Calibration and portfolio optimization issuesin asset price modelling driven by Levy processes

Jos e E. Figueroa-L opez

University of California, Santa Barabara

Department of Statistics and Applied Probability

ORIE Colloquium . Cornell University

January 23rd, 2007

Outline

I. Asset price modelling driven by Levy processes

II. Calibration issues

III. Portfolio optimization problems

1

Asset price modelling driven by L evy processes

What is a L evy process?

A random quantity Xt evolving in time in such a way that

• The increment, Xt+∆t −Xt, during a time period [t, t + ∆t] is both

(1) independent of the “past” Xs : s ≤ t and

(2) with distribution law depending only on the time span ∆t

• The process can exhibit sudden changes in magnitude (jumps), but these

occur at unpredictable times (no fixed jump times).

Conclusion: Levy process is the most natural generalization of Brownian motion,

where only continuity of paths is relaxed, while preserving the statistical qual-

ities of the increments.

2

A natural extension of the Black-Scholes model:

Geometric Levy models for an asset price process:

dSt = St−(bdt + d Zt︸︷︷︸Levy process

) ⇐⇒ St = S0 exp( Xt︸︷︷︸Levy process

),

Why a L evy Market?

• More flexible modelling of return distribution; e.g. heavy-tails, asymmetry,

and high kurtosis

• More consistency with the actual asset price evolution, which are made up

of discrete trades, and are exposed to sudden changes

3

Challenges and drawbacks of L evy-based modelling

Statistical issues

• Intractable or not explicit marginal probability density functions

• Too many parametric models around!! Normal, stable, hyperbolic distrib-

utions, tempered or truncated stable, etc. Which is better?

• Hard to guarantee convergence to the maximum likelihood estimators.

Conclusion: Computational demanding and numerically instable estimation

by traditional statistical methods.

Option pricing, hedging, and portfolio optimization:

(1) Incompleteness

(2) Infinitely many arbitrage free option prices.

(3) Lack of numerical schemes to find replication strategies and optimal (utility-

based) strategies.

4

Statistical properties of L evy processes

Characterization and parameters

• The law of the process is determined by the distribution of X1

• Three parameters, two reals σ2, µ, and a measure ν(dx), such that

(1) The counting process that counts the number of jumps with size falling

on an interval [a, b] is a Poisson process with intensity ν([a, b]).

(2) Xt = µt+σWt +limε↓0 Xεt − tmε, where Xε is a compound

Poisson process with mean EXεt = tmε.

(3) The jump evolution of the process is independent from the Brownian part.

A typical assumption: The Levy measure ν is determined by a “density func-

tion” s in the sense that

ν([a, b]) =∫ b

a

s(x) dx.

5

Remarks about L evy density functions:

Necessary and sufficient conditions to be a L evy density:

∫ 1

−1

s(x)x2dx < ∞ and∫

|x|>1

s(x)dx < ∞.

Consequences:

• A Levy density is not a probability density function:

∫ ∞

−∞s(x)dx = ∞⇒ s(x)

|x|→0−→ ∞⇐⇒ inf. many jumps of

arbitrarily small size

• It is easier to specify a Levy process via the Levy density of ν than via

the probability density of X1.

6

Nonparametric estimation of the L evy density

Problem: Estimate directly the Levy density s assuming only qualitative informa-

tion on s.

Objectives

• Avoid model biases: “Let the data speak”.

• Computational efficiency, and suitability to deal with high-frequency data.

7

Basic ideas of the model selection paradigm

• Approximation of the (infinite-dimensional) non-parametric model by finite-

dimensional linear models:

s(x) ≈ β1ϕ1(x) + · · ·+ βnϕn,

where the ϕ’s are known functions. The space

S := β1ϕ1(x) + · · ·+ βnϕn : β1, . . . , βn reals

is called an (approximating) linear model.

• Typical examples: Histograms, splines, trigonometric polynomials, wavelets.

• Two problems to solve:

Projection estimation: Estimating the element ofS that is “closest” to s.

Model selection: Determining a good approximating model from a col-

lection of linear models.

8

Implementation of the model selection approach

• Assumptions:

– Suppose that there exists a reference measure η such that, on a given es-

timation window D ⊂ R\0, s is bounded and∫

Ds2(x)η(dx) < ∞.

– Suppose one has a good estimator β(ϕ) of the inner product

β(ϕ) :=∫

D

ϕ(x)s(x)η(dx),

for any ϕ s.t.∫

Dϕ2(x)η(dx) < ∞.

• Implication: η introduce a natural distance between functions; namely,

‖s− p‖2η :=∫

D

(s(x)− p(x))2η(dx).

9

• Implementation:

– Take approximating linear models of the form

S := β1ϕ1(x) + · · ·+ βnϕn : β1, . . . , βn reals

with functions ϕ′is such that∫

Dϕ2

i (x) η(dx) < ∞. Without loose of

generality, one can always take ϕ1, . . . , ϕn orthonormal on D with re-

spect to η.

– The “closest” member of S to the function s is when

βi = β(ϕi) =∫

D

ϕi(x)s(x)η(dx).

– Natural projection estimator of s onS , with respect to the distance induce

by η, is

s(x) := β(ϕ1)ϕ1(x) + · · ·+ β(ϕn)ϕn(x).

10

• Data-driven Model Selection:

– Problem: Choose between two or more finite-dimensional linear models

– Goal: Accomplish a good trade off between the bias of the approximat-

ing model and the standard error of the estimation inside the model.

– A sensible solution: Choose the model that minimize an unbiased estima-

tor of the risk:

E[‖s− s‖2η

]= ‖s‖2η + E

[−‖s‖2η]+ β(ϕ2

S ).

where

ϕ2S (x) :=

∑

i

ϕ2i (x).

Choose the model that minimizes the observable statistic

−∑

i

β2(ϕi) + β(ϕ2S ).

11

A Numerical Illustration

Gamma Levy process with parameters α and β

• X(t) D∼ Gamma(αt, β), β - scale parameter and α - shape parameter

• No Brownian part and Levy density s(x) = αx e−x/β , x > 0:

Pure-jump increasing process with infinite jump-activity.

• Jump-behavioral interpretation:

α - overall jump activity and β - frequency of big-jumps

12

Model selection methods for the Gamma L evy process

I. Estimation outside the origin:

• On a window of estimation D = [a, b] away from the origin, s(x) =αx e−x/β is bonded and square integrable with respect to η(dx) = dx.

• Recall that

E∑

u≤T

χ(c,d)

(∆Xu) = T

∫χ(c,d)(x)s(x)dx.

Then,

E∑

u≤T

ϕ(∆Xu) = T

∫

D

ϕ(x)s(x)dx.

A natural estimator for β(ϕ) :=∫

Dϕ(x)s(x)dx is

β(ϕ) :=1T

∑

u≤T

ϕ(∆Xu).

13

• Example: Take histogram estimators on regular partitions.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

Penalized Estimation Projection

x

s(x

)

Sample Path Information: Gamma Process with α = 1 and β = 1(2000 jumps on [ 0 , 365 ])

Method of Estimation: Regular Histograms

c=2 Estimation window = [0.02, 1.0]

Best partition = 51 intervals

• How good is the estimator? Fit the model αx e−x/β (using least-squares)

to the histogram estimator: αMS = 0.93 and βMS = 1.055(vs. αMLE = 1.01 and βMLE = 0.94)

14

• Estimation based on discrete sampling:

– Problem: Estimate

β (ϕ) :=1T

∑

t≤T

ϕ(∆Xt),

based on n equally spaced observations on [0, T ]: XT/n

, . . . , XT .

– A natural solution: Approximate by

βn (ϕ) :=1T

n∑

k=1

ϕ(Xtk

−Xtk−1

),

where tk = T kn , for k = 1, . . . , n.

15

– Example: Gamma Levy process with α = 1 and β = 1:

Time span MS-LSF MLE

1 1.01 1.46 .997 .995

.5 1.03 1.09 .972 .978

.1 .944 .995 1.179 .837

.01 .969 .924 1.01 .98

(Based on 36500 jumps on [0, 365]≈ 100 jumps/ unit time)

16

– Sampling distribution

Means and standard errors of αMS and βMS based on 1000 repeti-

tions

∆t MS-LSF MLE

.1 0.81 (0.06) 1.40 (0.50) 1.001 (0.01) 0.99 (0.05)

.01 0.92 (0.08) 1.12 (0.31) 1.007 (0.07) 0.99 (0.08)

.001 0.93 (0.08) 1.13 (0.34) 1.007 (0.07) 0.99 (0.08)

(simulations based on equally-spaced observations, with time span ∆t,

during the time interval [0, 365])

17

Model selection methods for Tempered Stable Process

Model: [Rosinski 2005] No Brownian part with Levy density

s(x) =1

|x|α+1q(|x|), 0 < α < 2,

where q(x) and q(−x) (x > 0) are bounded “completely monotone func-

tions” (decreasing, convex; e.g. exp(−x)) vanishing for large x.

Appealing Property:

• h−1/αXhtD≈ Stable distribution with parameter α, for h small.

Stable-like high-frequency returns

• h−1/2XhtD≈ Normal distribution with parameter α, for h large.

Normal-like low-frequency returns

Goal: Estimate q.

18

Projection estimator:

• Approximating linear model for q:

S := β1ϕ1(x) + · · ·+ βdϕd(x),

where the ϕ’s are orthonormal functions defined on D = [0, h].

• Estimator of the projection of q on S :

q(x) = β(ϕ1)ϕ1(x) + · · ·+ β(ϕd)ϕd(x),

where

β (ϕ) :=1T

∑

t≤T

ϕ(∆Xt) with ϕ(x) := |x|α+1ϕ(x)

and α being a “good” estimate of α; e.g. using methods for stable processes.

19

• Intuition: If ϕ(x) ≈ xα+1ϕ(x) then

E [∑

t≤T

ϕ(∆Xt)] ≈ T

∫

D

ϕ(x)q(x)dx.

• Numerical illustration: One-sided TS, s(x) = axα+1 e−x/b, x > 0

– Histogram approximations of q(x) = ae−x/b on D = [0, 3]– Paths generated from 36500 jumps on [0, 365] with a = b = 1 and

α = .1.

– Jumps approximated by equally-spaced increments (time span ∆t)

– α estimated by Zolotarev method for stable distributions (valid for h ≈0) based on the moments of log |Xt|.

∆t Penalized Projection - Least-Squares Fit Misspecified Gamma MLE

.01 1.03 (0.15) 0.97 (0.14) 0.09 (0.0002) 1.2 (0.08) 0.89 (0.079)

20

Summary

We developed estimation and model selection schemes for the Levy density of a

Levy process:

• Flexible: it can be used histograms, splines, wavelets, etc.

• Model free

• Easily implementable

• Reliable and robust: Oracle inequality and adaptivity; i.e. asymptotically com-

parable to minimax estimators on classes of smooth Levy densities

21

The problem of portfolio optimization

• Formulation of the problem

• Solution using convex duality: advantages and drawbacks

• Improvements in jump-diffusion models driven by Levy processes

• Conclusions

22

Formulation of the problem

Set-up: A frictionless market consisting of a risky asset with price St and a risk-

free bond with value Bt defined on a probability space (Ω,F ,P).

Goal: Dynamically allocate an initial endowment w so that to maximize the agent’s

expected final utility during a fixed time horizon [0, T ].

State-dependent utility U(w,ω) : R+ × Ω → R: [Follmer & Leukert, 2000]

Increasing and concave in w and constant for w ≥ H(ω)︸ ︷︷ ︸“Contingent Claim”

.

Problem: MaximizeE U(VT, ω) over all self-financing trading strategies such

that the associated wealth process Vtt≥0 satisfies

V0 ≤ w︸ ︷︷ ︸Budget Constraint

and V· ≥ 0.︸ ︷︷ ︸Admissibility condition.

23

Motivation:

Optimal Hedging of Contingent Claims.

Setting: B−1t St is a semimartingale and the classM of Equivalent

Martingale Measures (EMM) is not empty. Then,

• Hedging away the risk of a contingent claim H is feasible: [Kramkov 97]

Exists admissible Vtt≤T with V0 = w s.t. VT≥ H,

for all

w ≥ w0 := supQ∈M

EQ[B−1

TH

].

• An initial endowment w smaller than w0 entails short-falling risk (sometimes

VT

< H), motivating the problem of minimizing the expected shortfall:

Minimize EL

((H − V

T)+

)s.t. V0 ≤ w and Vt ≥ 0,

for a Loss Function L : R+ → R (L(0) = 0, increasing, and convex).

24

• The shortfall minimization problem is equivalent to a utility maximization prob-

lem with (state-dependent) utility

U(z; ω) := L(H(ω))− L((H(ω)− z)+).

0 2 4 6 8 10 12 140

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

WEALTH

UT

ILIT

Y

State−Dependent Utility Function

H

U( ⋅,ω)

25

Convex Duality

• Basic idea: Upper bound a maximization problem with constraints, using a

convex minimization problem without constraints:

Primal Problem︷ ︸︸ ︷p∗ := max f(x)

s.t. h(x) ≤ 0=⇒

• f(x) ≤L(x,λ)︷ ︸︸ ︷

f(x)− λh(x), λ ≥ 0

• p∗ ≤ L(λ) := maxx L(x, λ)

Convex

• p∗ ≤ d∗ := minλ≥0

L(λ)︸ ︷︷ ︸

Dual Problem

.

• We say that strong duality holds if p∗ = d∗. This holds if for instance the

primal problem is “convex” and h(x) < 0 (Slater condition).

• Applications: Duality plays an important role in convex optimization:

(i) Perturbation and sensibility analysis; (ii) Interior point methods.

26

Convex duality in portfolio optimization problems[Karatzas et. al. 91, Schachermayer-Kramkov 99, Follmer-Leukert 2000]

Primal problem:

p∗(w) := supE U(VT, ω)

such that V0 ≤ w and V· ≥ 0,

Assumption: w < supQ∈M EQ[B−1

TH

]< ∞.

The dual domain Γ: Nonnegative supermartingales ξtt≥0 such that

• 0 ≤ ξ0 ≤ 1 and ξtB−1t Vtt≥0 is a supermaringale for all admissible

wealth processes Vtt≥0.

Consequences:

• E ξ

TB−1

TV

T

≤ ξ0B−10 V0 ≤w if V0 ≤ w.

• The pricing formula Eξ

TB−1

TH

is free of arbitrage.

• “ M⊂ Γ ” : The density processes of EMMs belong to Γ.

27

Road towards weak duality:

• For ξ ∈ Γ and any admissible wealth process V· with V0 ≤ w:

E U (VT , ω)≤E U (VT , ω) − λ(E

ξT B−1

TVT

− w)

= EU (V

T, ω)− λ ξ

TB−1

TV

T

+ λw

≤ E

supv≥0

U (v, ω)− λ ξT B−1

Tv

+ λw

= E

U(λξTB−1

T, ω)

+ λw,

where U(λ, ω) := supv≥0 U (v, ω)− λv [Convex Dual Function].

• Let Γ ⊂ Γ a subclass. Then,

p∗(w)= supE U(VT, ω) ≤ inf

ξ∈ΓE

U(λξ

TB−1

T, ω)

︸ ︷︷ ︸d∗Γ(λ)

+λw.

28

Dual problem associated to Γ ⊂ Γ:

d∗Γ(λ) := inf

ξ∈ΓE

U(λξT B−1

T, ω)

.

“Weak duality”: p∗(w) ≤ d∗Γ(λ) + λw.

0 0.5 1 1.5 2 2.5 3 3.5 40.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Dual domain

Convex Dual Function

Slope is −H

U(H)

U’(H−)U’(0+)

U(0)

Strictly Convex Part

29

Questions:

1. Does strong duality (p∗(w) = d∗Γ(λ) + λw) hold for some λ?

2. Is the dual problem infξ∈Γ E

U(λξTB−1

T, ω)

attainable?

3. Is the primal problem supV :V0≤w E U(VT , ω) attainable?

Solution when Γ = Γ: [Follmer-Leukert (2000) & Xu (2004)]

Yes, if−∞ < EU(0, ω) ≤ EU(H,ω) < ∞. Moreover, we have

a dual characterization of the optimal final wealth:

V ∗T

= I(λ∗ ξ∗

TB−1

T

),

where I(λ, ω) := inf z : U ′(z, ω) < λ .

Our problem:

• Can one specify further ξ∗ for a given market model and a utility function?

• Can one narrow down the dual domain Γ ⊂ Γ where to search ξ∗?

30

Portfolio optimization for geometric Levy models[Kallsen (2000), Nualart et. al. (2004), Kunita (2003), Figueroa & Ma (2007)]

Market model: Geometric Levy model and a constant interest rate bond:

(1) dSt = b St−dt + St− dZs, (2) dBt = rBtdt, B0 = 1,

(3) Zt = bt + σWt +∫ t

0

∫z (N(dt, dz)− ν(dz)ds)︸ ︷︷ ︸eN(dt,dz)

,

(4) F = Ftt≥0, where Ft := FZt ∧ P− null set.

Dual domain Γ:

S :=

Xt :=∫ t

0

G(s)dWs +∫ t

0

∫F (s, z)N(ds, dz) : F ≥ − 1

,

Γ := ξ· := ξ0 +∫ ·

0

ξs− d(Xs −As) s.t. ξ ∈ Γ, X ∈ S, A increasing.

31

Remark:

ξ = ξ0E(X −A) nonnegative with X ∈ S and A predictable increasing

belongs to Γ if and only if a.s.

b− r + α +∫

|z|≥1

zν(dz) + σG(t) +∫

RzF (t, z)ν(dz) ≤ at,

for almost every t ≤ τ(ω), where a· is the density of the absolutely contin-

uous part of A and τ is the time ξ hits 0.

Dual problem associated with Γ: There exist ξ∗ ∈ Γ and λ∗ > 0 such that

(1) The problem infλ>0

d∗

Γ(λ) + λw

is attained at λ∗

(2) The dual problem d∗Γ(λ∗) := infξ∈Γ E

U(λ∗ξ

TB−1

T, ω)

is attained

at ξ∗ ∈ Γ

(3) infλ>0

d∗

Γ(λ) + λw

= E

[U

(W∗Γ, ω

)], whereW∗

Γ= I

(λ∗ ξ∗

TB−1

T

)

(4) Eξ∗

TB−1

TW∗

Γ

= w.

32

Strategy to prove strong duality:

In view of the weak duality property (p∗(w) ≤ d∗(λ∗)), ifW∗Γ

is replicable

with an initial endowment of w, thenW∗Γ

will be the optimal final wealth.

Road towards replicability:

(1) By perturbating the “parameters” of ξ∗, obtain a “variational equality”:

Eξ∗

TB−1

TI

(λ∗B−1

Tξ∗

T

)Y

T

= 0,

for any (G, F , a) in a suitable convex set Λ∗, where

Yt :=∫ t

0

G(s)dWG∗s +

∫ t

0

∫

R0

F (s, z) NF∗(ds, dz)+∫ t

0

asds,

with

WG∗t := Wt −

∫ t

0

G∗(s)ds,

NF∗(dt, dz) := N(dt, dz)− (1 + F ∗(s, z))ν(dz)ds.

33

(2) Prove that A∗ in ξ∗ = E (X∗ −A∗) is identically zero, and hence, ξ∗

is a (local) martingale. By the Girsanov Theorem, WG∗ is a standard Brown-

ian Motion and NF∗ is a martingale measure with respect to the (finitely

additive) probability measure dQ∗ := ξ∗T dP (up to stopping times).

(3) Establish the decomposition

M2(Q∗) = L(R)⊕ L (Λ∗0) ,

where

– L(R) is the set of all M ∈M2(Q∗) which can be written as Mt =∫ t

0βsdRs, with dRt := S−1

t dSt, and S is the discounted price process.

– L (Λ∗0) is the stable subspace ofM2(Q∗) generated by the integrals∫ ·

0

G(s)dWG∗s +

∫ ·

0

∫F (s, z) NF∗(ds, dz)

with (G, F ) in an appropriate class Λ∗0.

34

(4) Show that theQ∗-martingale,

X∗t := EQ∗

B−1

TI

(λ∗ ξ∗

TB−1

T

) |Ft

admits the representation

X∗t = w +

∫ t

0

βsdRt.

Then, the portfolio with initial value w and trading strategy β∗t := Btβt

is admissible and its final wealth isW∗Γ

= I(λ∗ ξ∗

TB−1

T

).

Conclusions

• The method developed here is more explicit in the sense that the dual do-

main enjoys a concrete parametrization.

• Due to the explicit parametrization, discrete time approximations in particu-

lar cases might be possible to implement.

35

• It can accommodate much more general jump-diffusion models driven by Levy

processes such as

dSi(t) = Si(t−)bitdt +

d∑

j=1

σijt dW j

t +∫

Rd

h(t, z)N(dt, dz).

• Also suitable to solve optimum portfolio problems with consumption. The prob-

lem is then to maximize the utility coming from both consumption and final

wealth, utility determined as follows

E

U1(VT ) +

∫ T

0

U2(t, c(t))dt

,

under a budget constraint and a no bankruptcy condition (now the wealth is

determined by dVt := rVtdt + βtdSt − ctdt).

36