A pricing measure for non-tradable assets with mean-reverting dynamics

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1 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

2 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.

3 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 where τ = T t is the time to delivery.

4 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.

5 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 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 ].

6 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.

7 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

8 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.

9 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.

10 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

11 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

12 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.

13 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)

14 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),

15 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)

16 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

17 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 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!,

18 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 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).

19 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).

20 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 Benth, F.E. (2011) The stochastic volatility model of Barndor -Nielsen and Shepard in commodity markets. Math. Finance, 21, pp 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 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