![Motivation Model Solution Conclusion
Portfolio Objective
• Assume zero safe rate to simplify notation.
• Payoff Z = f1(X1) + · · · + fn(Xn) and price π.
• Maximize the Sharpe ratio, i.e., find the returns that
max
R
E[Z − π]
σ(Z)
• Payoff identified up to scaling and price.
Z optimal iff a + bZ optimal, with b > 0.
• Ubiquitous objective in performance evaluation.
• And tractable.](https://image.slidesharecdn.com/optionsfips-180912150337/85/Options-Portfolio-Selection-8-320.jpg)
![Motivation Model Solution Conclusion
Duality
• Maximixing Sharpe ratio equivalent to minimizing variance of SDF.
• Convex R ⊂ L2
(F, P) space of payoffs.
• Assume some SDF ˆM > 0 characterizes prices, and denote all SDFs by
M = {M ∈ L2
, E[RM] = E[R ˆM] 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:
sup
R∈R
σ(R)=0,E[MR]=0
E[R]
σ(R)
≤ inf
M∈M
σ(M)
• 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.](https://image.slidesharecdn.com/optionsfips-180912150337/85/Options-Portfolio-Selection-9-320.jpg)
![Motivation Model Solution Conclusion
Dual Problem
• To ease notation: two assets with payoffs X and Y. Solve
min
M∈M
E[M2
]
subject to the restrictions
E[M|X] =
qX (X)
pX (X)
, E[M|Y] =
qY (Y)
pY (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.](https://image.slidesharecdn.com/optionsfips-180912150337/85/Options-Portfolio-Selection-10-320.jpg)
![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
SR =
1
n
n
i=1 Ii
Φi (ξi )qi (ξi )dξi − 1. (2)
• (Regularity) Let (qi )n
i=1 ⊂ Ck
(R) with k ≥ 0. Denoting the continuous
partial derivatives by ∂β
ξi
p(ξ), 0 ≤ β ≤ k, if for any R > 0 there exists
α ∈ (1/2, 1] such that
sup
ξ: ξi ≤R
∂β
ξi
p(ξ)
(pc
i (ξc
i ))α
< ∞
Dc
i
(pc
i (ξc
i ))2α−1
dξc
i < ∞,
then m∗
(ξ) = 1
n
n
i=1 Φi (ξi ) is also in Ck
(R).](https://image.slidesharecdn.com/optionsfips-180912150337/85/Options-Portfolio-Selection-14-320.jpg)
![Motivation Model Solution Conclusion
Example: Variance Gamma Model
• Common wisdom on option portfolios:
Writing options profitable but risky. Diversify over many assets.
• Which strikes to write more? Impact of correlation?
• Example: Variance-Gamma model.
Combines no-arbitrage with different realized and implied volatilities.
Important to separate options’ risk-premia from assets’ risk premia.
• Two risky asset prices, both distributed as
Xt = X0eωt+Zt (σ,ν,θ)
,
where Zt has the characteristic function
E[eiuZt
] = (1 − iθνu +
σ2
2
u2
ν)−t/ν
, u ∈ R
• Marginal of a Levy process with jump measure kZ (x) = eθx/σ2
ν|x| e−
2
ν
+ θ2
σ2
σ |x|
.
• Dependence modeled through bivariate t-copula.
• Assets’ risk premia both zero.](https://image.slidesharecdn.com/optionsfips-180912150337/85/Options-Portfolio-Selection-19-320.jpg)
Options Portfolio Selection
- 1. Motivation Model Solution Conclusion Options Portfolio Selection Paolo Guasoni1,2 Eberhard Mayerhofer3 Boston University1 Dublin City University2 University of Limerick3 2018 Workshop on Finance, Insurance, Probability and Statistics September 11th 2018, King’s College London
- 2. 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.
- 3. 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.
- 4. Motivation Model Solution Conclusion One Asset • With one asset and one maturity, problem tractable. • X underlying asset price at maturity. cX (K) price of a call option on X with strike price K. pX (x) physical marginal density of X. • Assume that continuum of strikes is available. • Risk-neutral density qX (K) is (Breeden and Litzenberger, 1978) qX (K) := cX (K) (1) • Thus, the unique SDF is the random variable mX (x) = cX (x)/pX (X). • If the function mX is regular enough, the payoff decomposes as a portfolio of call and put options (Carr and Madan, 2001) mX (K) = mX (K0) + mX (K0)(K − K0) + K0 0 mX (κ)(κ − K)+ dκ + ∞ K0 mX (κ)(K − κ)+ dκ. • Payoffs with maximal Sharpe of the form R = a + b mX (X) with b < 0.
- 5. 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.
- 6. 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).
- 7. 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 X1, . . . , Xn. Random variables on a probability space (Ω, F, P), F = σ(X1, . . . , Xn). • By Carr-Madan formula, any smooth function f of Xi corresponds to a weighted average of options. • Define options portfolio as a n-tuple (f1(x1), . . . , fn(xn)) of L2 functions with finite price, defined as expecation under risk-neutral marginal. • Optimal payoffs regular if densities regular.
- 8. Motivation Model Solution Conclusion Portfolio Objective • Assume zero safe rate to simplify notation. • Payoff Z = f1(X1) + · · · + fn(Xn) and price π. • Maximize the Sharpe ratio, i.e., find the returns that max R E[Z − π] σ(Z) • Payoff identified up to scaling and price. Z optimal iff a + bZ optimal, with b > 0. • Ubiquitous objective in performance evaluation. • And tractable.
- 9. Motivation Model Solution Conclusion Duality • Maximixing Sharpe ratio equivalent to minimizing variance of SDF. • Convex R ⊂ L2 (F, P) space of payoffs. • Assume some SDF ˆM > 0 characterizes prices, and denote all SDFs by M = {M ∈ L2 , E[RM] = E[R ˆM] 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: sup R∈R σ(R)=0,E[MR]=0 E[R] σ(R) ≤ inf M∈M σ(M) • 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.
- 10. Motivation Model Solution Conclusion Dual Problem • To ease notation: two assets with payoffs X and Y. Solve min M∈M E[M2 ] subject to the restrictions E[M|X] = qX (X) pX (X) , E[M|Y] = qY (Y) pY (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.
- 11. Motivation Model Solution Conclusion Densities • Find m(x, y) that minimizes (interval (0, ∞) used for concreteness) ∞ 0 ∞ 0 m(x, y)2 p(x, y)dxdy subject to the constraints ∞ 0 m(x, y) p(x, y) pX (x) dy = qX (x) pX (x) ∞ 0 m(x, y) p(x, y) pY (y) dx = qY (y) pY (y) • Formally, rewrite as unconstrained problem: ∞ 0 ∞ 0 m(x, y)2 p(x, y)dxdy − ∞ 0 ΦX (x) ∞ 0 m(x, y)p(x, y)dy − qX (x) dx − ∞ 0 ΦY (y) ∞ 0 m(x, y)p(x, y)dx − qY (y) dy, • Functions ΦX (x) and ΦY (y) as infinite-dimensional Largrange multipliers.
- 12. Motivation Model Solution Conclusion Integral Equations • Eliminating constant terms, equivalent to: ∞ 0 ∞ 0 (m(x, y) − ΦX (x) − ΦY (y)) m(x, y)p(x, y)dxdy. • Setting first-order variation to zero leads to candidate solution m∗ (x, y) = 1 2 (ΦX (x) + ΦY (y)) where ΦX (x) and ΦY (y) are identified by the system of equations 1 2 ΦX (x)pX (x) + 1 2 ∞ 0 ΦY (y)p(x, y)dy =qX (x) x > 0, 1 2 ∞ 0 ΦX (x)p(x, y)dx + 1 2 ΦY (y)pY (y) =qY (y) y > 0. • Does this have a solution? • If (ΦX , ΦY ) works, then ΦX (x) = ΦX (x)+c, ΦY (y) = ΦY (y)−c also works. • Eliminate degree of freedom by setting ∞ 0 ΦX (x)pX (x)dx = ∞ 0 ΦY (y)pY (y)dy
- 13. Motivation Model Solution Conclusion Main Result (1/2) Theorem Assume that M = ∅ and pi pc i p 2 p < ∞, 1 ≤ i ≤ n. Then: • (Existence and Uniqueness) There exists a unique minimal SDF M∗ ∈ M. • (Linearity) There exist Φ := (Φ1, . . . , Φn), where each Φi ∈ L2 p for 1 ≤ i ≤ n, such that the SDF is of the form M∗ = m∗ (X), where m∗ (ξ) = 1 n n i=1 Φi (ξi ). • (Identification) Φ is the unique solution to the system of integral equations pi (ξi )Φi (ξi ) + j=i Dc i Φj (ξj )p(ξ)dξc i = nqi (ξi ) with the uniqueness constraints Ii Φi (ξi )pi (ξi )dξi = 1, 1 ≤ i ≤ n.
- 14. 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 SR = 1 n n i=1 Ii Φi (ξi )qi (ξi )dξi − 1. (2) • (Regularity) Let (qi )n i=1 ⊂ Ck (R) with k ≥ 0. Denoting the continuous partial derivatives by ∂β ξi p(ξ), 0 ≤ β ≤ k, if for any R > 0 there exists α ∈ (1/2, 1] such that sup ξ: ξi ≤R ∂β ξi p(ξ) (pc i (ξc i ))α < ∞ Dc i (pc i (ξc i ))2α−1 dξc i < ∞, then m∗ (ξ) = 1 n n i=1 Φi (ξi ) is also in Ck (R).
- 15. Motivation Model Solution Conclusion Sanity Checks • Risk-Neutrality: If options prices reflect zero risk premium qX /pX = qY /pY = 1, then we should neither buy nor sell them. • Indeed, in this case ΦX = ΦY = 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, ΦX (x) = 2qX (x) pX (x) − 1, ΦY (y) = 2qY (y) pY (y) − 1. No interaction. m∗ (x, y) = qX (x) pX (x) + qY (y) pY (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 qX,Y (x, y) = qX (x)pY (y) + qY (y)pX (x) − pX (x)pY (y). It does not correspond to any particular copula... • Nontrivial explicit solutions with dependence? • Tractability?
- 16. Motivation Model Solution Conclusion Mixture Distributions (1/2) • Solving integral equations is nontrivial. To break the spell, discretize. • (pi X )1≤i≤k , (pi Y )1≤i≤k strictly positive probability densities on (0, ∞). p(x, y) := 1 k k i=1 pi X (x)pi Y (y). (Remember the proof of Fubini-Tonelli theorem?) • Plug into integral equations. They become pX (x) 2 ΦX (x) = qX (x) − k i=1 ci Y pi X (x), pY (y) 2 ΦY (y) = qY (y) − k i=1 ci X pi Y (y), where the 2k constants (ci X )1≤i≤k , (ci Y )1≤i≤k are ci X = 1 2k ∞ 0 ΦX (x)pi X (x)dx, ci Y = 1 2k ∞ 0 ΦY (y)pi Y (y)dy. • Plug formulas for ΦX and ΦY again.
- 17. Motivation Model Solution Conclusion Mixture Distributions (2/2) • Obtain system of 2k equations in 2k unknowns ci Y = 1 k ∞ 0 qY (y) pi Y (y) pY (y) dy − 1 k k j=1 cj X ∞ 0 pY (y)j pi Y (y) pY (y) dy 1 ≤ i ≤ k ci X = 1 k ∞ 0 qX (x) pi X (x) pX (x) dx − 1 k n j=1 cj Y ∞ 0 pj X (x)pi X (x) pX (x) dx 1 ≤ i ≤ k. • But the rank is 2k − 1. • Drop one equation and replace it with the uniqueness constraint k i=1 ci X − k i=1 ci Y = 0. • Now system is invertible. • Note: k in mixture representation independent of number of assets n. (Independence corresponds to a minimal k = 1 regardless of n.) • No curse of dimensionality.
- 18. Motivation Model Solution Conclusion Discrete Densities • Another tractable discretization is with piecewise constant densities. • Two increasing finite sequences (xi )0≤i≤k and (yj )0≤j≤l . • Assume P(X ∈ [x0, xk ), Y ∈ [y0, yl )) = Q(X ∈ [x0, xk ), Y ∈ [y0, yl )) = 1. • Assume joint probability density p constant on each rectangle Ix i × Iy j , where Ix i = [xi−1, xi ), 1 ≤ i ≤ k, and Iy j = [yj−1, yj ), 1 ≤ j ≤ l. • Denote ˜pij = P(X ∈ Ix i , Y ∈ Iy j ), ˜pi X = P(X ∈ Ix i ), ˜pj Y = P(Y ∈ Iy j ), and ˜qi X = Q(X ∈ Ix i ), ˜qj Y = Q(Y ∈ Iy j ), 1 ≤ i ≤ k, 1 ≤ j ≤ l. • Any solution ΦX , ΦY piecewise constant on (Ix i )1≤i≤n and (Iy j )1≤j≤m. Set Φi X = ΦX (xi ) and Φi Y = ΦY (xj ). • Integral equations reduce to: Φi X ˜pi X + k j=1 Φj Y ˜pij = 2˜qi X , 1 ≤ i ≤ k, Φj Y ˜pj Y + l i=1 Φi X ˜pij = 2˜qj Y , 1 ≤ j ≤ l. • Uniqueness constraint n i=1 Φi X ˜pi X − m j=1 Φj Y ˜pj Y = 0. • Curse of dimensionality.
- 19. Motivation Model Solution Conclusion Example: Variance Gamma Model • Common wisdom on option portfolios: Writing options profitable but risky. Diversify over many assets. • Which strikes to write more? Impact of correlation? • Example: Variance-Gamma model. Combines no-arbitrage with different realized and implied volatilities. Important to separate options’ risk-premia from assets’ risk premia. • Two risky asset prices, both distributed as Xt = X0eωt+Zt (σ,ν,θ) , where Zt has the characteristic function E[eiuZt ] = (1 − iθνu + σ2 2 u2 ν)−t/ν , u ∈ R • Marginal of a Levy process with jump measure kZ (x) = eθx/σ2 ν|x| e− 2 ν + θ2 σ2 σ |x| . • Dependence modeled through bivariate t-copula. • Assets’ risk premia both zero.
- 20. Motivation Model Solution Conclusion σP X = 20%, σQ X = σQ Y = σP Y = 25% 80 90 100 110 120 −0.100.000.100.20 Underlying Asset 1 Payoff 80 90 100 110 120 −0.100.000.100.20 Underlying Asset 2 Payoff 80 90 100 110 120 0.0000.0100.0200.030 Density 80 90 100 110 120 0.0000.0100.0200.030 Density
- 21. Motivation Model Solution Conclusion σP X = 20%, σQ X = 25%, σP Y = 25%, σQ Y = 40% 80 90 100 110 120 −0.050.050.15 Underlying Asset 1 Payoff 80 90 100 110 120 −0.050.050.15 Underlying Asset 2 Payoff 80 90 100 110 120 0.0000.0100.0200.030 Density 80 90 100 110 120 0.0000.0100.0200.030 Density
- 22. Motivation Model Solution Conclusion Performance Figure 1 Figure 2 Correlation (annual) (monthly) (annual) (monthly) 0% 0.29 0.68 0.62 1.71 60% 0.31 0.74 0.58 1.63 75% 0.33 0.84 0.58 1.67 90% 0.43 1.17 0.63 1.99 • Annualized Sharpe ratios of optimal portfolios. • Trade annually (left) or monthly (right). • Higher correlation? Higher Sharpe ratio. Against intuition on diversification. • Reason: correlation is among assets, not all options. • Keeping the same marginals while increasing correlation increases the diversification and hedging opportunities among individual options.
- 23. Motivation Model Solution Conclusion Conclusion • Options portfolio selection. • Each option on one underlying asset. Market incomplete with multiple assets. • Maximize Sharpe ratio: system of linear integral equations. • Integral equations intractable virtually all nontrivial cases. Discretizations tractable in virtually all cases. • Optimal payoffs in one asset depend on options prices in all other assets. Except with independence. • It may be optimal to buy options in one asset, expecting to lose. Just to hedge more profitable options in another asset.
- 24. Motivation Model Solution Conclusion Thank You! Questions? http://ssrn.com/abstract=3075945