Schwartz 1997 Two-Factor Model Technical Document

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1 Schwartz 997 Two-Factor Model Technical Document Philipp Erb David Lüthi Simon Otziger February, 4 Abstract The purpose of this document is to give the formulas and relations needed to understand the Schwartz two-factor commodity model Schwartz, 997. This includes parameter estimation using the Kalman filter, pricing of European options as well as computation of risk measures. Introduction This document describes the Schwartz two-factor model to the extent which is necessary to understand the R package schwartz97. The two factors are the spot price of a commodity together with its instantaneous convenience yield. It was introduced in Gibson and Schwartz 99 and extended in Schwartz 997 for the pricing of futures contracts. Miltersen and Schwartz 998 and Hilliard and Reis 998 presented equations for arbitrage free prices of European options on commodity futures. In what follows we fully rely on the above mentioned articles and state the corresponding formulas. In addition we derive the transition density of the two state variables. Model The spot price of the commodity and the instantaneous convenience yield are assumed to follow the joint stochastic process: ds t = µ δ t S t dt + σ S S t dw S dδ t = α δ t dt + σ ɛ dw ɛ, with Brownian motions W S and W ɛ under the objective measure P and correlation dw S dw ɛ = ρdt. Under the pricing measure Q the dynamics are of the form ds t = r δ t S t dt + σ S S t d W S 3 dδ t = [α δ t λ]dt + σ ɛ d W ɛ,. 4 where the constant λ denotes the market price of convenience yield risk and W S and W ɛ are Q-Brownian motions. It may be handy to introduce a new mean-level for the convenience

2 yield process under Q which leads to the dynamics α = α λ/, 5 dδ t = α δ t dt + σ ɛ d W ɛ. 6 3 Distributions 3. Joint Distribution of State Variables The log-spot X t = logs t and the convenience yield δ t are jointly normally distributed. The transition density is X t µx t σ N, X t σ Xδ t δ t µ δ t σ Xδ t σδ t, 7 with parameters µ X t = X + µ σ S α t + α δ t µ δ t = e t δ + α t 9 σxt = σ ɛ t t + t + σ Sσ ɛ ρ t t + σ St σδ t = σ ɛ t σ Xδ t = { σ S σ ɛ ρ σ ɛ e t + σ ɛ t }. The mean-parameters given in 8 and 9 refer to the P-dynamics. To obtain the parameters under Q one can simply replace µ by r and α by α defined in equation 5. Let the Q-parameters be denoted by µ X t and µ δ t. 4 Futures Price It is worth to mention that the futures and forward price coincide since in our model the interest rate is assumed to be constant. In the rest of this document, the statements made about futures contracts therefore also hold for forward contracts. Let the futures price at time t with time to maturity τ = T t be GS t, δ t, t, T. For notational convenience we assume t = in what follows. At time zero the futures price is given by the Q-expectation of S T. { GS, δ,, T = E Q S T = exp µ X T + } σ XT 3 A derivation can be found in appendix A. 8 = S e AT +BT δ 4

3 with AT = r α + σɛ σ Sσ ɛ ρ T + T 4 σ ɛ 3 + α + σ S σ ɛ ρ σ ɛ T, 5 BT = T Distribution of Futures Prices According to 3 the futures price follows a log-normal law. That is, at time zero the T - futures price at time t has the following distribution under Q : where log GS t, δ t, t, T N µ G t, T, σ Gt, T, 7 µ G t, T = µ X t + AT t + BT t µ δ t 8 σ Gt, T = σ Xt + BT tσ Xδ t + BT t σ δ t. 9 5 European Commodity Options The fair price of a European call option on a commodity futures contract was derived as a special case of more general models in Miltersen and Schwartz 998 and Hilliard and Reis 998. Here we give the formula for the two-factor model. In this setting, the price of a European call option C at time zero with maturity t, exercise price K written on a commodity futures contract with maturity T is given by C G = E Q [ e r t GS t, δ t, t, T K +] Since the futures price GS t, δ t, t, T is log-normally distributed we obtain a Black-Scholes type formula for the call price C G. with C G = P, t {G, T Φd + KΦd } log G,T K ± σ d ± = σ σ = σst + σ Sσ ɛ ρ e T e t t + σ ɛ t + e T e t e T e t and Φ being the standard Gaussian distribution function. Note that the Q-dynamics is primarily interesting to value derivatives on the futures price. For simulation studies and dynamic financial analysis the real-world P dynamics is of relevance. To get the P-dynamics all tilde-parameters have to be replaced by the ones without tilde. 3

4 The following put-call parity is established C G P G = P, t {G, t K}. Thus, the price for a European put option P G at time zero with maturity t, exercise price K written on a commodity futures contract with maturity T becomes P G = P, t {KΦ d G, T Φ d + }. 3 Remark: For the special case when the exercise time t of the option and the maturity T of the futures contract coincide, formulas and 3 still hold. However, the options we price for t = T are no longer options written on futures contracts with maturity T but rather options with exercise time T, written on a commodity spot contract. 6 Parameter Estimation This section demonstrates an elegant way of estimating the Schwartz two-factor model. That is estimating the model parameters using the Kalman filter as in Schwartz 997. Subsection 6. shows how the Schwartz two-factor model can be expressed in state space form. Once the model has been cast in this form the likelihood can be computed and numerically maximized. 6. State Space Representation Let y t denote a n vector of futures prices observed at date t and α t denote the state vector of the spot price and the convenience yield. The state space representation of the dynamics of y is given by the linear system of equations y t = c t + Z t α t + G t η t 4 α t+ = d t + T t α t + H t ɛ t, 5 where ɛ t N, I and η t N, I n. G t and H t are assumed to be time-invariant. The errors innovations in the measurement equation 4 are further assumed to be independent in the implementation of this package G t G t = g gnn Using the functions A and B defined in 5 and 6 the components of the state space representation 4 and 5 are Xt+ t α t+ t = 7 δ t+ t T t = e t e t 8 µ d t = σ S α t + α t α t 9 H t H σ t = X t σ Xδ t σ Xδ t σδ t 3 4

5 log G t Bm t y t =. Z t =.. 3 log G t n Bm t n Am t g c t = G t G t = 3. Am t n... g nn where X t = log S t, t = t k+ t k and m t i denotes the remaining time to maturity of the i-th closest to maturity futures G t i. 5

6 A Derivation of the joint distribution The joint dynamics of the commodity log-price X t = log S t and the spot convenience yield δ t reads in an orthogonal decomposition of and dx t = µ δ t σ S dt + σ S ρ dwt + σ S ρdwt 33 dδ t = α δ t dt + σ ɛ dw t, 34 Equation 34 can be solved using the substitution δ t = e t δ t and Ito s lemma. Plugging 35 into 33 gives X t = X + = X + δ t = e t δ + α t + σ ɛ e t e u dwu 35 Let s have a look at the integral δ udu. δ u du = For the integral integration: dx u 36 µ σ S t δ u du + σ S ρ dwu + σ S ρdwu. 37 e u δ du + e u u α u du + u σ ɛ e u e s dws du 38 es dw s du we use Fubini s theorem to interchange the order of u e u e s dws du = = Plugging eq. 4 into eq. 38 and solving the Riemann integrals yields δ u du = δ t + αt α t + σ ɛ s e u e s du dws 39 t s dw s 4 t s dws. 4 This leaves us with the following expression for X t : X t = X + µ σ S α t + α δ t + { + σ S ρ dwu + σ S ρ + σ } ɛ e t u dw u. 4 The log-spot X t and the convenience yield δ t are jointly normally distributed with expectations EX t = µ X = X + µ σ S α t + α δ t 43 Eδ t = µ δ = e t δ + α t. 44 6

7 The variance are obtained using expectation rules for Ito integrals and the Ito isometry. { varx t = σx = σ ɛ t t } + t + σ Sσ ɛ ρ t t + σ St 45 varδ t = σδ = σ ɛ t 46 covx t, δ t = σ Xδ = [ σ S σ ɛ ρ σ ɛ e t + σ ɛ t ] 47 References Rajna Gibson and Eduardo S. Schwartz. Stochastic convenience yield and the pricing of oil contingent claims. The Journal of Finance, 453: , 99. Jimmy E. Hilliard and Jorge Reis. Valuation of commodity futures and options under stochastic convenience yields, interest rates, and jump diffusions in the spot. Journal of Financial and Quantitative Analysis, 33:6 86, 998. Kristian R. Miltersen and Eduardo S. Schwartz. Pricing of options on commodity futures with stochastic term structures of convenience yields and interest rates. Journal of Financial and Quantitative Analysis, 33:33 59, 998. Eduardo S. Schwartz. The stochastic behavior of commodity prices: Implications for valuation and hedging. Journal of Finance, 53:93 973,