Optimal Investment with State-Dependent Constraints Carole Bernard CMS 2011, Edmonton, June 2011. Carole Bernard Optimal Investment with State-Dependent Constraints 1/29
Introduction Cost-Efficiency Characterization Examples State-Dependent Constraints Conclusions ◮ This talk is joint work with Phelim Boyle (Wilfrid Laurier University, Waterloo, Canada) and with Steven Vanduffel (Vrije Universiteit Brussel (VUB), Belgium). ◮ Outline of the talk: Characterization of optimal investment strategies for an 1 investor with law-invariant preferences Extension to the case when investors have state-dependent 2 constraints . Carole Bernard Optimal Investment with State-Dependent Constraints 2/29
Introduction Cost-Efficiency Characterization Examples State-Dependent Constraints Conclusions ◮ This talk is joint work with Phelim Boyle (Wilfrid Laurier University, Waterloo, Canada) and with Steven Vanduffel (Vrije Universiteit Brussel (VUB), Belgium). ◮ Outline of the talk: Characterization of optimal investment strategies for an 1 investor with law-invariant preferences Extension to the case when investors have state-dependent 2 constraints . Carole Bernard Optimal Investment with State-Dependent Constraints 2/29
Introduction Cost-Efficiency Characterization Examples State-Dependent Constraints Conclusions Part I: Optimal portfolio selection for law-invariant investors Characterization of optimal investment strategies for an investor with law-invariant preferences and a fixed investment horizon • Optimal strategies are “cost-efficient”. • Cost-efficiency ⇔ Minimum correlation with the state-price process ⇔ Anti-monotonicity • In the Black-Scholes setting, ◮ Optimality of strategies increasing in S T . ◮ Suboptimality of path-dependent contracts. Carole Bernard Optimal Investment with State-Dependent Constraints 3/29
Introduction Cost-Efficiency Characterization Examples State-Dependent Constraints Conclusions What is “cost-efficiency”? Cost-Efficiency A strategy (or a payoff) is cost-efficient if any other strategy that generates the same distribution under P costs at least as much. This concept was originally proposed by Dybvig (1988). Carole Bernard Optimal Investment with State-Dependent Constraints 4/29
Introduction Cost-Efficiency Characterization Examples State-Dependent Constraints Conclusions Main Assumptions • Consider an arbitrage-free and complete market. • Given a strategy with final payoff X T at time T . There exists a unique probability measure Q , such that its price at 0 is c ( X T ) = E Q [ e − rT X T ] Distributional price of a cdf F under the physical measure P . PD ( F ) = { Y | Y ∼ F } c ( Y ) min • The strategy with payoff X T is cost-efficient if PD ( F ) = c ( X T ) Carole Bernard Optimal Investment with State-Dependent Constraints 5/29
Introduction Cost-Efficiency Characterization Examples State-Dependent Constraints Conclusions Traditional Approach to Portfolio Selection Consider an investor with increasing law-invariant preferences and a fixed horizon. Denote by X T the investor’s final wealth. • Optimize an increasing law-invariant objective function max X T ( E P [ U ( X T )]) where U is increasing. 1 Minimizing Value-at-Risk (a quantile of the cdf) 2 Probability target maximizing: max X T P ( X T > K ) 3 ... 4 • for a given cost (budget) cost at 0 = E Q [ e − rT X T ]. Find optimal strategy X ∗ ⇒ Optimal cdf F of X ∗ T T It is clear that the optimal strategy must be cost-efficient Carole Bernard Optimal Investment with State-Dependent Constraints 6/29
Introduction Cost-Efficiency Characterization Examples State-Dependent Constraints Conclusions Traditional Approach to Portfolio Selection Consider an investor with increasing law-invariant preferences and a fixed horizon. Denote by X T the investor’s final wealth. • Optimize an increasing law-invariant objective function max X T ( E P [ U ( X T )]) where U is increasing. 1 Minimizing Value-at-Risk (a quantile of the cdf) 2 Probability target maximizing: max X T P ( X T > K ) 3 ... 4 • for a given cost (budget) cost at 0 = E Q [ e − rT X T ]. Find optimal strategy X ∗ ⇒ Optimal cdf F of X ∗ T T It is clear that the optimal strategy must be cost-efficient Carole Bernard Optimal Investment with State-Dependent Constraints 6/29
Introduction Cost-Efficiency Characterization Examples State-Dependent Constraints Conclusions Assumptions To characterize cost-efficiency, we need to introduce the “state-price process” • Given a payoff X T at time T . P (“physical measure”) and Q (“risk-neutral measure”) satisfy � dQ � ξ T = e − rT c ( X T ) = E Q [ e − rT X T ] = E P [ ξ T X T ] . , dP T ξ T is called “state-price process” . Theorem (Sufficient condition for cost-efficiency) Any random payoff X T with the property that ( X T , ξ T ) is anti-monotonic is cost-efficient . X T and ξ T are anti-monotonic : “When ξ T increases, then X T decreases”. Carole Bernard Optimal Investment with State-Dependent Constraints 7/29
Introduction Cost-Efficiency Characterization Examples State-Dependent Constraints Conclusions Idea of the proof Minimizing the price c ( X T ) = E [ ξ T X T ] when X T ∼ F amounts to finding the dependence structure that minimizes the correlation between the strategy and the state-price process min E [ ξ T X T ] X T � X T ∼ F subject to ξ T ∼ G Recall that corr( X T , ξ T ) = E [ ξ T X T ] − E [ ξ T ] E [ X T ] . std( ξ T ) std( X T ) When the distributions for both X T and ξ T are fixed, we have ( X T , ξ T ) is anti-monotonic ⇒ corr[ X T , ξ T ] is minimal. Carole Bernard Optimal Investment with State-Dependent Constraints 8/29
Introduction Cost-Efficiency Characterization Examples State-Dependent Constraints Conclusions Explicit Representation for Cost-efficiency Assume ξ T is continuously distributed (for example a Black-Scholes market) Theorem (Necessary and sufficient Condition) The cheapest strategy that has cdf F is given explicitly by X ⋆ T = F − 1 (1 − F ξ ( ξ T )) . Note that X ⋆ T ∼ F and X ⋆ T is a.s. unique such that PD ( F ) = c ( X ⋆ T ) = E [ ξ T X ⋆ T ] where F − 1 is defined as follows: F − 1 ( y ) = min { x / F ( x ) � y } . Carole Bernard Optimal Investment with State-Dependent Constraints 9/29
Introduction Cost-Efficiency Characterization Examples State-Dependent Constraints Conclusions Idea of the proof Solving this problem amounts to finding bounds on copulas! min E [ ξ T X T ] X T � X T ∼ F subject to ξ T ∼ G The distribution G is known and depends on the financial market. Let C denote a copula for ( ξ T , X ) . � � E [ ξ T X ] = (1 − G ( ξ ) − F ( x ) + C ( G ( ξ ) , F ( x ))) dxd ξ, (1) Bounds for E [ ξ T X ] are derived from bounds on C max( u + v − 1 , 0 ) � C ( u , v ) � min( u , v ) (Fr´ echet-Hoeffding Bounds for copulas) ( anti-monotonic copula) Carole Bernard Optimal Investment with State-Dependent Constraints 10/29
Introduction Cost-Efficiency Characterization Examples State-Dependent Constraints Conclusions Black-Scholes Model Under the physical measure P , dS t = µ dt + σ dW P t S t Then � − b � dQ � � S T ξ T = e − rT = a dP S 0 σ ( µ − σ 2 2 ) t − ( r + θ 2 2 ) t and b = µ − r θ where a = e σ 2 . Theorem (Cost-efficiency in Black-Scholes model) To be cost-efficient, the contract has to be a European derivative written on S T and non-decreasing w.r.t. S T (when µ > r). In this case, X ⋆ T = F − 1 ( F S T ( S T )) Carole Bernard Optimal Investment with State-Dependent Constraints 11/29
Introduction Cost-Efficiency Characterization Examples State-Dependent Constraints Conclusions Geometric Asian contract in Black-Scholes model Assume a strike K . The payoff of the Geometric Asian call is given by � T � + � 0 ln( S t ) dt − K 1 X T = e T � + � 1 ��� n n − K which corresponds in the discrete case to k =1 S kT . n The efficient payoff that is distributed as the payoff X T is a power call option � + √ � − K X ⋆ S 1 / 3 T = d T d � �� µ − σ 2 � � 1 − 1 1 1 2 − T √ 3 2 3 where d := S e . 0 Similar result in the discrete case. Carole Bernard Optimal Investment with State-Dependent Constraints 12/29
Introduction Cost-Efficiency Characterization Examples State-Dependent Constraints Conclusions Example: Discrete Geometric Option 120 100 80 Payoff 60 * Z T 40 * Y T 20 0 40 60 80 100 120 140 160 180 200 220 240 260 Stock Price at maturity S T With σ = 20% , µ = 9% , r = 5% , S 0 = 100, T = 1 year, K = 100. C ( X ⋆ T ) = 5 . 3 < Price ( geometric Asian ) = 5 . 5 < C ( Z ⋆ T ) = 8 . 4 . Carole Bernard Optimal Investment with State-Dependent Constraints 13/29
Introduction Cost-Efficiency Characterization Examples State-Dependent Constraints Conclusions Put option in Black-Scholes model Assume a strike K . The payoff of the put is given by L T = ( K − S T ) + . The payout that has the lowest cost and that has the same distribution as the put option payoff is given by + µ − σ 2 � � 0 e 2 T K − S 2 2 Y ⋆ T = F − 1 ( F S T ( S T )) = . L S T This type of power option “dominates” the put option. Carole Bernard Optimal Investment with State-Dependent Constraints 14/29
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