Exponentially concave functions and multiplicative cyclical monotonicity

Transcription

1 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: )

2 Exponential concavity Fix a convex subset U R n. Typically U will equal the unit simplex n {(p 1,..., p n) [0, 1] n : or a convex subset of n, e.g. int( n ). Definition: n p i 1} 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. i1

3 Exponential concavity Fix a convex subset U R n. Typically U will equal the unit simplex n {(p 1,..., p n) [0, 1] n : or a convex subset of n, e.g. int( n ). Definition: n p i 1} 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. i1

4 Exponential concavity Fix a convex subset U R n. Typically U will equal the unit simplex n {(p 1,..., p n) [0, 1] n : or a convex subset of n, e.g. int( n ). Definition: n p i 1} 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. i1

5 Lemma: Let U be a convex subset of R n. A concave function ϕ : U R is exponentially concave iff for every µ 0, µ 1 U the function f (t) ϕ(tµ 0 + (1 t)µ 1 ) satisfies for almost all t in [0, 1]. f (t) (f (t)) 2, Indeed, for F (t) exp(f (t)) we have F (t) F (t)[f (t) + (f (t)) 2 ].

6 Lemma: Let U be a convex subset of R n. A concave function ϕ : U R is exponentially concave iff for every µ 0, µ 1 U the function f (t) ϕ(tµ 0 + (1 t)µ 1 ) satisfies for almost all t in [0, 1]. f (t) (f (t)) 2, Indeed, for F (t) exp(f (t)) we have F (t) F (t)[f (t) + (f (t)) 2 ].

7 Definition: Let U R n and T : U R n a (possibly multi-valued) map (interpreted as a transport map). We call T multiplicatively cyclically monotone if, for all µ 1,..., µ m, µ m+1 µ 1 U and all values T (µ 1 ),..., T (µ m ) we have T (µ j ), µ j+1 µ j > 1 and or, equivalently m (1 + T (µ j ), µ j+1 µ j ) 1, (1) j1 n log(1 + T (µ j ), µ j+1 µ j ) 0. (2) j1 Recall that T is cyclically monotone if n T (µ j ), µ j+1 µ j 0. (3) j1

8 Theorem [PW14]: Let U R n 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 or Φ(µ j+1 ) Φ(µ j ) + S(µ j ), µ j+1 µ j, Φ(µ j+1 ) Φ(µ j ) 1 + S(µ j ) Φ(µ j ), µj+1 µ j.

9 Theorem [PW14]: Let U R n 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 or Φ(µ j+1 ) Φ(µ j ) + S(µ j ), µ j+1 µ j, Φ(µ j+1 ) Φ(µ j ) 1 + S(µ j ) Φ(µ j ), µj+1 µ j.

10 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(µ j+1 ) Φ(µ j ) 1 + T (µ j ), µ j+1 µ j. If µ 1, µ 2,..., µ m, µ m+1 µ 1 is a roundtrip we have 1 Φ(µm+1 ) Φ(µ 1 ) m (1 + T (µ j ), µ j+1 µ j ). i1 Hence T is multiplicatively cyclically monotone.

11 Applications in Finance Relative Capital distribution of NY stock exchange:

12 Stochastic portfolio Theory R. Fernholz (1999), Karatzas & Fernholz (2009),... We consider n stocks with relative market capitalization at time t µ(t) (µ 1 (t),..., µ n (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) ( π 1(µ) µ 1,..., π n(µ) µ n ) R n +.

13 Stochastic portfolio Theory R. Fernholz (1999), Karatzas & Fernholz (2009),... We consider n stocks with relative market capitalization at time t µ(t) (µ 1 (t),..., µ n (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) ( π 1(µ) µ 1,..., π n(µ) µ n ) R n +.

14 Example: a) the market portfolio: b) The equal weight portfolio: c) Let 0 < p < 1 and π(µ) µ w(µ) (1,..., 1) π(µ) ( 1 n,..., 1 n ) w(µ) 1 n ( 1 µ 1,..., 1 µ n ) ( π(µ) µ p 1 n i1 µp i,..., µ p ) ( n n, w(µ) i1 µp i µ p p n i1 µp i,..., µ p 1 n 1 n p i1 µp i )

15 Example: a) the market portfolio: b) The equal weight portfolio: c) Let 0 < p < 1 and π(µ) µ w(µ) (1,..., 1) π(µ) ( 1 n,..., 1 n ) w(µ) 1 n ( 1 µ 1,..., 1 µ n ) ( π(µ) µ p 1 n i1 µp i,..., µ p ) ( n n, w(µ) i1 µp i µ p p n i1 µp i,..., µ p 1 n 1 n p i1 µp i )

16 Example: a) the market portfolio: b) The equal weight portfolio: c) Let 0 < p < 1 and π(µ) µ w(µ) (1,..., 1) π(µ) ( 1 n,..., 1 n ) w(µ) 1 n ( 1 µ 1,..., 1 µ n ) ( π(µ) µ p 1 n i1 µp i,..., µ p ) ( n n, w(µ) i1 µp i µ p p n i1 µp i,..., µ p 1 n 1 n p i1 µp i )

17 Question: Can you beat the market portfolio? Given the portfolio map π : int( n ) n and a sequence (µ(t)) m t1, we obtain for the relative wealth (in terms of the market portfolio) V (t + 1) V (t) n π i (µ(t)) i1 µ i (t + 1) µ i (t) n w i (µ(t)) µ i (t + 1). i1 w(µ(t)), µ(t + 1) 1 + w(µ(t)), µ(t + 1) µ(t)

18 Question: Can you beat the market portfolio? Given the portfolio map π : int( n ) n and a sequence (µ(t)) m t1, we obtain for the relative wealth (in terms of the market portfolio) V (t + 1) V (t) n π i (µ(t)) i1 µ i (t + 1) µ i (t) n w i (µ(t)) µ i (t + 1). i1 w(µ(t)), µ(t + 1) 1 + w(µ(t)), µ(t + 1) µ(t)

19 Question: Can you beat the market portfolio? Given the portfolio map π : int( n ) n and a sequence (µ(t)) m t1, we obtain for the relative wealth (in terms of the market portfolio) V (t + 1) V (t) n π i (µ(t)) i1 µ i (t + 1) µ i (t) n w i (µ(t)) µ i (t + 1). i1 w(µ(t)), µ(t + 1) 1 + w(µ(t)), µ(t + 1) µ(t)

20 Suppose that the market makes a round trip µ 1, µ 2,..., µ m, µ m+1 µ 1. Then V (m + 1) V (1) m t1 V (t + 1) V (t) m [1 + w(µ(t)), µ(t + 1) µ(t) ] (4) t1 is precisely the term appearing in the definition of multiplicative cyclical monotonocity. Taking logarithms yields [ m log t1 V (t + 1) ] V (t) m t1 [ ] log 1 + w(µ(t)), µ(t + 1) µ(t) (5) Note that w is multiplicatively cyclically monotone iff (4) is always 1 or, equivalently, (5) is always 0.

21 Suppose that the market makes a round trip µ 1, µ 2,..., µ m, µ m+1 µ 1. Then V (m + 1) V (1) m t1 V (t + 1) V (t) m [1 + w(µ(t)), µ(t + 1) µ(t) ] (4) t1 is precisely the term appearing in the definition of multiplicative cyclical monotonocity. Taking logarithms yields [ m log t1 V (t + 1) ] V (t) m t1 [ ] log 1 + w(µ(t)), µ(t + 1) µ(t) (5) Note that w is multiplicatively cyclically monotone iff (4) is always 1 or, equivalently, (5) is always 0.

22 Suppose that the market makes a round trip µ 1, µ 2,..., µ m, µ m+1 µ 1. Then V (m + 1) V (1) m t1 V (t + 1) V (t) m [1 + w(µ(t)), µ(t + 1) µ(t) ] (4) t1 is precisely the term appearing in the definition of multiplicative cyclical monotonocity. Taking logarithms yields [ m log t1 V (t + 1) ] V (t) m t1 [ ] log 1 + w(µ(t)), µ(t + 1) µ(t) (5) Note that w is multiplicatively cyclically monotone iff (4) is always 1 or, equivalently, (5) is always 0.

23 Suppose that the market makes a round trip µ 1, µ 2,..., µ m, µ m+1 µ 1. Then V (m + 1) V (1) m t1 V (t + 1) V (t) m [1 + w(µ(t)), µ(t + 1) µ(t) ] (4) t1 is precisely the term appearing in the definition of multiplicative cyclical monotonocity. Taking logarithms yields [ m log t1 V (t + 1) ] V (t) m t1 [ ] log 1 + w(µ(t)), µ(t + 1) µ(t) (5) Note that w is multiplicatively cyclically monotone iff (4) is always 1 or, equivalently, (5) is always 0.

24 Theorem: ((i) (ii): Fernholz (1999), (ii) (i): Pal, Wong (2014)) Fix a portfolio map π : int( n ) n and let w(µ) ( π1(µ) µ 1,..., πn(µ) 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. µ n )

25 Theorem: ((i) (ii): Fernholz (1999), (ii) (i): Pal, Wong (2014)) Fix a portfolio map π : int( n ) n and let w(µ) ( π1(µ) µ 1,..., πn(µ) 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. µ n )

26 Theorem: ((i) (ii): Fernholz (1999), (ii) (i): Pal, Wong (2014)) Fix a portfolio map π : int( n ) n and let w(µ) ( π1(µ) µ 1,..., πn(µ) 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. µ n )

27 Mass Transport Let P be a Probability measure on int( n ) (e.g. normalized Lebesgue) and let Q be a probability measure on R n +. We interpret Q as a probability measure on the weights w R n + which are not yet normalized. Note that for π : int( n ) n and we have w(µ) ( π1 (µ),..., π ) n(µ) µ 1 µ n µ, w(µ) 1 (6) Hence: If we associate (via the desired mass transport) µ ( n, P) with w(µ) R n + we have to make sure (via normalization) that (6) holds true.

28 Mass Transport Let P be a Probability measure on int( n ) (e.g. normalized Lebesgue) and let Q be a probability measure on R n +. We interpret Q as a probability measure on the weights w R n + which are not yet normalized. Note that for π : int( n ) n and we have w(µ) ( π1 (µ),..., π ) n(µ) µ 1 µ n µ, w(µ) 1 (6) Hence: If we associate (via the desired mass transport) µ ( n, P) with w(µ) R n + we have to make sure (via normalization) that (6) holds true.

29 Consider the cost function c(µ, w) log( µ, w ) µ int( n ), w R n +. For the probabilities P on int( n ) and Q on R n + we consider the optimal transport problem E[c(µ, w(µ))] min! Where we optimize over all (non-normalized) functions w : int( n ) R n + such that w # (P) Q (w.l.g. of Monge type).

30 Normalization Lemma: Let h : R n + R + be a function and S h : R n + R n + S h (y) h(y)y Given w : int( n ) R n + we define w h S h w. Then w is an optimal transport for the pair (P, Q) iff w h is an optimal transport for the pair (P, S h # (Q)). Proof. E P [log( µ, w h (µ) )] E P [log(h(w(µ)) µ, w(µ) ] E Q [log(h(w))] + E P [log( µ, w(µ) )]

31 Normalization Lemma: Let h : R n + R + be a function and S h : R n + R n + S h (y) h(y)y Given w : int( n ) R n + we define w h S h w. Then w is an optimal transport for the pair (P, Q) iff w h is an optimal transport for the pair (P, S h # (Q)). Proof. E P [log( µ, w h (µ) )] E P [log(h(w(µ)) µ, w(µ) ] E Q [log(h(w))] + E P [log( µ, w(µ) )]

32 Theorem [PW14]: Let P (a.c. with respect to Lebesgue) on int( n ) and Q on R n + be as above such that E P [log( µ, w(µ) )] min! has a finite value. Then there is an optimal transport ŵ : int( n ) R n +. Assuming (w.l.g.) that µ, ŵ(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.