Asymptotic Methods in Financial Mathematics Methods in Financial Mathematics José E. Figueroa-López1 1Department of Mathematics Washington University in St. Louis Statistics Seminar

Asymptotic Methods in Financial Mathematics

José E. Figueroa-López1

1Department of Mathematics

Washington University in St. Louis

Statistics Seminar

Washington University in St. Louis

February 17, 2017

(Joint work with Cecilia Mancini)

J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 1 / 29

Outline

1 Motivational Example

2 The Main Estimators

Multipower Variations and Truncated Realized Variations

3 Optimal Threshold Selection

via Expected number of jump misclassifications

via conditional Mean Square Error (cMSE)

4 Conclusions


Motivational Example

Outline







4 Conclusions



Merton Log-Normal Model

Consider the following model for the log-return process Xt = log StS0

of a

financial asset:

Xt = at + σWt +∑

i:τi≤t

ζi , ζii.i.d.∼ N(µjmp, σ

2jmp), {τi}i≥1 ∼ Poisson(λ)

Goal: Estimate σ based on a discrete record Xt1 , . . . ,Xtn .

0.00 0.02 0.04 0.06 0.08

1.0

1.1

1.2

1.3

Log-Normal Merton Model

Time in years (252 days)

Sto

ck P

rice

Price processContinuous processTimes of jumps



Model Simulation

1 Fix the time horizon T and the number of sampling points n. Set

the time mesh and the sampling times

h = hn =Tn, ti = ti,n = ihn, i = 0, . . . ,n.

2 Generate

Zii.i.d.∼ N(0,1), γi

i.i.d.∼ N(µjmp, σ2jmp), and Ii

i.i.d.∼ Bernoulli(λh)

3 Iteratively generate the process:

Xti = Xti−1 + ah + σ√

hZi + Iiγi


The Main Estimators

Outline







4 Conclusions


The Main Estimators Multipower Variations and Truncated Realized Variations

Two main classes of estimators

Precursor. Realized Quadratic Variation:

QV (X )n :=1T

n−1∑i=0

(Xti+1 − Xti

)2

QV (X )n n→∞−→ σ2 +∑

j:τj≤T ζ2j

1 Multipower Realized Variations (Barndorff-Nielsen and Shephard):

2 Truncated Realized Variations (Mancini):

TRVn[X ](ε) :=n−1∑i=0

(Xti+1 − Xti

)2 1{∣∣∣Xti+1−Xti

∣∣∣≤ε}, (ε ∈ [0,∞)).





QV (X )n :=1T

n−1∑i=0

(Xti+1 − Xti

)2

QV (X )n n→∞−→ σ2 +∑

j:τj≤T ζ2j


BPV (X )n :=n−1∑i=0

∣∣Xti+1 − Xti

∣∣ ∣∣Xti+2 − Xti+1

∣∣ ,


TRVn[X ](ε) :=n−1∑i=0

(Xti+1 − Xti

)2 1{∣∣∣Xti+1−Xti

∣∣∣≤ε}, (ε ∈ [0,∞)).





QV (X )n :=1T

n−1∑i=0

(Xti+1 − Xti

)2

QV (X )n n→∞−→ σ2 +∑

j:τj≤T ζ2j


MPV [X ](r1,...,rk )

n :=n−k∑i=0

∣∣Xti+1 − Xti

∣∣r1 . . .∣∣Xti+k − Xti+k−1

∣∣rk .


TRVn[X ](ε) :=n−1∑i=0

(Xti+1 − Xti

)2 1{∣∣∣Xti+1−Xti

∣∣∣≤ε}, (ε ∈ [0,∞)).





QV (X )n :=1T

n−1∑i=0

(Xti+1 − Xti

)2

QV (X )n n→∞−→ σ2 +∑

j:τj≤T ζ2j


MPV [X ](r1,...,rk )

n :=n−k∑i=0

∣∣Xti+1 − Xti

∣∣r1 . . .∣∣Xti+k − Xti+k−1

∣∣rk .


TRVn[X ](ε) :=n−1∑i=0

(Xti+1 − Xti

)2 1{∣∣∣Xti+1−Xti

∣∣∣≤ε}, (ε ∈ [0,∞)).



Calibration or Tuning of the Estimator

Problem: How do you choose the threshold parameter ε?

0.00 0.02 0.04 0.06 0.08

0.85

0.90

0.95

1.00



Sto

ck P

rice


0.00 0.01 0.02 0.03 0.040.0

0.1

0.2

0.3

0.4

0.5

Performace of Truncated Realized Variations

Truncation level (epsilon)

Tru

nca

ted

Re

aliz

ed

Va

ria

tio

n (

TR

V)

TRVTrue Volatility (0.4)Realized Variation (0.51)Bipower Variation (0.42)

Figure: (left) Merton Model with σ = 0.4, σjmp = 3(h), µjmp = 0, λ = 200;

(right) TRV performance wrt the truncation levelJ.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 8 / 29


Calibration or Tuning of the Estimator

Problem: How do you choose the threshold parameter ε?

0.00 0.02 0.04 0.06 0.08

0.800.850.900.951.00



Sto

ck P

rice


0.000 0.005 0.010 0.015 0.020 0.025 0.0300.0

0.1

0.2

0.3

0.4



Tru

nca

ted

Re

aliz

ed

Va

ria

tio

n (

TR

V)

TRVTrue Volatility (0.2)Realized Variation (0.4)Bipower Variation (0.26)Cont. Realized Variation (0.19)

Figure: (left) Merton Model with σ = 0.2, σjmp = 1.5(h), µjmp = 0, λ = 1000;

(right) TRV performance wrt the truncation levelJ.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 9 / 29


Popular truncations ε

Literature consists of mostly “ad hoc" selection methods for ε, aimed to

satisfy sufficient conditions for the consistency and asymptotic

normality of the associated estimators.

• Power Threshold (Mancini (2003))

εPrwα,ω := α hω, for α > 0 and ω ∈ (0,1/2).

• Bonferroni Threshold (Bollerslev et al. (2007) and Gegler &

Stadtmüller (2010))

BBFσ̃,C := σ̃h1/2Φ−1

(1− C h

2

), for C > 0 and σ̃ > 0.












(1− C h

2

), for C > 0 and σ̃ > 0.












(1− C h

2

), for C > 0 and σ̃ > 0.


Optimal Threshold Selection

Outline







4 Conclusions


Optimal Threshold Selection

Classical Approach

1 Fix a suitable and sensible metric for estimation error; typically,

the MSE:

MSE(ε) = E

[(1T

TRVn(ε)− σ2)2]

2 Show the existence of the optimal threshold ε?n minimizing the

error function;

3 Analyze the asymptotic behavior ε?n (when n→∞) to infer

• qualitative information such as rate of convergence on n and

dependence on the underlying parameters of the model (σ, σJ , λ)• Devise a plug-in type calibration of ε∗ by estimating those

parameters (if possible).


Optimal Threshold Selection via Expected number of jump misclassifications

Expected number of jump misclassifications

1 Notation:

Nt = # of jumps by time t

∆ni X = Xti − Xti−1

∆ni N = Nti − Nti−1 = # of jumps during (ti−1, ti ]

2 Estimation Error: (F-L & Nisen, 2013)

Lossn(ε) := En∑

i=1

(1[|∆n

i X |>ε,∆ni N=0] + 1[|∆n

i X |≤ε,∆ni N 6=0]

).

3 Motivation

The underlying principle is that the estimation error is directly link

to the ability of the threshold to detect jumps.



Expected number of jump misclassifications

1 Notation:

Nt = # of jumps by time t

∆ni X = Xti − Xti−1

∆ni N = Nti − Nti−1 = # of jumps during (ti−1, ti ]

2 Estimation Error: (F-L & Nisen, 2013)

Lossn(ε) := En∑

i=1

(1[|∆n

i X |>ε,∆ni N=0] + 1[|∆n

i X |≤ε,∆ni N 6=0]

).

3 Motivation

The underlying principle is that the estimation error is directly link

to the ability of the threshold to detect jumps.J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 13 / 29


Existence and Infill Asymptotic Characterization

Theorem (FL & Nisen (SPA, 2013))

1 For n large enough, the loss function Lossn(ε) is convex and,

moreover, possesses a unique global minimum ε?n.

2 As n→∞, the optimal threshold sequence (ε?n)n is such that

ε∗n =

√3σ2hn log

(1hn

)−

log(√

2πσλC(fζ))σh1/2

n√3 log(1/hn)

+ h.o.t.,

where C(fζ) = limδ→012δ

∫ δ−δ fζ(x)dx > 0 and fζ is the density of

the jumps ζi (h.o.t. refers to high order terms).








ε∗n =

√3σ2hn log

(1hn

)−

log(√


n√3 log(1/hn)

+ h.o.t.,











ε∗n =

√3σ2hn log

(1hn

)−

log(√


n√3 log(1/hn)

+ h.o.t.,






Remarks

1 The threshold sequences

ε?1n :=

√3σ2hn log

(1hn

), ε?2

n := ε?1n −

log(√


n√3 log(1/hn)

,

are the first and second-order approximations for B∗n , and the biases of

their corresponding TRV estimators attain the optimal rate of O(hn) as

n→∞.

3 They both provide “blueprints" for devising good threshold

sequences!



Remarks

1 The threshold sequences

ε?1n :=

√3σ2hn log

(1hn

), ε?2

n := ε?1n −

log(√


n√3 log(1/hn)

,

are the first and second-order approximations for B∗n , and the biases of

their corresponding TRV estimators attain the optimal rate of O(hn) as

n→∞.

3 They both provide “blueprints" for devising good threshold

sequences!



A Feasible Implementation based on ε?n

(i) Get a “rough" estimate of σ2 via, e.g., realized QV:

σ̂2n,0 :=

1T

n∑i=1

|Xti − Xti−1 |2

(ii) Use σ̂2n,0 to estimate the optimal threshold

ε̂?n,0 :=(

3 σ̂2n,0hn log(1/hn)

)1/2

(iii) Refine σ̂2n,0 using thresholding,

σ̂2n,1 =

1T

n∑i=1

|Xti − Xti−1 |21[|Xti−Xti−1 |≤ε̂

?n,0

](iv) Iterate Steps (ii) and (iii):

σ̂2n,0 → ε̂?n,0 → σ̂2

n,1 → ε̂?n,1 → σ̂2n,2 → · · · → σ̂2

n,∞





σ̂2n,0 :=

1T

n∑i=1

|Xti − Xti−1 |2


ε̂?n,0 :=(


)1/2


σ̂2n,1 =

1T

n∑i=1


?n,0


σ̂2n,0 → ε̂?n,0 → σ̂2

n,1 → ε̂?n,1 → σ̂2n,2 → · · · → σ̂2

n,∞





σ̂2n,0 :=

1T

n∑i=1

|Xti − Xti−1 |2


ε̂?n,0 :=(


)1/2


σ̂2n,1 =

1T

n∑i=1


?n,0

]

(iv) Iterate Steps (ii) and (iii):

σ̂2n,0 → ε̂?n,0 → σ̂2

n,1 → ε̂?n,1 → σ̂2n,2 → · · · → σ̂2

n,∞





σ̂2n,0 :=

1T

n∑i=1

|Xti − Xti−1 |2


ε̂?n,0 :=(


)1/2


σ̂2n,1 =

1T

n∑i=1


?n,0


σ̂2n,0 → ε̂?n,0 → σ̂2

n,1 → ε̂?n,1 → σ̂2n,2 → · · · → σ̂2

n,∞



Illustration I

0.00 0.02 0.04 0.06 0.08

0.85

0.90

0.95

1.00

1.05



Sto

ck P

rice

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

0.0

0.1

0.2

0.3

0.4



Tru

nca

ted

Re

aliz

ed

Va

ria

tio

n (

TR

V)


Figure: (left) Merton Model with σ = 0.4, σjmp = 3(h), µjmp = 0, λ = 200;

(right) TRV performance wrt the truncation level. Red dot is σ̂n,1 = 0.409,

while purple dot is the limiting estimator σ̂n,∞ = 0.405



Illustration II

0.00 0.02 0.04 0.06 0.08

0.96

1.00

1.04



Sto

ck P

rice

0.00 0.01 0.02 0.03 0.04

0.00

0.10

0.20

0.30



Tru

nca

ted

Re

aliz

ed

Va

ria

tio

n (

TR

V)


Figure: (left) Merton Model with σ = 0.2, σjmp = 1.5(h), µjmp = 0, λ = 1000;

(right) TRV performance wrt the truncation level. Red dot is σ̂n,1 = 0.336,

while purple dot is the limiting estimator σ̂n,∞ = 0.215


Optimal Threshold Selection via conditional Mean Square Error (cMSE)

Setting

1 Framework:

Xt = X0 +

∫ t

0σsdWs + Jt

where J is a pure-jump process.

2 Key Assumptions:

σt > 0 for all t , and σ and J are independent of W .

3 Notation:

mi = ∆ni J = Jti − Jti−1 , σ2

i =

∫ ti

ti−1

σ2s ds

4 Key Observation: Conditioning on σ and J, the increments ∆ni X

are i.i.d. normal with mean mi and variance σ2i



Key Relationships

Let

bi (ε) := E[(∆iX )21{|∆i X |≤ε}|σ, J

]

1 As it turns out,

bi (ε) = −(

e− (ε−mi )2

2σ2i (ε+ mi ) + e

− (ε+mi )2

2σ2i (ε−mi )

) σi√2π

+m2

i + σ2i√

π

(∫ ε−mi√2σi

0e−t2

dt +

∫ ε+mi√2σi

0e−t2

dt)

2 Furthermore,

dbi (ε)

dε= ε2ai (ε), with ai (ε) :=

e− (ε−mi )2

2σ2i + e

− (ε+mi )2

2σ2i

σi√

2π



Key Relationships

Let

bi (ε) := E[(∆iX )21{|∆i X |≤ε}|σ, J

]

1 As it turns out,

bi (ε) = −(

e− (ε−mi )2

2σ2i (ε+ mi ) + e

− (ε+mi )2

2σ2i (ε−mi )

) σi√2π

+m2

i + σ2i√

π

(∫ ε−mi√2σi

0e−t2

dt +

∫ ε+mi√2σi

0e−t2

dt)

2 Furthermore,

dbi (ε)


e− (ε−mi )2

2σ2i + e

− (ε+mi )2

2σ2i

σi√

2π



Key Relationships

Let

bi (ε) := E[(∆iX )21{|∆i X |≤ε}|σ, J

]

1 As it turns out,

bi (ε) = −(

e− (ε−mi )2

2σ2i (ε+ mi ) + e

− (ε+mi )2

2σ2i (ε−mi )

) σi√2π

+m2

i + σ2i√

π

(∫ ε−mi√2σi

0e−t2

dt +

∫ ε+mi√2σi

0e−t2

dt)

2 Furthermore,

dbi (ε)


e− (ε−mi )2

2σ2i + e

− (ε+mi )2

2σ2i

σi√

2π



Conditional Mean Square Error (MSEc)

Theorem (F-L & Mancini (2017))

1 Let IV :=∫ T

0 σ2s ds be the integrated variance of X and let

MSEc(ε) := E[

(TRVn(ε)− IV )2∣∣∣σ, J]

2 Then, MSEc(ε) is differential in (0,∞) and, furthermore,

d MSEc(ε)

dε= ε2G(ε),

where G(ε) :=∑

i ai(ε)(ε2 + 2

∑j 6=i bj(ε)− 2IV

).

2 Furthermore, there exists an optimal threshold ε??n that minimizes

MSEc(ε) and is such that G(ε??n ) = 0.





1 Let IV :=∫ T


MSEc(ε) := E[

(TRVn(ε)− IV )2∣∣∣σ, J]


d MSEc(ε)

dε= ε2G(ε),

where G(ε) :=∑

i ai(ε)(ε2 + 2


).







1 Let IV :=∫ T


MSEc(ε) := E[

(TRVn(ε)− IV )2∣∣∣σ, J]


d MSEc(ε)

dε= ε2G(ε),

where G(ε) :=∑

i ai(ε)(ε2 + 2


).







1 Let IV :=∫ T


MSEc(ε) := E[

(TRVn(ε)− IV )2∣∣∣σ, J]


d MSEc(ε)

dε= ε2G(ε),

where G(ε) :=∑

i ai(ε)(ε2 + 2


).





Asymptotics: FA process with constant variance


1 Suppose that σt ≡ σ is constant and J is a finite jump activity

process (with or without drift; not necessarily Lévy):

Xt = σWt +

Nt∑j=1

ζj

2 Then, as n→∞, the optimal threshold ε?n is such that

ε??n ∼

√2σ2hn log

(1hn

)







Xt = σWt +

Nt∑j=1

ζj


ε??n ∼

√2σ2hn log

(1hn

)







Xt = σWt +

Nt∑j=1

ζj


ε??n ∼

√2σ2hn log

(1hn

)



A Feasible Implementation based on ε??n

(i) Get a “rough" estimate of σ2 via, e.g., the QV:

σ̃2n,0 :=

1T

n∑i=1

|Xti − Xti−1 |2

(ii) Use σ̃2n,0 to estimate the optimal threshold

ε̂??n,0 :=(

2 σ̃2n,0hn log(1/hn)

)1/2

(iii) Refine σ̃2n,0 using thresholding,

σ̃2n,1 =

1T

n∑i=1


??n,0


σ̃2n,0 → ε̂??n,0 → σ̃2

n,1 → ε̂??n,1 → σ̃2n,2 → · · · → σ̃2

n,∞





σ̃2n,0 :=

1T

n∑i=1

|Xti − Xti−1 |2


ε̂??n,0 :=(


)1/2


σ̃2n,1 =

1T

n∑i=1


??n,0


σ̃2n,0 → ε̂??n,0 → σ̃2

n,1 → ε̂??n,1 → σ̃2n,2 → · · · → σ̃2

n,∞





σ̃2n,0 :=

1T

n∑i=1

|Xti − Xti−1 |2


ε̂??n,0 :=(


)1/2


σ̃2n,1 =

1T

n∑i=1


??n,0

]

(iv) Iterate Steps (ii) and (iii):

σ̃2n,0 → ε̂??n,0 → σ̃2

n,1 → ε̂??n,1 → σ̃2n,2 → · · · → σ̃2

n,∞





σ̃2n,0 :=

1T

n∑i=1

|Xti − Xti−1 |2


ε̂??n,0 :=(


)1/2


σ̃2n,1 =

1T

n∑i=1


??n,0


σ̃2n,0 → ε̂??n,0 → σ̃2

n,1 → ε̂??n,1 → σ̃2n,2 → · · · → σ̃2

n,∞



Illustration II. Continued...

0.00 0.02 0.04 0.06 0.08

0.96

1.00

1.04



Sto

ck P

rice

0.00 0.01 0.02 0.03 0.04

0.00

0.10

0.20

0.30



Tru

nca

ted

Re

aliz

ed

Va

ria

tio

n (

TR

V)


Figure: (left) Merton Model with λ = 1000. Red dot is σ̂n,1 = 0.336, while

purple dot is the limiting σ̂n,k = 0.215. Orange square is σ̃n,1 = 0.225, while

brown square is the limiting estimator σ̃n,∞ = 0.199



Monte Carlo Simulations

Estimator σ̂ std(σ̂) Loss ε N

RQV 0.3921 0.0279

σ̂n,1 0.29618 0.02148 70.1 0.01523 1

σ̂n,∞ 0.23 0.0108 49.8 0.00892 5.86

σ̃n,1 0.265 0.0163 62.6 0.0124 1

σ̃n,∞ 0.211 0.00588 39.1 0.00671 5.10

BPV 0.2664 0.0129

Table: Estimation of the volatility σ = 0.2 for a log-normal Merton model

based on 10000 simulations of 5-minute observations over a 1 month time

horizon. The jump parameters are λ = 1000, σJmp = 1.5√

h and µJmp = 0.

Loss is the number of jump misclassifications and N is the number of

iterations. bar is used to denote average.J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 25 / 29

Conclusions

Outline







4 Conclusions


Conclusions

Ongoing and Future Research

1 In principle, we can apply the proposed method for varying

volatility t → σt by localization; i.e., applying to periods where σ is

approximately constant.

2 However, is it possible to analyze the asymptotic behavior of

MSEc = ε2G(ε) in terms of certain estimable summary

measures? Say,

G(ε) ∼ G0(ε,m1, . . . ,mn, σ, σ̄),

where σ := inft≤T σt and σ̄ := supt≤T σt


Conclusions


1 In principle, we can apply the proposed method for varying

volatility t → σt by localization; i.e., applying to periods where σ is

approximately constant.

2 However, is it possible to analyze the asymptotic behavior of

MSEc = ε2G(ε) in terms of certain estimable summary

measures? Say,

G(ε) ∼ G0(ε,m1, . . . ,mn, σ, σ̄),

where σ := inft≤T σt and σ̄ := supt≤T σt


Conclusions


3 As it turns, for a Lévy process J and constant σ, the expected

MSE(ε) := E[(TRVn(ε)− IV )2

]is such that

ddε

MSE(ε) = nε2E[a1(ε)](ε2 + 2(n − 1)E[b1(ε)]− 2nhnσ

2)

Therefore, there exists a unique minimum point ε̄?n which is a

solution of the equation

ε2 + 2(n − 1)E[b1(ε)]− 2nhnσ2 = 0.


Conclusions


3 As it turns, for a Lévy process J and constant σ, the expected

MSE(ε) := E[(TRVn(ε)− IV )2

]is such that

ddε

MSE(ε) = nε2E[a1(ε)](ε2 + 2(n − 1)E[b1(ε)]− 2nhnσ

2)

Therefore, there exists a unique minimum point ε̄?n which is a

solution of the equation

ε2 + 2(n − 1)E[b1(ε)]− 2nhnσ2 = 0.


Conclusions


3 Furthermore,

• As expected, in the finite jump activity case, it turns out

ε̄?n ∼

√2σ2hn log

(1hn

)• But, surprisingly, if J is a Y -stable Lévy process (IA),

ε̄?n ∼

√(2− Y )σ2hn log

(1hn

)Thus the higher the jump activity is, the lower the optimal threshold

has to be to discard the higher noise represented by the jumps, in

order to catch information about IV.• Does this phenomenon holds for the minimizer ε?n of the cMSE?


Conclusions


3 Furthermore,


ε̄?n ∼

√2σ2hn log

(1hn

)

• But, surprisingly, if J is a Y -stable Lévy process (IA),

ε̄?n ∼


(1hn





Conclusions


3 Furthermore,


ε̄?n ∼

√2σ2hn log

(1hn

)• But, surprisingly, if J is a Y -stable Lévy process (IA),

ε̄?n ∼


(1hn





Documents

Asymptotic Methods in Financial Mathematics Methods in Financial Mathematics José E. Figueroa-López1 1Department of Mathematics Washington University in St. Louis Statistics Seminar