Optimal Portfolio Under Worst-Case Scenarios Carole Bernard (UW), Jit Seng Chen (UW) and Steven Vanduffel (Vrije Universiteit Brussel)
Rennes, March 2012.
Carole Bernard
Optimal Portfolio
1/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Contributions 1
A better understanding of the link between Growth Optimal Portfolio and optimal investment strategies
2
Understanding issues with traditional diversification strategies and how lowest outcomes of optimal strategies always happen in the worse states of the economy.
3
Develop innovative strategies to cope with this observation.
4
Implications in terms of assessing the risk and return of a strategy and in terms of reducing systemic risk
Carole Bernard
Optimal Portfolio
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Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Part I:
Traditional Diversification Strategies
Carole Bernard
Optimal Portfolio
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Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Growth Optimal Portfolio (GOP) • The Growth Optimal Portfolio (GOP) maximizes expected
logarithmic utility from terminal wealth. • It has the property that it almost surely accumulates more
wealth than any other strictly positive portfolios after a sufficiently long time. • Under general assumptions on the market, the GOP is a
diversified portfolio. • Details in Platen (2006).
Carole Bernard
Optimal Portfolio
4/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Growth Optimal Portfolio (GOP) • The Growth Optimal Portfolio (GOP) maximizes expected
logarithmic utility from terminal wealth. • It has the property that it almost surely accumulates more
wealth than any other strictly positive portfolios after a sufficiently long time. • Under general assumptions on the market, the GOP is a
diversified portfolio. • Details in Platen (2006).
Carole Bernard
Optimal Portfolio
4/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
For example, in the Black-Scholes model • A Black-Scholes financial market (mainly for ease of
exposition) • Risk-free asset {Bt = B0 e rt , t > 0} •
dSt1 St1 dSt2 St2
= µ1 dt + σ1 dWt1 = µ2 dt + σ2 dWt
,
(1)
where W 1 and W are two correlated Brownian motions under the physical probability measure P. p Wt = ρWt1 + 1 − ρ2 Wt2 where W 1 and W 2 are independent. Carole Bernard
Optimal Portfolio
5/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Constant-Mix Strategy • Dynamic rebalancing to preserve the initial target allocation • The payoff of a constant-mix strategy is
Stπ = S0π exp(Xtπ ) where Xtπ is normal. • For an initial investment V0 , VT is given by
VT = V0
STπ , S0π
where π is the vector of proportions.
Carole Bernard
Optimal Portfolio
6/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Growth Optimal Portfolio (GOP) In the 2-dimensional Black-Scholes setting, • The GOP is a constant-mix strategy with Xtπ = µπ − 12 σπ2 t + σπ Wtπ , that maximizes the expected growth rate µπ − 12 σπ2 . It is π ? = Σ−1 · (µ − r 1) .
(2)
• constant-mix portfolios given by π = απ ? with α > 0 and
where π ? is the optimal proportion for the GOP, are optimal strategies for CRRA expected utility maximizers. With a constant relative risk aversion coefficient η > 0, CRRA utility is ( 1−η x when η 6= 1 1−η U(x) = log(x) when η = 1, and α = 1/η. Carole Bernard
Optimal Portfolio
7/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Market Crisis The growth optimal portfolio S ? can also be interpreted as a major market index. Hence it is intuitive to define a stressed market (or crisis) at time T as an event where the market materialized through S ? - drops below its Value-at-Risk at some high confidence level. The corresponding states of the economy verify Crisis states = {ST? < qα } , (3) where qα is such that P(ST? < qα ) = 1 − α and α is typically high (e.g. α = 0.98).
Carole Bernard
Optimal Portfolio
8/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Srategy 1: GOP We invest fully in the GOP. In a crisis (GOP is low), our portfolio is low!
Carole Bernard
Optimal Portfolio
9/35
Strategy 1 vs the Growth Optimal Portfolio 200
180
Strategy 1
160
140
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100
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60 60
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100 120 140 160 Growth Optimal Portfolio, S ∗ (T )
180
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Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Srategy 2: Buy-and-Hold The buy-and-hold strategy is the simplest investment strategy. An initial amount V0 is used to purchase w0 units of the bank account and wi units of stock S i (i = 1, 2) such that V0 = w0 + w1 S01 + w2 S02 , and no further action is undertaken. Example with 1/3 invested in each asset (bank, S1 and S2 ) on next slide.
Carole Bernard
Optimal Portfolio
11/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Srategy 2: Buy-and-Hold The buy-and-hold strategy is the simplest investment strategy. An initial amount V0 is used to purchase w0 units of the bank account and wi units of stock S i (i = 1, 2) such that V0 = w0 + w1 S01 + w2 S02 , and no further action is undertaken. Example with 1/3 invested in each asset (bank, S1 and S2 ) on next slide.
Carole Bernard
Optimal Portfolio
11/35
Strategy 2 vs the Growth Optimal Portfolio 220 200 180
Strategy 2
160 140 120 100 80 60 60
80
100 120 140 160 Growth Optimal Portfolio, S ∗ (T )
180
200
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Strategy 3: Constant-Mix Strategy Example with 1/3 invested in each asset (bank, S1 and S2 ).
Carole Bernard
Optimal Portfolio
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Strategy 3 vs the Growth Optimal Portfolio 200
180
Strategy 3
160
140
120
100
80
60 60
80
100 120 140 160 Growth Optimal Portfolio, S ∗ (T )
180
200
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
I These three traditional diversification strategies do not offer protection during a crisis. I In a more general setting, optimal strategies share the same problem...
Carole Bernard
Optimal Portfolio
15/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Part II:
Optimal portfolio selection for law-invariant preferences
Carole Bernard
Optimal Portfolio
16/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Stochastic Discount Factor and Real-World Pricing: The GOP can be used as numeraire to price under P XT Price of −rT = EQ [e XT ] = EP [ξT XT ] = EP XT at 0 ST? where S0? = 1.
Carole Bernard
Optimal Portfolio
17/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Stochastic Discount Factor and Real-World Pricing: The GOP can be used as numeraire to price under P XT Price of −rT = EQ [e XT ] = EP [ξT XT ] = EP XT at 0 ST? where S0? = 1.
Carole Bernard
Optimal Portfolio
17/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Cost-efficient strategies (Dybvig (1988)) Optimal Portfolio Selection Problem: Consider an investor with fixed investment horizon: max U(XT ) XT
subject to a given “cost of XT ” (equal to initial wealth) • Law-invariant preferences XT ∼ YT ⇒ U(XT ) = U(YT ) • Increasing preferences XT ∼ F , YT ∼ G , ∀x, F (x) 6 G (x) ⇒ U(XT ) > U(YT ) A strategy (or a payoff) is cost-efficient if any other strategy that generates the same distribution under P costs at least as much. The optimal strategy for U must be cost-efficient. Carole Bernard
Optimal Portfolio
18/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Cost-efficient strategies (Dybvig (1988)) Optimal Portfolio Selection Problem: Consider an investor with fixed investment horizon: max U(XT ) XT
subject to a given “cost of XT ” (equal to initial wealth) • Law-invariant preferences XT ∼ YT ⇒ U(XT ) = U(YT ) • Increasing preferences XT ∼ F , YT ∼ G , ∀x, F (x) 6 G (x) ⇒ U(XT ) > U(YT ) A strategy (or a payoff) is cost-efficient if any other strategy that generates the same distribution under P costs at least as much. The optimal strategy for U must be cost-efficient. Carole Bernard
Optimal Portfolio
18/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Cost-efficient strategies (Dybvig (1988)) Optimal Portfolio Selection Problem: Consider an investor with fixed investment horizon: max U(XT ) XT
subject to a given “cost of XT ” (equal to initial wealth) • Law-invariant preferences XT ∼ YT ⇒ U(XT ) = U(YT ) • Increasing preferences XT ∼ F , YT ∼ G , ∀x, F (x) 6 G (x) ⇒ U(XT ) > U(YT ) A strategy (or a payoff) is cost-efficient if any other strategy that generates the same distribution under P costs at least as much. The optimal strategy for U must be cost-efficient. Carole Bernard
Optimal Portfolio
18/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Optimal Portfolio and Cost-efficiency Consider an investor with increasing law-invariant preferences and a fixed horizon. Denote by XT the investor’s final wealth. The optimal strategy solves a cost-efficiency problem XT min E ST? {XT | XT ∼F } Reciprocally a cost-efficient strategy with a continuous distribution F corresponds to the optimum of an expected utility investor for Z x U(x) = G −1 (1 − F (y ))dy 0
where G is the cdf of Carole Bernard
1 ST? . Optimal Portfolio
19/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Optimal Portfolio and Cost-efficiency Consider an investor with increasing law-invariant preferences and a fixed horizon. Denote by XT the investor’s final wealth. The optimal strategy solves a cost-efficiency problem XT min E ST? {XT | XT ∼F } Reciprocally a cost-efficient strategy with a continuous distribution F corresponds to the optimum of an expected utility investor for Z x U(x) = G −1 (1 − F (y ))dy 0
where G is the cdf of Carole Bernard
1 ST? . Optimal Portfolio
19/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Black-Scholes Model Theorem Consider the following optimization problem: XT PD(F ) := min E ST? {XT | XT ∼F } In a Black-Scholes model, the optimal strategy (cheapest way to get F ) is XT? = F −1 FST? (ST? ) . Note that XT? ∼ F and XT? is a.s. unique. Corollary A strategy with payoff XT = h(ST? ) is cost-efficient if and only if h is non-decreasing. Carole Bernard
Optimal Portfolio
20/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Black-Scholes Model Theorem Consider the following optimization problem: XT PD(F ) := min E ST? {XT | XT ∼F } In a Black-Scholes model, the optimal strategy (cheapest way to get F ) is XT? = F −1 FST? (ST? ) . Note that XT? ∼ F and XT? is a.s. unique. Corollary A strategy with payoff XT = h(ST? ) is cost-efficient if and only if h is non-decreasing. Carole Bernard
Optimal Portfolio
20/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Idea of the proof XT min E XT ST? ( XT ∼ F subject to 1 S? ∼ G T
Recall that corr XT ,
1 ST?
=
i h E XT S1? − E[ S1? ]E[XT ] T
T
std( S1? ) std(XT )
.
T
We can prove that when the distributions for both XT and S1? are T fixed, we have 1 ? (XT , ST ) is comonotonic ⇒ corr XT , ? is minimal. ST Carole Bernard
Optimal Portfolio
21/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Idea of the proof XT min E XT ST? ( XT ∼ F subject to 1 S? ∼ G T
Recall that corr XT ,
1 ST?
=
i h E XT S1? − E[ S1? ]E[XT ] T
T
std( S1? ) std(XT )
.
T
We can prove that when the distributions for both XT and S1? are T fixed, we have 1 ? (XT , ST ) is comonotonic ⇒ corr XT , ? is minimal. ST Carole Bernard
Optimal Portfolio
21/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Part III:
Investment under Worst-Case Scenarios
Carole Bernard
Optimal Portfolio
22/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Investment with State-Dependent Constraints Problem considered so far
min
{XT | XT ∼F }
XT E . ST?
A payoff that solves this problem is cost-efficient. New Problem
min
{VT | VT ∼F , S}
VT E . ST?
where S denotes a set of constraints. A payoff that solves this problem is called a S−constrained cost-efficient payoff.
Carole Bernard
Optimal Portfolio
23/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Type of Constraints We are able to find optimal strategies with final payoff VT I with an additional probability constraint P(ST? 6 s, VT 6 v ) = β I with a set of probability constraints ∀(s, v ) ∈ S, P(ST? 6 s, VT 6 v ) = Q(s, v ) where Q is an appropriate given function and S verifies some properties. I in particular, assuming that the final payoff of the strategy is independent of ST? during a crisis (defined as ST? 6 qα ), ∀s 6 qα , v ∈ R, P(ST? 6 s, VT 6 v ) = P(ST? 6 s)P(VT 6 v ) Carole Bernard
Optimal Portfolio
24/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Independence in the Tail - Strategy 4: Path-dependent Theorem The cheapest path-dependent strategy with a cumulative distribution F but such that it is independent of ST? when ST? 6 qα can be constructed as FS ? (ST? )−α −1 T F when ST? > qα , 1−α St? t ? ? VT = ln S ? t/T −(1− T ) ln(S0 ) ( ) T −1 q F Φ when ST? 6 qα , t2 σ t− ? T (4) where t ∈ (0, T ) can be chosen freely. (No uniqueness and path-independence anymore). Carole Bernard
Optimal Portfolio
25/35
Strategy 4 vs the Growth Optimal Portfolio 200
180
Strategy 4
160
140
120
100
80
60 60
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100 120 140 160 Growth Optimal Portfolio, S ∗ (T )
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200
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Independence in the Tail - Strategy 5: Path-independent In a financial market that contains at least two assets that are continuously distributed, the cheapest path-independent strategy with a cumulative distribution F but such that it is independent of ST? when ST? 6 qα can be constructed as ? −1 FST? (ST )−α F when ST? > qα 1−α ZT? = . (5) −1 ? F (Φ(A)) when ST 6 qα where A is explicitly known as a function of ST1 and ST? .
Carole Bernard
Optimal Portfolio
27/35
Strategy 5 vs the Growth Optimal Portfolio 200
180
Strategy 5
160
140
120
100
80
60 60
80
100 120 140 160 Growth Optimal Portfolio, S ∗ (T )
180
200
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Part IV:
Investment under Worst-Case Scenarios Some numerical examples
Carole Bernard
Optimal Portfolio
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Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Other Types of Dependence Recall that the joint cdf of a couple (ST? , X ) writes as P(ST? 6 s, XT 6 x) = C (H(s), F (x)) where • The marginal cdf of ST? : H • The marginal cdf of XT : F • A copula C Independence in the tail (independence copula C (u, v ) = uv ): ∀s 6 qα , v ∈ R, P(ST? 6 s, VT 6 v ) = P(ST? 6 s)P(VT 6 v ) I We were also able to derive formulas for optimal strategies that generate a Gaussian distribution in the tail with a correlation coefficient of -0.5. I Similarly for Clayton or Frank dependence. Carole Bernard
Optimal Portfolio
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Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Optimal Investment with a Clayton Tail Dependence
The cheapest strategy VT? with cdf F that verifies this Clayton dependence (with correlation -0.5) in the tail is h i−1/a −1 ? −a −a F (FST? (ST ) − α) − (1 − α) + 1 if ST? > qα ? VT = F −1 g 1 − FS ? (S ? ), jF ? (S ? ) (FZ (ZT )) if ST? 6 qα , T T S T T T
where ZT is such that (ST? , ZT ) is continuously distributed (with copula J) and where g is known explicitly: h i−1/a g (u, v ) = u −a v −a/(1+a) − 1 + 1 .
Carole Bernard
Optimal Portfolio
31/35
Strategy 8 vs the Growth Optimal Portfolio 200
180
Strategy 8
160
140
120
100
80
60 60
80
100 120 140 160 Growth Optimal Portfolio, S ∗ (T )
180
200
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Some numerical results We define two events related to the market, i.e. the market crisis ? C = {S and T? < qα?} rT a decrease in the market D = ST < S0 e . We further define two events for the portfolio value by A = VT < V0 e rT and B = VT < 75%V0 e rT
GOP Buy-and-Hold Independence Gaussian Clayton
Carole Bernard
T 5 5 5 5 5
Cost 100 100 101.67 103.40 102.35
Sharpe 0.266 0.239 0.214 0.159 0.193
P(A|C) 1.00 0.9998 0.46 0.12 0.24
P(A|D) 1.00 0.965 0.94 0.90 0.91
P(B|C) 1.00 0.99 0.13 0.01 0.02
Optimal Portfolio
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Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Conclusions • Cost-efficiency: a preference-free framework for ranking
different investment strategies. • Characterization of optimal portfolio strategies for
investors with law invariant preferences and a fixed horizon. I Lowest outcomes in worst states of the economy • Optimal investment choice under state-dependent
constraints.
• not always non-decreasing with the GOP ST? . • not anymore unique • could be path-dependent.
I Trade-off between losing “utility” and gaining from better fit of the investor’s preferences. Carole Bernard
Optimal Portfolio
34/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
More Implications I The new strategies do not incur their biggest losses in the worst states in the economy. I can be used to reduce systemic risk. • the idea of assessing risk and performance of a portfolio not
only by looking at its final distribution but also by looking at its interaction with the economic conditions is indeed related to the increasing concern to evaluate systemic risk. • Acharya (2009) explains that regulators should “be regulating each bank as a function of both its joint (correlated) risk with other banks as well as its individual (bank-specific) risk”. • An insight of this work is that if all institutional investors implement strategies that are resilient against crisis regimes, as we propose, then systemic risk can be diminished.
Do not hesitate to contact me to get updated working papers! Carole Bernard
Optimal Portfolio
35/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
References I Bernard, C., Boyle P., Vanduffel S., 2011, “Explicit Representation of Cost-efficient Strategies”, available on SSRN. I Bernard, C., Jiang, X., Vanduffel, S., 2012. “Note on Improved Frechet bounds and model-free pricing of multi-asset options”, Journal of Applied Probability. I Bernard, C., Maj, M., Vanduffel, S., 2011. “Improving the Design of Financial Products in a Multidimensional Black-Scholes Market,”, North American Actuarial Journal. I Bernard, C., Vanduffel, S., 2011. “Optimal Investment under Probability Constraints,” AfMath Proceedings. I Bernard, C., Vanduffel, S., 2012. “Financial Bounds for Insurance Prices,”Journal of Risk Insurance. I Cox, J.C., Leland, H., 1982. “On Dynamic Investment Strategies,” Proceedings of the seminar on the Analysis of Security Prices,(published in 2000 in JEDC). I Dybvig, P., 1988a. “Distributional Analysis of Portfolio Choice,” Journal of Business. I Dybvig, P., 1988b. “Inefficient Dynamic Portfolio Strategies or How to Throw Away a Million Dollars in the Stock Market,” Review of Financial Studies. I Goldstein, D.G., Johnson, E.J., Sharpe, W.F., 2008. “Choosing Outcomes versus Choosing Products: Consumer-focused Retirement Investment Advice,” Journal of Consumer Research. I Jin, H., Zhou, X.Y., 2008. “Behavioral Portfolio Selection in Continuous Time,” Mathematical Finance. I Nelsen, R., 2006. “An Introduction to Copulas”, Second edition, Springer. I Pelsser, A., Vorst, T., 1996. “Transaction Costs and Efficiency of Portfolio Strategies,” European Journal of Operational Research. I Platen, E., 2005. “A benchmark approach to quantitative finance,” Springer finance. I Tankov, P., 2011. “Improved Frechet bounds and model-free pricing of multi-asset options,” Journal of Applied Probability, forthcoming. I Vanduffel, S., Chernih, A., Maj, M., Schoutens, W. 2009. “On the Suboptimality of Path-dependent Pay-offs in L´ evy markets”, Applied Mathematical Finance.
∼∼∼ Carole Bernard
Optimal Portfolio
36/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Part V:
Proofs with Copulas Optimal Portfolio under Tail Dependence
Carole Bernard
Optimal Portfolio
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Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Copulas and Sklar’s theorem The joint cdf of a couple (ξT , X ) can be decomposed into 3 elements • The marginal cdf of ξT : G • The marginal cdf of XT : F • A copula C
such that P(ξT 6 ξ, XT 6 x) = C (G (ξ), F (x))
Carole Bernard
Optimal Portfolio
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Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Where do copulas appear?
in the derivation of “cost-efficient” strategies...
Solving the cost-efficiency problem amounts to finding bounds on copulas! min E [ξT XT ] XT XT ∼ F subject to ξT ∼ G
Carole Bernard
Optimal Portfolio
39/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Proof of the cost-efficient payoff min E [ξT XT ] XT XT ∼ F subject to ξT ∼ G The distribution G is known and depends on the financial market. Let C denote a copula for (ξT , X ). Z Z E[ξT X ] =
(1 − G (ξ) − F (x) + C (G (ξ), F (x)))dxdξ,
(6)
The lower bound for E[ξT X ] is derived from the lower bound on C max(u + v − 1, 0) 6 C (u, v ) (where max(u + v − 1, 0) corresponds to the anti-monotonic copula). E [ξ F −1 (1 − G (ξ ))] 6 E [ξ X ] T
T
T
T
then XT? = F −1 (1 − G (ξT )) has the minimum price for the cdf F . Carole Bernard
Optimal Portfolio
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Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Proof of the cost-efficient payoff min E [ξT XT ] XT XT ∼ F subject to ξT ∼ G The distribution G is known and depends on the financial market. Let C denote a copula for (ξT , X ). Z Z E[ξT X ] =
(1 − G (ξ) − F (x) + C (G (ξ), F (x)))dxdξ,
(6)
The lower bound for E[ξT X ] is derived from the lower bound on C max(u + v − 1, 0) 6 C (u, v ) (where max(u + v − 1, 0) corresponds to the anti-monotonic copula). E [ξ F −1 (1 − G (ξ ))] 6 E [ξ X ] T
T
T
T
then XT? = F −1 (1 − G (ξT )) has the minimum price for the cdf F . Carole Bernard
Optimal Portfolio
40/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Proof of the cost-efficient payoff min E [ξT XT ] XT XT ∼ F subject to ξT ∼ G The distribution G is known and depends on the financial market. Let C denote a copula for (ξT , X ). Z Z E[ξT X ] =
(1 − G (ξ) − F (x) + C (G (ξ), F (x)))dxdξ,
(6)
The lower bound for E[ξT X ] is derived from the lower bound on C max(u + v − 1, 0) 6 C (u, v ) (where max(u + v − 1, 0) corresponds to the anti-monotonic copula). E [ξ F −1 (1 − G (ξ ))] 6 E [ξ X ] T
T
T
T
then XT? = F −1 (1 − G (ξT )) has the minimum price for the cdf F . Carole Bernard
Optimal Portfolio
40/35
Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Sufficient condition for the existence Theorem Let t ∈ (0, T ). If there exists a copula L satisfying S such that L 6 C (pointwise) for all other copulas C satisfying S then the payoff YT? given by YT? = F −1 (f (ξT , ξt )) is a S-constrained cost-efficient payoff. Here f (ξT , ξt ) is given by f (ξT , ξt ) = `G (ξT )
−1
jG (ξT ) (G (ξt )) ,
where the functions ju (v ) and `u (v ) are defined as the first partial derivative for (u, v ) → J(u, v ) and (u, v ) → L(u, v ) respectively and where J denotes the copula for the random pair (ξT , ξt ). If (U, V ) has a copula L then `u (v ) = P(V 6 v |U = u). When S = ∅, f (ξt , ξT ) = F −1 (1 − G (ξT )). Carole Bernard
Optimal Portfolio
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Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Sufficient condition for the existence Theorem Let t ∈ (0, T ). If there exists a copula L satisfying S such that L 6 C (pointwise) for all other copulas C satisfying S then the payoff YT? given by YT? = F −1 (f (ξT , ξt )) is a S-constrained cost-efficient payoff. Here f (ξT , ξt ) is given by f (ξT , ξt ) = `G (ξT )
−1
jG (ξT ) (G (ξt )) ,
where the functions ju (v ) and `u (v ) are defined as the first partial derivative for (u, v ) → J(u, v ) and (u, v ) → L(u, v ) respectively and where J denotes the copula for the random pair (ξT , ξt ). If (U, V ) has a copula L then `u (v ) = P(V 6 v |U = u). When S = ∅, f (ξt , ξT ) = F −1 (1 − G (ξT )). Carole Bernard
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Introduction
Diversification Strategies
Cost-Efficiency
Tail Dependence
Numerical Example
Conclusions
Proofs
Existence of the optimum ⇔ Existence of minimum copula Theorem (Sufficient condition for existence of a minimal copula L) Let S be a rectangle [u1 , u2 ] × [v1 , v2 ] ⊆ [0, 1]2 . Then a minimal copula L(u, v ) satisfying S exists and is given by L(u, v ) = max {0, u + v − 1, K (u, v )} . where K (u, v ) = max(a,b)∈ S {Q(a, b) − (a − u)+ − (b − v )+ }. Proof in a note written with Xiao Jiang and Steven Vanduffel extending Tankov’s result. Consequently the existence of a S−constrained cost-efficient payoff is guaranteed when S is a rectangle. More generally it also holds when S ⊆ [0, 1]2 satisfies a “monotonicity property” of the upper and lower “boundaries” and v0 + v1 ∀ (u, v0 ) , (u, v1 ) ∈ S, u, ∈ S. (7) 2 Carole Bernard
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