On Asymptotic Power Utility-Based Pricing and Hedging

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

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

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?

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 ( )

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

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?

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

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?

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?

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?

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?

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?

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

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

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 )))

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!

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?

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 )))

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 )))

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?

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

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

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

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

Application to affine models Results in the NIG-Gamma-OU model Utility-based hedge ϕ : 1 0.9 0.8 Black Scholes p=2 p=150 0.7 Initial hedge ratio 0.6 0.5 0.4 0.3 0.2 0.1 0 80 85 90 95 100 105 110 115 120 Initial stock price

Application to affine models Results in the NIG-Gamma-OU model π 0 + qπ for risk aversion p = 150, endowment v = 241: Call price 25 20 15 10 Black Scholes q= 2 q= 1 q=0 q=1 q=2 5 0 80 85 90 95 100 105 110 115 120 Initial stock price

Application to affine models Results in the NIG-Gamma-OU model p = 150: Difference to Black-Scholes price 0.8 0.6 0.4 Difference to Black Scholes 0.2 0 0.2 0.4 0.6 0.8 q= 2 q= 1 q=0 q=1 q=2 1 80 85 90 95 100 105 110 115 120 Initial stock price

Application to affine models Results in the NIG-Gamma-OU model π 0 + qπ for risk aversion p = 2, endowment v = 241: Call price 25 20 15 10 Black Scholes q= 2 q= 1 q=0 q=1 q=2 5 0 80 85 90 95 100 105 110 115 120 Initial stock price

Application to affine models Results in the NIG-Gamma-OU model p = 2: Difference to Black-Scholes price 0.1 Difference to Black Scholes 0.05 0 0.05 0.1 q= 2 q= 1 q=0 q=1 q=2 0.15 80 85 90 95 100 105 110 115 120 Initial stock price

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

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

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?

Outlook Example: NIG v.s. NIG-Gamma-OU Optimal hedging strategies in NIG resp. NIG-Gamma-OU models for calibrated parameters from Schoutens (2003): 1 0.9 0.8 0.7 NIG variance optimal NIG utility based BS NIG OU variance optimal NIG OU utility based Initial hedge ratio 0.6 0.5 0.4 0.3 0.2 0.1 0 80 85 90 95 100 105 110 115 120 Initial spot price