dz i dz j = ρ ijdt, where ρ ij is the correlation coefficient between the two stochastic processes i and j. With the consideration of the three stochastic processes, the futures price formula must satisfy the following partial differential equation: 1 2 σ2 1S 2 F SS σ2 2F δδ σ2 3F rr + σ 1 σ 2 ρ 12 SF Sδ + σ 1 σ 3 ρ 13 SF Sr + σ 2 σ 3 ρ 23 F δr + (r δ)sf S + κ( α δ)f δ + α(m r)f r F T = 0 Schwartz proposed the solution of the above PDE in 1997, which is as follows: where F (S, δ, r, T ) = Sexp( δ 1 e κt κ C(T ) = (κ α+σ1σ2ρ12)((1 e κt ) κt ) κ 2 ) + r 1 e αt α + C(T )) σ2 2 (4(1 e κt ) (1 e 2κT ) 2κT 4κ 3 (αm +σ 1σ 3ρ 13)((1 e αt ) αt ) α 2 σ2 3 (4(1 e αt ) (1 e 2αT ) 2αT ) 4α 3 + σ 2 σ 3 ρ 23 ( (1 e κt ) (1 e (κ+α)t )+(1 e αt ) κα(κ+α) + κ2 (1 e αt )+α 2 (1 e κt ) κα 2 T ακ 2 T κ 2 α 2 (κ+α) ) The volatility of the spot price is considered as a parameter (or, say, a constant) in the above three-factor model. Since the spot price is the most important index of the futures price, the volatility of the spot price should get more attention in dealing with the model system. In our previous work, we have tested the two-factor model with the stochastic process of the volatility of the spot price, and the results were very good. Here, the above three-factor model will be expanded with the stochastic volatility of the spot price. 2.2 The Developed Four-Factor Model The four factor model can be described as follows The four stochastic processes are: with ds = (r δ)sdt + σ 1 SdW 1 dδ = κ 1 (α δ)dt + σ 2 dw 2 dσ 1 = κ 2 (θ σ 1 )dt + σ 3 dw 3 dr = κ 3 (γ r)dt + σ 4 dw 4 5

6 dw 1 dw 2 = ρ 12 dt dw 1 dw 3 = ρ 13 dt dw 1 dw 4 = ρ 14 dt dw 2 dw 3 = ρ 23 dt dw 2 dw 4 = ρ 24 dt dw 3 dw 4 = ρ 34 dt where r, α, θ and γ representing the long-term return on investing in oil, the long-term convenience yield, the volatility of the spot price and the interest rate, respectively; κ 1, κ 2 and κ 3 are the coefficient of reverting in the stochastic processes, respectively; similarly, σ 1, σ 2, σ 3 and σ 4 are the volatilities of spot price, convenience yield, the volatility of the spot price and the interest rate, respectively; dw 1, dw 2, dw 3 and dw 4 are increments of Brownian motions and following normal distribution N(0, dt 1/2 ) with dw i dw j = ρ ij dt, where ρ ij is the correlation coefficient between the two stochastic processes i and j. With the consideration of the four stochastic processes, the futures price formula must satisfy the following partial differential equation: 1 2 σ2 1S 2 F SS σ2 2F δδ σ2 3F σ1σ σ2 4F rr + Sσ 1 σ 2 ρ 12 F Sδ + Sσ 1 σ 3 ρ 13 F Sσ1 + Sσ 1 σ 4 ρ 14 F Sr + σ 2 σ 3 ρ 23 F δσ1 + σ 2 σ 4 ρ 24 F δr + σ 3 σ 4 ρ 34 F σ1r + (r δ)sf S + κ 1 (α δ)f δ + κ 2 (θ σ 1 )F σ1 + κ 3 (γ r)f r F τ = 0 Similarly, as previous studies (Heston,1993 and Chen, Ewald and Zong, 2014) guessed, the solution of the PDE is F = Sexp(A + Bδ + Cσ 1 + Dr) then the PDE can be reduced to four ODEs(Ordinary Differential Equation) as follows: 1 κ 1 B = B τ 1 κ 3 D = D τ σ 2 ρ 12 B + σ 3 ρ 13 C + σ 4 ρ 14 D κ 2 C = C τ 1 2 σ2 2B σ2 3C σ2 4D 2 + σ 2 σ 3 ρ 23 BC + σ 2 σ 4 ρ 24 BD + σ 3 σ 4 ρ 34 CD + κ 1 αb + κ 2 θc + κ 3 γd = A τ B, C,D and A then can be easily solved. Last, the solution of the PDE can be expressed as follows: B = e κ 1 τ 1 κ 1 6

7 D = 1 e κ 3 τ κ 3 C = ρ 14 σ 4 /(κ 3 ( ρ 13 σ 3 + κ 2 )) ρ 12 σ 2 /(κ 1 ( ρ 13 σ 3 + κ 2 )) + ρ 12 σ 2 exp( ( ρ 13 σ 3 + κ 2 )τ τ(ρ 13 σ 3 + κ 1 κ 2 ))/(κ 1 ( ρ 13 σ 3 κ 1 + κ 2 )) ρ 14 σ 4 exp( ( ρ 13 σ 3 + κ 2 )τ + τ( ρ 13 σ 3 + κ 2 κ 3 ))/(κ 3 ( ρ 13 σ 3 + κ 2 κ 3 )) + exp( ( ρ 13 σ 3 + κ 2 )τ)( ρ 14 σ 4 /(κ 3 ( ρ 13 σ 3 + κ 2 )) + ρ 12 σ 2 /(κ 1 ( ρ 13 σ 3 + κ 2 )) ρ 12 σ 2 /(κ 1 ( ρ 13 σ 3 κ 1 + κ 2 )) + σ 4 ρ 14 /(( ρ 13 σ 3 + κ 2 κ 3 )κ 3 )) A = 0.5(1/(κ 2 3( ρ 13 σ 3 + κ 2 ) 2 κ 2 1(ρ 13 σ 3 + κ 1 κ 2 ) 2 ( ρ 13 σ 3 + κ 2 κ 3 ) 2 ))((1/(ρ 13 σ 3 κ 1 κ 2 ))(2( ρ 13 σ 3 + κ 2 )(κ 1 ρ 23 + (ρ 13 ρ 23 ρ 12 )σ 3 κ 2 ρ 23 )(σ 4 ρ 14 κ 1 σ 2 ρ 12 κ 3 +(ρ 12 σ 2 ρ 14 σ 4 )( ρ 13 σ 3 +κ 2 ))κ 2 3σ 3 κ 1 ( ρ 13 σ 3 +κ 2 κ 3 )σ 2 exp(τ(ρ 13 σ 3 κ 1 κ 2 ))) (1/(ρ 13 σ 3 κ 2 κ 3 ))(2σ 4 ( ρ 13 σ 3 +κ 2 )(ρ 13 σ 3 + κ 1 κ 2 )(σ 4 ρ 14 κ 1 σ 2 ρ 12 κ 3 + (ρ 12 σ 2 ρ 14 σ 4 )( ρ 13 σ 3 + κ 2 ))κ 3 σ 3 ( κ 3 ρ 34 + ( ρ 13 ρ 34 + ρ 14 )σ 3 + κ 2 ρ 34 )κ 2 1exp(τ(ρ 13 σ 3 κ 2 κ 3 ))) + (σ 4 ρ 14 κ 1 σ 2 ρ 12 κ 3 + (ρ 12 σ 2 ρ 14 σ 4 )( ρ 13 σ 3 +κ 2 )) 2 κ 2 3σ 2 3κ 2 1exp( (2( ρ 13 σ 3 +κ 2 ))τ)/(2ρ 13 σ 3 2κ 2 )+ (1/(ρ 13 σ 3 κ 2 ))(2(ρ 13 σ 3 +κ 1 κ 2 )(σ 4 ρ 14 κ 1 σ 2 ρ 12 κ 3 +(ρ 12 σ 2 ρ 14 σ 4 )( ρ 13 σ 3 + κ 2 ))κ 3 κ 1 ( ρ 13 σ 3 + κ 2 κ 3 )((θκ 2 ( ρ 13 σ 3 + κ 2 )κ 3 + σ 4 (( ρ 13 ρ 34 + ρ 14 )σ 3 + κ 2 ρ 34 )σ 3 )κ 1 (( ρ 13 ρ 23 +ρ 12 )σ 3 +κ 2 ρ 23 )κ 3 σ 3 σ 2 )exp((ρ 13 σ 3 κ 2 )τ)) (1/( κ 1 κ 3 ))(2σ 4 ( ρ 13 σ 3 + κ 2 ) 2 (ρ 13 σ 3 + κ 1 κ 2 )(( ρ 24 κ 3 + ( ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 + κ 2 ρ 24 )κ 1 + ((ρ 12 ρ 34 ρ 13 ρ 24 )σ 3 + κ 2 ρ 24 )κ 3 + ( ρ 24 ρ (ρ 12 ρ 34 + ρ 14 ρ 23 )ρ 13 ρ 12 ρ 14 )σ 2 3 κ 2 (ρ 12 ρ 34 2ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 κ 2 2ρ 24 )κ 3 κ 1 ( ρ 13 σ 3 + κ 2 κ 3 )σ 2 exp( τ(κ 1 +κ 3 )))+(1/κ 1 )(2( ρ 13 σ 3 +κ 2 )(ρ 13 σ 3 +κ 1 κ 2 )( κ 3 α( ρ 13 σ 3 + κ 2 )κ (α( ρ 13 σ 3 + κ 2 ) 2 κ 3 σ 4 (( ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 + κ 2 ρ 24 )σ 2 )κ ((( θκ 2 ρ 12 ρ 13 +σ 2 (ρ 12 ρ 23 ρ 13 ))σ 3 +κ 2 (κ 2 ρ 12 θ+σ 2 ))κ 3 +σ 4 ((ρ 24 ρ 2 13+( ρ 12 ρ 34 ρ 14 ρ 23 )ρ 13 + ρ 12 ρ 14 )σ κ 2 (ρ 12 ρ 34 2ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 + κ 2 2ρ 24 ))σ 2 κ 1 (( 2ρ 12 ρ 13 ρ 23 + ρ ρ 2 13)σ κ 2 (ρ 12 ρ 23 ρ 13 )σ 3 + κ 2 2)κ 3 σ 2 2)κ 3 ( ρ 13 σ 3 + κ 2 κ 3 ) 2 exp( κ 1 τ)) + (1/κ 3 )(2( ρ 13 σ 3 + κ 2 )(ρ 13 σ 3 + κ 1 κ 2 ) 2 κ 1 ( ρ 13 σ 3 + κ 2 κ 3 )(( γ( ρ 13 σ 3 + κ 2 )κ γ( ρ 13 σ 3 + κ 2 ) 2 κ σ 4 (( θκ 2 ρ 13 ρ 14 + σ 4 ( ρ 14 ρ 34 + ρ 13 ))σ 3 + κ 2 (κ 2 ρ 14 θ σ 4 ))κ 3 + σ 2 4(( 2ρ 13 ρ 14 ρ 34 + ρ ρ 2 14)σ 2 3 2κ 2 ( ρ 14 ρ 34 + ρ 13 )σ 3 + κ 2 2))κ 1 σ 4 κ 3 ((( ρ 12 ρ 34 + ρ 13 ρ 24 )σ 3 κ 2 ρ 24 )κ 3 + (ρ 24 ρ ( ρ 12 ρ 34 ρ 14 ρ 23 )ρ 13 + ρ 12 ρ 14 )σ κ 2 (ρ 12 ρ 34 2ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 + κ 2 2ρ 24 )σ 2 )exp( κ 3 τ)) 0.5(1/κ 1 )(( ρ 13 σ 3 + κ 2 ) 2 (κ (( 2ρ 12 ρ ρ 13 )σ 3 2κ 2 )κ 1 +( 2ρ 12 ρ 13 ρ 23 +ρ 2 12+ρ 2 13)σ κ 2 (ρ 12 ρ 23 ρ 13 )σ 3 +κ 2 2)κ 2 3( ρ 13 σ 3 +κ 2 κ 3 ) 2 σ 2 2exp( 2κ 1 τ)) + 2(ρ 13 σ 3 + κ 1 κ 2 ) 2 ( 0.25(1/κ 3 )(σ 2 4( ρ 13 σ 3 + κ 2 ) 2 (κ (( 2ρ 14 ρ ρ 13 )σ 3 2κ 2 )κ 3 + ( 2ρ 13 ρ 14 ρ 34 + ρ ρ 2 14)σ 2 3 2κ 2 ( ρ 14 ρ 34 + ρ 13 )σ 3 + κ 2 2)κ 2 1exp( 2κ 3 τ)) + (( ( ρ 13 σ 3 + κ 2 ) 2 (α γ)κ θκ 2 σ 4 ρ 14 ( ρ 13 σ 3 + κ 2 )κ 3 + (1/2)σ 2 4(( 2ρ 13 ρ 14 ρ 34 + ρ ρ 2 14)σ 2 3 2κ 2 ( ρ 14 ρ 34 + ρ 13 )σ 3 + κ 2 2))κ 2 1 (θκ 2 ρ 12 ( ρ 13 σ 3 + κ 2 )κ 3 + σ 4 ((ρ 24 ρ ( ρ 12 ρ 34 ρ 14 ρ 23 )ρ 13 + ρ 12 ρ 14 )σ κ 2 (ρ 12 ρ 34 2ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 + κ 2 2ρ 24 ))κ 3 σ 2 κ (( 2ρ 12 ρ 13 ρ 23 + ρ ρ 2 13)σ κ 2 (ρ 12 ρ 23 ρ 13 )σ 3 + κ 2 2)κ 2 3σ 2 2)( ρ 13 σ 3 + κ 2 κ 3 ) 2 τ)) 0.5(1/(κ 2 3( ρ 13 σ 3 + κ 2 ) 2 κ 2 1(ρ 13 σ 3 + κ 1 κ 2 ) 2 ( ρ 13 σ 3 + κ 2 κ 3 ) 2 ))((1/(ρ 13 σ 3 κ 1 κ 2 ))(2( ρ 13 σ 3 + κ 2 )(κ 1 ρ 23 + (ρ 13 ρ 23 ρ 12 )σ 3 κ 2 ρ 23 )(σ 4 ρ 14 κ 1 σ 2 ρ 12 κ 3 + (ρ 12 σ 2 ρ 14 σ 4 )( ρ 13 σ 3 + κ 2 ))κ 2 3σ 3 κ 1 ( ρ 13 σ 3 + κ 2 κ 3 )σ 2 ) (1/(ρ 13 σ 3 κ 2 κ 3 ))(2σ 4 ( ρ 13 σ 3 + κ 2 )(ρ 13 σ 3 + κ 1 κ 2 )(σ 4 ρ 14 κ 1 σ 2 ρ 12 κ 3 + (ρ 12 σ 2 ρ 14 σ 4 )( ρ 13 σ 3 + κ 2 ))κ 3 σ 3 ( κ 3 ρ 34 + ( ρ 13 ρ 34 + ρ 14 )σ 3 + κ 2 ρ 34 )κ 2 1) + (σ 4 ρ 14 κ 1 σ 2 ρ 12 κ 3 + (ρ 12 σ 2 ρ 14 σ 4 )( ρ 13 σ 3 + κ 2 )) 2 κ 2 3σ 2 3κ 2 1/(2ρ 13 σ 3 2κ 2 ) + (1/(ρ 13 σ 3 7

8 κ 2 ))(2(ρ 13 σ 3 + κ 1 κ 2 )(σ 4 ρ 14 κ 1 σ 2 ρ 12 κ 3 + (ρ 12 σ 2 ρ 14 σ 4 )( ρ 13 σ 3 + κ 2 ))κ 3 κ 1 ( ρ 13 σ 3 + κ 2 κ 3 )((θκ 2 ( ρ 13 σ 3 + κ 2 )κ 3 + σ 4 (( ρ 13 ρ 34 + ρ 14 )σ 3 + κ 2 ρ 34 )σ 3 )κ 1 (( ρ 13 ρ 23 +ρ 12 )σ 3 +κ 2 ρ 23 )κ 3 σ 3 σ 2 )) (1/( κ 1 κ 3 ))(2σ 4 ( ρ 13 σ 3 + κ 2 ) 2 (ρ 13 σ 3 + κ 1 κ 2 )(( ρ 24 κ 3 + ( ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 + κ 2 ρ 24 )κ 1 + ((ρ 12 ρ 34 ρ 13 ρ 24 )σ 3 +κ 2 ρ 24 )κ 3 +( ρ 24 ρ (ρ 12 ρ 34 +ρ 14 ρ 23 )ρ 13 ρ 12 ρ 14 )σ 2 3 κ 2 (ρ 12 ρ 34 2ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 κ 2 2ρ 24 )κ 3 κ 1 ( ρ 13 σ 3 + κ 2 κ 3 )σ 2 ) + (1/κ 1 )(2( ρ 13 σ 3 + κ 2 )(ρ 13 σ 3 +κ 1 κ 2 )( κ 3 α( ρ 13 σ 3 +κ 2 )κ 3 1+(α( ρ 13 σ 3 +κ 2 ) 2 κ 3 σ 4 (( ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 + κ 2 ρ 24 )σ 2 )κ ((( θκ 2 ρ 12 ρ 13 + σ 2 (ρ 12 ρ 23 ρ 13 ))σ 3 + κ 2 (κ 2 ρ 12 θ + σ 2 ))κ 3 + σ 4 ((ρ 24 ρ ( ρ 12 ρ 34 ρ 14 ρ 23 )ρ 13 + ρ 12 ρ 14 )σ κ 2 (ρ 12 ρ 34 2ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 + κ 2 2ρ 24 ))σ 2 κ 1 (( 2ρ 12 ρ 13 ρ 23 + ρ ρ 2 13)σ κ 2 (ρ 12 ρ 23 ρ 13 )σ 3 + κ 2 2)κ 3 σ 2 2)κ 3 ( ρ 13 σ 3 + κ 2 κ 3 ) 2 ) + (1/κ 3 )(2( ρ 13 σ 3 + κ 2 )(ρ 13 σ 3 + κ 1 κ 2 ) 2 κ 1 ( ρ 13 σ 3 + κ 2 κ 3 )(( γ( ρ 13 σ 3 + κ 2 )κ γ( ρ 13 σ 3 + κ 2 ) 2 κ σ 4 (( θκ 2 ρ 13 ρ 14 +σ 4 ( ρ 14 ρ 34 +ρ 13 ))σ 3 +κ 2 (κ 2 ρ 14 θ σ 4 ))κ 3 +σ 2 4(( 2ρ 13 ρ 14 ρ 34 + ρ ρ 2 14)σ 2 3 2κ 2 ( ρ 14 ρ 34 + ρ 13 )σ 3 + κ 2 2))κ 1 σ 4 κ 3 ((( ρ 12 ρ 34 + ρ 13 ρ 24 )σ 3 κ 2 ρ 24 )κ 3 + (ρ 24 ρ ( ρ 12 ρ 34 ρ 14 ρ 23 )ρ 13 + ρ 12 ρ 14 )σ κ 2 (ρ 12 ρ 34 2ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 + κ 2 2ρ 24 )σ 2 )) 0.5(1/κ 1 )(( ρ 13 σ 3 + κ 2 ) 2 (κ (( 2ρ 12 ρ ρ 13 )σ 3 2κ 2 )κ 1 + ( 2ρ 12 ρ 13 ρ 23 + ρ ρ 2 13)σ κ 2 (ρ 12 ρ 23 ρ 13 )σ 3 + κ 2 2)κ 2 3( ρ 13 σ 3 + κ 2 κ 3 ) 2 σ 2 2) 0.5(1/κ 3 )((ρ 13 σ 3 + κ 1 κ 2 ) 2 σ 2 4( ρ 13 σ 3 + κ 2 ) 2 (κ (( 2ρ 14 ρ ρ 13 )σ 3 2κ 2 )κ 3 +( 2ρ 13 ρ 14 ρ 34 +ρ ρ 2 14)σ 2 3 2κ 2 ( ρ 14 ρ 34 +ρ 13 )σ 3 +κ 2 2)κ 2 1)) Then, C and A can be simplified as: C = and A = ρ 14σ 4 κ ρ 12σ 2 3( ρ 13 σ 3+κ 2) κ + 1( ρ 13σ 3+κ 2) ρ 12σ 2e ( ρ 13 σ 3 +κ 2 )τ τ(ρ 13 σ 3 +κ 1 κ 2 ) ρ 14σ 4e ( ρ 13 σ 3 +κ 2 )τ+τ( ρ 13 σ 3 +κ 2 κ 3 ) κ 1( ρ 13σ 3 κ 1+κ 2) κ 3( ρ 13σ 3+κ 2 κ 3) + e ( ρ13σ3+κ2)τ ρ ( 14σ 4 κ + ρ 12σ 2 3( ρ 13σ 3+κ 2) κ ρ 12σ 2 1( ρ 13σ 3+κ 2) κ + 1( ρ 13σ 3 κ 1+κ 2) σ 4ρ 14 κ ) 3( ρ 13σ 3+κ 2 κ 3) κ 2 3 ( ρ13σ3+κ2)2 κ 2 1 (ρ13σ3+κ1 κ2)2 ( ρ 13σ 3+κ 2 κ 3) [ 2 ρ 13σ 3 κ 1 κ 2 (2( ρ 13 σ 3 + κ 2 )(κ 1 ρ 23 + (ρ 13 ρ 23 ρ 12 )σ 3 κ 2 ρ 23 )(σ 4 ρ 14 κ 1 σ 2 ρ 12 κ 3 + (ρ 12 σ 2 ρ 14 σ 4 )( ρ 13 σ 3 + κ 2 ))κ 2 3σ 3 κ 1 ( ρ 13 σ 3 + κ 2 κ 3 )σ 2 e τ(ρ13σ3 κ1 κ2) ) 1 ρ 13σ 3 κ 2 κ 3 (2σ 4 ( ρ 13 σ 3 + κ 2 )(ρ 13 σ 3 + κ 1 κ 2 )(σ 4 ρ 14 κ 1 σ 2 ρ 12 κ 3 + (ρ 12 σ 2 ρ 14 σ 4 )( ρ 13 σ 3 + κ 2 ))κ 3 σ 3 ( κ 3 ρ 34 + ( ρ 13 ρ 34 + ρ 14 )σ 3 + κ 2 ρ 34 )κ 2 1e τ(ρ13σ3 κ2 κ3) ) + (σ 4 ρ 14 κ 1 σ 2 ρ 12 κ 3 + (ρ 12 σ 2 ρ 14 σ 4 )( ρ 13 σ 3 + κ 2 )) 2 κ 2 3σ3κ e 2τ( ρ 13 σ 3 +κ 2 ) 1 2ρ 13σ 3 2κ 2 + ρ 13σ 3 κ 2 (2(ρ 13 σ 3 + κ 1 κ 2 )(σ 4 ρ 14 κ 1 σ 2 ρ 12 κ 3 + (ρ 12 σ 2 ρ 14 σ 4 )( ρ 13 σ 3 + κ 2 ))κ 3 κ 1 ( ρ 13 σ 3 + κ 2 κ 3 )((θκ 2 ( ρ 13 σ 3 + κ 2 )κ 3 + σ 4 (( ρ 13 ρ 34 + ρ 14 )σ 3 + κ 2 ρ 34 )σ 3 )κ 1 (( ρ 13 ρ 23 + ρ 12 )σ 3 + κ 2 ρ 23 )κ 3 σ 3 σ 2 )e (ρ13σ3 κ2)τ 1 ) κ 1 κ 3 (2σ 4 ( ρ 13 σ 3 + κ 2 ) 2 (ρ 13 σ 3 + κ 1 κ 2 )(( ρ 24 κ 3 + ( ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 + κ 2 ρ 24 )κ 1 + ((ρ 12 ρ 34 ρ 13 ρ 24 )σ 3 + κ 2 ρ 24 )κ 3 + ( ρ 24 ρ (ρ 12 ρ 34 + ρ 14 ρ 23 )ρ 13 ρ 12 ρ 14 )σ3 2 κ 2 (ρ 12 ρ 34 2ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 κ 2 2ρ 24 )κ 3 κ 1 ( ρ 13 σ 3 + κ 2 κ 3 )σ 2 e τ(κ1+κ3) ) + 1 κ 1 (2( ρ 13 σ 3 + κ 2 )(ρ 13 σ 3 +κ 1 κ 2 )( κ 3 α( ρ 13 σ 3 +κ 2 )κ 3 1+(α( ρ 13 σ 3 +κ 2 ) 2 κ 3 σ 4 (( ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 + κ 2 ρ 24 )σ 2 )κ ((( θκ 2 ρ 12 ρ 13 + σ 2 (ρ 12 ρ 23 ρ 13 ))σ 3 + κ 2 (κ 2 ρ 12 θ + σ 2 ))κ 3 + σ 4 ((ρ 24 ρ ( ρ 12 ρ 34 ρ 14 ρ 23 )ρ 13 + ρ 12 ρ 14 )σ3 2 + κ 2 (ρ 12 ρ 34 2ρ 13 ρ

9 ρ 14 ρ 23 )σ 3 + κ 2 2ρ 24 ))σ 2 κ 1 (( 2ρ 12 ρ 13 ρ 23 + ρ ρ 2 13)σ κ 2 (ρ 12 ρ 23 ρ 13 )σ 3 + κ 2 2)κ 3 σ2)κ 2 3 ( ρ 13 σ 3 + κ 2 κ 3 ) 2 e κ1τ ) + 1 κ 3 (2( ρ 13 σ 3 + κ 2 )(ρ 13 σ 3 + κ 1 κ 2 ) 2 κ 1 ( ρ 13 σ 3 + κ 2 κ 3 )(( γ( ρ 13 σ 3 + κ 2 )κ γ( ρ 13 σ 3 + κ 2 ) 2 κ σ 4 (( θκ 2 ρ 13 ρ 14 +σ 4 ( ρ 14 ρ 34 +ρ 13 ))σ 3 +κ 2 (κ 2 ρ 14 θ σ 4 ))κ 3 +σ4(( 2ρ 2 13 ρ 14 ρ 34 + ρ ρ 2 14)σ3 2 2κ 2 ( ρ 14 ρ 34 + ρ 13 )σ 3 + κ 2 2))κ 1 σ 4 κ 3 ((( ρ 12 ρ 34 + ρ 13 ρ 24 )σ 3 κ 2 ρ 24 )κ 3 + (ρ 24 ρ ( ρ 12 ρ 34 ρ 14 ρ 23 )ρ 13 + ρ 12 ρ 14 )σ3 2 + κ 2 (ρ 12 ρ 34 2ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 + κ 2 2ρ 24 )σ 2 )e κ3τ ) κ 1 (( ρ 13 σ 3 + κ 2 ) 2 (κ (( 2ρ 12 ρ ρ 13 )σ 3 2κ 2 )κ 1 + ( 2ρ 12 ρ 13 ρ 23 + ρ ρ 2 13)σ κ 2 (ρ 12 ρ 23 ρ 13 )σ 3 + κ 2 2)κ 2 3( ρ 13 σ 3 + κ 2 κ 3 ) 2 σ2e 2 2κ1τ ) + 2(ρ 13 σ 3 + κ 1 κ 2 ) 2 ( κ 3 (σ4( ρ 2 13 σ 3 + κ 2 ) 2 (κ (( 2ρ 14 ρ ρ 13 )σ 3 2κ 2 )κ 3 + ( 2ρ 13 ρ 14 ρ 34 + ρ ρ 2 14)σ3 2 2κ 2 ( ρ 14 ρ 34 + ρ 13 )σ 3 + κ 2 2)κ 2 1e 2κ3τ ) + (( ( ρ 13 σ 3 + κ 2 ) 2 (α γ)κ θκ 2 σ 4 ρ 14 ( ρ 13 σ 3 + κ 2 )κ σ2 4(( 2ρ 13 ρ 14 ρ 34 + ρ ρ 2 14)σ3 2 2κ 2 ( ρ 14 ρ 34 + ρ 13 )σ 3 + κ 2 2))κ 2 1 (θκ 2 ρ 12 ( ρ 13 σ 3 + κ 2 )κ 3 + σ 4 ((ρ 24 ρ ( ρ 12 ρ 34 ρ 14 ρ 23 )ρ 13 + ρ 12 ρ 14 )σ3 2 + κ 2 (ρ 12 ρ 34 2ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 + κ 2 2ρ 24 ))κ 3 σ 2 κ (( 2ρ 12ρ 13 ρ 23 + ρ ρ 2 13)σ κ 2 (ρ 12 ρ 23 ρ 13 )σ 3 + κ 2 2)κ 2 3σ2)( ρ 2 13 σ 3 + κ 2 κ 3 ) 2 τ)] κ 2 3 ( ρ13σ3+κ2)2 κ 2 1 (ρ13σ3+κ1 κ2)2 ( ρ 13σ 3+κ 2 κ 3) [ 2 ρ 13σ 3 κ 1 κ 2 (2( ρ 13 σ 3 + κ 2 )(κ 1 ρ 23 + (ρ 13 ρ 23 ρ 12 )σ 3 κ 2 ρ 23 )(σ 4 ρ 14 κ 1 σ 2 ρ 12 κ 3 + (ρ 12 σ 2 ρ 14 σ 4 )( ρ 13 σ 3 + κ 2 ))κ 2 3σ 3 κ 1 ( ρ 13 σ 3 + κ 2 κ 3 )σ 2 ) 1 ρ 13σ 3 κ 2 κ 3 (2σ 4 ( ρ 13 σ 3 + κ 2 )(ρ 13 σ 3 + κ 1 κ 2 )(σ 4 ρ 14 κ 1 σ 2 ρ 12 κ 3 + (ρ 12 σ 2 ρ 14 σ 4 )( ρ 13 σ 3 + κ 2 ))κ 3 σ 3 ( κ 3 ρ 34 + ( ρ 13 ρ 34 + ρ 14 )σ 3 + κ 2 ρ 34 )κ 2 1) + (σ 4ρ 14κ 1 σ 2ρ 12κ 3+(ρ 12σ 2 ρ 14σ 4)( ρ 13σ 3+κ 2)) 2 κ 2 3 σ2 3 κ ρ 13σ 3 2κ 2 + ρ 13σ 3 κ 2 (2(ρ 13 σ 3 + κ 1 κ 2 )(σ 4 ρ 14 κ 1 σ 2 ρ 12 κ 3 + (ρ 12 σ 2 ρ 14 σ 4 )( ρ 13 σ 3 + κ 2 ))κ 3 κ 1 ( ρ 13 σ 3 + κ 2 κ 3 )((θκ 2 ( ρ 13 σ 3 + κ 2 )κ 3 + σ 4 (( ρ 13 ρ 34 + ρ 14 )σ 3 + κ 2 ρ 34 )σ 3 )κ 1 (( ρ 13 ρ ρ 12 )σ 3 + κ 2 ρ 23 )κ 3 σ 3 σ 2 )) κ 1 κ 3 (2σ 4 ( ρ 13 σ 3 + κ 2 ) 2 (ρ 13 σ 3 + κ 1 κ 2 )(( ρ 24 κ 3 + ( ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 + κ 2 ρ 24 )κ 1 + ((ρ 12 ρ 34 ρ 13 ρ 24 )σ 3 + κ 2 ρ 24 )κ 3 + ( ρ 24 ρ (ρ 12 ρ 34 + ρ 14 ρ 23 )ρ 13 ρ 12 ρ 14 )σ3 2 κ 2 (ρ 12 ρ 34 2ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 κ 2 2ρ 24 )κ 3 κ 1 ( ρ 13 σ 3 + κ 2 κ 3 )σ 2 ) + 1 κ 1 (2( ρ 13 σ 3 + κ 2 )(ρ 13 σ 3 + κ 1 κ 2 )( κ 3 α( ρ 13 σ 3 + κ 2 )κ (α( ρ 13 σ 3 + κ 2 ) 2 κ 3 σ 4 (( ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 + κ 2 ρ 24 )σ 2 )κ ((( θκ 2 ρ 12 ρ 13 + σ 2 (ρ 12 ρ 23 ρ 13 ))σ 3 + κ 2 (κ 2 ρ 12 θ + σ 2 ))κ 3 + σ 4 ((ρ 24 ρ ( ρ 12 ρ 34 ρ 14 ρ 23 )ρ 13 + ρ 12 ρ 14 )σ3 2 + κ 2 (ρ 12 ρ 34 2ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 + κ 2 2ρ 24 ))σ 2 κ 1 (( 2ρ 12 ρ 13 ρ 23 + ρ ρ 2 13)σ κ 2 (ρ 12 ρ 23 ρ 13 )σ 3 + κ 2 2)κ 3 σ2)κ 2 3 ( ρ 13 σ 3 + κ 2 κ 3 ) 2 ) + 1 κ 3 (2( ρ 13 σ 3 + κ 2 )(ρ 13 σ 3 + κ 1 κ 2 ) 2 κ 1 ( ρ 13 σ 3 + κ 2 κ 3 )(( γ( ρ 13 σ 3 + κ 2 )κ γ( ρ 13 σ 3 + κ 2 ) 2 κ σ 4 (( θκ 2 ρ 13 ρ 14 +σ 4 ( ρ 14 ρ 34 +ρ 13 ))σ 3 +κ 2 (κ 2 ρ 14 θ σ 4 ))κ 3 +σ4(( 2ρ 2 13 ρ 14 ρ 34 + ρ ρ 2 14)σ3 2 2κ 2 ( ρ 14 ρ 34 + ρ 13 )σ 3 + κ 2 2))κ 1 σ 4 κ 3 ((( ρ 12 ρ 34 + ρ 13 ρ 24 )σ 3 κ 2 ρ 24 )κ 3 + (ρ 24 ρ ( ρ 12 ρ 34 ρ 14 ρ 23 )ρ 13 + ρ 12 ρ 14 )σ3 2 + κ 2 (ρ 12 ρ 34 2ρ 13 ρ 24 + ρ 14 ρ 23 )σ 3 + κ 2 2ρ 24 )σ 2 )) κ 1 (( ρ 13 σ 3 + κ 2 ) 2 (κ (( 2ρ 12 ρ ρ 13 )σ 3 2κ 2 )κ 1 + ( 2ρ 12 ρ 13 ρ 23 + ρ ρ 2 13)σ κ 2 (ρ 12 ρ 23 ρ 13 )σ 3 + κ 2 2)κ 2 3( ρ 13 σ 3 + κ 2 κ 3 ) 2 σ2) 2 0.5(1/κ 3 )((ρ 13 σ 3 + κ 1 κ 2 ) 2 σ4( ρ 2 13 σ 3 + κ 2 ) 2 (κ (( 2ρ 14 ρ ρ 13 )σ 3 2κ 2 )κ 3 +( 2ρ 13 ρ 14 ρ 34 +ρ ρ 2 14)σ3 2 2κ 2 ( ρ 14 ρ 34 +ρ 13 )σ 3 +κ 2 2)κ 2 1)] 9

10 Figure 1: The Basic Priciple Behind Filtering Technology 3 Methodology 3.1 The Kalman Filter Algorithm In the past decades, filtering technology has become an established framework in which a state-space model can be well analysed. Indeed, filter techniques have been used in many domains of communication technology, radar tracking, satellite navigation and applied physics, signal processing, economics, econometrics, and finance. In the following, the Kalman Filter algorithm will be explained. The basic principles behind filtering technology are not very complicated: using Bayes theory, filters can use the information about current observation to predict the values of unobservable variables at next time point, and then update the information and forecast the situation at next time point (Pasricha, 2006). To be more specific, the process of filter technology can be described in Figure 1. Any state space model contains two parts that the state variable x k for k = 1, 2..., K and the observations y k for k = 1, 2..., K, where K is the number of observations of the time variable. Normally, x k = f k (x k 1, v k 1 ), where x k and x k 1 are the state at time point k and its previous time point, and assume the x k is following a first-order Markov process as: x k x k 1 p xk x k 1 (x k x k 1 ). As for the observations, the relationship between state variable(s) and the observations can be described as: z k = h k (x k, n k ), where x k is the state variable(s) and n k is the measurement noise at time point k. Denote z 1:k as the estimates of the x k from the start of the time series to the updated time point k, and the observations are conditionally independently provided x k as: p = (z k x k ). Here, p(x 0 ) can be either given or be obtained 10

11 as an assumption. When k 1, denote p(x k x k 1 ) as the state transition probability. Let f k be any integrable function that depends on all the state and the whole trajectory in state space; then, the expectation of f k (x 0:k ) can be calculated as: E(f k (x 0:k )) = f(x 0:k )p(x 0:k z 1:k )dx 0:k Essentially, the recursive filters consist of two steps: The first step is the named as prediction step, which spreads the state probability density function because of noise; the second step is the update step, which combines the likelihood of the current measurement with the predicted state. Their mathematic expressions are p(x k 1 z 1:k 1 ) p(x k z 1:k 1 ) and p(x k z 1:k 1 ), z k p(x k z 1:k ), respectively. Then there are two probability density functions for the above steps. For the prediction step, assume the probability density function p(x k 1 z 1:k 1 ) is available at time point k-1, using the Chapman-Kolmogoroff equation, the prior probability of the state at time point k can be expressed as: p(x k z 1:k 1 ) = p(x k x k 1 )p(x k 1 z 1:k 1 )dx k 1 As for the update step, the posterior probability density function is Then, the p(x k z 1:k ) = p(z k x k )p(x k z 1:k 1 ) p(z k z 1:k 1 ) can be obtained (Muehlich, 2003). p(z k z 1:k 1 ) = p(z k x k )p(x k z 1:k 1 )dx k When this recursive system is considered in practice, in general, the recursive propagation of the posterior density is only a conceptual solution, but solutions definitely exist in some restrictive cases. For example, based on this recursive process, the Kalman Filter was developed by Kalman in 1960, and then rapidly became a widely used method in state-space models to calculate optimal estimates of unobservable state variables. In other words, the Kalman Filter can be seen as an optimal algorithm. Recalling the measurement equation z k = h k (x k, n k ) and the state equation x k = f k (x k 1, v k 1 ), in a linear system, those two equations can be defined as: { xk = F k x k 1 + v k 1 z k = H k x k + n k (1) where the random variables v and n represent the noises that are assumed to be independent and with normal probability distributions p(v) N(0, Q) and p(n) N(0, R). If F k and H k are assumed to be constants, to simplify the system, the two equations can be then rewritten as: 11

12 { xk = Ax k 1 + v k 1 z k = Hx k + n k (2) where A and H are known matrices. Denote x k and x k to be prior and posterior state estimates at time point k, respectively. Then the prior and posterior estimate errors can be defined as e k x k x k and e k x k x k at the time point k, respectively. In a similar way, the prior and posterior estimated error covariance can be obtained as P k = E[e k e T k ] and P k = E[e k e T k ]. Based on z k = H k x k + n k, x k = x k + K(z k H) x k can be obtained, where K = P k HT (HP k HT + R) 1 (Jacobs, 1993). The Kalman Filter algorithm is an optimal algorithm to solve a system with state variables. However, there is a limitation that cannot be ignored. As described, the traditional Kalman Filter algorithm needs a strict Gaussian assumption for the posterior density at each time point. p(x k z 1:k ) is proved to be Gaussian as p(x k 1 z 1:k 1 ) is assumed to be Gaussian. Regarding the Kalman Filter algorithm as a recursive process and connected with and p(x k z 1:k 1 ) = p(x k x k 1 )p(x k 1 z 1:k 1 )dx k 1 p(x k z 1:k ) = p(z k x k )p(x k z 1:k 1 ) p(z k z 1:k 1 ) the prior and the posterior density probabilities can be written as: and with and p(x k 1 z 1:k 1 ) = N(x k 1 ; m k 1 k 1, P k 1 k 1 ) p(x k z 1:k 1 ) = N(x k ; m k k 1, P k k 1 ) p(x k z 1:k ) = N(x k ; m k k, P k k ) m k k 1 = F k m k 1 k 1, P k k 1 = Q k 1 + F k P k 1 k 1 F T k, m k k = m k k 1 + K k (z k H k m k k 1 ) P k k = P k k 1 K k H k P k k 1 = (I K k H k )P k k 1 where N(x; m, P ) is a Gaussian density with argument x, mean m and covariance P. Since is known, K = P k HT (HP k HT + R) 1 K k = P k k 1 H T k (H kp k k 1 H T k + R k) 1 12

13 can be obtained. In this case of implementing of the Kalman Filter, there are unknown parameters needed to be estimated based on the initial set. According to Harvey (1989), the joint density can be written as: L(z; Ψ) = K k=1 p(z k), where p(z k ) is the (joint) probability density function of t-th set of observations, and Ψ is the set of unknown parameters, when the T sets of observations z 1,...z K are independently and identically distributed. However, the sets of observations are not independent; therefore, the aforementioned L(z; Ψ) cannot be applied. The probability density needs to be set conditionally as: L(z; Ψ) = K k=1 p(z k Z k 1 ), where the capital Z k 1 is denoted as Z k 1 = {z k 1, z k 2,..., z 1 }. The distribution of z k conditional on z k is itself normal, if the initial state vector and the disturbances have multivariate normal distributions. Since the expectation of the z k at time point k 1 is based only on the information at k 1, the likelihood function can be finally written as: log L = NK 2 log 2π 1 K 2 k=1 log D k 1 2 log K where v k = z k z k k 1 and D k = H k P k k 1 H k + R 3.2 The Extended Kalman Filter Algorithm k=1 v k D 1 k The extended Kalman Filter algorithm is a very useful development. By using this extended Kalman Filter algorithm, the measurement function or/and the state function does not need to be linear anymore. Hence, the measurement equation z k = h k (x k, n k ) and the state equation x k = f k (x k 1, v k 1 ) cannot be expressed as v k { xk = F k x k 1 + v k 1 z k = H k x k + n k (3) anymore. To run the filter algorithm, a local linearisation of the aforementioned equations might be a description of the nonlinear system. Then the p(x k 1 z 1:k 1 ), p(x k z 1:k 1 ) and p(x k z 1:k ) are approximated by a Gaussian distributions as and p(x k 1 z 1:k 1 ) N(x k 1 ; m k 1 k 1, P k 1 k 1 ) p(x k z 1:k 1 ) N(x k ; m k k 1, P k k 1 ) p(x k z 1:k ) N(x k ; m k k, P k k ) m k k 1 = f k (m k 1 k 1 ) P k k 1 = Q k 1 + F k P k 1 k 1 Fk T 13

14 m k k = m k k 1 + K k (z k h k (m k k 1 )) and P k k = P k k 1 K k Ĥ k P k k 1 = (I K k Ĥ k )P k k 1 where is known as gain, and and K k = P k k 1 Ĥ k T ( Ĥ k P k k 1 Ĥ k T + Rk ) 1 F k = df k(x) dx Ĥ k = dh k(x) dx x=mk 1 k 1 x=mk 1 k 1 are Jacobian matrices. This process is known as the extended Kalman filter algorithm. The traditional Kalman Filter algorithm does not work in this paper, based on the aforementioned model. Hence, the model will be run with the extended Kalman Filter. The principles of the Kalman Filter argorithm and the extended Kalman Filter argorithm are the same, except for the linearization, hence, in this paper, the likelihood function and the maximum likelihood value of the Kalman Filter is seen reasonable approximations of the likelihood function and the maximum likelihood value of the extended Kalman Filter argorithm. 4 Data and Assumptions In this paper, to simplify the complex state space model, weekends and other non-trading days can be ignored, meaning that trading days are considered to be continuous. Based on the efficient market hypothesis, powerful information from non-trading days (e.g. weekends and the Christmas holiday) can be immediately reflected in prices after a non-trading day. Hence, this is a reasonable and popular assumption in the financial world. The next assumption is that each futures contract is immediately executed on the first day they mature. This is an assumption about the length of the maturity. Based on this assumption, the length of the maturity can be measured more accurately. The third assumption is a measurement of the level of the drift of crude oil. Since the interest rate r is considered to be instantaneous, and based on the no-arbitrage assumption, the drift of crude oil equals the instantaneous interest rate, but not a constant any more. In addition, in this paper, instead of a single average T, the dynamics of the T over time are added to the model, which is expected to provide a better fit for the data. As has been explained in the previous sections, since the spot price of crude 14

15 oil is set as an unobservable state variable, the observable futures prices are the only collected data when the model is implemented. In this paper, the data of futures prices of WTI crude oil were collected from Bloomberg. Specifically, in this paper, 12 future contracts will be used, their maturities are from Feb 2013 to Jan 2014, respectively, and the test period is an entire year of Empirical Results The estimated state variables of the three-factor model are shown within Figures 2-5. To be specific, Figure 2 exhibits the estimated spot price of WTI crude oil, which fluctuates between about 115 dollars and about 80 dollars per barrel in the test period. The estimated spot price shows a downward trend in the test period. It fluctuates over 100 dollars per barrel, but then plummets to about 80 dollars per barrel. In the second half of the test period, it rebounds a little bit and then decreases again. Figure 3 shows the estimated convenience yield of the three-factor model. The estimated convenience yield also shows a downward trend in the entire test period. It rapidly decreases from 0.02 to -0.04, and, after a rebound to about 0.02, the estimated convenience yield keeps moving down to about at the end of the test year. Figure 4 exhibits the estimated interest rate, which fluctuates at around zero during the entire test period. Specifically, it fluctuates in an interval between 0.04 and Furthermore, Figure 5 shows the implied stochastic volatility of the spot price in the three factor model. After a stable fluctuation at around 0.2, the implied sigma1 starts to move upward to 0.5 at the end of the test period. However, the estimated state variables of the four-factor model are shown within Figures 6-9. First, the estimated spot price and the convenience yield of WTI crude oil are shown in Figures 6 and 7, respectively. Compared with the estimated spot price from the three-factor model, the estimated spot price from the four-factor model looks extremely similar to the one shown in Figure 2, except the peak of the estimated spot price from the four-factor model is slightly lower; and the valley of the estimated spot price from the four-factor model is also slightly lower than the estimated spot price from the three-factor model. However, the estimated convenience yields of WTI crude oil from the both models are significantly different. Similar to the one estimated from the three-factor model, the estimated convenience yield from the four-factor model keeps moving downward after the beginning of the entire test period. To be more specific, at the beginning, the estimated convenience yield decreases from 0.02 to about After that, it rapidly increases to about 0. Then, it fluctuates around Then, the interval of the fluctuation of the convenience yield suddenly invreases. To be more specific, the estimated convenience yield drops to about -0.12, which is the valley during the entire period. Then, it fluctuates around Moreover, the estimated interest rate is shown in Figure 8. Unlike the estimated interest rate from the three-factor model which fluctuates around zero, 15

16 Figure 2: The Estimated Spot Price OfFigure 3: The Estimated Convenience WTI Crude Oil From The Three-Factor Yield Of WTI Crude Oil From The Model Three-Factor Model Figure 4: The Estimated Interest RateFigure 5: The Estimated Sigma 1 Of Of WTI Crude Oil From The Three- WTI Crude Oil From The Three-Factor Factor Model Model 16

17 the estimated interest rate from the four-factor model is actually above zero for most trading days. To be more specific, the estimated interest rate first decreases down to about and then increases to about 0.08 in the first half of the test period. In the second half of the test period, the estimated interest rate keeps fluctuating around 0.04 at the end of the test period. Last, the estimated volatility of the spot price from the four-factor model is shown in Figure 9. The estimated volatility of the spot price is very stable. It fluctuates in a small interval between and To be more specific, the estimated σ 1 drops to about at the beginning of the test period, and then it fluctuates for a while. After the fluctuation, the estimated σ 1 suddenly decreases to about After the drop to about -0.1, it keeps moving upto a relatively high value at the end of the test period. This is also considerably different from the implied volatility of the spot price from the three-factor model, which is significant higher and fluctuates in a larger interval. Estimated Parameters κ α α Three factor model (0.0853) (0.0011) (0.0175) Estimated Parameters m σ 2 σ 3 Three factor model (SE1) (SE2) (SE3) Estimated Parameters ρ 12 ρ 13 ρ 23 Three factor model (SE4) (0.1225) (SE5) Note:SE1 = e 03 + ComputationalError1 SE2 = e 05 + ComputationalError2 SE3 = e 04 + ComputationalError3 SE4 = e 02 + ComputationalError4 SE5 = e 01 + ComputationalError5 T able1 The estimated parameters from the three-factor model and the four factor model are shown in Tables 1 and 2, respectively. Since they are estimated from two totally different models, comparing the estimated parameters might be meaningless. Hence, in this paper, we compare the effectiveness and usefulness of the parameters rather than compare them to each other. Because of the limitation of space in this paper, all 12 futures contracts will not be shown their fittings, but the first six, which mature in February, March, April, May, June and July in 2013, will be shown for both models. To be more specific, there will 17

18 Figure 6: The Estimated Spot Price OfFigure 7: The Estimated Convenience WTI Crude Oil From The Four-Factor Yield Of WTI Crude Oil From The Model Four-Factor Model Figure 8: The Estimated Interest RateFigure 9: The Estimated Sigma 1 Of Of WTI Crude Oil From The Four- WTI Crude Oil From The Four-Factor Factor Model Model 18

19 be six pictures appropriate for each model. Estimated Parameters κ 1 κ 2 κ 3 Four factor model (0.0917) ( ) (0.0445) Estimated Parameters σ 2 σ 3 σ 4 Four factor model (0.0753) (1.7320) (0.0201) Estimated Parameters ρ 12 ρ 13 ρ 14 Four factor model (NA) (NA) (NA) The fitting results of the three-factor model is shown from Figure 10 to Figure 15. The rank of the title of each picture corresponds to the rank of its maturity. For example, the title Compare with real futures three-factor model 1 and the title Compare with real futures three-factor model 2 correspond to the future contracts that matured in February and March 2013, respectively, and so forth. On the other side, the fitting results of the four-factor model is shown from Figure 16 to Figure 21. Similarly with the three-factor model, the rank of the title of each picture also corresponds to the rank of its maturity. For example, the title Compare with real futures four-factor model 1 and the title Compare with real futures four-factor model 2 correspond to the future contracts that matured in February and March 2013, respectively, and so forth. It is not hard to say that both models are useful in pricing a futures contract with a particular maturity, because all 12 pictures are show that the estimated model price of each futures contract is very close to the real observed price of the futures contract. Then, the six chosen futures contracts are shown together from Figure 22 to Figure 24 to see if there are significant differences in the effectiveness and usefulness of the three-factor and the four-factor models. To be specific, the real observed futures prices of the six chosen futures contracts are shown in Figure 22. Then, the estimated model futures prices of the six chosen future contracts, which are estimated from the three-factor model, are shown in FIgure 23. Last, the estimated model futures prices of the six chosen futures contracts, which are estimated from the four-factor model, are shown in Figure 24. To see the trend clearly, in Figure 23 and Figure 24, only 26 points are chosen in each figure, and those were picked the first trading day in each of the 10 trading days. There are not significant differences among the three figures. However, WTI crude oil did not follow a strict backwardation or a strict contango during However, it is not hard to see that the trend of backwardation and contango is easier to find 19

20 Figure 10: Fitting Results Of The Figure 11: Fitting Results Of The Three-Factor Model For The First Chosen Futures Contract Chosen Futures Three-Factor Model For The Second Contract Figure 12: Fitting Results Of The Figure 13: Fitting Results Of The Three-Factor Model For The Third Three-Factor Model For The Fourth Chosen Futures Contract Chosen Futures Contract Figure 14: Fitting Results Of The Figure 15: Fitting Results Of The Three-Factor Model For The Fifth Chosen Futures Contract sen Futures Three-Factor Model For The Sixth Cho- Contract 20

21 Figure 16: Fitting Results Of The Four- Figure 17: Fitting Results Of The Four- Factor Model For The First Chosen Fu-Factotures Contract Futures Model For The Second Chosen Contract Figure 18: Fitting Results Of The Four- Figure 19: Fitting Results Of The Four- Factor Model For The Third Chosen Fu-Factotures Contract Futures Model For The Fourth Chosen Contract Figure 20: Fitting Results Of The Four- Figure 21: Fitting Results Of The Four- Factor Model For The Fifth Chosen Futures Contract tures Factor Model For The Sixth Chosen Fu- Contract 21

22 Estimated Parameters ρ 23 ρ 24 ρ 34 Four factor model (NA) (NA) (0.2374) Estimated Parameters α θ γ Four factor model (0.0131) (0.6096) (0.0194) T able2 when WTI crude oil stably fluctuates, while all future prices seem to be close when the price of WTI crude oil rapidly increases or rapidly decreases. By comparing Figure 23 and Figure 24 carefully, the interval of the differences between different futures contracts by the four-factor model is narrower when WTI crude oil shows a trend of backwardation at the beginning of the test period. This might also imply that the three-factor model with the implied volatility of the spot price is a better choice when the underlying asset of the future contracts stably fluctuates,if a research aims to price a futures contract with a particular maturity. Last, the forward curves from both three-factor and four-factor models are shown within Figures 25 to Figure 30. The first three pictures (from Figure 25 to Figure 27) show the forward curvs on the 50th, 100th and 200th trading days during the test period by the three-factor model. The three following pictures (from Figure 28 to Figure 30) show the forward curves on the same trading days by the four-factor model. First, on the 50th trading day, the estimated prices of both three-factor and four-factor models are slightly higher than the observed prices. Second, on the 100th trading day, the estimated prices of the three-factor model are still slightly higher than the observed prices, while the estimated prices of the four-factor model are almost the same as the observed prices. Last, on the 200th trading day, the estimated prices of both three-factor and four-factor models are crossed with the observed prices. As expected, the estimated prices from both models are close to the observed prices. When on compares the two models, the differences between the estimated prices and the observed prices from the four-factor model seem to be slightly larger. 6 Conclusion Based on the previous analysis and exhibited pictures, the original Schwartz three-factor model with the implied stochastic volatility of the spot price of the underlying asset and the expanded four-factor model are both useful in the domain of WTI crude oil futures markets. However, they have different features for different purposes of research or for use in real life. This paper not only 22

23 Figure 22: The Real Observed Futures Prices Of The Six Chosen Futures Contracts Figure 23: The Estimated Model Futures Prices Of The Six Chosen Future Contracts From The Three-Factor Model 23

24 Figure 24: The Estimated Model Futures Prices Of The Six Chosen Future Contracts From The Four-Factor Model provides a new, expanded four-factor model, but also points out the features of the applications of the Schwartz three-factor model and the new four-factor model. However, this paper does not explain why some of the estimated state variables based on the same data are significantly different in the three-factor model and the four-factor model. For example, the estimated convenience yield from the four-factor model is positive, while the estimated convenience yield from the three-factor model is negative for most trading days in the test period. In the future, people might explain why adding the new factor of the volatility of the spot price to the model system can cause changes in the estimated state variables. 24

25 Figure 25: Figure 26: Figure 27: Figure 28: Figure 29: Figure 30: 25

26 References Alizadeh, A, H. Lin S. and Nomikos, N. (2004) Effectiveness of oil futures contracts for Hedging international crude oil prices, Faculty of Finance, Cass business school, City University. Bjerksund P. (1991) Contingent claims evaluation when the convenience yield is stochastic: Analytical results, Working Paper, Norwegian School of Economics and Business Administration. Brennan, Michael J. (1958) The supply of storage, American Economic Review 48, Brennan, Michael, J. and Schwartz, Eduardo, S. (1979) A continuous time approach to the pricing of bonds. Journal of Banking and Finance 3 (1979) North-Holland Publishing Company. Carmona, R. and Ludkovski, M. (2004) Spot convenience yield models for the Energy Markets, Contemporary Mathematics, Volume 351, Casassus, J. and Collin-Dufresne, P. (2005) Stochastic Convenience Yield Implied From Commodity Futures and Interest Rates, Journal of Finance, 60, Chen, Jilong. Ewald, Christian,Ewald, O. and Zong, Zhe. (2014) Pricing Gold Futires with Three Factor Models in Stochastic Volatility Case. Working Paper. Deaton, A. and Laroque, G. (1992) On the behavior of commodity prices, Review of Economic Studies 59, 123. Fan, J. (2008) A selective overview of nonparametric methods in financial econometrics (with discussion), Department of Operation Research and Financial Engineering, Princeton University, Statistical Science 20, Geman, H. (2003) Commodities and commodity derivatives, Modeling and pricing for Agricultural, Metals and Energy, Press: John Wiley & Sons, Ltd. Gibson, R. and Schwartz, E, S. (1990) Stochastic convenience yield and the pricing of oil contingent claims, Journal of Finance, 45,