Motivation Model Solution Conclusion Options Portfolio Selection Paolo Guasoni 1 , 2 Eberhard Mayerhofer 3 Boston University 1 Dublin City University 2 University of Limerick 3 2018 Workshop on Finance, Insurance, Probability and Statistics September 11 th 2018, King’s College London
Motivation Model Solution Conclusion Outline • Problem: Optimal Investment in Options. Multiple Assets, Dependence. • Model: One-Period Model. Infinitely Many Securities. • Results: Optimal Portfolios and Performance.
Motivation Model Solution Conclusion The Problem • Options: Available on stocks, bonds, indices, futures, commodities. Usually available on dozens of strikes and a handful of maturities. • S&P 500 index options returns: approximately -3% a week . • Potentially high returns from selling options. Certainly high risks. • How to construct optimal portfolios? • High dimensional problem. Example: 10 assets × 20 strikes = 200 options. With a single maturity. • Markowitz? Problematic. Options with only a small strike difference are nearly collinear. Nearly singular covariance matrix.
Motivation Model Solution Conclusion One Asset • With one asset and one maturity, problem tractable. • X underlying asset price at maturity. c X ( K ) price of a call option on X with strike price K . p X ( x ) physical marginal density of X . • Assume that continuum of strikes is available. • Risk-neutral density q X ( K ) is (Breeden and Litzenberger, 1978) q X ( K ) := c ′′ X ( K ) (1) • Thus, the unique SDF is the random variable m X ( x ) = c ′′ X ( x ) / p X ( X ) . • If the function m X is regular enough, the payoff decomposes as a portfolio of call and put options (Carr and Madan, 2001) m X ( K ) = m X ( K 0 ) + m ′ X ( K 0 )( K − K 0 ) � ∞ � K 0 m ′′ X ( κ )( κ − K ) + d κ + m ′′ X ( κ )( K − κ ) + d κ. + 0 K 0 • Payoffs with maximal Sharpe of the form R = a + b m X ( X ) with b < 0.
Motivation Model Solution Conclusion Incompleteness with Multiple Assets • Call and Put options available on all sorts of underlying assets. • But each option depends only on one asset. • Option prices identify risk-neutral marginals, but not the risk-neutral dependence structure. • Infinitely many risk-neutral laws consistent with market marginals. • Market incomplete. • High dimensional problem, but not high enough to complete market... • Which risk neutral law to use? • It depends on the investor’s objective.
Motivation Model Solution Conclusion Literature • Significant (negative) risk premia in options: Coval and Shumway (2001), Bakshi and Kapadia (2003), Santa-Clara and Saretto (2009), Schneider and Trojani (2015). • Optimal payoff as weighted sum of calls and puts on all strikes. Carr and Madan (2001), Carr, Jin, Madan (2001). • Performance manipulation with options on one asset: Goetzmann, Ingersoll, Spiegel, Welch (2007), Guasoni, Huberman, Wang (2011). • Dynamic portfolio choice with options on one asset and one or two strikes: Liu and Pan (2003), Eraker (2013), Faias and Stanta Clara (2011). • “Greek efficient” portfolios with multiple assets: Malamud (2014).
Motivation Model Solution Conclusion The Model • Simplifications: one maturity, continuum of strikes. Shortest maturity options are most liquid. Strikes very numerous. Over 200 for the S&P 500 index, over 100 for large stocks. • One period. Underlying asset prices at end of period X 1 , . . . , X n . Random variables on a probability space (Ω , F , P ) , F = σ ( X 1 , . . . , X n ) . • By Carr-Madan formula, any smooth function f of X i corresponds to a weighted average of options. • Define options portfolio as a n -tuple ( f 1 ( x 1 ) , . . . , f n ( x n )) of L 2 functions with finite price, defined as expecation under risk-neutral marginal. • Optimal payoffs regular if densities regular.
Motivation Model Solution Conclusion Portfolio Objective • Assume zero safe rate to simplify notation. • Payoff Z = f 1 ( X 1 ) + · · · + f n ( X n ) and price π . • Maximize the Sharpe ratio, i.e., find the returns that E [ Z − π ] max σ ( Z ) R • Payoff identified up to scaling and price. Z optimal iff a + bZ optimal, with b > 0. • Ubiquitous objective in performance evaluation. • And tractable.
Motivation Model Solution Conclusion Duality • Maximixing Sharpe ratio equivalent to minimizing variance of SDF. • Convex R ⊂ L 2 ( F , P ) space of payoffs. • Assume some SDF ˆ M > 0 characterizes prices, and denote all SDFs by M = { M ∈ L 2 , E [ RM ] = E [ R ˆ M ] for all R ∈ R} . • Implies that for any excess return: 0 = E [ RM ] = cov( R , M ) + E [ R ] E [ M ] ≥ − σ ( R ) σ ( M ) + E [ R ] • Whence Hansen-Jagannathan bound: E [ R ] sup σ ( R ) ≤ inf M ∈M σ ( M ) R ∈R σ ( R ) � = 0 , E [ MR ]= 0 • Morale: instead of looking for R , look for SDF M ∗ with minimal variance. • If M ∗ is a payoff, R = − M ∗ + E [( M ∗ ) 2 ] spans all optimal returns.
Motivation Model Solution Conclusion Dual Problem • To ease notation: two assets with payoffs X and Y . Solve M ∈M E [ M 2 ] min subject to the restrictions E [ M | X ] = q X ( X ) E [ M | Y ] = q Y ( Y ) p X ( X ) , p Y ( Y ) . • To guess solution, consider SDF of the form M = m ( X , Y ) . (Intuitively, other sources of randomness would only increase variance.) • Two families of infinitely many constraints: Lagrange multipliers? • Reformulate problem in terms of densities.
Motivation Model Solution Conclusion Densities • Find m ( x , y ) that minimizes (interval ( 0 , ∞ ) used for concreteness) � ∞ � ∞ m ( x , y ) 2 p ( x , y ) dxdy 0 0 subject to the constraints � ∞ � ∞ m ( x , y ) p ( x , y ) p X ( x ) dy = q X ( x ) m ( x , y ) p ( x , y ) p Y ( y ) dx = q Y ( y ) p X ( x ) p Y ( y ) 0 0 • Formally, rewrite as unconstrained problem: ∞ ∞ ∞ ∞ � � � � m ( x , y ) 2 p ( x , y ) dxdy − dx m ( x , y ) p ( x , y ) dy − q X ( x ) Φ X ( x ) 0 0 0 0 ∞ ∞ � � dy , − Φ Y ( y ) m ( x , y ) p ( x , y ) dx − q Y ( y ) 0 0 • Functions Φ X ( x ) and Φ Y ( y ) as infinite-dimensional Largrange multipliers.
Motivation Model Solution Conclusion Integral Equations • Eliminating constant terms, equivalent to: � ∞ � ∞ ( m ( x , y ) − Φ X ( x ) − Φ Y ( y )) m ( x , y ) p ( x , y ) dxdy . 0 0 • Setting first-order variation to zero leads to candidate solution m ∗ ( x , y ) = 1 2 (Φ X ( x ) + Φ Y ( y )) where Φ X ( x ) and Φ Y ( y ) are identified by the system of equations � ∞ 1 2 Φ X ( x ) p X ( x ) + 1 Φ Y ( y ) p ( x , y ) dy = q X ( x ) x > 0 , 2 0 � ∞ 1 Φ X ( x ) p ( x , y ) dx + 1 2 Φ Y ( y ) p Y ( y ) = q Y ( y ) y > 0 . 2 0 • Does this have a solution? • If (Φ X , Φ Y ) works, then Φ ′ X ( x ) = Φ ′ X ( x )+ c , Φ ′ Y ( y ) = Φ Y ( y ) − c also works. • Eliminate degree of freedom by setting � ∞ � ∞ Φ X ( x ) p X ( x ) dx = Φ Y ( y ) p Y ( y ) dy 0 0
Motivation Model Solution Conclusion Main Result (1/2) Theorem 2 � p i p c � Assume that M � = ∅ and p < ∞ , 1 ≤ i ≤ n . Then: i � � p � � • (Existence and Uniqueness) There exists a unique minimal SDF M ∗ ∈ M . • (Linearity) There exist Φ := (Φ 1 , . . . , Φ n ) , where each Φ i ∈ L 2 p for 1 ≤ i ≤ n, such that the SDF is of the form M ∗ = m ∗ ( X ) , where � n m ∗ ( ξ ) = 1 i = 1 Φ i ( ξ i ) . n • (Identification) Φ is the unique solution to the system of integral equations � � Φ j ( ξ j ) p ( ξ ) d ξ c p i ( ξ i )Φ i ( ξ i ) + i = nq i ( ξ i ) D c j � = i i � with the uniqueness constraints I i Φ i ( ξ i ) p i ( ξ i ) d ξ i = 1 , 1 ≤ i ≤ n .
Motivation Model Solution Conclusion Main Result (2/2) Theorem • (Performance) Optimal excess returns are of the form a ( m ∗ − E [( m ∗ ) 2 ]) for a < 0 , and their common maximum Sharpe ratio is � n � � 1 � � � SR = Φ i ( ξ i ) q i ( ξ i ) d ξ i − 1 . (2) n I i i = 1 • (Regularity) Let ( q i ) n i = 1 ⊂ C k ( R ) with k ≥ 0 . Denoting the continuous partial derivatives by ∂ β ξ i p ( ξ ) , 0 ≤ β ≤ k, if for any R > 0 there exists α ∈ ( 1 / 2 , 1 ] such that ∂ β � � ξ i p ( ξ ) � � � ( p c i ( ξ c i )) 2 α − 1 d ξ c sup � < ∞ i < ∞ , � � ( p c i ( ξ c i )) α � � D c ξ : � ξ i �≤ R � i � n then m ∗ ( ξ ) = 1 i = 1 Φ i ( ξ i ) is also in C k ( R ) . n
Motivation Model Solution Conclusion Sanity Checks • Risk-Neutrality: If options prices reflect zero risk premium q X / p X = q Y / p Y = 1, then we should neither buy nor sell them. • Indeed, in this case Φ X = Φ Y = 1, whence m ∗ = 1, which has zero variance. • Independence: If X and Y are independent under p , then the optimization problem should separate across assets. • Indeed, Φ X ( x ) = 2 q X ( x ) p X ( x ) − 1, Φ Y ( y ) = 2 q Y ( y ) p Y ( y ) − 1. No interaction. m ∗ ( x , y ) = q X ( x ) p X ( x ) + q Y ( y ) p Y ( y ) − 1. • Trivial example, nontrivial message. If options on multiple underlyings are not traded, the risk-neutral density consistent with independence and the maximization of the Sharpe ratio is q X , Y ( x , y ) = q X ( x ) p Y ( y ) + q Y ( y ) p X ( x ) − p X ( x ) p Y ( y ) . It does not correspond to any particular copula... • Nontrivial explicit solutions with dependence? • Tractability?
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