Optimal discretization of hedging strategies rosenbaum

Introduction
Asymptotically optimal strategies
Microstructure effects
Optimal discretization of hedging strategies
M. Fukasawa1 C.Y. Robert2 M. Rosenbaum3 P. Tankov4
1Department of Mathematics, Osaka University
2ISFA, Université Lyon 1
3LPMA, Université Pierre et Marie Curie (Paris 6)
4LPMA, Université Paris Diderot (Paris 7)
8 July 2014
Fukasawa, Robert, Rosenbaum, Tankov Optimal discretization of hedging strategies 1

Introduction
Outline
1 Introduction
2 Asymptotically optimal strategies
3 Microstructure eﬀects

Introduction
Outline
1 Introduction

Introduction
Hedging
Classical problem
A basic problem in mathematical finance is to replicate a
FT -measurable payoff HT with a portfolio involving only the
underlying asset Y and cash.
When HT = H(YT ) and Y follows a diffusion process of the
form
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 at
time t is equal to Xt = ∂P(t,Yt )
∂Y , where P(t, Yt) is the price of
the option, which is uniquely defined in such model.

Introduction
Discrete hedging
Hedging in practice
However, to implement such strategy, the hedging portfolio
must be readjusted continuously.
This is of course physically impossible and anyway irrelevant
because of the presence of microstructure eﬀects and
transaction costs.
Thus the theoretical strategy is always replaced by a piecewise
constant one, leading to a discretization error.
In practice, traders rebalance their portfolio about once per
day.
Nevertheless, (relatively) high frequency trading technologies
oﬀer the possibility to rebalance much more often, which
could reduce the risk.

Introduction
Computing the discretization error
Deterministic rebalancing strategies
The discretization error associated to given deterministic
rebalancing 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),. . .

Introduction
Regular rebalancings
Hedging error
Assume ﬁrst that the hedging portfolio is readjusted at regular
time intervals of length h = T
n . Zhang (1999) (see also
Bertsimas et al ; Hayashi and Mykland) shows that the
discretization error
En
T =
T
0
XtdYt −
T
0
Xh t/h dYt
essentially satisﬁes
lim
h→0
nE[(En
T )2
] =
T
2
E
T
0
∂2P
∂Y 2
2
σ(s, Ys)4
ds .

Introduction
General strategies
Optimal discretization
It is intuitively clear that readjusting the portfolio at regular
deterministic times is not optimal.
Therefore, we want to consider adaptive, path dependent
strategies. The relevant questions are then : how big is the
discretization error and what are the right rebalancing times
for the hedging portfolio ?
However, the optimal strategy for fixed n (or fixed cost) is
very difficult to compute.

Introduction
Outline
1 Introduction

Introduction
Asymptotic approach
Asymptotic criterion
We consider an asymptotic approach which enables us to
derive asymptotically optimal strategies in quite an easy way.
More precisely, the performances of different discretization
strategies are compared based on their asymptotic behavior as
the number of readjustment dates n tends to infinity, rather
than for fixed n.

Introduction
Asymptotic approach
Discretization
A discretization rule is a family of stopping times
0 = Tn
0 < Tn
1 < · · · < Tn
j < . . . ,
with supj |Tn
j+1 ∧ T − Tn
j ∧ T| → 0 as n → ∞.
These stopping times represent the rebalancing dates of the
portfolio.
Let Xn
s = XTn
j ∧T for s ∈ (Tn
j ∧ T − Tn
j+1 ∧ T]. The hedging
error En
T is given by En
T =
T
0 XsdYs −
T
0 Xn
s dYs, with
T
0
Xn
s dYs =
∞
j=0
XTn
j ∧T (YTn
j+1∧T − YTn
j ∧T ).

Introduction
Asymptotic approach
Asymptotic criterion
Let Nn
T := max{j ≥ 0; Tn
j ≤ T}. To compare diﬀerent
discretization strategies in terms of their asymptotic behavior,
one can use the following criterion :
lim
n→∞
E[Nn
T ]E[(En
T )2
].
A discretization rule A is considered better than a
discretization rule B if the value of the criterion for A is
smaller than for B.
This criterion is quite natural. The quantity Nn
T is viewed as a
proxy for transaction costs and so we want it small. On the
other hand we also want the error small.

Introduction
Dynamics
Notation and assumptions
We consider the following formalism (we express everything in
term of σX
t ) for the discretization of
T
0 XsdYs :
dXt = ψt(σX
t )2
dt + σX
t dWt, dYt = KtσX
t dWt,
with all the coeﬃcients bounded (locally).
We deﬁne admissibility conditions for the rules :
supj |Tn
j+1 ∧ T − Tn
j ∧ T| → 0, supt≤T |Xt − Xn
t | ≤ cvn,
with vn → 0.

Introduction
Lower bound
Theorem
For any admissible discretization rule,
liminf
n→+∞
E[Nn
T ]E[(En
T )2
] ≥
1
6
E
T
0
Kt(σX
t )2
dt
2
.

Introduction
Upper bound
Theorem
The strategy
Tn
0 = 0, Tn
j+1 = inf t > Tn
j , (Xt − Xn
Tn
j
)2
≥
hn
KTn
j
,
with hn a positive deterministic sequence going to zero, is
asymptotically optimal.
For the chosen criterion, this improves the regular
deterministic rebalancing times strategy by at least a factor 3.

Introduction
Upper bound : example
Example
In the Black-Scholes model :
Xt =
∂P(t, Yt)
∂Y
, σX
t =
∂2P(t, Yt)
∂Y 2
σYt = ΓtσYt.
Thus,
Kt = 1/Γt
and
Tn
0 = 0, Tn
j+1 = inf t > Tn
j , (Xt − Xn
Tn
j
)2
≥ hnΓTn
j
,

Introduction
About the preceding approach
Limitations
The above approach is quite natural and provides very explicit
results. However, it fails to take into account important factors of
market reality :
The criterion is somewhat ad hoc, and does not reflect any
specific 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.

Introduction
Outline
1 Introduction

Introduction
Transaction price and efficient price
High frequency effects
In the classical mathematical finance theory, the transaction
price 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 differ
from observations of a semi-martingale.
We have to check high frequency effects are negligible when
using results obtained in the classical setting.

Introduction
Bund contract, one and a half year, one data every hour
0.0 0.2 0.4 0.6 0.8 1.0
112114116118120122124126
Time
Bund
.

Introduction
Bund contract, whole day and one hour, one data every
second
time
value
0 5000 10000 15000 20000 25000 30000
115.40115.45115.50115.55115.60115.65
time
value
0 500 1000 1500 2000 2500 3000 3500
115.46115.48115.50115.52115.54115.56

Introduction
Our setting
We will work in a model that accommodates the stylized facts of
ultra high frequency prices and durations together with a
semi-martingale efficient price. In particular, transaction prices will
belong to the tick grid. Consequently :
Impossibility to buy or sell a share at the efficient price : the
microstructure leads to a cost (possibly negative).
The transaction price changes a finite number of times on a
given time period. Therefore, it is reasonable to assume that
one waits for a price change before rebalancing the hedging
portfolio.

Introduction
Microstructure modeling
We want to study microstructure eﬀects. So we need a model
describing prices and durations in the high frequencies. In
particular, 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.

Introduction
Aversion for price changes
Aversion for price changes
In an idealistic framework, transactions would occur when the
efficient price crosses the tick grid.
In practice, uncertainty about the efficient price and aversion
for price changes of the market participants.
The price changes only when market participants are
convinced that the efficient price is far from the last traded
price.
We introduce a parameter η quantifying this aversion for price
changes.

Introduction
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.

Introduction
Model with uncertainty zones
Model with uncertainty zones : dynamics
d log Xu = audu + σu−dWu.
τi+1 = inf t > 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
.

Introduction
Model with uncertainty zones101.00101.01101.02101.03101.04101.05101.06
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
101.00101.01101.02101.03101.04101.05101.06
Time
Price
2ηα
α
τ0 τ1 τ2 τ3 τ4 τ5
L0 = 1
L1 = 1
L2 = 1
L3 = 1
L4 = 1
L5 = 2
Xτ 0
(α)
Xτ 1
(α)
Xτ 2
(α)
101.00101.01101.02101.03101.04101.05101.06
Time
Price
Observed Price
Theoretical Price
Tick
Mid Tick
Uncertainty Zone Limit
Barriers To Reach
101.00101.01101.02101.03101.04101.05101.06
Time
Price

Introduction
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 small
η (< 1/2) means that for market participants, the tick size is
too large and conversely. It can be seen as an implicit spread.
The parameter η can be very easily estimated from market
data.

Introduction
Some properties : Durations
q q q q q q q q q
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
0.0000.0050.0100.015
Volatility
q q q q q q q q q
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
050010002000
Duration

Introduction
Some properties : The price
q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q
10:00:00 10:04:00 10:08:00 10:12:00 10:16:00 10:20:00 10:24:00 10:28:00
99.9100.2100.5
Observedprice
q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q
10:00:00 10:04:00 10:08:00 10:12:00 10:16:00 10:20:00 10:24:00 10:28:00
99.9100.2100.5
Theoreticalpricee

Introduction
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 0.33
5 Oct. 0.15 0.37 19 Oct. 0.13 0.37
6 Oct. 0.15 0.37 20 Oct. 0.13 0.33
7 Oct. 0.15 0.38 21 Oct. 0.15 0.33
8 Oct. 0.15 0.41 22 Oct. 0.11 0.33
11 Oct. 0.14 0.36 25 Oct. 0.12 0.31
12 Oct. 0.14 0.36 26 Oct. 0.14 0.31
13 Oct. 0.14 0.32 27 Oct. 0.14 0.32
14 Oct. 0.16 0.35 28 Oct. 0.14 0.32
15 Oct. 0.16 0.35 29 Oct. 0.14 0.34

Introduction
Benchmark frictionless hedging strategy
Benchmark strategy
The benchmark frictionless hedging strategy is those of an
agent deciding (possibly wrongly) that the volatility of the
efficient price at time t is equal to σ(t, Xt).
It leads to a benchmark frictionless hedging portfolio whose
value Πt satisfies
Πt = C (0, X0) +
t
0
˙Cx (u, Xu) dXu.
Note that, if the model is misspecified, Πt is different from
the model price C (t, Xt).

Introduction
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 hedging
portfolio may be rebalanced are the times where the
transaction price moves. Thus, the hedging portfolio can only
be rebalanced at the transaction times τi .
In this setting, we consider strategies such that, if τi is a
rebalancing time, the number of shares in the risky asset at
time τi is ˙Cx (τi , Xτi ).

Introduction
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 the
transaction price moves.
The hedging portfolio is rebalanced only once the transaction
price has varied by more than a selected value.

Introduction
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 finite
number 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.

Introduction
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 the
price although the trading times are endogenous (for all i,
Pτi = Xτi ).
Second, we assume the presence of the microstructure eﬀects
and discussed the two hedging strategies.

Introduction
Hedging error without microstructure eﬀects on the price
Discrete trading error
Let φ (t) = sup{τi : τi < t}. In the absence of microstructure
eﬀects on the price, the hedging error is given by
L
(1)
α,t =
t
0
[ ˙Cx (u, Xu) − ˙Cx (φ (u) , Xφ(u))]dXu.

Introduction
Hedging error without microstructure effects on the price
Theorem
As α tends to 0,
N
1/2
α,t L
(1)
α,t
I−Ls
→ L
(1)
t := f
1/2
t
t
0
c
(1)
fs
dW
(1)
fs
,
in D[0, T], where W(1) is a Brownian motion defined on an
extension of the filtered probability space (Ω, (Ft)t≥0, P) and
independent of all the preceding quantities, and
(c
(1)
s )2
=
1
6
¨C2
xx (θs, Xθs ) µ4 (χθs ) ,
ft =
t
0
m
j=1
pj (χu)j(j − 1 + 2η)
−1
σ2
uX2
u du.

Introduction
Hedging error with microstructure effects
Total hedging error
In the presence of microstructure effects on the price, the
transaction prices differ from the efficient prices. The hedging error
is now given by
L
(2)
α,t =
t
0
˙Cx (u, Xu) dXu −
t
0
˙Cx (φ (u) , Xφ(u))dPu.

Introduction
Theorem
As α tends to 0,
L
(2)
α,t
I−Ls
→ L
(2)
t :=
t
0
b
(2)
fs
dXs +
t
0
c
(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)2 ˙C2
x (θs, Xθs ) ϕ (χθs )
πa (χθs ) ϕ−1
(χθs ) − (µ∗
1,a (χθs ))2
.

Introduction
Comments
The microstructural hedging error process is not renormalized
as in the previous case.
It means that the hedging error does not vanish even if the
number of rebalancing transactions goes to inﬁnity.
If η = 1/2, the error due to the microstructure eﬀects on the
price vanishes.

Introduction
Optimal rebalancing
Total hedging error
The hedging portfolio is now rebalanced only once the price has
varied by lα ticks. The hedging error is given by
L
(3)
α,t =
t
0
˙Cx (u, Xu) dXu −
t
0
˙Cx (φ(l)
(u) , Xφ(l)(u))dPu.
with φ(l) (t) = sup{τ
(l)
i : τ
(l)
i < t} and the τ
(l)
i are stopping times
associated to moves of lα ticks.

Introduction
Optimal rebalancing
Theorem
Let lα = α−1/2. As α tends to 0, we have,
(N
(l)
α,t)1/4
L
(3)
α,t
I−Ls
→ L
(3)
t := (f
(l)
t )1/4
t
0
b
(3)
f
(l)
s
dXs +
t
0
c
(3)
f
(l)
s
dW
(3)
f
(l)
s
in D[0, T], with
b
(3)
s =
(1 − 2η)
2
˙Cx (θs, Xθs )
and
(c
(3)
s )2
=
(1 − 2η)2
4
˙C2
x (θs, Xθs ) +
1
6
¨C2
xx (θs, Xθs ) .

Introduction
Optimal rebalancing
Comments
This optimal strategy allows to reduce signiﬁcantly the
hedging error in the presence of microstructure eﬀects.
The asymptotic variance of the hedging error now depends
both on the delta and on the gamma of the derivative security.
The optimal lα is of the same order of magnitude as the
square root of the number of times where the hedging
portfolio is rebalanced.

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