An overview of some financial models using BSDE with enlarged filtrations

An overview of some financial models using BSDE with enlarged filtrations Anne EYRAUD-LOISEL Workshop : Enlargement of Filtrations and Applications to Finance and Insurance May 31st - June 4th, 2010, Jena (Germany) 1 / 40

Outline Hedging problems for small investors Option hedging with asymmetrical information : insider trading Defaultable contingent claims : the progressive enlargement case 1 Hedging problems for small investors Option hedging with asymmetrical information : insider trading Defaultable contingent claims : the progressive enlargement case 2 3 2 / 40

General hedging problem Option hedging with asymmetrical information : insider trading Defaultable contingent claims : the progressive enlargement case Consider a financial market with 1 risky asset, whose price is driven by a Brownian motion t t P t = P 0 + P s b(s, P s )ds + P s σ(s, P s )dw s 0 0 (standard hypotheses on σ, b to have complete market). An agent wants to hedge against a contingent claim ξ in this market with maturity T. 3 / 40

General hedging problem written as a BSDE Option hedging with asymmetrical information : insider trading Defaultable contingent claims : the progressive enlargement case The self-financing hypothesis can be written as dx t = X t r t dt + π t (b t r t )dt + π }{{} t σ t dw }{{} t f (t,x t,z t)dt Z t The problem can be represented by the BSDE (integrating from t to T ) : T T X t = ξ + f (s, X s, Z s )ds Z s dw s t t where X T = ξ, Z s = σ s π s and the driver is f (x, X s, Z s ) = X s r s + Z s σ 1 (r s b s ). 5 / 40

Information Problem Option hedging with asymmetrical information : insider trading Defaultable contingent claims : the progressive enlargement case The influential trader is supposed to possess an additional information = Initial Enlargement of the Brownian Filtration Adding L to the initial filtration Y t = s>t(f s σ(l)) Hypothesis (H 3 ) (Jacod, Jeulin 1985) There exists a probability Q equivalent to P under which F t and σ(l) are independent, t < T. 6 / 40

First results Hedging problems for small investors Option hedging with asymmetrical information : insider trading Defaultable contingent claims : the progressive enlargement case Theorem ([EL05]) Under Hypothesis (H 3 ), and standard Lipschitz and Q-integrability hypotheses on driver f, there exists a unique solution of the BSDE T T X t = ξ + f (s, X s, Z s )ds Z s dw s t t in the enlarged space (Ω, Y, Q). Which financially means Proposition ([EL05]) The insider trader has a unique solution of the hedging problem, which is the same as the solution in the non informed trader hedging problem. 9 / 40

Option hedging with asymmetrical information : insider trading Defaultable contingent claims : the progressive enlargement case Extension to BSDE with random terminal time The previous results can be extended to the case of BSDE with random terminal time (Joint work with M. Royer-Carenzi) Theorem ([ELRC10]) Under Hypothesis (H 3 ), and standard Lipschitz hypotheses and integrability conditions on driver f, if τ is a stopping time, there exists a unique solution of the BSDE T τ T τ X t = ξ + f (s, X s, Z s )ds Z s dw s t t in the enlarged space (Ω, Y, Q). The consequences and financial interpretation are the same for hedging of options with random terminal horizon (American-style options, Lookback options,...). 10 / 40

Option hedging with asymmetrical information : insider trading Defaultable contingent claims : the progressive enlargement case The case of defaultable contingent claims : BSDE with uncertain time horizon When terminal time is random, but not a stopping time (ex. default time, or death time for life-insurance contracts). General payoff ξ = V 1 τ>t + C τ 1 τ T where V is a standard option payoff (e.g. V = (S T K) + ), and C is a compensation. Hedging modeled with : T τ T τ X t = ξ + f (s, X s, Z s )ds Z s dw s t t where τ is not a stopping time. A new approach to this well-known problem using BSDE with enlarged filtration : joint work with C. Blanchet-Scalliet and M. Royer-Carenzi [BSELRC09] 12 / 40

Model and notations Option hedging with asymmetrical information : insider trading Defaultable contingent claims : the progressive enlargement case Progressive enlargement of filtration to be used (makes τ a stopping time) : G t = F t σ(1 τ t ) We will assume Density Hypothesis, or stronger Hypothesis (H). Hypothesis (Density Hypothesis) a F t B(R + )-measurable function α : (ω, θ) α t (ω, α) which satisfies P(τ dθ F t ) := α t (θ)dθ, P a.s. Hypothesis (H)(Immersion Property) Any square integrable (F, IP)-martingale is a square integrable (G, IP)-martingale 13 / 40

Model and notations Option hedging with asymmetrical information : insider trading Defaultable contingent claims : the progressive enlargement case ρ t defaultable zero-coupon bond Defaultable Process H t = 1 τ t Conditional Probability F t = IP(τ t F t ) = t 0 α t(s)ds Under Density Hypothesis, M t = H t t τ 0 (1 H s ) α s(s) 1 F s ds is a (G, IP)-martingale. and any (F, P)-martingale X is a (G, P), and the process X defined by X t = X t t τ 0 d X, F s 1 F s is a (G, IP)-martingale. t t τ d X, α(u) α s (u), 0 t T u=τ 14 / 40

Results Hedging problems for small investors Option hedging with asymmetrical information : insider trading Defaultable contingent claims : the progressive enlargement case We use a Representation Theorem by Jeanblanc and Le Cam under Density Hypothesis Theorem ([JLC07]) For every (G, IP) martingale X, there exist two G-predictable process β and γ such that d X t = γ t d W t + β t dm t 15 / 40

Results Hedging problems for small investors Option hedging with asymmetrical information : insider trading Defaultable contingent claims : the progressive enlargement case Obtained result on the general BSDE under Density Hypothesis Theorem ([BSELRC09], 2010) Let ξ L 2 (Ω, G T τ, IP) and f be G-measurable. Under standard hypotheses on f, there exists a unique G-adapted triple (Y, Z, U) solution of the BSDE : T τ T τ T τ Y t τ = ξ+ f (s, Y s, Z s, U s ) ds Z s d W s U s dm s. t τ t τ t τ Remark : we only needs Density Hypothesis to prove this Theorem (Immersion Property is not necessary). 16 / 40

Explicit solution for the hedging strategy Option hedging with asymmetrical information : insider trading Defaultable contingent claims : the progressive enlargement case Theorem ([BSELRC09], 2010) Let f be the G-measurable generator defined by f (t, y, z, u) = r t y θ t z + (1 H t ) ψ t λ t u, From previous Theorem, under Hypothesis (H), there exists a unique G-adapted triple (Y, Z, U) solution of the BSDE. Moreover and U t = C t R 1 t Z t = ac t + a V t R t (1 F ψ t ), IE IP ψ(r τ C τ G t ) R 1 IE IP ψ(r T V 1 T <τ G t ), From which we derive explicit expressions of the hedging portfolio in the different available assets. t 17 / 40

The case of quadratic BSDE Option hedging with asymmetrical information : insider trading Defaultable contingent claims : the progressive enlargement case When considering an optimization problem under constraints, using indifference principles = Quadratic BSDE to be studied under progressive enlargement of filtration Joint work with S. Ankirchner and C. Blanchet-Scalliet [ABSEL10], to be presented into more details this afternoon 18 / 40

Outline Hedging problems for small investors 1 Hedging problems for small investors Option hedging with asymmetrical information : insider trading Defaultable contingent claims : the progressive enlargement case 2 3 19 / 40

Risky asset price is supposed to be driven by t t P t = P 0 + b(s, P s, X s, π s )ds + σ(s, P s, X s, π s )dw s 0 0 Additional information : initial enlargement of filtration with L satisfying Hypothesis (H 3 ). Influence hypothesis : the informed trader may influence asset prices dynamics : Large investor : his wealth X may influence drift b and volatility σ of price dynamics. Influential investor : his investment strategy π may influence drift b and volatility σ of price dynamics. 20 / 40

Influential Problem Fundamental Property : Martingale Representation Theorem under (Y, Q) = Complete market for the informed trader [EL09a]. A non informed agent investing on the market has the information F P filtration generated by prices, who satisfies : F F P Y No martingale representation Theorem under F P, = [EL09b]. 21 / 40

Solving FBSDE under initially enlarged filtration The problem leads to a coupling between the forward equation of prices and the backward equation of wealth { P t = P 0 + t 0 b(s, P s, X s, Z s )ds + t 0 σ(s, P s, X s, Z s )dw s X t = ξ T t f (s, P s, X s, Z s )ds T t Z s, dw s Under Lipschitz, linear growth and integrability hypotheses on b, σ, f and ξ (cf Pardoux-Tang [PT99]), and additional integrability hypotheses under probability Q, existence and uniqueness can be proved 3 cases where results are obtained Weak influence : b and σ weakly depend on X and Z, The agent wants to hedge a finite value a.s. : ξ does not depend on price P, The portfolio does not influence volatility of prices : σ is independent of Z. 25 / 40

Results Hedging problems for small investors Proposition Under hypothesis (H 3 ), for any functions f, b, σ satisfying previous hypotheses and one of the 3 influence cases, the FBSDE in the enlarged space (Ω, Y, Q) has a unique solution. Financial application : the influent agent has a unique admissible hedging strategy. The strategy is adapted to Y enlarged filtration. Comparison with the strategy of a non informed trader investing on this market? 26 / 40

Bound on the wealth process Additional result Proposition Under the same hypotheses, if f does not depend on p and if f (s, 0, 0, 0) is bounded, then for any bounded payoff ξ L 2 (Ω, Y, Q), the process X is also bounded. Ex : European put option ξ = (K P T ) +, no influence, Black and Scholes model, f (s, p, x, z) = xr + σ 1 (b r)z. Then we know a bound on the wealth of the hedging portfolio between 0 and T. 27 / 40

Incompleteness of the non informed trader market A non informed agent investing on the influenced market has the information F filtration generated by prices, who satisfies : F F Y No Martingale Representation Theorem under F, = No general solution of the hedging BSDE under filtration F. Methods used : Kunita-Watanabe Decomposition Quadratic Hedging in incomplete markets (Föllmer-Schweizer 1991) 29 / 40

Kunita-Watanabe Decomposition Q (resp. Q N ) the set of Y (resp. F)-martingale measures (risk-neutral probabilities for the insider (resp. non insider)) Definition If N, M M 2 (G, P) then the unique Kunita-Watanabe decomposition of N w.r.t. M is : t N t = N 0 + θ u dm u + L t, P p.s. 0 where θ L 2 (M), L M 2 0 (G, P) is orthogonal to M in (G, P). Let Q Q, and ξ L 2 (Ω, F, Q). The martingale representation Theorem under filtration Y gives T ξ = E Q (ξ σ(l)) + φ L s dp s 0 30 / 40

Kunita-Watanabe Decomposition (2) Under any Q Q N, we have the K-W decomposition of ξ w.r.t. ( F, Q) and the martingale P : t V t := E Q (ξ F t ) = E Q (ξ) + φ Q s dp s + L Q t And from Föllmer and Schweizer, φ Q s Proposition 0 = d<v,p>s d<p> s Under any Q Q, the integrand of this decomposition may be written φ Q ( ) s = E Q φ L s F s 31 / 40

Kunita-Watanabe Decomposition (3) Proof : L t = E Q (ξ F t ) E Q (ξ) t 0 E Q (φl s F s )dp s is a ( F, Q)-martingale, and E Q (L t) = 0 Prove that [ L t is orthogonal to the stable space generated by P t. ] T L t = E Q 0 φl s dp s F t t 0 E Q (φl s F s )dp s + N t ( where N t = E Q E Q (ξ σ(l)) ) F t E Q (ξ). From a filtering lemma from Pardoux (1989), L t = N t. We can write E Q (ξ σ(l)) = f (L) where f is a borelian function, and we show that Q = f (L) Q E Q (f (L)) Q We deduce for any θ F-adapted bounded process, E Q t ] [N t θ s dp s = E Q 0 [ t ] [ t ] f (L) θ s dp s = E Q (f (L))E Q θ s dp s = 0 0 0 N so L is orthogonal to P. So it is the unique Kunita-Watanabe expected decomposition. 32 / 40

Clark-Ocone Formula An expression of φ L s may be derived from the Malliavin derivative of ξ. Proposition If ξ L 2 (Ω, Y, Q) and if x, ξ(., x) D 1,2, then φ L s = (σ L s ) 1 (E Q [D s (ξ(., x)) F t ] x=l ) 33 / 40

Expression of the quadratic residual risk From the problem introduced by Schweizer on the quadratic risk, we study the variance of L Q T, and we obtain an expression of the quadratic residual risk which measures the risk taken by an agent who doesn t know information L. For a probability measure Q Q N Q, ( ( ) Var Q L Q ( T T = E Q (ξ E Q (ξ))2) E Q 0 ) 2 φ Q s dp s For a probability measure Q Q, we obtain a more simple expression Var Q (L Q T ) = E [ ] 2 ) Q (E Q EQ (ξ σ(l)) F T E Q (ξ) 2 35 / 40

Expression of the quadratic residual risk (2) A minimum risk exists, it corresponds to the minimum risk linked to un-information : the agent does not have all the information driving the market. Measure of the lack of information. But measuring a risk under a probability measure is arbitrary. The evaluated risk is the risk of the model under the chosen risk-neutral information, and not the intern risk of the model : the historical probability measure is not taken into account. 36 / 40

Influence model example Example of influence model that satisfies our hypotheses : f (s, p, x, z) = xr + σ 1 (b r)z ( ) b 1 b(t, x, p, π) = p b 0 + (1 + p)(1 + π 2 ) σ(t, x, p) = p ( σ 0 1 [0,η[ (t) + σ 1 1 [η,t ] (t) ), η [0, T + ε] ξ = (P T K) + Strong initial information : L = η jump time of volatility. Lipschitz, linear growth. Integrability : all coefficients are zero whenever p, x, z are zero. 3rd influence case : σ independent of z. 37 / 40

BSDE and FBSDE give a useful tool to model hedging in a large number of problems. As soon as there exists representation properties, these tools may be studied under enlarged filtration, modeling from additional information and insider trading in financiel markets, to credit risk derivatives, or other uncertain asymmetrical information. A more difficult problem is when no representation property exist, when the asymmetry makes the market incomplete. Further research : study of the incompleteness of the market, and in particular link between F P and the lack of information. Question : which part of the information is transferred to the market? and what kind of information is sufficient to complete this incomplete market, due to the lack of information? 38 / 40

Thank you for your attention! 39 / 40

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