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Basis risk with random parameters Michael Monoyios University of Oxford Princeton University, 28 March, 2009 Michael Monoyios (University of Oxford) Basis risk with random parameters Princeton University, 28 March, 2009 1 / 45

Outline Basis risk with random parameters 1 Exponential utility-based pricing and hedging Optimal hedging theorem The case Q E = Q M 2 Payoff decompositions and residual risk Asymptotic expansions Malliavin method for asymptotic expansion Lognormal basis risk with partial information 3 Restoring a full information model via filtering Exponential hedging with partial information 4 Numerical example of hedging performance Michael Monoyios (University of Oxford) Basis risk with random parameters Princeton University, 28 March, 2009 2 / 45

Related literature Basis risk models: ◮ Davis [3], Henderson [5], MM [8, 9, 10], Musiela and Zariphopoulou [11], Ankirchner, Imkeller and Reis [1] Exponential utility maximisation with random endowment: ◮ Delbaen et al [4], Mania and Schweizer [7] Asymptotic analysis of utility-based prices for small numbers of claims: ◮ Kramkov and Sˆ ırbu [6] Partial information models: ◮ Rogers [13], Bj¨ ork, Davis and Land´ en [2] Michael Monoyios (University of Oxford) Basis risk with random parameters Princeton University, 28 March, 2009 3 / 45

Random parameter basis risk model (Ω , F , P ), F := ( F t ) 0 ≤ t ≤ T ( S , Y ) = ( S t , Y t ) 0 ≤ t ≤ T dS t = σ S t S t ( λ S t dt + dB S dY t = σ Y t Y t ( λ Y t dt + dB Y t ) , t ) where � B Y = ρ t B S + 1 − ρ 2 t Z S , ρ t ∈ [ − 1 , 1] F -adapted parameters r = 0 σ S , σ Y , ρ also F -adapted Markovian model: ν t ≡ ν ( t , S t , Y t ), for ν ∈ { σ S , σ Y , λ S , λ Y , ρ } Claim C ( Y T ) ≥ 0, bounded, F T -measurable Michael Monoyios (University of Oxford) Basis risk with random parameters Princeton University, 28 March, 2009 4 / 45

Exponential valuation and hedging Fot t ∈ [0 , T ], given X t = x , portfolio wealth process is � T � T π u ( λ S u du + dB S X u = x + θ u dS u = x + u ) , t ≤ u ≤ T t t where π := θ S . Denote by Θ (or Π) the set of admissible θ (or π ) U ( x ) := exp( − α x ) , x ∈ R , α > 0 Given ( X t , S t , Y t ) = ( x , s , y ), primal value function is u C ( t , x , s , y ) := sup E t , x , s , y [ U ( X T − C ( Y T ))] (1) π ∈ Π Indifference price p ( t , s , y ) defined by u C ( t , x + p ( t , s , y ) , s , y ) = u 0 ( t , x , s , y ) Denote optimal strategy for (1) by π C . Optimal hedging strategy π ( H ) defined by π ( H ) := π C − π 0 Michael Monoyios (University of Oxford) Basis risk with random parameters Princeton University, 28 March, 2009 5 / 45

Entropy and admissibility P e := { Q ∼ P | S is a local ( Q , F )-martingale } P e , f := { Q ∈ P e | H ( Q , P ) < ∞} � = ∅ � dQ � dP log dQ (if finite, else H ( Q , P ) := ∞ ) H ( Q , P ) := E , dP For Q ∈ P e , f , define P -martingale Γ Q by � � t := dQ = E ( − λ S · B S − ψ · Z S ) , � Γ Q 0 ≤ t ≤ T � dP F t Admissible strategies Θ := { θ | ( θ · S ) is a ( Q , F )-martingale for all Q ∈ P e , f } Michael Monoyios (University of Oxford) Basis risk with random parameters Princeton University, 28 March, 2009 6 / 45

Duality Dual value function � � � � η Γ Q − η Γ Q � T T u ( t , η, s , y ) := ˜ inf C ( Y T ) Q ∈ P e , f E t , s , y U Γ Q Γ Q t t where, for exponential utility � � η � � U = η � log − 1 α α Hence U ( η ) + η u ( t , η, s , y ) = � α H C ( t , s , y ) ˜ where � � � � Γ Q H C ( t , s , y ) := Q ∈ P e , f E Q T inf log − α C ( Y T ) t , s , y Γ Q t Primal and dual value functions conjugate, so primal problem has representation � � u C ( t , x , s , y ) = − exp − α x − H C ( t , s , y ) (2) Michael Monoyios (University of Oxford) Basis risk with random parameters Princeton University, 28 March, 2009 7 / 45

Dual control problem � t � t B S , Q Z S , Q := B S λ S := Z S 0 ≤ t ≤ T t + u du , t + ψ u du , t t 0 0 ( ψ ≡ 0 corresponds to Q M ). Then, for Q ∈ P e , f , � T t , s , y log Γ Q � � 1 u ) 2 + ψ 2 E Q T = E Q ( λ S du < ∞ (3) t , s , y u Γ Q 2 t t Let Ψ denote set of ψ such that (3) is satisfied. Then H C is value function of stochastic control problem � � � T � � 1 u ) 2 + ψ 2 H C ( t , s , y ) := inf ψ ∈ Ψ E Q ( λ S du − α C ( Y T ) (4) t , s , y u 2 t where, under Q ∈ P e , f , state variables S , Y follow t S t dB S , Q σ S = dS t t � � � t ψ t ) dt + dB Y , Q σ Y ( λ Y t − ρ t λ S 1 − ρ 2 dY t = t Y t t − t � B Y , Q ρ t B S , Q t Z S , Q 1 − ρ 2 = + t t t Michael Monoyios (University of Oxford) Basis risk with random parameters Princeton University, 28 March, 2009 8 / 45

Dual representation of indifference price For C ≡ 0, H 0 (0 , · , · ) = H ( Q E , P ) Write H 0 ( t , s , y ) ≡ H E ( t , s , y ), the “minimal entropy” value function. Then � � u 0 ( t , x , s , y ) = − exp − α x − H E ( t , s , y ) (5) Indifference price definition plus (2) and (5) give u C ( t , x , s , y ) = u 0 ( t , x , s , y ) exp ( α p ( t , s , y )) and indifference price has entropic representation − α p ( t , s , y ) = H C ( t , s , y ) − H E ( t , s , y ) Michael Monoyios (University of Oxford) Basis risk with random parameters Princeton University, 28 March, 2009 9 / 45

Optimal hedging strategy Theorem Optimal hedge: hold θ ( H ) shares of S t at t ∈ [0 , T ] , where t � ∂ p � ∂ s ( t , S t , Y t ) + ρ ( t , S t , Y t ) σ Y ( t , S t , Y t ) Y t ∂ p θ ( H ) = ∂ y ( t , S t , Y t ) t σ S ( t , S t , Y t ) S t Remark Additional term p s ( t , S t , Y t ) compared with earlier studies. Partial information model of Section 3 is of this form, and p s ( t , S t , Y t ) reflects additional risk induced by parameter uncertainty. Michael Monoyios (University of Oxford) Basis risk with random parameters Princeton University, 28 March, 2009 10 / 45

Proof. Use HJB equation associated with primal the value function ∂ u C A X , S , Y u C = 0 ∂ t + max π Compute optimal Markov control and use separable form (2) of value function, to obtain optimal strategy as π C t = π C ( t , S t , Y t ), where � � π C ( t , s , y ) = λ S s − α p s ) + ρσ Y σ S α − 1 s ( H E σ S y ( H E y − α p y ) α with similar form for C = 0. Apply definition of optimal hedging strategy: θ ( H ) S ≡ π ( H ) = π C − π 0 , to get result (Smoothness of value function proven using methods in Pham [12]) Michael Monoyios (University of Oxford) Basis risk with random parameters Princeton University, 28 March, 2009 11 / 45

Stochastic control problem for p ( t , s , y ) when Q E = Q M Suppose λ S t ≡ λ S ( t , S t ), and σ S t ≡ σ S ( t , S t ) Then infimum in (4) for C = 0 is achieved by ψ = 0, so Q E = Q M and H 0 ( t , s , y ) = H E ( t , s ) is given by � � � T 1 H E ( t , s ) = E Q M ( λ S u ) 2 du t , s 2 t Indifference price p ( t , s , y ) has stochastic control representation � � � T C ( Y T ) − 1 E Q ψ 2 p ( t , s , y ) = sup u du t , s , y 2 α ψ ∈ Ψ t subject to state dynamics σ S ( t , S t ) S t dB S , Q dS t = t � � � σ Y ( λ Y t − ρ t λ S t ψ t ) dt + dB Y , Q 1 − ρ 2 dY t = t Y t t − t Michael Monoyios (University of Oxford) Basis risk with random parameters Princeton University, 28 March, 2009 12 / 45

Indifference price PDE � 1 − ρ 2 σ Y yp y . HJB equation for p ( t , s , y ) is Define φ ( t , s , y ) = � � − 1 2 αψ 2 − φψ p t + A Q M S , Y p + max = 0 , p ( T , s , y ) = C ( y ) ψ Optimal Markov control is ψ C t ≡ ψ C ( t , S t , Y t ), where ψ C ( t , s , y ) = − αφ ( t , s , y ) Note that α → 0 ⇒ ψ C → ψ 0 = 0 Hence p solves semi-linear PDE S , Y p + 1 2 αφ 2 = 0 , p t + A Q M p ( T , s , y ) = C ( y ) For α → 0, obtain linear PDE for marginal price p M , and α → 0 p ( t , s , y ) = E Q M p M ( t , s , y ) := lim t , s , y C ( Y T ) Michael Monoyios (University of Oxford) Basis risk with random parameters Princeton University, 28 March, 2009 13 / 45

Hedging error (residual risk) process � t θ ( H ) R t := p (0 , S 0 , Y 0 ) + dS u − p ( t , S t , Y t ) , R 0 = 0 , 0 ≤ t ≤ T u 0 Itˆ o and PDE for p ( t , s , y ) gives, with φ t ≡ φ ( t , S t , Y t ), dR t = 1 2 αφ 2 t dt − φ t dZ S t Define A t := − exp( − α R t ) , 0 ≤ t ≤ T Then A t = −E ( αφ · Z S ) t , 0 ≤ t ≤ T so A is a Q M -martingale (and also a P -martingale) Michael Monoyios (University of Oxford) Basis risk with random parameters Princeton University, 28 March, 2009 14 / 45

Payoff decomposition and price representation Integrate SDE for R over [ t , T ] and use definition of R (equivalently, directly use Itˆ o and PDE for p ( t , s , y ) over [ t , T ]): � T � T � T C ( Y T ) = p ( t , S t , Y t ) − 1 φ 2 φ u dZ S θ ( H ) 2 α u du + u + dS u (6) u t t t Expectation under Q M given ( S t , Y t ) = ( s , y ) gives Lemma Indifference price satisfies � T p ( t , s , y ) = p M ( t , s , y ) + 1 2 α E Q M φ 2 ( u , S u , Y u ) du (7) t , s , y t with � t σ Y 1 − ρ 2 φ ( t , S t , Y t ) = t Y t p y ( t , S t , Y t ) , 0 ≤ t ≤ T Michael Monoyios (University of Oxford) Basis risk with random parameters Princeton University, 28 March, 2009 15 / 45