Kim Weston (Carnegie Mellon University) Market Stability and Indifference Prices. 1st Eastern Conference on Mathematical Finance.

Transcription

1 1st Eastern Conference on Mathematical Finance March 216 Based on Stability of Utility Maximization in Nonequivalent Markets, Finance & Stochastics (216)

2 Basic Problem Consider a financial market consisting of Bank account with r = Traded risky asset S Derivative security φ(s ) with non-traded underlying S with corr(s, S) 1 Question: corr(s,s) 1? = price(φ(s )) price(φ(s))? Examples: Derivatives on oil Hedging with an index

3 Related Work Exponential investor where S, S are correlated geometric Brownian motions: Davis (working paper 2) - value function Monoyois (QF 24) - indifference prices Utility Maximization Stability Literature: Larsen-Žitković (SPA 27) Bayraktar-Kravitz (SPA 213) Kardaras-Žitković (MF 211) Main hurdle: All previous stability work crucially relies on a fixed volatility structure.

4 Counterexample (I) Set-up Let B and W be two independent Brownian motions. Financial Market: Bank account with r = Stock (indexed by ρ ( 1,1)) dst ρ = St (dt ρ + ) 1 ρ 2 db t + ρdw t, S ρ := 1 Financial derivative with payoff φ(b T ): Let φ min := inf x R φ(x) φ : R R is bounded, continuous, not constant Unless ρ =, the payoff φ(b T ) cannot be replicated.

5 Counterexample (II) Utility Maximization Problem Consider an investor with x > φ min : Initial wealth U(x) = xp p : Utility function (p < 1, p = corresponds to log) Objective: u(x,ρ) := [ ( T )] sup E U x + HdS ρ + φ(b T ), H A(ρ) where A(ρ) := {H : K = K(H), t HdS ρ K t}. Question: Does u(x,ρ) converge to u(x,) as ρ?

6 Counterexample (III) Admissibility Constraints Let H A(ρ) such that Two Cases: x + T HdS ρ + φ(b T ). ρ = : We can replicate φ(b T ) = p + T ds T T x + HdS + φ(b T ) = x + p + (H + ) ds }{{} H ρ : Cannot replicate φ(b T ) by trading in S ρ. It can be shown (see El Karoui & Quenez, 1995) that T x + HdS ρ + φ min Yet φ min = inf x φ(x) < p. Therefore, the ρ markets are strictly more restrictive than the ρ = market.

7 Counterexample (IV) Results Theorem (W. 216) u(x,ρ) does not converge to u(x,) as ρ. For x >, consider the value function without random endowment [ ( T )] w(x,ρ) := sup E U x + HdS ρ. H A(ρ) We define the utility indifference price of φ(b T ) by p = p(x,ρ) such that u(x,ρ) = w(x + p,ρ). For ρ =, p(x,) is the unique arbitrage-free price. Corollary (W. 216) Indifference prices do not converge: lim sup p(x,ρ) < p(x,). ρ

8 Positive Result (I) Problems arise from the property that U(x) = for x <. Real-line utility function: U : R R (main example is U(x) = e ax ) In a Brownian framework, for 1 n, we consider ds n = dm n + λ n d M n, S n R. Suppose S n S and λn dm n λ dm in the semimartingale topology. Financial derivative payoff: V T L. Theorem (W. 216) Under appropriate integrability and nondegeneracy conditions, the value functions and indifference prices converge as n.

9 Positive Result (II) Main Difficulty: Because of the changing volatility structure, small perturbations in the limiting market s investment strategies are not consistent with strategies in the pre-limiting markets. Primal Problem: Suppose (H S ) is S -admissible. Is (H S n ) S n -admissible? Is H even S n -integrable? Dual Problem: Consider a dual element E( λ M ) T E(L) T where L, (λ M ). But L, (λ n M n ) =?

10 THANK YOU!