On Asymptotic Power Utility-Based Pricing and Hedging

Transcription

1 On Asymptotic Power Utility-Based Pricing and Hedging Johannes Muhle-Karbe TU München Joint work with Jan Kallsen and Richard Vierthauer Workshop "Finance and Insurance", Jena

2 Overview Introduction Utility-based pricing and hedging Application to affine models Outlook

3 Introduction Pricing and hedging in incomplete markets? Complete markets: Any contingent claim is replicable perfect hedging strategy Unique price compatible with No Arbitrage Incomplete markets: Incompleteness caused by e.g. jumps, stochastic volatility, market frictions Replication no longer possible Many different prices consistent with No Arbitrage How to price and hedge contingent claims?

4 Utility-based pricing and hedging Idea Preferences modelled by utility function u : R + R Investor maximizes expected utility from terminal wealth Without contingent claims: U(v) := sup E(u(v + ϕ S T )) ϕ ( ) After selling q contingent claims H for π q each: U q (v + qπ q ) := sup E(u(v + qπ q + ϕ S T qh)) ϕ ( ) Indifference price π q : U(v) = U q (v + qπ q ) Hedge: Difference between optimizers ϕ q in ( ) and ϕ 0 in ( )

5 Utility-based pricing and hedging Existence and uniqueness? NFLVR, finite utility, claim can be superhedged: Unique indifference price π q for sufficiently small number q of contingent claims But: No existence in general, unlike for exponential utility Reason: Different notion of admissibility! Hugonnier & Kramkov (2004): Utility-based hedge ϕ q ϕ 0 exists for small enough q Problem: Computation impossible even in simple concrete models

6 Utility-based pricing and hedging Feasible alternative: First-order approximations for q 0, i.e. π q = π 0 + qπ + o(q 2 ) ϕ q = ϕ 0 + qϕ + o(q 2 ) Mania & Schweizer (2005), Becherer (2006), Kallsen & Rheinländer (2008): Exponential utility Henderson (2002), Henderson & Hobson (2002), Kramkov & Sîrbu (2006, 2007): Utility functions on R + Computation of π 0, π, ϕ 0, ϕ?

7 Computation of ϕ 0 Optimizer in pure investment problem General duality principle: ϕ 0 optimal u (V T (ϕ 0 )/v) C 1 density of dual EMM Q 0 with C 1 := E(u (V T (ϕ 0 )/v) : finite discrete time, : generally For power utility u(x) = x 1 p /(1 p): V T (ϕ 0 )/v and hence Q 0 independent of endowment v Computation through ansatz for ϕ 0 and density process of Q 0 via control theory or martingale methods

8 Computation of π 0, π, ϕ Idea: Taylor expansion for small q. Maximize g(ϕ ) := E ( u(v + qπ q + ϕ q ST qh) ) = E ( u(v T (ϕ 0 ) + q(π 0 + qπ + (ϕ + o(1)) S T H) + o(q 2 )) ) = E ( u(v T (ϕ 0 )) ) + qe ( u (V T (ϕ 0 ))(π 0 + qπ + (ϕ + o(1)) S T H) ) + q2 2 E ( u (V T (ϕ 0 ))(π 0 + ϕ ST H) 2) + o(q 2 ) Here: Power utility u(x) = x 1 p /(1 p) Neglect o(q 2 ). What about the o(q)-term?

9 Computation of π 0, π, ϕ Idea: Taylor expansion for small q. Maximize g(ϕ ) := E ( u(v + qπ q + ϕ q ST qh) ) = E ( u(v T (ϕ 0 ) + q(π 0 + qπ + (ϕ + o(1)) S T H) + o(q 2 )) ) = E ( u(v T (ϕ 0 )) ) + qe ( u (V T (ϕ 0 ))(π 0 + qπ + (ϕ + o(1)) S T H) ) + q2 2 E ( u (V T (ϕ 0 ))(π 0 + ϕ ST H) 2) + o(q 2 ) Here: Power utility u(x) = x 1 p /(1 p) Neglect o(q 2 ). What about the o(q)-term?

10 Computation of π 0, π, ϕ Idea: Taylor expansion for small q. Maximize g(ϕ ) := E ( u(v + qπ q + ϕ q ST qh) ) = E ( u(v T (ϕ 0 ) + q(π 0 + qπ + (ϕ + o(1)) S T H) + o(q 2 )) ) = E ( u(v T (ϕ 0 )) ) + qc 1 v p E Q0 ( (π 0 + qπ + (ϕ + o(1)) S T H) ) + q2 2 E ( u (V T (ϕ 0 ))(π 0 + ϕ ST H) 2) + o(q 2 ) Here: Power utility u(x) = x 1 p /(1 p) Neglect o(q 2 ). What about the o(q)-term?

11 Computation of π 0, π, ϕ Idea: Taylor expansion for small q. Maximize g(ϕ ) := E ( u(v + qπ q + ϕ q ST qh) ) = E ( u(v T (ϕ 0 ) + q(π 0 + qπ + (ϕ + o(1)) S T H) + o(q 2 )) ) = E ( u(v T (ϕ 0 )) ) + qc 1 v p E Q0 ( (π 0 + qπ H) ) + q2 2 E ( u (V T (ϕ 0 ))(π 0 + ϕ ST H) 2) + o(q 2 ) Here: Power utility u(x) = x 1 p /(1 p) Neglect o(q 2 ). What about the o(q)-term?

12 Computation of π 0, π, ϕ Idea: Taylor expansion for small q. Maximize g(ϕ ) := E ( u(v + qπ q + ϕ q ST qh) ) = E ( u(v T (ϕ 0 ) + q(π 0 + qπ + (ϕ + o(1)) S T H) + o(q 2 )) ) = E ( u(v T (ϕ 0 )) ) + qc 1 v p E Q0 ( (π 0 + qπ H) ) + q2 2 E ( u (V T (ϕ 0 ))(π 0 + ϕ ST H) 2) + o(q 2 ) Here: Power utility u(x) = x 1 p /(1 p) Neglect o(q 2 ). What about the o(q)-term?

13 Computation of π 0, π, ϕ Idea: Taylor expansion for small q. Maximize g(ϕ ) := E ( u(v + qπ q + ϕ q ST qh) ) = E ( u(v T (ϕ 0 ) + q(π 0 + qπ + (ϕ + o(1)) S T H) + o(q 2 )) ) = E ( u(v T (ϕ 0 )) ) + qc 1 v p E Q0 ( (π 0 + qπ H) ) + q2 2 E ( pv T (ϕ 0 ) 1 p ( π(0) + ϕ ST H ) 2 ) + o(q 2 ) Here: Power utility u(x) = x 1 p /(1 p) Neglect o(q 2 ). What about the o(q)-term?

14 Computation of π 0, π, ϕ Idea: Taylor expansion for small q. Maximize g(ϕ ) := E ( u(v + qπ q + ϕ q ST qh) ) = E ( u(v T (ϕ 0 ) + q(π 0 + qπ + (ϕ + o(1)) S T H) + o(q 2 )) ) = E ( u(v T (ϕ 0 )) ) + qc 1 v p ( E Q0 (π 0 + qπ H) ) q 2 C 1 v p p ( 2 E 1 ( ) ) 2 Q 0 V T (ϕ 0 π 0 + ϕ ST H + o(q 2 ) ) Here: Power utility u(x) = x 1 p /(1 p) Neglect o(q 2 ). What about the o(q)-term?

15 Computation of π 0, π, ϕ Idea: Taylor expansion for small q. Maximize g(ϕ ) := E ( u(v + qπ q + ϕ q ST qh) ) = E ( u(v T (ϕ 0 ) + q(π 0 + qπ + (ϕ + o(1)) S T H) + o(q 2 )) ) = E ( u(v T (ϕ 0 )) ) + qc 1 v p ( E Q0 (π 0 + qπ H) ) q 2 C 1 v p p ( 2 E 1 ( ) ) 2 Q 0 V T (ϕ 0 π 0 + ϕ ST H + o(q 2 ) ) Here: Power utility u(x) = x 1 p /(1 p) Neglect o(q 2 ). What about the o(q)-term?

16 Computation of π 0, π, ϕ Idea: Taylor expansion for small q. Maximize g(ϕ ) := E ( u(v + qπ q + ϕ q ST qh) ) = E ( u(v T (ϕ 0 ) + q(π 0 + qπ + (ϕ + o(1)) S T H) + o(q 2 )) ) = E ( u(v T (ϕ 0 )) ) + qc 1 v p ( E Q0 (π 0 + qπ H) ) q 2 C 1 v p p 2 E Q 0 V T (ϕ 0 ) v 2 Here: Power utility u(x) = x 1 p /(1 p) Neglect o(q 2 ). What about the o(q)-term? ( π 0 + ϕ ) 2 ST H V T (ϕ 0 )/v

17 Change of numeraire EMM Q $ relative to numeraire V (ϕ 0 )/v: dq $ := V T (ϕ 0 ) dq 0 v Discounted values relative to V (ϕ 0 )/v: S $ := S V (ϕ 0 )/v, π0$ := Self-financing condition: π 0 + ϕ ST H V T (ϕ 0 )/v π 0 V (ϕ 0 )/v, H$ := = π 0$ + ϕ S $ T H $ H V (ϕ 0 )/v

18 Indifference criterion Maximize g(ϕ ) = E(u(V T (ϕ 0 ))) + q C 1 v p E ( Q 0 π 0 + qπ H ) q 2 C 1 p v p 2 E V ( T (ϕ) π 0 + ϕ ) 2 ST H Q 0 v 2 V T (ϕ)/v = E(u(V T (ϕ 0 ))) + q C ( ) 1 v p π 0 E Q0 (H) + q 2 C ( 1 v p π p ) 2v ε2 $ (ϕ ) with ε 2 $ (ϕ ) := E Q $ ( ( π 0$ + ϕ S $ T H$) 2 ). Indifference criterion: Has to equal E(u(V T (ϕ 0 )))

19 Indifference criterion Maximize g(ϕ ) = E(u(V T (ϕ 0 ))) + q C 1 v p E ( Q 0 π 0 + qπ H ) q 2 C 1 p v p 2 E V ( T (ϕ) π 0 + ϕ ) 2 ST H Q 0 v 2 V T (ϕ)/v = E(u(V T (ϕ 0 ))) + q C ( ) 1 v p π 0 E Q0 (H) + q 2 C ( 1 v p π p ) 2v ε2 $ (ϕ ) with ε 2 $ (ϕ ) := E Q $ ( ( π 0$ + ϕ S $ T H$) 2 ). Indifference criterion: Has to equal E(u(V T (ϕ 0 )))

20 Indifference criterion Maximize g(ϕ ) = E(u(V T (ϕ 0 ))) + q C 1 v p E ( Q 0 π 0 + qπ H ) q 2 C 1 p v p 2 E V ( T (ϕ) π 0 + ϕ ) 2 ST H Q 0 v 2 V T (ϕ)/v = E(u(V T (ϕ 0 ))) + q C ( ) 1 v p π 0 E Q0 (H) + q 2 C ( 1 v p π p ) 2v ε2 $ (ϕ ) with ε 2 $ (ϕ ) := E Q $ ( ( π 0$ + ϕ S $ T H$) 2 ). Indifference criterion: Has to equal E(u(V T (ϕ 0 )))

21 Characterization of π 0, π, ϕ Kramkov and Sîrbu (2006, 2007): This indeed works under weak technical conditions! π 0 = E Q0 (H), i.e. expectation under dual EMM ϕ : variance-optimal hedge for H under measure Q $ relative to numeraire V (ϕ 0 )/v π p : 2v -fold of corresponding hedging error ε2 $ Solution to quadratic hedging problem under EMM Q $? Fölmer & Sondermann (1986) Galtchouk-Kunita-Watanabe (GKW) decomposition But: Numeraire no longer normalized, dimensionality increased!

22 Quadratic hedging under EMM Approaches in the literature: Cont et al. (2007): PDE methods Hubalek et al. (2006), Pauwels (2007): Laplace transform approach In both cases: Additional dimension increases computational complexity E.g. quintuple instead of triple integral for hedging error in stochastic volatility models Representation in terms of original numeraire?

23 Quadratic hedging under EMM Approaches in the literature: Cont et al. (2007): PDE methods Hubalek et al. (2006), Pauwels (2007): Laplace transform approach In both cases: Additional dimension increases computational complexity E.g. quintuple instead of triple integral for hedging error in stochastic volatility models Representation in terms of original numeraire?

24 An alternative representation Reconsider Taylor expansion: Maximize g(ϕ ) := E(u(v + qπ q + ϕ q ST qh)) = E ( u(v T (ϕ 0 )) ) + qc 1 v p E Q0 ( (π 0 + qπ H) ) q 2 p 2 E (V T (ϕ 0 ) 1 p ( π 0 + ϕ ST H Instead of new numeraire, introduce measure P e with ) 2 ) + o(q 2 ) dp e dp = (V T (ϕ 0 )/v) 1 p C 2, C 2 := E((V T (ϕ 0 )/v) 1 p ) Indifference criterion: g(ϕ ) = E(u(V T (ϕ 0 )))

25 An alternative representation Reconsider Taylor expansion: Maximize g(ϕ ) := E(u(v + qπ q + ϕ q ST qh)) = E ( u(v T (ϕ 0 )) ) + qc 1 v p E Q0 ( (π 0 + qπ H) ) q 2 p 2 E (V T (ϕ 0 ) 1 p ( π 0 + ϕ ST H Instead of new numeraire, introduce measure P e with ) 2 ) + o(q 2 ) dp e dp = (V T (ϕ 0 )/v) 1 p C 2, C 2 := E((V T (ϕ 0 )/v) 1 p ) Indifference criterion: g(ϕ ) = E(u(V T (ϕ 0 )))

26 An alternative representation Reconsider Taylor expansion: Maximize g(ϕ ) := E(u(v + qπ q + ϕ q ST qh)) = E ( u(v T (ϕ 0 )) ) + qc 1 v p E Q0 ( (π 0 + qπ H) ) q 2 p 2 C 2v 1 p E P e ( ( π 0 + ϕ ST H Instead of new numeraire, introduce measure P e with ) 2 ) + o(q 2 ) dp e dp = (V T (ϕ 0 )/v) 1 p C 2, C 2 := E((V T (ϕ 0 )/v) 1 p ) Indifference criterion: g(ϕ ) = E(u(V T (ϕ 0 )))

27 An alternative representation Reconsider Taylor expansion: Maximize g(ϕ ) := E(u(v + qπ q + ϕ q ST qh)) = E ( u(v T (ϕ 0 )) ) + qc 1 v p E Q0 ( (π 0 + qπ H) ) q 2 p 2 C 2v 1 p E P e ( ( π 0 + ϕ ST H Instead of new numeraire, introduce measure P e with ) 2 ) + o(q 2 ) dp e dp = (V T (ϕ 0 )/v) 1 p C 2, C 2 := E((V T (ϕ 0 )/v) 1 p ) Indifference criterion: g(ϕ ) = E(u(V T (ϕ 0 )))

28 Heuristic derivation of asymptotic expansions An alternative representation Consequence: ϕ : Variance-optimal hedge of H under P e w.r.t. original numeraire π = C 2 p C 1 2v ε2 e for squared hedging error ε 2 e under Pe Advantage: Original numeraire Disadvantage: P e typically not an EMM Correspondence to mean-variance hedging: Gourieroux et al. (1998): Hedging under specific EMM after change of numeraire Černý & Kallsen (2007): Hedging w.r.t. to original numeraire But does alternative approach really work?

29 An alternative approach Idea: Start with the results of Kramkov and Sîrbu Switch measure and numeraire Technical problem: Admissible strategies for mean-variance hedging not invariant under change of numeraire Transformation only works if optimal ϕ 0 from pure investment problem is admissible in the sense of Černý & Kallsen (2007) Hard to check even in concrete models (martingale property of ϕ 0 S under all EMMs!) No reason why this should hold in general

30 Precise mathematical results An alternative approach Černý & Kallsen (2007): Mean-variance hedging via Local drift conditions Global admissibility condition Idea: Local conditions still lead to solution without admissibility here Key tool: Differential semimartingale calculus which only depends on local characteristics Leads to formulas of complexity as for mean-variance hedging through use of original numeraire

31 Application to affine models Computation of asymptotic expansions To compute utility-based prices and hedges: 1. Solve pure investment problem 2. Use optimal wealth process to compute model dynamics under P e 3. Apply formulas from mean-variance hedging under P e Need structure under P to carry over to P e : Some, but not all affine stochastic volatility models are P e -affine, too Mean-variance hedging studied by Kallsen & Vierthauer (2009) in this case

32 Application to affine models Example: Time-changed Lévy models Consider time-changed Lévy models by Carr et al. (2003): X t = X 0 + L t 0 vsds dv t = λv t dt + dz λt NIG Lévy process L, Lévy-driven OU process v with stationary Gamma distribution Heavy tails, skewness, volatility clustering as observed in data Parameters estimated from DAX time series via first 4 moments and autocorrelation function For numerical example: European call with strike 100 and maturity 0.25

33 Application to affine models Results in the NIG-Gamma-OU model Utility-based hedge ϕ : Black Scholes p=2 p= Initial hedge ratio Initial stock price

34 Application to affine models Results in the NIG-Gamma-OU model π 0 + qπ for risk aversion p = 150, endowment v = 241: Call price Black Scholes q= 2 q= 1 q=0 q=1 q= Initial stock price

35 Application to affine models Results in the NIG-Gamma-OU model p = 150: Difference to Black-Scholes price Difference to Black Scholes q= 2 q= 1 q=0 q=1 q= Initial stock price

36 Application to affine models Results in the NIG-Gamma-OU model π 0 + qπ for risk aversion p = 2, endowment v = 241: Call price Black Scholes q= 2 q= 1 q=0 q=1 q= Initial stock price

37 Application to affine models Results in the NIG-Gamma-OU model p = 2: Difference to Black-Scholes price 0.1 Difference to Black Scholes q= 2 q= 1 q=0 q=1 q= Initial stock price

38 Outlook Present methodology 1. Input: Model under physical measure P, parameters via estimation Preferences over expected terminal values, here CRRA p 2. Computations: Pure investment problem Change of measure to P e P e -mean-variance hedging problem 3. Output: 1st order hedges and prices

39 Outlook Alternative approach proposed by Kramkov 1. Input: Model under dual measure Q 0, parameters via calibration Preferences over indirect utility (e.g. CRRA), Risk-tolerance wealth process 2. Computations: Change of measure to P e P e -mean-variance hedging problem 3. Output: 1st order hedges and prices

40 Outlook Alternative approach proposed by Kramkov Advantage: Extends industry practice Uses information from traded options Open problems: Consistency of Q 0 and P-models? E.g. P-model and preferences for Q 0 -Heston? Liquidly traded options How to choose risk-tolerance wealth process? How to hedge with them? How to calibrate in order to avoid excessive model risk?

41 Outlook Example: NIG v.s. NIG-Gamma-OU Optimal hedging strategies in NIG resp. NIG-Gamma-OU models for calibrated parameters from Schoutens (2003): NIG variance optimal NIG utility based BS NIG OU variance optimal NIG OU utility based Initial hedge ratio Initial spot price