Optimal discretization of hedging strategies rosenbaum

IntroductionAsymptotically optimal strategies

Microstructure effects

Optimal discretization of hedging strategies

M. Fukasawa1 C.Y. Robert2 M. Rosenbaum3 P. Tankov4

1Department of Mathematics, Osaka University

2ISFA, Universite Lyon 1

3LPMA, Universite Pierre et Marie Curie (Paris 6)

4LPMA, Universite Paris Diderot (Paris 7)

8 July 2014

Fukasawa, Robert, Rosenbaum, Tankov Optimal discretization of hedging strategies 1



Outline

1 Introduction

2 Asymptotically optimal strategies

3 Microstructure effects




Outline

1 Introduction






Hedging

Classical problem

A basic problem in mathematical finance is to replicate aFT -measurable payoff HT with a portfolio involving only theunderlying asset Y and cash.

When HT = H(YT ) and Y follows a diffusion process of theform

dYt = µ(t,Yt)dt + σ(t,Yt)dWt ,

HT can be replicated with the so-called delta hedging strategy.

This means that the number of units of underlying to hold attime t is equal to Xt = ∂P(t,Yt)

∂Y , where P(t,Yt) is the price ofthe option, which is uniquely defined in such model.




Discrete hedging

Hedging in practice

However, to implement such strategy, the hedging portfoliomust be readjusted continuously.

This is of course physically impossible and anyway irrelevantbecause of the presence of microstructure effects andtransaction costs.

Thus the theoretical strategy is always replaced by a piecewiseconstant one, leading to a discretization error.

In practice, traders rebalance their portfolio about once perday.

Nevertheless, (relatively) high frequency trading technologiesoffer the possibility to rebalance much more often, whichcould reduce the risk.




Computing the discretization error

Deterministic rebalancing strategies

The discretization error associated to given deterministicrebalancing strategies has been widely studied, see Zhang (1999),Bertsimas, Kogan and Lo (2000), Gobet and Temam (2001), Geiss(2002), Hayashi and Mykland (2005), Geiss and Geiss (2006),Geiss and Toivola (2007), Tankov and Voltchkova (2008),. . .




Regular rebalancings

Hedging error

Assume first that the hedging portfolio is readjusted at regulartime intervals of length h = T

n . Zhang (1999) (see alsoBertsimas et al ; Hayashi and Mykland) shows that thediscretization error

EnT =

∫ T

0XtdYt −

∫ T

0Xhbt/hcdYt

essentially satisfies

limh→0

nE [(EnT )2] =T

2E[ ∫ T

0

(∂2P

∂Y 2

)2

σ(s,Ys)4ds].




General strategies

Optimal discretization

It is intuitively clear that readjusting the portfolio at regulardeterministic times is not optimal.

Therefore, we want to consider adaptive, path dependentstrategies. The relevant questions are then : how big is thediscretization error and what are the right rebalancing timesfor the hedging portfolio ?

However, the optimal strategy for fixed n (or fixed cost) isvery difficult to compute.




Outline

1 Introduction






Asymptotic approach

Asymptotic criterion

We consider an asymptotic approach which enables us toderive asymptotically optimal strategies in quite an easy way.

More precisely, the performances of different discretizationstrategies are compared based on their asymptotic behavior asthe number of readjustment dates n tends to infinity, ratherthan for fixed n.




Asymptotic approach

Discretization

A discretization rule is a family of stopping times

0 = T n0 < T n

1 < · · · < T nj < . . . ,

with supj |T nj+1 ∧ T − T n

j ∧ T | → 0 as n→∞.

These stopping times represent the rebalancing dates of theportfolio.

Let X ns = XT n

j ∧T for s ∈ (T nj ∧ T − T n

j+1 ∧ T ]. The hedging

error EnT is given by EnT =∫ T

0 XsdYs −∫ T

0 X ns dYs , with∫ T

0X ns dYs =

∞∑j=0

XT nj ∧T (YT n

j+1∧T − YT nj ∧T ).




Asymptotic approach

Asymptotic criterion

Let NnT := maxj ≥ 0;T n

j ≤ T. To compare differentdiscretization strategies in terms of their asymptotic behavior,one can use the following criterion :

limn→∞

E [NnT ]E [(EnT )2].

A discretization rule A is considered better than adiscretization rule B if the value of the criterion for A issmaller than for B.

This criterion is quite natural. The quantity NnT is viewed as a

proxy for transaction costs and so we want it small. On theother hand we also want the error small.




Dynamics

Notation and assumptions

We consider the following formalism (we express everything in

term of σXt ) for the discretization of∫ T

0 XsdYs :

dXt = ψt(σXt )2dt + σXt dWt , dYt = Ktσ

Xt dWt ,

with all the coefficients bounded (locally).

We define admissibility conditions for the rules :

supj |T nj+1 ∧ T − T n

j ∧ T | → 0, supt≤T |Xt − X nt | ≤ cvn,

with vn → 0.




Lower bound

Theorem

For any admissible discretization rule,

liminfn→+∞

E[NnT ]E[(EnT )2] ≥ 1

6

(E[ ∫ T

0Kt(σ

Xt )2dt

])2.




Upper bound

Theorem

The strategy

T n0 = 0, T n

j+1 = inft > T n

j , (Xt − X nT nj

)2 ≥ hnKT n

j

,

with hn a positive deterministic sequence going to zero, isasymptotically optimal.

For the chosen criterion, this improves the regulardeterministic rebalancing times strategy by at least a factor 3.




Upper bound : example

Example

In the Black-Scholes model :

Xt =∂P(t,Yt)

∂Y, σXt =

∂2P(t,Yt)

∂Y 2σYt = ΓtσYt .

Thus,Kt = 1/Γt

and

T n0 = 0, T n

j+1 = inft > T n

j , (Xt − X nT nj

)2 ≥ hnΓT nj

,




About the preceding approach

Limitations

The above approach is quite natural and provides very explicitresults. However, it fails to take into account important factors ofmarket reality :

The criterion is somewhat ad hoc, and does not reflect anyspecific model for the transaction costs or market impact.

The continuity assumption is arguable.

What about high frequency microstructure effects ?

Further approaches have been proposed to remedy these issues.We discuss here the issue of microstructure effects.




Outline

1 Introduction






Transaction price and efficient price

High frequency effects

In the classical mathematical finance theory, the transactionprice is assumed to be equal to the efficient/theoretical price,typically modeled by a Brownian semi-martingale.

However, in the high frequencies, observed prices largely differfrom observations of a semi-martingale.

We have to check high frequency effects are negligible whenusing results obtained in the classical setting.




Bund contract, one and a half year, one data every hour

0.0 0.2 0.4 0.6 0.8 1.0

112

114

116

118

120

122

124

126

Time

Bund

.




Bund contract, whole day and one hour, one data everysecond

time

va

lue

0 5000 10000 15000 20000 25000 30000

11

5.4

01

15

.45

11

5.5

01

15

.55

11

5.6

01

15

.65

time

va

lue

0 500 1000 1500 2000 2500 3000 3500

11

5.4

61

15

.48

11

5.5

01

15

.52

11

5.5

41

15

.56




Our setting


We will work in a model that accommodates the stylized facts ofultra high frequency prices and durations together with asemi-martingale efficient price. In particular, transaction prices willbelong to the tick grid. Consequently :

Impossibility to buy or sell a share at the efficient price : themicrostructure leads to a cost (possibly negative).

The transaction price changes a finite number of times on agiven time period. Therefore, it is reasonable to assume thatone waits for a price change before rebalancing the hedgingportfolio.




Microstructure modeling

We want to study microstructure effects. So we need a modeldescribing prices and durations in the high frequencies. Inparticular, we want :

Properties of the microstructure model

Discrete prices.

Bid-Ask bounce.

Stylized facts of returns, durations and volatility.

A diffusive behavior at large sampling scales.

An interpretation of the model.




Aversion for price changes

Aversion for price changes

In an idealistic framework, transactions would occur when theefficient price crosses the tick grid.

In practice, uncertainty about the efficient price and aversionfor price changes of the market participants.

The price changes only when market participants areconvinced that the efficient price is far from the last tradedprice.

We introduce a parameter η quantifying this aversion for pricechanges.




Model with uncertainty zones

Model with uncertainty zones : notation

Efficient price : Xt .

α : tick value.

τi : time of the i-th transaction with price change.

Pτi : transaction price at time τi .

Uncertainty zones : Uk = [0,∞)× (dk , uk) with

dk = (k + 1/2− η)α and uk = (k + 1/2 + η)α.




Model with uncertainty zones

Model with uncertainty zones : dynamics

d logXu = audu + σu−dWu.

τi+1 = inft > τi ,Xt = X (α)

τi± α(

1

2+ η)

,

with X(α)τi the value of Xτi rounded to the nearest multiple of α.

Pτi = X (α)τi.




Model with uncertainty zones10

1.00

101.0

110

1.02

101.0

310

1.04

101.0

510

1.06

Time

Price

10:00:00 10:01:07 10:01:52 10:02:41 10:03:32 10:04:5010:00:00 10:01:00 10:02:00 10:03:00 10:04:00 10:05:00

101.0

010

1.01

101.0

210

1.03

101.0

410

1.05

101.0

6

Time

Price

2ηα

α

τ0 τ1 τ2 τ3 τ4 τ5

L0 = 1L1 = 1

L2 = 1L3 = 1

L4 = 1L5 = 2

Xτ0

(α)

Xτ1

(α)

Xτ2

(α)

101.0

010

1.01

101.0

210

1.03

101.0

410

1.05

101.0

6

Time

Price

Observed PriceTheoretical Price

TickMid Tick

Uncertainty Zone LimitBarriers To Reach

101.0

010

1.01

101.0

210

1.03

101.0

410

1.05

101.0

6

Time

Price




Discussion

Comments on the model

The model reproduces (almost) all the stylized facts of (ultra)high to low frequency financial data.

The parameter η quantifies the tick size of the market. A smallη (< 1/2) means that for market participants, the tick size istoo large and conversely. It can be seen as an implicit spread.

The parameter η can be very easily estimated from marketdata.




Some properties : Durations

09:00:00 10:00:00 11:00:00 12:00:00 13:00:00 14:00:00 15:00:00 16:00:00 17:00:00

0.00

00.

005

0.01

00.

015

Vol

atili

ty

09:00:00 10:00:00 11:00:00 12:00:00 13:00:00 14:00:00 15:00:00 16:00:00 17:00:00

050

010

0020

00

Dur

atio

n




Some properties : The price

10:00:00 10:04:00 10:08:00 10:12:00 10:16:00 10:20:00 10:24:00 10:28:00

99

.91

00

.21

00

.5

Ob

se

rve

d p

rice

10:00:00 10:04:00 10:08:00 10:12:00 10:16:00 10:20:00 10:24:00 10:28:00

99

.91

00

.21

00

.5

Th

eo

retica

l p

rice

10:00:00 10:04:00 10:08:00 10:12:00 10:16:00 10:20:00 10:24:00 10:28:00

−2

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04

1e

−0

4

Mic

rostr

uctu

re n

ois

e

00:00:00 00:00:06 00:00:12 00:00:18 00:00:24 00:00:30 00:00:36 00:00:42 00:00:48

0.0

0.4

0.8

AC

F f

or

the

no

ise




Bund and DAX, estimation of η, October 2010

Day η (Bund) η (FDAX) Day η (Bund) η (FDAX)

1 Oct. 0.18 0.41 18 Oct. 0.16 0.335 Oct. 0.15 0.37 19 Oct. 0.13 0.376 Oct. 0.15 0.37 20 Oct. 0.13 0.33



15 Oct. 0.16 0.35 29 Oct. 0.14 0.34




Benchmark frictionless hedging strategy

Benchmark strategy

The benchmark frictionless hedging strategy is those of anagent deciding (possibly wrongly) that the volatility of theefficient price at time t is equal to σ(t,Xt).

It leads to a benchmark frictionless hedging portfolio whosevalue Πt satisfies

Πt = C (0,X0) +

∫ t

0Cx (u,Xu) dXu.

Note that, if the model is misspecified, Πt is different fromthe model price C (t,Xt).




Hedging strategies in the model with uncertainty zones

Hedging strategies in the model with uncertainty zones (1)

We naturally impose that the times when the hedgingportfolio may be rebalanced are the times where thetransaction price moves. Thus, the hedging portfolio can onlybe rebalanced at the transaction times τi .

In this setting, we consider strategies such that, if τi is arebalancing time, the number of shares in the risky asset attime τi is Cx (τi ,Xτi ).




Hedging strategies in the model with uncertainty zones

Hedging strategies in the model with uncertainty zones (2)

We will consider two hedging strategies :

The hedging portfolio is rebalanced every time that thetransaction price moves.

The hedging portfolio is rebalanced only once the transactionprice has varied by more than a selected value.




Components of the hedging error

Components of the hedging error

In our setting, the microstructural hedging error is due to :

Discrete trading : the hedging portfolio is rebalanced a finitenumber of times.

Microstructure on the price : between two rebalancing times,the variation of the market price (multiple of the tick size)differs from the variation of the efficient price.




Analysis of the hedging error

Two steps analysis

We analyse this microstructure hedging error in two steps.

First, we assume that there is no microstructure effects on theprice although the trading times are endogenous (for all i ,Pτi = Xτi ).

Second, we assume the presence of the microstructure effectsand discussed the two hedging strategies.




Hedging error without microstructure effects on the price

Discrete trading error

Let φ (t) = supτi : τi < t. In the absence of microstructureeffects on the price, the hedging error is given by

L(1)α,t =

∫ t

0[Cx (u,Xu)− Cx(φ (u) ,Xφ(u))]dXu.




Hedging error without microstructure effects on the price

Theorem

As α tends to 0,

N1/2α,t L

(1)α,tI−Ls→ L

(1)t := f

1/2t

∫ t

0c

(1)fs

dW(1)fs,

in D[0,T ], where W(1) is a Brownian motion defined on anextension of the filtered probability space (Ω, (Ft)t≥0,P) andindependent of all the preceding quantities, and

(c(1)s )2 =

1

6C 2xx (θs ,Xθs )µ4 (χθs ) ,

ft =

∫ t

0

( m∑j=1

pj(χu)j(j − 1 + 2η))−1

σ2uX

2u du.




Hedging error with microstructure effects

Total hedging error

In the presence of microstructure effects on the price, thetransaction prices differ from the efficient prices. The hedging erroris now given by

L(2)α,t =

∫ t

0Cx (u,Xu) dXu −

∫ t

0Cx(φ (u) ,Xφ(u))dPu.





Theorem

As α tends to 0,

L(2)α,tI−Ls→ L

(2)t :=

∫ t

0b

(2)fs

dXs +

∫ t

0c

(2)fs

dW(2)fs,

in D[0,T ], with

b(2)s = (1− 2η)Cx (θs ,Xθs )µ

∗1,a (χθs )ϕ (χθs )

and

(c(2)s )2 = (1− 2η)2C 2

x (θs ,Xθs )ϕ (χθs )(πa (χθs )ϕ

−1 (χθs )− (µ∗1,a (χθs ))2).





Comments

The microstructural hedging error process is not renormalizedas in the previous case.

It means that the hedging error does not vanish even if thenumber of rebalancing transactions goes to infinity.

If η = 1/2, the error due to the microstructure effects on theprice vanishes.




Optimal rebalancing

Total hedging error

The hedging portfolio is now rebalanced only once the price hasvaried by lα ticks. The hedging error is given by

L(3)α,t =

∫ t

0Cx (u,Xu) dXu −

∫ t

0Cx(φ(l) (u) ,Xφ(l)(u))dPu.

with φ(l) (t) = supτ (l)i : τ

(l)i < t and the τ

(l)i are stopping times

associated to moves of lα ticks.




Optimal rebalancing

Theorem

Let lα = α−1/2. As α tends to 0, we have,

(N(l)α,t)

1/4L(3)α,tI−Ls→ L

(3)t := (f

(l)t )1/4

( ∫ t

0b

(3)

f(l)s

dXs +

∫ t

0c

(3)

f(l)s

dW(3)

f(l)s

)in D[0,T ], with

b(3)s =

(1− 2η)

2Cx (θs ,Xθs )

and

(c(3)s )2 =

(1− 2η)2

4C 2x (θs ,Xθs ) +

1

6C 2xx (θs ,Xθs ) .




Optimal rebalancing

Comments

This optimal strategy allows to reduce significantly thehedging error in the presence of microstructure effects.

The asymptotic variance of the hedging error now dependsboth on the delta and on the gamma of the derivative security.

The optimal lα is of the same order of magnitude as thesquare root of the number of times where the hedgingportfolio is rebalanced.


Optimal discretization of hedging strategies rosenbaum

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