Exponentially concave functions and multiplicative cyclical - - PowerPoint PPT Presentation

Exponentially concave functions and multiplicative cyclical monotonicity

• W. Schachermayer

University of Vienna Faculty of Mathematics

11th December, RICAM, Linz Based on the paper S. Pal, T.L. Wong: The Geometry of relative arbitrage. (ArXiv:1402.3720)

Exponential concavity

Fix a convex subset U ⊆ Rn. Typically U will equal the unit simplex ∆n = {(p1, . . . , pn) ∈ [0, 1]n :

n

• i=1

pi = 1}

• r a convex subset of ∆n, e.g. int(∆n).

Definition: A function ϕ : U → R is called exponentially concave if Φ = exp(ϕ) is a concave function. An exponentially concave function is concave but not vice versa. For example, consider U = (0, 1). An affine function ϕ(x) = ax + b on U is concave but not exponentially concave. On the other hand, the function ϕ(x) = log(x) is the arch-example of an exponentially concave function on U.

Exponential concavity

Fix a convex subset U ⊆ Rn. Typically U will equal the unit simplex ∆n = {(p1, . . . , pn) ∈ [0, 1]n :

n

• i=1

pi = 1}

• r a convex subset of ∆n, e.g. int(∆n).

Definition: A function ϕ : U → R is called exponentially concave if Φ = exp(ϕ) is a concave function. An exponentially concave function is concave but not vice versa. For example, consider U = (0, 1). An affine function ϕ(x) = ax + b on U is concave but not exponentially concave. On the other hand, the function ϕ(x) = log(x) is the arch-example of an exponentially concave function on U.

Exponential concavity

Fix a convex subset U ⊆ Rn. Typically U will equal the unit simplex ∆n = {(p1, . . . , pn) ∈ [0, 1]n :

n

• i=1

pi = 1}

• r a convex subset of ∆n, e.g. int(∆n).

Definition: A function ϕ : U → R is called exponentially concave if Φ = exp(ϕ) is a concave function. An exponentially concave function is concave but not vice versa. For example, consider U = (0, 1). An affine function ϕ(x) = ax + b on U is concave but not exponentially concave. On the other hand, the function ϕ(x) = log(x) is the arch-example of an exponentially concave function on U.

Lemma: Let U be a convex subset of Rn. A concave function ϕ : U → R is exponentially concave iff for every µ0, µ1 ∈ U the function f (t) = ϕ(tµ0 + (1 − t)µ1) satisfies f ′′(t) ≤ −(f ′(t))2, for almost all t in [0, 1]. Indeed, for F(t) = exp(f (t)) we have F ′′(t) = F(t)[f ′′(t) + (f ′(t))2].

Lemma: Let U be a convex subset of Rn. A concave function ϕ : U → R is exponentially concave iff for every µ0, µ1 ∈ U the function f (t) = ϕ(tµ0 + (1 − t)µ1) satisfies f ′′(t) ≤ −(f ′(t))2, for almost all t in [0, 1]. Indeed, for F(t) = exp(f (t)) we have F ′′(t) = F(t)[f ′′(t) + (f ′(t))2].

Definition: Let U ⊆ Rn and T : U → Rn a (possibly multi-valued) map (interpreted as a transport map). We call T multiplicatively cyclically monotone if, for all µ1, . . . , µm, µm+1 = µ1 ∈ U and all values T(µ1), . . . , T(µm) we have T(µj), µj+1 − µj > −1 and

m

• j=1

(1 + T(µj) , µj+1 − µj) ≥ 1, (1)

• r, equivalently

n

• j=1

log(1 + T(µj) , µj+1 − µj) ≥ 0. (2) Recall that T is cyclically monotone if

n

• j=1

T(µj), µj+1 − µj ≥ 0. (3)

Theorem [PW14]: Let U ⊆ Rn be a convex set and ϕ : U → R a concave function. Then ϕ is exponentially concave iff its super-differential T := ∂ϕ on U is multiplicatively cyclically monotone. In this case, ϕ is unique up to an additive constant. Sketch of proof Assume that ϕ is exponentially concave so that Φ(µ) = exp(ϕ(µ)) is a concave function on U. Denote by S(µ) the super-differential of Φ for which we have Φ(µj+1) ≤ Φ(µj) + S(µj), µj+1 − µj,

• r

Φ(µj+1) Φ(µj) ≤ 1 + S(µj) Φ(µj), µj+1 − µj .

Theorem [PW14]: Let U ⊆ Rn be a convex set and ϕ : U → R a concave function. Then ϕ is exponentially concave iff its super-differential T := ∂ϕ on U is multiplicatively cyclically monotone. In this case, ϕ is unique up to an additive constant. Sketch of proof Assume that ϕ is exponentially concave so that Φ(µ) = exp(ϕ(µ)) is a concave function on U. Denote by S(µ) the super-differential of Φ for which we have Φ(µj+1) ≤ Φ(µj) + S(µj), µj+1 − µj,

• r

Φ(µj+1) Φ(µj) ≤ 1 + S(µj) Φ(µj), µj+1 − µj .

Sketch of proof contd. Assuming that Φ is differentiable, the super-differential S(µ) equals ∇Φ(µ) so that T := ∇ϕ = ∇(log(Φ) = ∇Φ

Φ = S Φ. Therefore

Φ(µj+1) Φ(µj) ≤ 1 + T(µj), µj+1 − µj. If µ1, µ2, . . . , µm, µm+1 = µ1 is a roundtrip we have 1 = Φ(µm+1) Φ(µ1) ≤

m

• i=1

(1 + T(µj), µj+1 − µj). Hence T is multiplicatively cyclically monotone.

Applications in Finance

Relative Capital distribution 1929-1999 of NY stock exchange:

Stochastic portfolio Theory

• R. Fernholz (1999), Karatzas & Fernholz (2009), . . .

We consider n stocks with relative market capitalization at time t µ(t) = (µ1(t), . . . , µn(t)) ∈ int(∆n). A portfolio is a map π : int(∆n) → ∆n. Interpretation: The agent invests her wealth V (t) according to π(µ(t)) during ]t, t + 1]. We associate to π the weights w(µ) = (π1(µ) µ1 , . . . , πn(µ) µn ) ∈ Rn

+.

Stochastic portfolio Theory

• R. Fernholz (1999), Karatzas & Fernholz (2009), . . .

We consider n stocks with relative market capitalization at time t µ(t) = (µ1(t), . . . , µn(t)) ∈ int(∆n). A portfolio is a map π : int(∆n) → ∆n. Interpretation: The agent invests her wealth V (t) according to π(µ(t)) during ]t, t + 1]. We associate to π the weights w(µ) = (π1(µ) µ1 , . . . , πn(µ) µn ) ∈ Rn

+.

Example: a) the market portfolio: π(µ) = µ w(µ) = (1, . . . , 1) b) The equal weight portfolio: π(µ) = (1 n, . . . , 1 n) w(µ) = 1 n( 1 µ1 , . . . , 1 µn ) c) Let 0 < p < 1 and π(µ) =

• µp

1

n

i=1 µp i

, . . . , µp

n

n

i=1 µp i

• , w(µ) =
• µp−1

1 1 p

n

i=1 µp i

, . . . , µp−1

n 1 p

n

i=1 µp i

Example: a) the market portfolio: π(µ) = µ w(µ) = (1, . . . , 1) b) The equal weight portfolio: π(µ) = (1 n, . . . , 1 n) w(µ) = 1 n( 1 µ1 , . . . , 1 µn ) c) Let 0 < p < 1 and π(µ) =

• µp

1

n

i=1 µp i

, . . . , µp

n

n

i=1 µp i

• , w(µ) =
• µp−1

1 1 p

n

i=1 µp i

, . . . , µp−1

n 1 p

n

i=1 µp i

Example: a) the market portfolio: π(µ) = µ w(µ) = (1, . . . , 1) b) The equal weight portfolio: π(µ) = (1 n, . . . , 1 n) w(µ) = 1 n( 1 µ1 , . . . , 1 µn ) c) Let 0 < p < 1 and π(µ) =

• µp

1

n

i=1 µp i

, . . . , µp

n

n

i=1 µp i

• , w(µ) =
• µp−1

1 1 p

n

i=1 µp i

, . . . , µp−1

n 1 p

n

i=1 µp i

Question: Can you beat the market portfolio? Given the portfolio map π : int(∆n) → ∆n and a sequence (µ(t))m

t=1, we

• btain for the relative wealth (in terms of the market portfolio)

V (t + 1) V (t) =

n

• i=1

πi(µ(t)) µi(t + 1) µi(t) =

n

• i=1

wi(µ(t)) µi(t + 1). = w(µ(t)) , µ(t + 1) = 1 + w(µ(t)) , µ(t + 1) − µ(t)

Question: Can you beat the market portfolio? Given the portfolio map π : int(∆n) → ∆n and a sequence (µ(t))m

t=1, we

• btain for the relative wealth (in terms of the market portfolio)

V (t + 1) V (t) =

n

• i=1

πi(µ(t)) µi(t + 1) µi(t) =

n

• i=1

wi(µ(t)) µi(t + 1). = w(µ(t)) , µ(t + 1) = 1 + w(µ(t)) , µ(t + 1) − µ(t)

Question: Can you beat the market portfolio? Given the portfolio map π : int(∆n) → ∆n and a sequence (µ(t))m

t=1, we

• btain for the relative wealth (in terms of the market portfolio)

V (t + 1) V (t) =

n

• i=1

πi(µ(t)) µi(t + 1) µi(t) =

n

• i=1

wi(µ(t)) µi(t + 1). = w(µ(t)) , µ(t + 1) = 1 + w(µ(t)) , µ(t + 1) − µ(t)

Suppose that the market makes a “round trip” µ1, µ2, . . . , µm, µm+1 = µ1. Then V (m + 1) V (1) =

m

• t=1

V (t + 1) V (t) =

m

• t=1

[1 + w(µ(t)), µ(t + 1) − µ(t)] (4) is precisely the term appearing in the definition of multiplicative cyclical

• monotonocity. Taking logarithms yields

log m

• t=1

V (t + 1) V (t)

• =

m

• t=1

log

• 1 + w(µ(t)), µ(t + 1) − µ(t)
• (5)

Note that w is multiplicatively cyclically monotone iff (4) is always ≥ 1

• r, equivalently, (5) is always ≥ 0.

Suppose that the market makes a “round trip” µ1, µ2, . . . , µm, µm+1 = µ1. Then V (m + 1) V (1) =

m

• t=1

V (t + 1) V (t) =

m

• t=1

[1 + w(µ(t)), µ(t + 1) − µ(t)] (4) is precisely the term appearing in the definition of multiplicative cyclical

• monotonocity. Taking logarithms yields

log m

• t=1

V (t + 1) V (t)

• =

m

• t=1

log

• 1 + w(µ(t)), µ(t + 1) − µ(t)
• (5)

Note that w is multiplicatively cyclically monotone iff (4) is always ≥ 1

• r, equivalently, (5) is always ≥ 0.

Suppose that the market makes a “round trip” µ1, µ2, . . . , µm, µm+1 = µ1. Then V (m + 1) V (1) =

m

• t=1

V (t + 1) V (t) =

m

• t=1

[1 + w(µ(t)), µ(t + 1) − µ(t)] (4) is precisely the term appearing in the definition of multiplicative cyclical

• monotonocity. Taking logarithms yields

log m

• t=1

V (t + 1) V (t)

• =

m

• t=1

log

• 1 + w(µ(t)), µ(t + 1) − µ(t)
• (5)

Note that w is multiplicatively cyclically monotone iff (4) is always ≥ 1

• r, equivalently, (5) is always ≥ 0.

Suppose that the market makes a “round trip” µ1, µ2, . . . , µm, µm+1 = µ1. Then V (m + 1) V (1) =

m

• t=1

V (t + 1) V (t) =

m

• t=1

[1 + w(µ(t)), µ(t + 1) − µ(t)] (4) is precisely the term appearing in the definition of multiplicative cyclical

• monotonocity. Taking logarithms yields

log m

• t=1

V (t + 1) V (t)

• =

m

• t=1

log

• 1 + w(µ(t)), µ(t + 1) − µ(t)
• (5)

Note that w is multiplicatively cyclically monotone iff (4) is always ≥ 1

• r, equivalently, (5) is always ≥ 0.

Theorem: ((i) ⇒ (ii): Fernholz (1999), (ii) ⇒ (i): Pal, Wong (2014)) Fix a portfolio map π : int(∆n) → ∆n and let w(µ) = ( π1(µ)

µ1 , . . . , πn(µ) µn )

be the weight function. TFAE (i) There is an exponentially concave function ϕ : int(∆n) → R such that w(µ) is in the super-gradient of ϕ. (ii) The map w is multiplicatively cyclically monotone.

Theorem: ((i) ⇒ (ii): Fernholz (1999), (ii) ⇒ (i): Pal, Wong (2014)) Fix a portfolio map π : int(∆n) → ∆n and let w(µ) = ( π1(µ)

µ1 , . . . , πn(µ) µn )

be the weight function. TFAE (i) There is an exponentially concave function ϕ : int(∆n) → R such that w(µ) is in the super-gradient of ϕ. (ii) The map w is multiplicatively cyclically monotone.

Theorem: ((i) ⇒ (ii): Fernholz (1999), (ii) ⇒ (i): Pal, Wong (2014)) Fix a portfolio map π : int(∆n) → ∆n and let w(µ) = ( π1(µ)

µ1 , . . . , πn(µ) µn )

be the weight function. TFAE (i) There is an exponentially concave function ϕ : int(∆n) → R such that w(µ) is in the super-gradient of ϕ. (ii) The map w is multiplicatively cyclically monotone.

Mass Transport

Let P be a Probability measure on int(∆n) (e.g. normalized Lebesgue) and let Q be a probability measure on Rn

+. We interpret Q as a

probability measure on the weights w ∈ Rn

+ which are not yet normalized.

Note that for π : int(∆n) → ∆n and w(µ) = π1(µ) µ1 , . . . , πn(µ) µn

• we have

µ, w(µ) = 1 (6) Hence: If we associate (via the desired mass transport) µ ∈ (∆n, P) with w(µ) ∈ Rn

+ we have to make sure (via normalization) that (6) holds true.

Mass Transport

Let P be a Probability measure on int(∆n) (e.g. normalized Lebesgue) and let Q be a probability measure on Rn

+. We interpret Q as a

probability measure on the weights w ∈ Rn

+ which are not yet normalized.

Note that for π : int(∆n) → ∆n and w(µ) = π1(µ) µ1 , . . . , πn(µ) µn

• we have

µ, w(µ) = 1 (6) Hence: If we associate (via the desired mass transport) µ ∈ (∆n, P) with w(µ) ∈ Rn

+ we have to make sure (via normalization) that (6) holds true.

Consider the cost function c(µ, w) = log(µ, w) µ ∈ int(∆n), w ∈ Rn

+.

For the probabilities P on int(∆n) and Q on Rn

+ we consider the optimal

transport problem E[c(µ, w(µ))] → min! Where we optimize over all (non-normalized) functions w : int(∆n) → Rn

+ such that w#(P) = Q (w.l.g. of Monge type).

Normalization Lemma: Let h : Rn

+ → R+ be a function and

Sh : Rn

+ → Rn +

Sh(y) = h(y)y Given w : int(∆n) → Rn

+ we define w h = Sh ◦ w.

Then w is an optimal transport for the pair (P, Q) iff w h is an optimal transport for the pair (P, Sh

#(Q)).

Proof. EP[log(µ, w h(µ))] = EP[log(h(w(µ))µ, w(µ)] = EQ[log(h(w))] + EP[log(µ, w(µ))]

Normalization Lemma: Let h : Rn

+ → R+ be a function and

Sh : Rn

+ → Rn +

Sh(y) = h(y)y Given w : int(∆n) → Rn

+ we define w h = Sh ◦ w.

Then w is an optimal transport for the pair (P, Q) iff w h is an optimal transport for the pair (P, Sh

#(Q)).

Proof. EP[log(µ, w h(µ))] = EP[log(h(w(µ))µ, w(µ)] = EQ[log(h(w))] + EP[log(µ, w(µ))]

Theorem [PW14]: Let P (a.c. with respect to Lebesgue) on int(∆n) and Q on Rn

+ be as

above such that EP[log(µ, w(µ))] → min! has a finite value. Then there is an optimal transport w : int(∆n) → Rn

+.

Assuming (w.l.g.) that µ, w(µ) = 1, for all µ, the function w is in the supergradient of an exponentially concave function ϕ and therefore w is multiplicatively cyclically monotone.