A pricing measure for non-tradable assets with mean-reverting dynamics Salvador Ortiz-Latorre, UiO joint work with Fred E. Benth AMAMEF & Swiss Quote Conference EPFL, Laussane 7-10 September 2015 Support from the projects EMMOS and MAWREM is gratefully acknowledged
Non tradable assets I An asset is non tradable if we cannot build a portfolio with it. I Energy related examples: non-storable commodities, weather indices, electricity... I Interested in pricing forwards contracts on these assets. I There is no buy and hold strategy =) classical non-arbitrage arguments break down. I Any probability measure Q equivalent to the historical measure P is a valid risk neutral pricing measure. I The forward price with time to delivery 0 < T < T at time 0 < t < T is given by F Q (t, T ) = E Q [S(T )jf t ] where F t is the information in the market up to time t. Assuming deterministic interest rates and r = 0.
Risk premium pro le I The risk premium for forward prices is de ned by RQ F (t, T ), E Q [S(T )jf t ] E P [S(T )jf t ]. I Goal: To be able to obtain more realistic risk premium pro les. For instance: R 4 2 2 4 50 100 150 200 250 300 350 where τ = T t is the time to delivery.
Mathematical model I Let (Ω, F, ff t g t2[0,t ], P) be a ltered probability space satisfying the usual hypothesis, where T > 0 is a xed nite time horizon. I Consider a standard Brownian motion W and a pure jump Lévy subordinator Z L(t) = zn L (ds, dz), t 2 [0, T ], 0 0 where N L (ds, dz) is a Poisson random measure with Lévy measure ` satisfying R 0 z`(dz) <. I Let κ L (θ), log E P [e θl(1) ], and Θ L, supfθ 2 R + : E P [e θl(1) ] < g. I A minimal assumption is that Θ L > 0.
Mathematical model I Consider the Ornstein-Uhlenbeck processes X with Barndor -Nielsen & Shephard stochastic volatility σ X (t) = X (0) α X (s)ds + σ(s)w (t), 0 0 σ 2 (t) = σ 2 (0) ρ 0 σ2 (s))ds + L(t) Z = σ 2 (0) + 0 (κ0 L (0) ρσ2 (s))ds + 0 0 with t 2 [0, T ], α, ρ > 0, X (0) 2 R, σ 2 (0) > 0 and zñ L (ds, dz), Ñ L (ds, dz) = N L (ds, dz) `(dz)ds. I We model the spot price process by S(t) = Λ g exp (X (t)), t 2 [0, T ].
The change of measure I Let β = (β 1, β 2 ) 2 [0, 1] 2 and θ = (θ 1, θ 2 ) 2 D L, R D L, where D L, (, Θ L /2). I Consider the following family of kernels G θ1,β 1 (t), σ 1 (t) (θ 1 + αβ 1 X (t)), t 2 [0, T ], H θ2,β 2 (t, z), e θ 2z 1 + ρβ 2 κl 00(θ 2) zσ2 (t ), t 2 [0, T ], z 2 R +. I Next, de ne the following family of Wiener and Poisson integrals G θ1,β 1 (t), G θ 1,β 1 (s)dw (s), t 2 [0, T ], 0 Z H θ2,β 2 (t), H θ2,β 2 (s, z) 1 Ñ L (ds, dz), t 2 [0, T ], 0 0 associated to the kernels G θ1,β 1 and H θ2,β 2, respectively.
The change of measure I The family of measure changes is given by Q θ, β P, β 2 [0, 1] 2, θ 2 D L, with dq θ, β dp, E( G θ1,β 1 + H θ2,β 2 )(t), t 2 [0, T ], Ft where E() denotes the stochastic exponential. I Recall that, if M is a semimartingale, the stochastic exponential of M is the unique strong solution of de(m)(t) = E(M)(t )dm(t), t 2 [0, T ], E(M)(0) = 1, which is given by E(M)(t) = exp M(t) 1 2 hmc, M c i(t) (1 + M(s)) e M (s). 0st
The change of measure I Yor s Addition Formula: Let M 1 and M 2 two semimartingales starting at 0. Then, E(M 1 + M 2 + [M 1, M 2 ])(t) = E(M 1 )(t)e(m 2 )(t), 0 t T, where [M 1, M 2 ](t) = hm1 c, Mc 2 i(t) + M 1 (s) M 2 (s). st I As [ G θ1,β 1, H θ2,β 2 ] 0, we can write E( G θ1,β 1 + H θ2,β 2 )(t) = E( G θ1,β 1 )(t)e( H θ2,β 2 )(t), t 2 [0, T ]. I Conditioning on F L T, we have E P [E( G θ1,β 1 + H θ2,β 2 )(T )] = E P [E P [E( G θ1,β 1 )(T )jf L T ]E( H θ2,β 2 )(T )]. I Hence the problem is reduced to show that E( G θ1,β 1 ) and E( H θ2,β 2 ) are true martingales with respect to appropriate ltrations.
Sketch of the proof that E(M) is a martingale I M can be G θ1,β 1 or H θ2,β 2. I Localize E(M) using a reducing sequence fτ n g n1. I Check that 1 = lim n! E P [E(M) τn (T )] = E P [E(M)(T )]. I Test the uniform integrability of fe(m) τ n (T )g n1 with F (x) = x log(x), i.e. sup E P [F (E(M) τ n (T ))] <. (1) n I For any n 1, fe(m) τ n (t)g t2[0,t ] is a true martingale and induces a change of measure dq n dp = E(M) τ n (t). Ft I Condition (1) can be rewritten as sup n E Q n [log(e(m) τ n (T ))] <. I We can get rid of the ordinary exponential in E(M) τ n (T ). I The problem is reduced to nd a uniform bound for the second moment of X and σ 2 under Q n.
The dynamics under the new pricing measure I By Girsanov s theorem for semimartingales, we can write X (t) = X (0) + BQ X (t) + σ(s)dw θ, Q (t), t 2 [0, T β 0 θ, β ], Z σ 2 (t) = σ 2 (0) + BQ σ2 (t) + zñ θ, β Q L (ds, dz), t 2 [0, T ], 0 0 θ, β where B X Q θ, β (t) = 0 (θ 1 α(1 β 1 )X (s))ds, t 2 [0, T ], and BQ σ2 (t) = κ 0 θ, β L (θ 2) ρ(1 β 2 )σ 2 (s) ds, t 2 [0, T ]. 0
The dynamics under the new pricing measure I The Q θ, β -compensator measure of σ2 is given by vq σ2 (dt, dz) = θ, β eθ 2z 1 + ρβ 2 κl 00(θ 2) zσ2 (t ) `(dz)dt. I Using integration by parts, we get X (T ) = X (t)e α(1 β 1)(T t) θ + 1 α(1 β 1 ) (1 e α(1 β 1)(T t) ) Z T + σ(u)e α(1 β 1)(T t u) dw Q (u), θ, β σ 2 (T ) = σ 2 (t)e α(1 β 2)(T t) + κ0 L (θ 2) where 0 t T T. α(1 β 2 ) (1 e α(1 β 2)(T t) ) Z T Z + e ρ(1 β 2)(T u) zñq L (du, dz), t 0 θ, β
Moment condition under the historical measure I A su cient condition for S to have nite expectation under P is the following: Assumption (P) We assume that α, ρ > 0 and Θ L satisfy for some δ > 0. 1 ρ 2ρ 2α 1 ρ 1 2α Θ L δ, I If Θ L = then assumption P is satis ed. I If Θ L <, then if we choose ρ close to zero the value of α must be bounded away from zero, and vice versa, for assumption P to be satis ed.
Forward price formula Proposition The forward price F Q θ, (t, T ) is given by β F Q (t, T ) θ, β! = Λ g (T ) exp X (t)e α(1 β 1)(T t) + σ 2 (t)e ρ(1 β 2)(T t) 1 e (2α ρ(1 β 2))(T t) 2(2α ρ(1 β 2 )) κ 0 exp L (θ!! 2) 1 e 2α(T t) e ρ(1 β 2)(T t) 1 e (2α ρ(1 β 2))(T t) 2ρ(1 β 2 ) 2α (2α ρ(1 β 2 )) θ exp 1 α(1 β 1 ) (1 e α(1 β 1)(T t) ) " e 2αT Z T Z s Z! # E Q θ, exp e (2α ρ(1 β 2))s e ρ(1 β2)u zñ L β 2 Q (du, dz) ds jf t t t 0 In the particular case Q θ, β = P, it holds that! F P (t, T ) = Λ g (T ) exp X (t)e α(t t) + σ 2 ρ(t t) 1 e (2α ρ)(t t) (t)e 2(2α ρ)!! Z T t ρs 1 e (2α ρ)s exp κ L e ds. 0 2(2α ρ)
A ne transform formula Theorem Let β = (β 1, β 2 ) 2 [0, 1] 2, θ = (θ 1, θ 2 ) 2 D L and T > 0. Suppose there exist functions Ψ θ, β i, i = 0, 1, 2 belonging to C 1 ([0, T ]; R), satisfying the generalised Riccati equation d dt Ψ θ, β 0 (t) = θ 1Ψ θ, β 2 (t) + κ L d dt Ψ θ, β 1 (t) = ρψ θ, β 1 (t) + (Ψ θ, β d dt Ψ θ, β 2 (t) = α(1 β 1)Ψ θ, β 2 (t), Ψ θ, β 1 (t) + θ 2 2 (t))2 2 + ρβ 2 κl 00(θ 2) κ 0 L κ L (θ 2 ), Ψ θ, β 1 (t) + θ 2 with initial conditions Ψ θ, β 0 (0) = Ψ θ, β 1 (0) = 0 and Ψ θ, β 2 (0) = 1. Moreover, suppose that the integrability condition sup κl 00 θ 2 + Ψ θ, β 1 (t) <, t2[0,t ] holds. κl 0 (θ 2),
A ne transform formula Theorem (Continue) Then, and E Q [exp(x (T ))jf θ, β t ] = exp Ψ θ, β 0 (T t) + Ψ θ, β 1 (T t)σ2 (t) + Ψ θ, β 2 (T t)x (t), RQ F (t, T ) = E θ, P [S(T )jf β t ] ( for t 2 [0, T ]. exp Ψ θ, Z T t β 0 (T t) κ L e 0 + Ψ θ, β ρ(t t) 1 e (2α ρ)(t t) 1 (T t) e 2(2α ρ) + Ψ θ, β 2 (T t) e α(t t) X (t)! ρs 1 e (2α ρ)s ds 2(2α ρ)! o 1, σ 2 (t)
Some remarks on the theorem I The proof follows by applying a result by Kallsen and Muhle-Karbe (2010). I Limited applicability as it is stated. I Reduction to a one dimensional non autonomous ODE. I I We have that for any θ 2 D L, β 2 [0, 1] 2, the solution of the last equation is given by Ψ θ, β 2 (t) = exp( α(1 β 1)t). Plugging this solution to the rst equation we get the following equation to solve for Ψ θ, β 1 (t) d dt Ψ θ, β 1 (t) = ρψ θ, β 1 (t) + e 2α(1 β1)t 2 + ρβ 2 κl 00(θ 2) (κ0 L (Ψ θ, β 1 (t) + θ 2) κl 0 (θ 2)), I with initial condition Ψ θ, β 1 (0) = 0. θ, β The equation for Ψ 0 (t) is solved by integrating Λ θ, β 0 (Ψ θ, β 1 (t), Ψ θ, β 2 (t)), i.e., Ψ θ, β 0 (t) = 0 1 e α(1 β1)t = θ 1 α(1 β 1 ) fθ 1 Ψ θ, β 2 (s) + κ L(Ψ θ, β 1 (s) + θ 2) κ L (θ 2 )gds + fκ L (Ψ θ, β 1 (s) + θ 2) κ L (θ 2 )gds. 0
Su cient conditions for existence of global solutions I Study of the equation d dt ˆΨ θ 2,β 2 (t) = Λ θ 2,β 2 ( ˆΨ θ 2,β 2 ), where I Let Λ θ 2,β 2 (u) = ρu + 1 2 + ρβ 2 κ 00 L (θ 2) (κ0 L (u + θ 2) κ 0 L (θ 2)). D b = f(θ 2, β 2 ) 2 D L (0, 1) : 9u 2 [0, Θ L θ 2 ) s.t. Λ θ 2,β 2 (u) 0g. Theorem If (θ 2, β 2 ) 2 D b and (θ 1, β 1 ) 2 R [0, 1) then Ψ θ, β 0 (t), Ψ θ, β 1 (t) and Ψ θ, β 2 (t)) are C 1 ([0, T ]; R) for any T > 0. Moreover, Ψ θ, β 0 (t)! θ Z 1 nκ α(1 β 1 ) + L Ψ θ, β 1 (s) + θ 2 0 and 1 (t), Ψ θ, β 2 (t))! (0, 0), t!, (Ψ θ, β t 1 log (Ψ θ, β for some negative constant γ. 1 (t), Ψ θ, β 2 (t))! γ, t!, o κ L (θ 2 ) ds, t!,
Risk premium analysis in the geometric BNS model Lemma If (θ 2, β 2 ) 2 D b and (θ 1, β 1 ) 2 R [0, 1), the sign of the risk premium RQ F (t, T ) is the same as the sign of the function θ, β Σ(t, T ), Ψ θ, β 0 (T t) Ψ0,0 Moreover, + (Ψ θ, β 2 lim Σ(t, T ) T t! = 0 (T t) + (Ψ θ, β 1 (T t) Ψ0,0 2 (T t))x (t). θ 1 α(1 β 1 ) + Z Z Z 1 0 0 Z 1 0 κ0 L λψ θ, β 1 (s) + θ 2! 0 κ0 L λe ρs 1 e (2α ρ)s 2(2α ρ) (T t) Ψ0,0 1 (T t))σ2 (t) dλψ θ, β 1 (s)ds dλe ρs 1 e (2α ρ)s ds, 2(2α ρ) and d lim T t!0 dt Σ(t, T ) = θ 1 + αβ 1 X (t). I Analysis of possible risk pro les in Benth and O.-L. (2015).
Conclusions I A pricing measure for the BNS model for non-tradable assets with mean reversion extending Esscher s transform. I Preserves a ne structure of the model. I Gives control on the speed and level of mean reversion. I Provides more realistic risk premium pro les. Further work I Study the BNS model with jumps I Calibration. X (t) = X (0) σ 2 (t) = σ 2 (0) I Pricing more complex derivatives. I Other models with mean reversion. α X (s)ds + σ(s)w (t) + ηl(t), 0 0 ρ 0 σ2 (s)ds + L(t).
References Barndor -Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in nancial economics. Journal of the Royal Statistical Society Series B, Royal Statistical Society, Vol. 63(2), pp. 167-241. Benth, F.E. (2011) The stochastic volatility model of Barndor -Nielsen and Shepard in commodity markets. Math. Finance, 21, pp. 595-625. Benth, F.E. and Ortiz-Latorre, S. (2014) A pricing measure to explain the risk premium in power markets. SIAM Journal on Financial Mathematics, Vol. 5, No. 1, pp. 685-728. Benth, F.E. and Ortiz-Latorre, S. (2015) A change of measure preserving the a ne structure in the BNS model for commodity markets. To appear in International Journal of Theoretical and Applied Finance. Kallsen, J. and Muhle-Karbe, J. (2010). Exponentially a ne martingales, a ne measure changes and exponential moments of a ne processes. Stochastic Processes and their Applications 120, pp. 163 181.