Around Model Risk in Finance
Denis TALAY
INRIA Sophia Antipolis
– Joint work with J. JACOD, A. LEJAY –
– Joint work with Z. ZHENG –
– Joint work with C. BLANCHET, M. BOSSY, A. DIOP, R. GIBSON,
B. de SAPORTA, É. TANRÉ –
– Funded by NCCR FINRISK, Swiss Banking Institute, University of Zürich
April 18, 2007
1/111
Contents
I
II
An Example of Model Risk: Hedging With Misspecified Securities
A Few Reminders
3
11
III The Brownian Dimension of a Stochastic Model (Jacod, Lejay,
Talay (2006))
16
IV
V
VI
Introduction To The Statistics of Diffusions Processes
On Calibration Methods in Finance
Approximation of Quantiles of Diffusion Processes
23
52
61
2/111
VII
VIII
A Stochastic Game to Face Model Risk
When One Does Not Control Model Risk
82
85
3/111
Part I
An Example of Model Risk: Hedging
With Misspecified Securities
4/111
Our Market Model
Consider a primary asset with price process S and a saving account with price process
F defining a no-arbitrage and complete market. To the saving account corresponds
a change of numeraire . We denote by StF the price of the primary asset expressed
in this numeraire and suppose that, up to a change in probability from P to a new
probability PF , the price StF is a F-martingale which defines a complete and perfect
market in the Harrison and Kreps (1979) and Harrison and Pliska (1981) sense.
For example, within the Black and Scholes paradigm, S F is the discounted stock
price which is a martingale under the risk neutral probability.
5/111
Consider an option on the primary asset with maturity T O and payoff function φ.
At all time 0 ≤ t ≤ T O , the perfectly hedging portfolio consists of Ht0 units of the
saving account and Ht units of the primary asset:
Vt = Ht0 Ft + Ht St .
In the numeraire F this can be rewritten as
VtF = Ht0 + Ht StF .
As this means that we hold Ht0 units of an asset with the constant value 1, the
self-financing condition implies
Z t
F
F
Vt = V0 +
Hθ dSθF .
0
Suppose that the process (Ht ) is non anticipating and the preceding stochastic integral
is well defined.
6/111
Since the martingale S F is supposed to satisfy the martingale representation property, the process H is fully characterized. However, without further constraints on the
model, H can only be implicitly defined, or represented, by means of quantities such
as conditional expectations of Malliavin derivatives as in the Clark–Ocone formula,
which are not easily tractable from a numerical point of view.
Usually, traders consider a Markovian environment, that is, an observed (e.g., one
dimensional) process (ρt ) which they decide to model as a Markov process satisfying
Ft = h̄(t, ρt ) and StF = ḡ(t, ρt ), for some functions ḡ and h̄. Thus, to design the
hedging strategy, the trader considers a market governed by a probability P and a
Brownian motion W̄ and its Brownian filtration F̄, and (ρt ) is supposed to solve
dρt = β̄(t, ρt )dt + γ̄(t, ρt )dW̄t ,
for some smooth enough functions β̄, γ̄.
7/111
Models which are commonly used satisfy the following constraint: there exists a
F
probability P equivalent to P under which S F is a martingale satisfying the martingale representation property and solving
(1)
dStF =
∂ḡ
(t, ρt )γ̄(t, ρt )dW̄tF
∂x
for some a P̄F -Brownian motion W̄ F and functions ḡ and γ̄. Notice that, under
appropriate conditions on all the functions involved in the model,
dρt = β̄˜(t, ρt )dt + γ̄(t, ρt )dW̄tF ,
where the new drift coefficient β̄˜ can be expressed in terms of β̄, γ̄ and h̄.
In order to obtain a representation for (Ht ) suitable for numerical approximations,
we seek a smooth function π̄(t, x) solution to
∂ π̄
(t, x) + L̄ρt π̄(t, x) = 0
∂t
with boundary condition (remember that VTFO =
1
O
FT O φ(FT
STFO ))
φ(h̄(T O , x)ḡ(T O , x))
π̄(T , x) =
,
h̄(T O , x)
O
where L̄ρt is the infinitesimal operator of (ρt ).
8/111
Within the classical Black and Scholes paradigm we have ρt = StF = e−rt St and
π̄(t, x) = e−rt v(t, ert x), where v(t, x) is the solution of the standard Black and Scholes
PDE. More generally, if one chooses to use a model with constant interest rate and
stock price with a volatility of the type σ̄(t, St ) then ρ is chosen as S F itself.
In order to deduce H̄t , as shown in Revuz and Yor (2005), a tractable sufficient
condition to ensure that the process S F solution to (1) defines a complete market is
that
∂ḡ
∀0 ≤ t ≤ T O ,
(t, ρt )γ̄(t, ρt ) > 0 a.s.
∂x
Then
Z TO
∂ π̄
O
π̄(T , ρT O ) = π̄(0, ρ0 ) +
(θ, ρθ )γ̄(θ, ρθ )dW̄θF ,
∂x
0
and
−1
∂ π̄
∂ḡ
H̄t =
.
(t, ρt )
(t, ρt )
∂x
∂x
Thus
(2)
F
dV t = H t dStF .
9/111
Now define the model risk P&L function as
F
P &LFt = V t − VtF .
(3)
Suppose that, in the true world , the process (ρt ) is a (not necessarily Markov) Itô
process whose dynamics under PF are given by
dρt = βt dt + γt dWtF
for some adapted processes β and γ. Set
Lρt π̄(t, ρt ) := βt
(4)
1
∂ 2π
∂π
(t, ρt ) + (γt )2 2 (t, ρt ).
∂x
2
∂x
Applying Itô’s theorem to π(t, ρt ):
∂π
∂π
(t, ρt )dt + Lρ π(t, ρt )dt +
(t, ρt )γt dWtF
∂t
∂x
ρ
F
ρ
= (Lt − Lt )π̄(t, ρt )dt + dV t ,
dπ(t, ρt ) =
so that
V
F
t
−V
F
0
Z
= π(t, ρt ) − π(0, ρ0 ) +
0
t
ρ
(Lθ − Lρθ )π(θ, ρθ )dθ.
10/111
Thus we have
(5)
P &LFt
=V
F
t
−
VtF
=V
F
0
− π(0, ρ0 ) + π(t, ρt ) −
VtF
Z
t
(Lθ − Lθ )π(θ, ρθ )dθ.
+
0
At maturity T O , this equality simplifies to
(6)
P &LFT O
=V
F
TO
−
VTFO
=V
F
0
Z
− π(0, ρ0 ) +
0
TO
ρ
(Lθ − Lρθ )π(θ, ρθ )dθ.
Notice that, if (ρt ) is a Markov process, that is, if βt = β(t, ρt ) and γt = γ(t, ρt )
for some functions β and γ, then Lρt is the classical infinitesimal generator of (ρt ) and
VtF = π(t, ρt )/Ft where π(t, x) solves a PDE driven by Lρt replacing L̄ρt .
In particular, if the model error only affects the volatility term in StF , the above
representation of the P & L function reduces to an expression involving the position’s
gamma.
We finally emphasize that the above equality holds true under PF almost surely,
and therefore almost surely under the historical probability as well.
11/111
Part II
A Few Reminders
12/111
The Girsanov theorem for one dimensional S.D.E.s
Consider the S.D.E.
dXt = α(Xt )dt + dWt .
Suppose that the drift α is such that there is no explosion (i.e., Xt is finite for all
t). Then the Girsanov theorem holds true: for all T > 0, the law P X of (Xt , t ≤ T )
is absolutely continuous w.r.t. the law P W of (Wt , t ≤ T ), and the Radon–Nikodym
density is defined by
Z T
Z
dP X
1 T 2
ZT (W• (ω)) :=
(W• (ω)) = exp{
α(Ws )dWs (ω) −
α (Ws (ω))ds}.
dP W
2
0
0
We now integrate by parts to get rid of the stochastic integral.
Rx
Set A(x) := 0 α(z)dz. We have
Z
1 T 2
(α (Ws ) + α0 (Ws ))ds},
ZT (W• ) = exp{A(WT ) − A(0) −
2 0
and thus one can extend ZT on the whole space of continuous functions ξ:
Z
1 T 2
(α (ξs ) + α0 (ξs ))ds}.
ZT (ξ) = exp{A(ξT ) − A(ξ0 ) −
2 0
13/111
Time change for stochastic differential equations
Let (Yt ) be a d dimensional solution to a stochastic differential equation with coefficients b and σ. Let λ be a continuous function such that
0 < λ1 < λ(y) < λ2 , ∀y ∈ Rd
for some real numbers λ1 and λ2 . Set
Z
Λ(t) :=
t
λ(Ys )ds.
0
Theorem.
Consider the time change t → Λ−1 (t), where Λ−1 is the inverse map of Λ, and the
time changed process
Xt := YΛ−1 (t) .
Define also the time changed filtration (FtΛ ) by
FtΛ := FΛ−1 (t) .
Then
Z tp
WtΛ :=
λ(Xs )dWΛ−1 (s)
0
14/111
is a Brownian motion on the space (Ω, F, P , (FtΛ )), and one has
Z t
Z t
b(Xs )
σ(X )
p s dWsΛ P − a.s.
Xt = X0 +
ds +
λ(Xs )
0 λ(Xs )
0
15/111
Viscosity Solutions
(P)
F (t, x, v(t, x), Dt v(t, x), Dv(t, x), D2 v(t, x)) = 0
Definition.
• v is a viscosity sub-solution of (P) if
F (t̄, x̄, v(t̄, x̄), Dt ϕ(t̄, x̄), Dϕ(t̄, x̄), D2 ϕ(t̄, x̄)) ≤ 0
for all (t̄, x̄) and all functions ϕ C 1,2 such that (t̄, x̄) is a local maximum of v − ϕ.
• v is a viscosity super-solution of (P) if
F (t̄, x̄, v(t̄, x̄), Dt ϕ(t̄, x̄), Dϕ(t̄, x̄), D2 ϕ(t̄, x̄)) ≥ 0
for all (t̄, x̄) and all functions ϕ C 1,2 such that (t̄, x̄) is a local minimum of v − ϕ.
16/111
Part III
The Brownian Dimension of a Stochastic
Model (Jacod, Lejay, Talay (2006))
17/111
Statement of the problem
Consider
Z
Xt = X0 +
t
Z
as ds +
0
t
σs dWs ,
0
where
• W is a standard q–dimensional Brownian Motion,
• a is a predictable Rd –valued locally bounded process,
• σ is a d × q matrix–valued adapted and càdlàg processes.
We are in the “pure diffusion case” when σs = σ(Xs ).
Set cs := σs σs? . Thus cs = c(Xs ), where c := σσ ? in the pure diffusion case.
Aim: to estimate the maximal rank of cs on the basis of the observation of XiT /n
for i = 0, 1, . . . , n.
18/111
Linear Algebra
Let Ar be the family of all subsets of {1, . . . , d} with r elements. For all K ∈ Ar and all
d × d symmetric nonnegative matrix Σ, denote by determinantK (Σ) the determinant
of the r × r sub–matrix (Σkl : k, l ∈ K), and set
X
determinant(r; Σ) :=
determinantK (Σ).
K∈Ar
Observe that determinant(d; Σ) = determinant(Σ), and determinant(1; Σ) is the trace
of Σ.
Lemma.
The matrix Σ has eigenvalues
λ(1) ≥ . . . λ(d) ≥ 0,
and, for r = 1, . . . , d:
1
determinant(r; Σ)
d(d − 1) . . . (d − r + 1)
≤ λ(1)λ(2) . . . λ(r) ≤ determinant(r; Σ).
19/111
In addition,
1≤r≤d
=⇒
(
r ≤ rank(Σ) =⇒ determinant(r; Σ) > 0
r > rank(Σ) =⇒ determinant(r; Σ) = 0,
and
2 ≤ r ≤ d =⇒
r! determinant(r; Σ)
d!
determinant(r; Σ)
≤ λ(r) ≤
.
d! determinant(r − 1; Σ)
(r − 1)! determinant(r − 1; Σ)
20/111
An Example of Estimator
Set
Z
t
L(r)t =
λ(r)s ds.
0
Determinants are much easier to estimate than eigenvalues. So we propose to measure
the distance between X and the set of models with multiplicity r , over the time
interval [0, t], by
Z
t
L(r)t :=
determinant(r; cs ) ds.
0
Up to multiplicative constants, this more or less amounts to replace the “natural”
distance L(r)t by
Z t
L̄(r)t :=
λ(1)s . . . λ(r)s ds.
0
Set
R(ω)t := sup rank(cs (ω)).
s∈[0,t]
21/111
Discretization and Limit Theorem
We discretize by using the observations at times iT /n: Set
ζ(r)ni
=
r
X
(∆ni+j−1 X) (∆ni+j−1 X)∗ , where ∆ni X = XiT /n − X(i−1)T /n .
j=1
Then we set ([x] being the integer part of x):
L(r)nt
nr−1
:= r−1
T
r
[nt/T ]−r+1
X
determinant(r; ζ(r)ni ).
i=1
Theorem.
The variables L(r)nt converge in probability to L(r)t uniformly in t ∈ [0, T ].
The processes
V (r)nt :=
√
n (L(r)nt − L(r)t )
converge stably in law to a limiting process (V (r)t )1≤r≤d , which is defined on an
extension of the original space and is a non-homogeneous Wiener process with an
“explicit” quadratic variation process.
22/111
Estimator based on a relative threshold
Define the scale invariant estimator
n
−1/r
n (r+1)/r
Rn,t := inf r ∈ {0 . . . , d − 1} : L(r + 1)t < ρn t
(L(r)t )
.
Theorem.
For all r, r0 in {1, . . . , d}, provided P(Rt = r0 ) > 0, we have
(
1 if r 6= r0 ,
0
P(Rn,t 6= r | Rt = r ) −→
0 if r = r0 .
23/111
Part IV
Introduction To The Statistics of
Diffusions Processes
24/111
Estimators and statistical structures
Definition.
A statistical structure (or a statistical model) is defined as a measurable space (S, S)
and a collection of probability measures {Qθ , θ ∈ Θ} on that space. The set Θ of the
possible values of the parameter θ is an open set in R` .
An estimation procedure is an algorithm allowing to select one the possible values of
θ from the observation of one particular element π of S. In other terms, the algorithm
aims to select one of the probability measures Qθ in view of π.
An estimator is a measurable functional defined on (S, S). The application of an
estimator to the observation provides an estimation procedure.
25/111
A naive estimation procedure is as follows. Given an observation π, one selects
one of the θ’s for which Qθ (π) is maximum. Of course, this ‘method’ needs to be
improved when the Qθ (π)’s are identically 0 for all π in S, which is the case in the
two statistical structures below.
• The parameter θ appears in the diffusion coefficient only, that is, one is given
the drift coefficient b(x) and a family of functions {σ(θ, x), θ ∈ Θ}. We suppose
that we observe a single trajectory π := (πt ), 0 ≤ t ≤ T .
• The parameter θ appears in the drift coefficient only: one is given the drift
coefficient σ(x) and a family of functions {b(θ, x), θ ∈ Θ}. We suppose that the
observation time interval is either [0, T ] or [0, +∞). The corresponding statistical
structure is: The set S := C([0, T ], R) equipped with the Borel sigma–field BT
θ
and the collection of probability measures is {P X
T , θ ∈ Θ}
26/111
In both cases, the statistical structure is: The set S := C([0, T ], R) equipped with
θ
its Borel sigma–field BT and the collection of probability measures {P X
T , θ ∈ Θ},
θ
Xθ
where P X
on the time interval [0, T ] of the unique solution
T denotes the law P
of
Z t
Z t
Xt = X0 +
b(Xs )ds +
σ(θ, Xs )dWs .
0
0
In practice, often all the parameters of a stochastic differential equation are unknown, which means that one is given a a family of functions
{b(θ1 , x), σ(θ2 , x), (θ1 , θ2 ) ∈ Θ1 × Θ2 }.
27/111
Estimation of Constant Volatilities
Given a time interval [0, T ] and a subdivision Tin := iTn with step
quadratic variation of (Wt ) between 0 and T is defined as
VTn
:=
n−1
X
T
n,
the discrete
n
(WTi+1
− WTin )2 .
i=0
Proposition.
For all T > 0, VTn tends to T almost surely when n tends to infinity.
Proof
In view of the Borel–Cantelli lemma, it is enough to prove that
∞
X
E (VTn − T )4 < ∞.
n=1
28/111
Let (Gi ) be a sequence of independent Gaussian random variables with zero mean
and unit variance. Let αi := G2i − 1.
It holds:
4
E (VTn − T )4 =
"
T
E
n4

n
X
#4
αi
i=1

n
n
X
X

T4 
4
2
2

= 4
E αi +
E αi E αj 
n
i=1
i,j=1
i6=j
T4
= 4 nE α14 + n(n − 1)(E α12 )2
n 1
1
≤ CT 4
+
.
n2 n3
29/111
Consider the Lognormal model
1 2
Xt = log(S0 ) + µ − σ t + σWt .
2
Choose an arbitrary time interval [0, T ]. Suppose that one observes (St ) at times
iT
Tin := . Set
n
2
n−1
X
n
STi+1
n
RT :=
log
STin
i=0
=
n−1
X
n
(XTi+1
− XTin )2
i=0
=
σ2
µ−
2
2
T2
σ2
T
+2 µ−
σWT + σ 2 VTn .
n
2
n
One defines an approximation Σ̄nT of σ 2 by
Σ̄nT
RTn
:=
.
T
For all T > 0, the sequence (ΣnT ) converges almost surely to σ 2 when n tends to
infinity.
30/111
Now, the statistical structure is the space C([0, T ], R) of continuous functions
equipped with its Borel sigma–field and the family of probability measures
o
n θ
X µ,σ
X
P T := P T , θ := (µ, σ) ∈ Θ := R × R+ .
We denote by E X
T
µ,σ
µ,σ
the expectation under the probability law P X
.
T
Proposition.
For all function π in C([0, T ], R), set
( n−1
)
n 2
n
2
X
(π(Ti+1 ) − π(Ti ))
1
(π(T ) − π(0))
,
ΣnT (π) :=
−
n − Tn
n − 1 i=0
Ti+1
T
i
iT
.
n
The estimator ΣnT is unbiased , that is,
where Tin :=
µ,σ
∀(µ, σ) ∈ Θ, E X
(ΣnT ) = σ 2 .
T
2σ 2
In addition, the variance
,
− σ | of the estimation error is equal to
n−1
which is the minimal error variance within the class of the estimators of σ 2 requiring
n observations in the time interval [0, T ].
µ,σ
EX
|ΣnT
T
2 2
µ,σ
-almost all continuous map
Finally, the estimation is strongly consistent : for P X
T
π,
lim ΣnT (π) = σ 2 .
n→∞
31/111
Conclusion.
Thus, for any value of σ, if the observed path π is a trajectory of a diffusion process
then the estimator ΣnT (π) provides a good estimator of the squared volatility when n is
large enough, and this approximation becomes better and better when the frequency
of the observations increases.
Remark.
An estimator of the volatility is the square root of ΣnT (π).
Exercise.
Suppose that the model is the one dimensional stochastic differential system
Z t
Z t
(7)
Xt = X0 +
b(s, Xs )ds + θ
σ(Xs )dWs ,
0
0
where the parameter θ varies in an open set of R+ and the Brownian motion (Wt ) is
real valued. Suppose that σ(x) 6= 0 for all x. Construct an estimator of θ.
32/111
Estimation of Non Constant Volatilities
Consider
Z
Xt = X0 +
t
Z
b(s, Xs )ds +
0
t
σ(Xs )dWs ,
0
where the solution (Xt ) is Rd valued and the Brownian motion (Wt ) is Rr valued.
From the observation of a single trajectory in a fixed time interval [0, T ], we aim
to approximate the function σ(·)2 .
Let (hn ) be a sequence of real numbers tending to 0. D. Florens’ estimator is
ΣnT (π, x)
Pn−1
∗
i=0 I[kπ(iT /n)−xk<hn ] [π((i + 1)T /n) − π(iT /n)] [π((i + 1)T /n) − π(iT /n)]
:=
,
P
T n−1
I
i=0 [kπ(iT /n)−xk<hn ]
where
∗
stands for the transposition of a vector in Rd .
Theorem.
Suppose that b and σ are Lipschitz, kσ(x)σ(x)∗ k is bounded, and σ(x)σ(x)∗ is an
invertible matrix for all x ∈ Rd . Suppose that
nh4n
= +∞.
n→∞ (log(hn ))2
lim
33/111
Set
Tx (ω) := inf{t > 0; Xt (ω) = x}.
Then
∀δ > 0, lim P ω ∈ [Tx < T ]; |ΣnT (X· (ω), x) − |σ(x)|2 | > δ = 0.
n→∞
The estimator is weakly consistent .
For proofs and Limit Theorems for fluctuations, see Brugière (1991, 1993), Florens–
Zmirou (1993).
34/111
Maximum Likelihood Estimators
Preliminary remark.
We will not consider minimum contrast estimators developed by, e.g., Génon–
Catalot and Jacod (1993), or wavelet estimators, etc.
One is given an open set Θ ⊂ R` , a function b defined on Θ × R+ × R, a function σ
defined on R+ × R, a real random variable X0 and a standard Wiener process (Wt ),
and
Z t
Z t
θ
θ
Xt = X0 +
b(θ, s, Xs )ds +
σ(s, Xsθ )dWs , t ≥ 0.
0
0
Remark.
The diffusion coefficient σ does not depend on the parameter θ. This hypothesis
is necessary to the construction of maximum likelihood estimators because of the use
of Girsanov’s theorem.
35/111
The statistical structure is
θ
{C(0, T ; R), BT , {P X
T }, θ ∈ Θ}.
The construction of the maximum likelihood estimator of θ is based upon a reference probability measure for the model which is often chosen as follows. One chooses
a particular value θ0 ∈ Θ and, setting Yt := Xtθ0 , one defines the reference probability
measure as the law PTY . By definition, (Yt ) is the solution of
Z t
Z t
Yt = X0 +
b(θ0 , s, Ys )ds +
σ(s, Ys )dWs .
0
0
More generally one may choose (Yt ) as the solution of
Z t
Z t
Yt = X0 +
β(s, Ys )ds +
σ(s, Ys )dWs ,
0
0
where β is a function (not necessarily of the form b(θ0 , s, x)) chosen so that the corresponding maximum likelihood estimator has a representation so simple as possible
(for example, one can choose β ≡ 0).
One cannot choose a diffusion coefficient different from σ to define (Yt ) since Girsanov’s theorem will be used.
36/111
Definition.
Suppose that X0 and the functions bY (t, x) := β(t, x), bX (t, x) := b(θ, t, x) and
σX (t, x) := σ(x) satisfy the hypotheses of Girsanov’s Theorem for all θ in Θ. Then
θ
Y
PX
T is absolutely continuous with respect to P T . The model is said dominated by
P YT .
For all fixed continuous function π of [0, T ] to R, the (possibly empty) set of θ in
Θ where the likelihood ratio
θ
dP X
T
(π)
dP YT
reaches it maximum value, is denoted by
)
(
Xθ
dP T
(π) .
Argmax
dP YT
θ∈Θ
By definition, a likelihood estimator is a functional θ̂T from C(0, T ; R) to Θ such
that
(
)
Xθ
dP T
∀π ∈ C(0, T ; R), θ̂T (π) ∈ Argmax
(π) .
dP YT
37/111
Remark.
When the set
(
Argmax
θ∈Θ
)
θ
dP X
T
(π)
dP YT
is not reduced to a single element, the definition leads to different maximum likelihood
estimators.
From Girsanov’s theorem, for almost every π
θ
dP X
T
(π)
dP YT
is strictly positive. Therefore it holds that
(
)
θ
dP X
T
θ̂T (π) ∈ Argmax log
(π) .
dP YT
θ∈Θ
Definition. The map
θ
dP X
T
π −→ LT (π) := log
(π)
dP YT
is called the log–likelihood ratio .
38/111
In practice one can construct maximum likelihood estimators for diffusion processes
as follows. Set
b(θ, s, x) − β(s, x)
,
σ(s, x)
Z x
b(θ, s, y) − β(s, y)
Ψσ (θ, s, x) :=
dy.
|σ(s, y)|2
0
ψ(θ, s, x) :=
Then
θ
dP X
T
LT (Ys (ω), s ≤ T ) := log
(Ys (ω), s ≤ T ).
dP YT
Suppose that Ψσ (θ, ·, ·) is of class C([0, T ] × R) and use Itô’s formula. For every
path π := (Ys (ω), s ≤ T ),
Z T
∂Ψσ
(θ, s, π(s))ds
LT (π) = Ψσ (θ, T, π(T )) − Ψσ (θ, 0, π(0)) −
∂s
0
Z T
∂Ψσ
−
β(s, π(s))
(θ, s, π(s))ds
∂x
0
Z
Z
2
1 T
1 T
2 ∂ Ψσ
−
|σ(s, π(s))|
(θ, s, π(s))ds −
|ψ(θ, s, π(s))|2 ds.
2
2 0
∂x
2 0
39/111
Thus
Z
T
∂Ψ
LT (π) = Ψ(θ, T, π(T )) − Ψ(θ, 0, π(0)) −
(θ, s, π(s))ds
0 ∂s
Z
2
1 T
2∂ Ψ
−
|σ(s, π(s))|
(θ, s, π(s))ds
2 0
∂y 2
Z T
−
|σ ] (s, π(s)(ω))|2 (|b(θ, s, π(s))|2 − |β(s, π(s))|2 )ds
0
Now, for all continuous function π one seeks the values of θ(π) for which LT (π) is
maximum. The selection of a solution defines a maximum likelihood estimator of θ.
40/111
Example 1: Lognormal model.
Consider
1
Xt = log(S0 ) + (µ − σ 2 )t + σWt .
2
Suppose that the volatility σ is known. In view of an observation of the stock
price (or, equivalently, of the logarithm of the stock price), one wants to estimate the
1
parameter θ := µ − σ 2 . The statistical structure is
2
θ
{C(0, T ; R), BT , {P X
T , θ ∈ R}},
θ
where P X denotes the law of the process X θ :
Xtθ = X0 + θt + σWt .
A natural reference law is the law P YT of
Yt = X0 + σWt .
Here
θ
,
σ
θ
Ψσ (θ, s, x) = 2 x.
σ
ψ(θ, s, x) =
41/111
The likelihood ratio is
LT (π) =
θ
θ2T
(π(T
)
−
π(0))
−
,
σ2
2σ 2
since
θ
Z
1 T θ2
θ
dYs (ω) −
ds P − a.s.
2
2 0 σ2
0 σ
θ
θ2T
= exp
(YT (ω) − Y0 ) − 2 P − a.s.
σ2
2σ
dP X
T
(Y· (ω)) = exp
dP YT
Z
T
Thus
π(T ) − π(0)
T
2
is a maximum likelihood estimator of θ = µ − σ2 .
θ̂T (π) :=
42/111
θ
Proposition. For P X -almost all application π, θ̂T (π) tends to θ when T tends to
infinity: the estimation is strongly consistent .
Proof.
One has
P
Xθ
h
i
π ∈ C(0, T ; R); θ̂T (π) −→ θ
T →∞
π(T ) − π(0)
Xθ
=P
π ∈ C(0, T ; R);
−→ θ
T →∞
T
θ
XT − X0θ
=P
−→ θ
T →∞
T
WT
−→ θ ,
=P θ+σ
T T →∞
and the latter probability is 1 in view of the Iterated Logarithm Law for the Brownian
Motion:
(8)
lim sup √
T →∞
WT
= 1 P − a.s.,
2T log log T
and
(9)
lim inf √
T →∞
WT
= −1 P − a.s.
2T log log T
43/111
Example 2: Simplified Vasicek model.
Consider
Xtθ
t
Z
Xsθ ds + dWt .
=x−θ
0
Set (Yt ) := (Wt ). The corresponding log-likelihood ratio satisfies
Z T
Z
1 T 2 2
θ Ys (ω)ds
LT (Y· (ω)) = −θ
Ys dYs (ω) −
2 0
0
Z T
Z
θ2 T 2
Ws (ω)ds P − a.s.
= −θ
Ws dWs (ω) −
2 0
0
Consequently,
−
θ̂T (W· (ω)) =
R
T
0
Ws dWs (ω)
RT
Ws2 (ω)ds
T − WT2 (ω)
= RT
P − a.s.,
2 0 Ws2 (ω)ds
0
and
T − π(T )2
θ̂T (π) := R T
.
2 0 π(s)2 ds
44/111
Therefore,
RT
T − π(T )2 − 2θ 0 π(s)2 ds
θ̂T (π) − θ =
.
RT
2 0 π(s)2 ds
45/111
Convergence of the Estimators
Consider models with small volatilities . Let Θ be an interval (α, β), and let (Xtθ ())
be the one dimensional process solution of
Z t
θ
Xt () = x +
b(θ, s, Xsθ ())ds + Wt ,
0
where (Wt ) is a one dimensional Wiener process and θ belongs to Θ.
Let (Yt ()) be the solution to
Yt () = x + Wt .
Denote by θ̂T (, π) the maximum likelihood estimator defined as a value of θ which
maximizes
X θ ()
dP T
(π).
Y ()
dP T
Denote by (Xtθ (0)) the solution to the ordinary differential equation
Z t
θ
Xt (0) = x +
b(θ, s, Xsθ (0))ds,
0
and set
JTθ :=
Z
0
T
2
∂
b(θ, s, Xsθ (0)) ds.
∂θ
46/111
Theorem.
Suppose that the functions b(θ, t, x), θ → b(θ, t, x), and
addition, suppose
∂
∂θ b(θ, t, x)
are smooth. In
(i) For all θ ∈ Θ, JT (θ) > 0.
(ii) For all θ1 , θ2 ∈ Θ, if θ1 6= θ2 then
Z T
|b(θ1 , s, Ys ()) − b(θ2 , s, Ys ())|2 ds > 0.
0
Then any maximum likelihood estimator of θ in Θ is weakly consistent : δ > 0 and
for all θ ∈ Θ,
h
i
X θ ()
(10)
lim P T
π ∈ C([0, T ]; R); |θ̂T (, π) − θ| > δ = 0.
→0
In addition, for all y ∈ R and for all θ ∈ Θ,
"
#
θ
θ̂
(,
π)
−
θ
T
X ()
(11)
lim P T
π ∈ C([0, T ]; R);
≤ y = FTθ (y),
→0
where FTθ is the distribution function of the Gaussian law with zero mean and variance
equal to 1/JTθ .
The proof relies on the convergence of normalized Brownian stochastic integrals.
47/111
Theorem.
Let (Wt ) be a standard Wiener process on a filtered probability space (Ω, F, P , (Ft )).
Let (ft ) be a (Ft )-adapted process satisfying
Z T
ft2 dt < ∞ P − a.s for all T > 0,
0
and
Z
∞
ft2 dt = ∞ P − a.s.
0
Then
Rt
fu dWu
= 0 P − a.s.
lim R0 t
2 du
t→∞
f
u
0
Theorem. Let (Wt ) be a m-dimensional standard Wiener process on a filtered
probability space (Ω, F, P , (Ft )). Let (Ft ) be a (Ft )-adapted process taking values in
the set of the matrices of dimension n × m and satisfying
Z T
E
|Ftik |2 dt < ∞ for all T > 0, i = 1, . . . , n, k = 1, . . . , m.
0
Suppose that there exist constants cij such that
Z TX
m
P
−1
T
Ftik Ftjk dt −→ cij for all i, j = 1, . . . , n.
0
k=1
T →∞
48/111
Then
lim T
T →∞
−1/2
Z
T
Fs dWs = N ((0, . . . , 0), C := (cij )n,n ),
0
where N ((0, . . . , 0), C) denotes the Gaussian law on Rn of zero mean and covariance
matrix equal to C.
49/111
Cramer–Rao Lower Bounds
We thus are given an open set Θ ⊂ R and a family of functions {b(θ, ·), θ ∈ Θ} from
R to R. We suppose that, for each θ ∈ Θ, the function x ∈ R → b(θ, x) is Lipschitz.
Consider the model
(12)
Xtθ
Z
= X0 +
t
b(θ, Xsθ )ds + Wt ,
0
where (Wt ) is a standard one dimensional Brownian motion.
Remark.
Here the diffusion coefficient is identically equal to 1. This usually is not restrictive
since, other wise, one can use a time change or transform the original problem and
data by means of the function
Z x
1
dz.
φ(x) :=
σ(z)
0
50/111
Theorem.
Suppose that, for each θ ∈ Θ, the function x ∈ R → b(θ, x) is Lipschitz.
θ
θ
Let P X be the law of (Xtθ ) and let E X
T denote the expectation corresponding to
Xθ
P T . Suppose that the function
2
Z T
∂b
X θ1
(θ2 , π(s)) ds
IT (θ1 , θ2 ) := E T
∂θ
0
is strictly positive and continuous in (θ, θ) for all θ in Θ.
Under weak conditions, for all estimator θ̂T of θ based upon an observation between
times 0 and T such that the function
θ
2
QT (θ) := E X
T (θ̂T − θ)
is bounded on compact sets, the bias
θ
βT (θ) := E X
T (θ̂T − θ)
is differentiable w.r.t. Θ, and the quadratic estimation error is bounded from below:
θ
2
EX
T (θ̂T − θ) ≥
(1 + βT0 (θ))2
+ βT (θ)2 for all θ ∈ Θ.
IT (θ, θ)
The right hand side is the Cramer–Rao lower bound .
For a proof, see, e.g., Kutoyants (1984).
51/111
Example.
Consider
Xtθ = X0 + θt + σWt .
Set Ztθ :=
Xtθ
σ ,
that is,
Ztθ =
X0 θ
+ t + Wt .
σ
σ
The Cramer–Rao lower bound implies that any estimator of σθ based upon the
observation of one trajectory of (Ztθ ) in the time interval [0, T ], has a quadratic esti2
mation error larger than σT .
Suppose that the volatility is equal to 0.1 and that the instantaneous yield θ is
0.1. If the unit of time is one year and if one observes the stock prices during one
trimester, then the quadratic estimation error of any estimator of the instantaneous
yield cannot be lower than 0.04: the best possible standard deviation of the error is
around 0.2 (remember that the exact value is 0.1).
Which value of T makes the best possible standard deviation of the error equal to
0.01?
52/111
Part V
On Calibration Methods in Finance
The calibration of stochastic models is a hard task, especially in Finance .
Data: sample χ of a random vector ξ which represents market prices of products
related to the asset under consideration (e.g., forward contracts, derivatives,. . . ).
Let X be the state space of ξ. The set of calibration measures is
Pχ := Q probability on X equivalent to P , E Q [ξ] = χ .
Problem: choose an ‘optimal’ element of Pχ .
Different approaches developed by various authors:
• Use PDEs for option prices and inverse problem techniques,
• Use Dupire PDEs and optimization techniques,
• Use optimisation techniques to fit the data,
• Use entropy minimization techniques, etc.
53/111
Calibration from Dupire PDE (from Achdou and Pironneau’s
book (2005))
Let C(T, K) denote the price of a Call European option with maturity T and strike
K.
Suppose than an asset has a local volatility σ(t, x).
The Dupire PDE is
2
∂C
∂C 1 2
2∂ C
− σ (T, K)K
+r K
=0
2
∂T
2
∂K
∂K
with boundary condition
C(0, K) = (S0 − K)+ .
Given observed option prices Pj , the choice of a local volatility model may result
from a minimizing procedure involving a parameter α and weights ωj :
(
)
X
min
ωj |C(Tj , Kj ) − Pj |2 , C solution of the Dupire PDE governed by σα2 (t, x) .
α
j
54/111
Example.
Fourier decomposition:
σα2 (t, x) = σ0 +
X
Re(σ` e2iπt ) e−λ(x−ρ` ) ,
`
where σ0 , λ and ρj are suitably chosen constants.
Gradient methods.
Consider a functional J from RN to R. When J is lower semicontinuous, bounded
from below and coercive (i.e, lim|x|→∞ J(x) = +∞), J admits at least one minimum.
The minimum is unique when J is strictly convex.
Suppose that J is differentiable. Then each minimum a∗ satisfies ∇J(a∗ ) = 0. The
standard Gradient method is the induction
an+1 = an − ρn ∇J(an ),
where the weights ρn are chosen such that
J(an − ρn ∇J(an ))
is as small as possible : for example, one can solve a new minimization problem at
each step n. . .
55/111
Conjugate Gradient methods.
an+1 = an + ρn dn ,
where
dn = −∇J(an ) + γn dn−1
and ρn minimizes J(an + ρdn ).
Possible choices of γn are:
|∇J(an )|2
Fletcher–Reeves: γn :=
|∇J(an−1 )|2
Polak–Ribières: γn :=
∇J(an ) · (∇J(an ) − ∇J(an−1 ))
|∇J(an−1 )|2
Hestenes–Stiefel: γn :=
∇J(an ) · (∇J(an ) − ∇J(an−1 ))
dn−1 (∇J(an ) − ∇J(an−1 ))
56/111
Avellaneda and Samperi Approach for Volatility Calibration
(from Achdou and Pironneau’s book (2005))
Consider an asset whose volatility process (σt ) is progressively measurable and satisfies
0 < σ ≤ σt ≤ σ
for some deterministic constants σ and σ. The set of all such processes is denoted by
H.
Suppose that the market is complete and that various European options are priced
on the market, all the maturities belonging to the time interval [0, T ].
Avellaneda’s approach for cailibration volatility consists in
• Choosing a smooth and strictly convex function H defined on R+ with minimal
value 0 at at given value σ0 (resulting, e.g., from statistics based on historical
data),
• Searching the process (σt ) which solves
Z T
σ
exp(−rθ)H((σθ )2 )dθ.
sup −E
(σt )∈H
0
57/111
Denote the observed option prices by Pk , their maturities by Tk , and their payoff
functions by Φk . Then set
Z T
σ
f (σ· ) := −E
exp(−rθ)H((σθ )2 )dθ,
0
σ
gk (σ· ) := E (exp(−rTk )Φk (STk )).
One aims to solve
sup inf (f (σ· ) +
(σt )∈H µk
X
µk (gk (σ· ) − Pk )).
k
Of course, one has
X
X
sup inf (f (σ· ) +
µk (gk (σ· ) − Pk )) ≤ inf sup (f (σ· ) +
µk (gk (σ· ) − Pk )).
(σt )∈H µk
k
µk (σ )∈H
t
k
The left hand-side is the primal problem , the right hand-side is the dual problem.
The difference is the duality gap .
So far, the dual program only is well understood and solved by relying it to a
family of stochastic control problems (for fixed µ’s), and an optimisation algorithm
to minimize in µ.
58/111
On El Karoui and Hounkpatin’s Calibration Method
Consider the entropy
Z
H(Q, P ) :=
dQ
log
dQ if Q << P , + ∞ otherwise.
dP
Observe that H(Q, P ) is positive, and that H(Q, P ) = 0 iff P = Q.
Reminder on the total variation norm.
|P − Q|Var
Z dP
dQ
dR,
−
:= dR dR where
R :=
P +Q
.
2
Observe that P << R and Q << R; if Q ∼ P , then
Z dQ
|P − Q|Var = − 1 dP .
dP
In addition, if Q and P are probabilities such that Q << P , then
|P − Q|2Var ≤ 2H(Q, P ).
59/111
Theorem (Csiszar).
Let A be a convex set of probabilities on X. Suppose that A is closed for the total
variation norm topology, and
∃Q0 ∈ A, H(Q0 , P ) < ∞.
Then there exists a unique Q∗ ∈ A such that
H(Q∗ , P ) = inf H(Q, P ).
Q∈A
In addition, if
A :=
Z
Q;
fi dQ = ai for all 1 ≤ i ≤ N
for some prescribed N , ai , fi , and if ∃Q ∈ A, Q ∼ P , then
P
∗
dQ∗
exp( N
i=1 λi fi )
=R
,
P
∗ f )dP
dP
exp( N
λ
i
i=1 i
where λ∗ solves
max
λ∈RN
( N
X
i=1
Z
λi ai − log
N
X
exp(
λi fi )dP
)
.
i=1
60/111
Application of Csiszar’s theorem.
Suppose that the asset price solves
dXt = b(t, Xt )dt + σ(t, Xt )dBt .
For ξ of the form φ(XT ), set
"
h(t, x, λ) := E P
#
PN
i
exp( i=1 λi ξT ) Xt = x .
P
i)
E P exp( N
λ
ξ
i=1 i T
Under Q∗ , the dynamics of (Xt ) is
dXt = (b(t, Xt ) + σ(t, Xt )2 ∂x log h(t, Xt , λ∗ ))dt + σ(t, Xt )dBt∗ , t ≤ T,
where (Bt∗ ) is a Brownian motion under Q∗ , and λ∗ solves
( N
)
N
X
X
λi χi − log E P exp(
λi ξTi ) .
maxN
λ∈R
i=1
Numerical approximation of λ∗ , h(t, x, λ∗ ),
to Monte Carlo methods; practically?
i=1
∂h
∗
∂x (t, x, λ )
: theoretically possible owing
61/111
Part VI
Approximation of Quantiles of Diffusion
Processes
62/111
Introduction
Consider the stochastic differential equation
Z t
r Z t
X
Xt (x) = x +
Ai (s, Xs (x))dWsi ,
A0 (s, Xs (x))ds +
0
i=1
0
and the Euler scheme
n
n
n
X(p+1)T
/n (x) = XpT /n (x) + A0 (pT /n, XpT /n (x))
+
r
X
T
n
n
i
i
Ai (pT /n, XpT
/n (x))(W(p+1)T /n − WpT /n ).
i=1
Perturbed Euler scheme:
X̃Tn (x) = XTn (x) + Z n .
Here, (Ws ) is a r-dimensional Brownian motion, and the functions A0 , A1 , . . . , Ar
are smooth with bounded derivatives.
63/111
Our aim today: estimates on the approximation by the Euler scheme of the quantile
of level δ, ρ(x, δ), of the law of XTd (x).
Constraint: we do not suppose that the Malliavin covariance matrix of (Xt (x)) is
invertible.
Organization:
• Quantiles of uniformly hypoelliptic diffusions
• Quantiles of diffusions satisfying Condition (M)
• Verification of Condition (M): The partially strictly elliptic case
• Verification of Condition (M): The VaR of a portfolio
• Verification of Condition (M): a model risk problem
• A lower bound for a marginal density
64/111
Quantiles of Uniformly Hypoelliptic Diffusions
Time homogeneous coefficients:
Z t
r Z t
X
(13)
Xt (x) = x +
Ai (Xs (x))dWsi .
A0 (Xs (x))ds +
0
i=1
0
For multi-indices α = (α1 , . . . , αk ) ∈ {0, 1, . . . r}k set A∅i = Ai and, for 0 ≤ j ≤ r,
(α,j)
Ai
:= [Aj , Aαi ].
Also set
VL (x, η) :=
r
X
X
< Aαi (x), η >2 ,
i=1 |α|≤L−1
and
VL (x) := 1 ∧ inf VL (x, η).
kηk=1
65/111
Suppose
(UH) CL := inf x∈Rd VL (x) > 0 for some integer L,
and
(C) The coefficients Aji , i = 0, . . . , r, j = 1, . . . , d are of class Cb∞ (Rd ) (the Aji ’s may
be unbounded).
Let (fn ) be measurable and bounded functions. Then (cf. Bally & T.)
E fn (XT (x)) − E fn (XTn (x)) = −
and
|Cfn (T, x)| + sup |Qn (fn , T, x)| ≤
n
Cfn (T, x) Qn (fn , T, x)
+
,
n
n2
K(T )
(1 + kxkQ ) sup kfn k∞ .
q
T
n
66/111
Under (UH) and (C), the law of XT (x) has a smooth density pT (x, x0 ), so that
the d-th marginal distribution of XT (x) also has a smooth density pdT (x, y). strictly
positive at all point y in the interior of its support (cf. Nualart).
For 0 < δ < 1 set
ρ(x, δ) := inf{ρ ∈ R; P [XTd (x) ≤ ρ] = δ}
and
ρ̃n (x, δ) := inf{ρ ∈ R; P [X̃Tn,d (x) ≤ ρ] = δ}.
Theorem.
Under Conditions (UH) and (C) we have
(14)
|ρ(x, δ) − ρ̃n (x, δ)| ≤
K(T ) 1 + kxkQ 1
· d
· ,
Tq
pT (ρ(x, δ)) n
where
pdT (ρ(x, δ)) =
inf
y∈(ρ(x,δ)−1,ρ(x,δ)+1)
pdT (x, y).
67/111
The global error on the quantile is of order
!
1
1
O
,
+O
√
d
n,d
pT (ρ(x, δ))n
p̃T (x, ρ(x, δ)) N
n,d
where p̃n,d
T (x, ξ) denotes the density of X̃T (x).
d
One has (cf. Bally & T.) that p̃n,d
T (x, ξ) − pT (x, ξ) is of order 1/n. For practical
applications, one thus needs accurate estimates from below of pdT (x, ρ(x, δ)). Such
estimates are available when the generator of (Xt ) is strictly uniform elliptic (see,
e.g., Azencott) and, in the hypoelliptic case, under restrictive assumptions on b (the
generator more or less needs to be in divergence form: see Kusuoka & Stroock). At
the end we address this question in particular cases.
68/111
Diffusions Satisfying Condition (M)
We now return to general inhomogeneous stochastic differential equation. Let (Xst (x0 ), 0 ≤
s ≤ T − t) be a smooth version of the flow solution to
Z s
r Z s
X
i
Ai (t + θ, Xθt (x0 ))dWt+θ
.
Xst (x0 ) = x0 +
A0 (t + θ, Xθt (x0 ))dθ +
0
i=1
0
We denote by M (t, s, x0 ) the Malliavin covariance matrix of Xst (x0 ).
We now suppose:
(C’) The functions Aji , i = 0, . . . , r, j = 1, . . . , d are of class Cb∞ ([0, T ] × Rd ) (the
Aji ’s may be unbounded).
69/111
(M) For all p ≥ 1 there exist a non decreasing function K, a positive real number r,
and a positive Borel measurable function Ψ such that
1
≤ K(T ) Ψ(t, x0 )
M d (t, s, x0 ) sr
d
p
for all t in [0, T ) and s in (0, T − t]. In addition, Ψ satisfies: for all λ ≥ 1, there
exists a function Ψλ such that
sup E [Ψ(t, Xt (x))λ ] < Ψλ (x),
t∈[0,T ]
and
sup sup E [Ψ(t, Xtn (x))λ ] < Ψλ (x).
n>0 t∈[0,T ]
70/111
Theorem.
Let (fn ) be bounded functions of class C ∞ (R) such that
sup kfn k∞ < ∞.
n
Suppose that Conditions (M) and (C’) hold. Then
|E fn (XTd (x)) − E fn (XTn,d (x))| ≤
1
K(T )
Q
(1
+
kxk
)Ψ
(x)
sup
kf
k
·
.
λ
n
∞
Tq
n
n
Here we suppose that f is smooth as in T. & Tubaro who obtain an expansion
of the error. What is new here, and technically demanding, is the control of the
error in terms of kf k∞ . In Bally & T., the function f can be supposed bounded and
measurable only because Condition (UH) is much more restrictive than Condition
(M). This explains why an expansion might not hold under Condition (M) only.
In spite of the limitation to smooth functions f and an inequality instead of an
expansion, the theorem provides the key result to get the desired convergence rate for
the approximation of quantiles.
71/111
Under Condition (M), the d-th marginal distribution of XT (x) has a smooth density
is strictly positive at all point y in the interior of its support.
pdT (x, y)
Theorem.
Under Conditions (M) and (C’), we have
K(T ) 1 + kxkQ
1
|ρ(x, δ) − ρ̃ (x, δ)| ≤
·
·
Ψ
(x)
·
,
λ
Tq
n
pdT (ρ(x, δ))
n
where
pdT (ρ(x, δ)) =
inf
y∈(ρ(x,δ)−1,ρ(x,δ)+1)
pdT (x, y).
72/111
Let
u(t, x0 ) := E [f (XTt,d−t (x0 ))].
Key Lemma 1. It holds that
T −T /n,d
n,d
n,d
d
n
E f XT (x) − E f XT (x) = E f XT (x) − E f XT /n
(XT −T /n (x))
X
n−2
n−2
T2 X
n
kT
Rkn ,
E Φ n , X kT (x) +
+ 2
2n
n
k=0
k=0
where Φ is a sum of terms, each of them being of the form ϕ[β (t, x)∂β u(t, x), and Rkn
is a sum of terms, each of them being of the form
"
#
Z
Z
Z
(k+1)T /n
E ϕ\α (kT /n, xnkT /n (x))
kT /n
s1
s2
kT /n
kT /n
ϕ]α (s3 , Xsn3 (x))∂α u(s3 , Xsn3 (x))ds3 ds2 ds1 .
73/111
Key Lemma 2. For all multiindex α whose order w.r.t t is no more than 3, and
order w.r.t x is no more than 6, and for any smooth function g with polynomial
growth,
∀t ∈ [0, T ], |E [g(Xt (x))∂α u(t, Xt (x))]| ≤
K(T )
(1 + kxkQ )Ψλ (x)kf k∞
q
T
and
K(T )
T
, |E [g(Xtn (x))∂α u(t, Xtn (x))]| ≤
(1 + kxkQ )Ψλ (x)kf k∞ .
∀t ∈ 0, T −
q
n
T
The same statements appear in Bally & T. but here we need to take into account
the fact that Condition (M) does not allow us to control the inverse of the Malliavin
covariance matrix of (Xt (x)), which make less easy the integrations by parts.
74/111
Sketch of the proof. We observe that
h
n
o
i
t,d
n
n
n
0
E [g(Xt (x))∂α u(t, Xt (x))] = E g(Xt (x)) ∂α E f (XT −t (x )) x0 =X n (x) ,
t
and
∂α E [f (XTt,d−t (x0 ))]
=
|α|
X
h
E f
(i)
XTt,d−t (x0 )
i
Θi (T − t, x ) ,
0
i=1
where f (i) is the i-th order derivative of f , and Θi (T − t, x0 ) are sums of products of
∂β (XTt,d−t (x0 )) where |β| ≤ |α| − i + 1. Let XTt,d−t (Xtn (x)) denote the d-th component
of the image of Xtn (x) by the flow X·t at time T − t, and let Mdd (t, T − t; n, x) denote
the Malliavin covariance of XTt,d−t (Xtn (x)):
Mdd (t, T − t; n, x) :=< D(XTt,d−t (Xtn (x)), D(XTt,d−t (Xtn (x)) > .
75/111
Using standard inequalities (see, e.g., Nualart) one gets
"
#
α
h
i X
t,d
n
(i)
0
0
E f (XT −t (x ))Θi (T − t, x )
E g(Xt (x))
n
0
x =Xt (x) i=1
"
#
α
X
t,d
n
(i)
n
n
= E g(Xt (x))
f (XT −t (Xt (x)))Θi (T − t, Xt (x)) i=1
`
1
≤ K(T )(1 + kxkQ )kf k∞ M d (t, T − t; n, x) d
k
for some integers Q, k and `. As XTt,d−t (Xtn (x)) is a good approximation of XTd (x), we
can adapt the technique used in Bally & T. and make use of Condition (M).
76/111
Verification of Condition (M): The Partially Strictly Elliptic
Case
Theorem.
P
Suppose that ri=1 |Adi (t, x)|2 ≥ a > 0 for some t in [0, T ] and x in Rd . Then the
d-th marginal law of Xt (x) has a smooth density, and Condition (M) is satisfied.
Corollary.
One has
K(T ) 1 + kxkQ 1
· d
· ,
|ρ(x, δ) − ρ̃ (x, δ)| ≤
Tq
pT (ρ(x, δ)) n
n
where
pdT (ρ(x, δ)) =
inf
y∈(ρ(x,δ)−1,ρ(x,δ)+1)
pdT (x, y).
77/111
Verification of Condition (M): The VaR of a Portfolio
(
Rt
P Rt
Xtj (x) = xj + 0 σ0j (s, Xs (x))Xsj (x)ds + ri=1 0 σij (s, Xs (x))Xsj (x)dWsi , j = 1, . . . , d
Pr Pd−1 R t d
P Rt d
k
k
(x)ds
+
σ
(s,
X
(x))X
Xtd (x) = xd + d−1
s
s
k=1 0 σi,k (s, Xs (x))Xs (x)dW
i=1
k=1 0 0,k
d
Suppose that k[σi,k
(t, x0 )]ξk2 ≥ akξk2 for all ξ in Rr , and [x1 , . . . , xd−1 ] 6= 0 .
Then Condition (M) holds true if one uses the ‘modified Euler scheme’

Pr 1 j 2
T

n,j
n,j
j
n

−
)
)(pT
/n,
X
(x))
X
(x)
=
X
(x)
exp
(σ
(σ

0
i=1
pT /n
(p+1)T /n
pT /n


2 i

n

Pr

n
i
i
+ i=1 σij (pT /n, XpT
/n (x))(W(p+1)T /n − WpT /n ) , j = 1, . . . , d − 1,

Pd−1 d
T
n,d
n,d
n,k

n

X
(x)
=
X
(x)
+
σ
(pT
/n,
X
(x))X
(x)

k=1 0,k
pT /n
(p+1)T /n
pT /n
pT /n

n

Pr Pd−1 d

n,k

n
i
i
+ i=1 k=1 σi,k (pT /n, XpT /n (x))XpT /n (x)(W(p+1)T /n − WpT
/n ).
78/111
Verification of Condition (M): a Model Risk Problem
The trader wants to hedge a European option Φ(B(T O , T )). To hedge the trader
uses the bond of maturity T O and the bond of maturity T . When the HJM model is
governed by a deterministic function σ, the exact hedging strategy is
Ht =
∂πσ
(t, xt (x)),
∂x
where πσ solves

2
1 02 ∗
 ∂πσ
0
O
∗
2 ∂ πσ
(t, x ) + x (σ (t, T ) − σ (t, T ))
(t, x0 ) = 0,
02
∂t
2
∂x
π (T, x0 ) = Φ(x0 ).
σ
Suppose that the trader chooses a deterministic model structure σ(s, T ).
79/111
For suitable u1 (s), u2 (s), ϕ(s) consider
(
Rt
Rt
Xt1 (x1 )
= x1 + 0 Xs1 (x1 )u1 (s)ds + 0 Xs1 (x1 )u2 (s)dWs ,
Rt
Rt
Xt2 (x1 , x2 ) = x2 + 0 ϕ(s, Xs1 (x1 ))Xs1 (x1 )u1 (s)ds + 0 ϕ(s, Xs1 (x1 ))Xs1 (x1 )u2 (s)dWs ,
Then (see Bossy et al.) Xt2 (x1 , x2 ) is the forward value of the trader’s Profit & Loss.
If
|ϕ(t, x1 )u2 (t)| ≥ a > 0 ∀t, ∀x1 > 0,
then Condition (M) is satisfied.
80/111
A Lower Bound for a Marginal Density
Suppose that
|ϕ(t, x)| ≥ a > 0 and |u2 (t)| ≥ a > 0 ∀t ∈ [0, T O ], ∀x1 > 0.
In addition, admit technical assumptions on u1 , u2 and ϕ.
Then, for some ‘explicit’ constant K and all ρ(x, δ) > K, the density of the law of
satisfies
p2T (ρ(x, δ)) ≥ E gΛ(T ) (H−1 (ρ(x, δ)))J (ρ(x, δ)) .
XT2 (x)
81/111
In the preceding formula, g denotes the Gaussian density N (0, ), and we have set
Z t
Z z
2
Λ(t) :=
u2 (s)ds, Υ(s, z) :=
ϕ(Λ−1 (s), α)dα,
0
0
WtΛ :=
h(s, z) :=
p
Λ−1 (t)WΛ−1 (t) ,
∂Υ
1 ∂ϕ −1
(s, z) +
(Λ (s), z),
∂s
2 ∂z
H(x, z, ω) := x2 − Υ(0, x1 ) + Υ Λ(t), x1 exp U t + z
Z Λ(t) s
zs
1
Λ
Λ
−
h s, x exp U s + W̃s −
W̃
+
ds,
Λ(t) Λ(t) Λ(t)
0
and J is the Jacobian matrix of H−1 (x, ·, ω).
82/111
Part VII
A Stochastic Game to Face Model Risk
83/111
The Market Model
Consider the market

Pd
ij
j
i i
i

dt
+
[b
=
S
dS

t
t
t
j=1 σt dWt ] for 0 ≤ i ≤ n,



dPt
= Pt
Pn
i
i=1 πt
h
bit dt
+
Pd
ij
j
j=1 σt dWt
i
+ rPt 1 −
Pn
i
i=1 πt
dt.
Here {π i } = set of prescribed strategies. Consider u(·) := (b(·), σ(·)) as the market’s
control process.
The trader acts as a minimizer of the risk; on the other hand, the market systematically behaves against the interest of the trader, and acts as a maximizer of the risk.
Thus the model risk control problem can be set up as a two players (Trader versus
Market) zero-sum stochastic differential game problem. Notice that we include model
risk on volatilities, stock appreciation rates, yield curves, etc.
We aim to compute the minimal amount of money that the financial institution
needs to contain the worst possible damage.
84/111
Given a suitable function F the cost function is
J(t, x, p, Π, u(·)) := E t,x,p F (ST , PT ),
and the value function is
V (t, x, p) :=
inf
sup
J(t, x, p, Π, u(·)).
Π∈AdΠ (t) u(·)∈Adu (t)
Theorem.
Under an appropriate locally Lipschitz condition on F , the value function V (t, x, p)
is the unique viscosity solution in the space
S := {ϕ(t, x, p) is continuous on [0, T ] × Rn × R; ∃A > 0,
lim
|p|2 +x2 →∞
ϕ(t, x, p) exp(−A| log(|p|2 + x2 )|2 ) = 0 for all t ∈ [0, T ]}
to the Hamilton-Jacobi-Bellman-Isaacs equation

∂v
−
2
n+1


 ∂t (t, x, p) + H (D v(t, x, p), Dv(t, x, p), x, p) = 0 in [0, T ) × R ,



v(T, x, p) = F (x, p),
where
1
H− (A, z, x, p) := max min Tr (a(x, p, σ, π)A) + z · q(x, p, b, π) .
u∈Ku π∈Kπ 2
85/111
Part VIII
When One Does Not Control Model Risk
86/111
Introduction
In the financial industry one follows
• the fundamental approach , where strategies are based on fundamental economic
principles,
• the technical analysis approach , where strategies are based on past prices behavior,
• the mathematical approach , where strategies are based on mathematical models.
The main advantage of technical analysis is that it avoids model specification. On
the other hand, technical analysis techniques have limited theoretical justifications. It
might be useful to compare the performances obtained by using erroneously calibrated
mathematical models and the performances obtained by technical analysis techniques.
Our purpose is to present the mathematical complexity of this question, and some
preliminary results.
87/111
Mathematical Analysis of Chartists Techniques: Description
of the Setting
Here we consider the case of an asset whose instantaneous expected rate of return
changes at an unknown random time. We compare the performances of traders who
have a logarithmic utility function (!) and respectively use:
• a strategy which is optimal when the model is perfectly specified and calibrated,
• mathematical strategies for misspecified situations,
• a technical analysis technique.
The real market is described by
(
dSt0 = St0 rdt,
dSt = St µ2 + (µ1 − µ2 )I(t≤τ ) dt + σSt dBt .
88/111
One has
σ2
0
St = S exp σBt + (µ1 − )t + (µ2 − µ1 )
2
Z
t
I(τ ≤s) ds
=: S 0 exp(Rt ),
0
where the process (Rt )t≥0 is defined as
Z t
σ2
Rt = σBt + µ1 −
t + (µ2 − µ1 )
I(τ ≤s) ds.
2
0
This model was considered by Shiryaev who studied the problem of detecting the
change time τ as early and reliably as possible when one only observes the process
(St )t≥0 .
89/111
Assumptions and Notation
• FtS := σ (Su , 0 ≤ u ≤ t) is the σ algebra generated by the observations.
• (Bt )t≥0 and τ are independent.
• The change time τ follows an exponential law with parameter λ.
• The value of the portfolio at time t is denoted by Wt .
• Finally,
µ1 −
σ2
σ2
< r < µ2 − .
2
2
90/111
Moving Average Based on the Prices
Consider a regular the time partition
0 = t0 < t1 < . . . < tN = T,
tn = n∆t.
We denote by πt ∈ {0, 1} the proportion of the agent’s wealth invested in the risky
asset at time t, and by Mtδ the moving average indicator of the prices. Therefore,
Z t
1
Mtδ =
Su du.
δ t−δ
At time 0, the agent knows the history before time 0 and has enough data to compute
M0δ .
At each tn , n ∈ [1 · · · N ], the agent invests all his/her wealth into the risky asset if Stn > Mtδn . Otherwise, he/she invests all the wealth into the riskless asset.
Consequently,
πtn = I(St ≥M δ ) .
n
tn
The wealth at time tn+1 is
Wtn+1 = Wtn
!
0
S
Stn+1
t
πtn + n+1
(1 − πtn ) ,
S tn
St0n
91/111
from which (W0 = x)
WT = x
N
−1
Y
πtn exp(Rtn+1 − Rtn ) − exp(r∆t) + exp(r∆t) ,
n=0
and
σ2
(1)
E log(WT ) = log(x) + rT + µ2 −
− r T pδ
2
2
σ
1 − e−λT (2)
(1) λδ
(3)
+ ∆t µ2 −
−r
(pδ − pδ )e + pδ
2
1 − e−λ∆t
1 − e−λT (3)
−λ∆t
− ∆t(µ2 − µ1 )(e
− λ∆t)
p ,
1 − e−λ∆t δ
92/111
where
(1)
pδ
Z ∞Z
=
0
y
Z δZ
∞
2
z µ2 −3/2 − (µ2 /σ−σ/2)
2
e
2y
µ −3/2
δ
(1+z 2 )
− 2σ 2 y
i
σ 2 δ/2
2
z
σ2y
dzdy,
2
(µ2 /σ−σ/2) (δ−v) (1+z2 )
z2 2
−
− 2σ 2 y
(
)
2
2
=
I
e
z

2y2
δy ≥ +z
0
R4
y
µ1 −3/2 (µ1 /σ−σ/2)2 v (1+z 2 )
1
z1
z
z2
−
−
1
2
2
2σ y1 i 2
e
iσ2 (δ−v)/2
σ v/2
σ 2 y2
2y1
σ 2 y1
e−λv dy1 dz1 dy2 dz2 dv,
Z ∞Z ∞ µ1 −3/2 (µ1 /σ−σ/2)2 δ (1+z 2 )
z
z
−
−
(3)
2
2
2σ y i 2
e
dzdy,
pδ =
σ δ/2
2y
σ2y
0
y
Z
2
zeπ /4y ∞ −z cosh(u)−u2 /4y
e
sinh(u) sin(πu/2y)du.
iy (z) = √
π πy 0
(2)
pδ
R tThe tedious calculation involves an explicit formula, due to Yor, for the density of
( 0 exp(2Bs )ds, Bt ) (see below).
93/111
The Optimal Portfolio Allocation Strategy
Our aim is to explicit the optimal wealth and strategy of a trader who perfectly knows
the parameters µ1 , µ2 , λ and σ, and thus can get optimal financial performances. We
impose constraints: a technical analyst is only allowed to invest all his/her wealth in
the stock or the bond. Therefore the proportions of the trader’s wealth invested in
the stock are constrained to lie within the interval [0, 1].
We use the martingale approach developed by Karatzas, Shreve, Cvitanić, etc.
• The drift coefficient of the dynamics of the risky asset is not constant over time
(since it changes at the random time τ ).
• Here we must face some subtle measurability issues since the trader’s strategy
needs to be adapted with respect to the filtration generated by (St ) which, because of τ , is different from the filtration generated by (Bt ).
94/111
Let πt be the proportion of the trader’s wealth invested in the stock at time t;
denotes the corresponding wealth process. Let A(x) denote the set of admissible
strategies , that is,
W·x,π
A(x) := {π· − FtS − progressively measurable process s.t.
W0x,π = x, Wtx,π > 0 for all t > 0, π· ∈ [0, 1]}.
The value function thus is
V (x) := sup E U (WTπ ).
π· ∈A(x)
95/111
As in Karatzas-Shreve, we introduce an auxiliary unconstrained market defined as
follows. We first decompose the process R in its own filtration as
σ2
dRt = (µ1 − ) + (µ2 − µ1 )Ft dt + σdB t ,
2
where B · is the innovation process, i.e., the FtS - Brownian motion defined as
Z t
1
σ2
Bt =
Rt − (µ1 − )t − (µ2 − µ1 )
Fs ds , t ≥ 0,
σ
2
0
where F is the conditional a posteriori probability (given the observation of S) that
τ has occurred within [0, t]:
Ft := P τ ≤ t/FtS .
Let D the subset of the {FtS }− progressively measurable processes ν : [0, T ] × Ω →
R such that
Z
T
ν − (t)dt < ∞ , where ν − (t) := − inf(0, ν(t)).
E
0
The bond price process S 0 (ν) and the stock price S(ν) satisfy
Z t
0
St (ν) = 1 +
Su0 (ν)(r + ν − (u))du,
Z0 t
St (ν) = S0 +
Su (ν) (µ1 + (µ2 − µ1 )Fu + ν(u)− + ν(u))du + σdB u .
0
96/111
For each auxiliary unconstrained market driven by a process ν, the value function is
V (ν, x) :=
sup Ex U (WTπ (ν)),
π· ∈A(ν,x)
where
dWtπ (ν) = Wtπ (ν) (r + ν − (t))dt + πt (ν(t)dt + (µ2 − µ1 )Ft dt
+(µ1 − r)dt + σdB t .
97/111
Karatzas and Shreve have proven: If there exists νe such that
V (e
ν , x) = inf V (ν, x)
ν∈D
then there exists an optimal portfolio π ∗ for which the optimal wealth (for the constrained admissible strategies ) is
∗
Wt∗ = Wtπe (e
ν ).
An optimal portfolio allocation strategy is
πt∗ := σ −1
µ1 − r + (µ2 − µ1 )Ft + νe(t)
φt
+
Rt
−
σ
HtνeWt∗ e−rt− 0 νe (s)ds
!
,
98/111
where Ft satisfies
Rt
e−λs L−1
s ds
,
Ft =
Rt
ds
1 + λeλt Lt 0 e−λs L−1
s
λeλt Lt
0
and Htνe is the exponential process defined by
Z t
µ
−
r
+
ν
e
(s)
(µ
−
µ
)F
1
2
1
s
Htνe = exp −
dB s
+
σ
σ
0
2 !
Z t
µ1 − r + νe(s) (µ2 − µ1 )Fs
1
+
ds ,
−
2 0
σ
σ
and φ is a FtS adapted process which satisfies
E
RT
RT
0 −1
νe −rT − 0 νe− (t)dt
νe −rT − 0 νe− (t)dt
)
(U ) (υHT e
HT e
/ FtS
Z
=x+
t
φs dB s .
0
Here, v is the Lagrange multiplier which makes the expectation of the left hand side
equal to x for all x.
99/111
If U (·) = log(·) then
RT
−
xer(T −t)+ t νe (t)dt
,
Wt∗,x =
νe
H
t
µ
−
r
+
(µ
−
µ
)F
+
ν
e
(t)
1
2
1
t
πt∗ =
,
σ2
where
νe(t) =




 − (µ1 − r + (µ2 − µ1 )Ft )




µ1 − r + (µ2 − µ1 )Ft
if
< 0,
σ2
µ1 − r + (µ2 − µ1 )Ft
0
if
∈ [0, 1],
σ2
σ 2 − (µ1 − r + (µ2 − µ1 )Ft )
otherwise,
and, as above,
νe− (t) = − inf (0, νe(t)) .
The optimal strategies for the constrained problem are the projections on [0, 1] of
the optimal strategies for the unconstrained problem. One can obtain an (horrible)
explicit formula for the value function corresponding to the optimal strategy:
100/111
Let B be a real Brownian Motion. Let σ > 0 and ν be in R. Let V be a geometric
Brownian Motion:
2
Vs = eσ νs+σBs .
Then (cf. Yor)
Z t
2
2 2
)
z
z ν−1 − ν 2σ t − (1+z
2
2σ y i 2
P
Vs ds ∈ dy ; Vt ∈ dz =
e
dydz,
σ t
2y
σ2y
0
2
where
2
zeπ /4y
iy (z) := √
π πy
Thus the law of Φt :=
Z
∞
e−z cosh u−u
2
/4y
sinh u sin(πu/2y)du.
0
Ft
1−Ft
is explicitly known since
µ 2 − µ1
(µ2 − µ1 )2
Φt = λ exp
B̃t + λ −
t
σ
σ2
Z t
µ2 − µ 1
(µ2 − µ1 )2
exp −
×
B̃u − λ −
u du.
2
σ
σ
0
101/111
As
V (x) = V (e
ν , x)
= log(x) + rT


Z 2
 T

σ

 dt
µ
−
r
+
(µ
−
µ
)F
−
+ EP 
I
1
2
1
t


σ  − (µ − r)  
2
0
Ft >

µ − µ  
"Z 2
T
1
µ1 − r + (µ2 − µ1 )Ft
+ EP
2
σ
0


 dt ,
I
 µ − r
σ  − (µ − r)  
<Ft <
µ − µ
µ − µ 


102/111
we deduce, denoting by g(a, t) the density of
Φt
λ,
V (x) = log(x) + rT
Z T Z ∞"
σ2 
a
I σ  − µ + r 
−
+
µ1 − r + (µ2 − µ1 )


1+a
2
0
0
a>
 µ − σ + r 

#
2
1
a

I
µ
−
r
+
(µ
−
µ
)

1
2
1

 µ − r
2
σ
−
µ
+
r

σ
1+a
<a<
−
 µ −r
µ − σ  + r 

e−λt (1 + a)g(λa, t)λdadt.
103/111
A Model and Detect Strategy
We now consider the case of a trader who chooses a mathematical model and wants
to reinvest the portfolio only once, namely at the time where the change time τ is
optimally detected owing to the price history. We suppose that the reinvestment rule
is the same as the technical analyst’s one: at the detected change time from µ1 to µ2 ,
all the portfolio is reinvested in the risky asset.
We consider Karatzas’ optimal stopping rule ΘK which minimizes the expected
miss
R(Θ) := E|Θ − τ |
over all stopping rules Θ, where τ is a positive random variable. One has:
We prove: The stopping rule ΘK which minimizes the expected miss E|Θ − τ | over
all the stopping rules Θ with E(Θ) < ∞ is
Z t
∗
p
,
ΘK = inf t ≥ 0 λeλt Lt
e−λs L−1
s ds ≥
∗
1
−
p
0
104/111
where (Lt )t≥0 is the exponential likelihood-ratio process
2
1
σ
µ2 − µ1
Lt = exp
Rt − 2 (µ2 − µ1 )2 + 2(µ2 − µ1 )(µ1 − ) t ,
2
σ
2σ
2
and p∗ is the unique solution in ( 21 , 1) of the equation
Z
0
1/2
(1 − 2s)e−β/s 2−β
s ds =
(1 − s)2+β
Z
p∗
1/2
(2s − 1)e−β/s 2−β
s ds
(1 − s)2+β
with β = 2λσ 2 /(µ2 − µ1 )2 .
The value of the portfolio at maturity T is
xSθ0K
WT =
ST I(θK ≤T ) + xST0 I(θK >T ) .
Sθ K
For a logarithmic utility function, one can again exhibit an (horrible) exact formula
for E(log(WT )).
105/111
4.85
Optimal allocation
Model and detect
Technical Analyst
E[log(Wt )]
4.8
4.75
4.7
4.65
4.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time
Figure 1: Comparison
However we can numerically compare the performances of the two change time
detection strategies (one based on technical analysis, the other one based on a mathematical model), and the optimal portfolio allocation strategy. The figure illustrates
that the methods using mathematical models have better performances than the technical analyst method.
106/111
The Performances of the Strategies Based on Misspecified
Models
In practice, it is extremely difficult to know parameters exactly. If one may hope to
calibrate µ1 and σ relatively well owing to historical data, the value of µ2 cannot be
determined a priori, and data concerning τ miss.
Consider a trader who believes that the stock price is
dSt = St (µ2 + (µ1 − µ2 )It≤τ ) dt + σSt dBt ,
where the law of τ is exponential with parameter λ.
Notation.
1
1
Lt = exp
2 (µ2 − µ1 )Rt −
σ
2σ 2
Rt
−1
λeλt Lt 0 e−λs Ls ds
Ft =
.
Rt
−1
−λs
λt
1 + λe Lt 0 e Ls ds
2
σ
(µ2 − µ1 )2 + 2(µ2 − µ1 )(µ1 − ) t ,
2
107/111
On the Misspecified Optimal Allocation Strategy
The trader computes a pseudo optimal allocation by using the erroneous parameters
µ1 , µ2 , σ and τ . Thus the value of his/her misspecified optimal allocation strategy is
π ∗t = proj[0,1]
(µ1 − r + (µ2 − µ1 )F t )
,
σ2
and the corresponding wealth is
∗
Wt
rt
Z
= e exp
t
π ∗u d(e−ru Su )
.
0
Set µ1 = −0.2, σ = 0.15, r = 0.0 and λ = 2.0. Assume that they are perfectly
known by the trader. A contrario µ2 is misspecified. Its true value is µ2 = 0.2.
108/111
On Misspecified Model and Detect Strategies
The erroneous stopping rule is
Z t
K
−1
λt
Θ = inf t ≥ 0, λe Lt
e−λs Ls ds ≥
0
p∗
1 − p∗
,
1
where p∗ is the unique solution in ( , 1) of
2
Z 1/2
Z p∗
(1 − 2s)e−β/s 2−β
(2s − 1)e−β/s 2−β
s ds =
s ds,
2+β
(1 − s)2+β
0
1/2 (1 − s)
with β = 2λσ 2 /(µ2 − µ1 )2 .
The value of the corresponding portfolio is
W T = xS 0 K
Θ
ST
I(ΘK ≤T ) + xST0 I(ΘK >T ) .
SΘK
109/111
A Comparison Between Misspecified Strategies and the Technical Analysis
Technique
Our main question is: Is it better to invest according to a mathematical strategy
based on a misspecified model, or according to a strategy which does not depend on
any mathematical model?
Study case for Monte Carlo simulations:
Parameters of the model
µ1 µ 2 λ
σ
r
True values
−0.2 0.2 2 0.15 0.0
Parameters used by the trader µ1 µ2 λ
σ
r
Misspecified values (case I)
−0.3 0.1 1.0 0.25 0.0
Misspecified values (case II) −0.3 0.1 3.0 0.25 0.0
110/111
We have looked for other cases where the technical analyst is able to overperform
the misspecified optimal allocation strategies. Consider the case where the true values
of the parameters are in Table 1. Table 2 summarizes our results. It must be read as
follows. For the misspecified values µ2 = 0.1, σ = 0.25, λ = 1, if the trader chooses
µ1 in the interval (−0.5, −0.05) then the misspecified optimal strategy is worse than
the technical analyst’s one.
Table 1: True values of the parameters
Parameter
µ1
µ2
σ
λ
True Value
-0.2
0.2
0.15
2
Table 2: Misspecified values and range of the parameters
µ1
µ2
σ
λ
(-0.5,-0.05)
0.1
0.25
1
µ1
µ2
σ
λ
-0.3
(0,0.13)
0.25
1
µ1
µ2
σ
λ
-0.3
0.1
(0.2,→)
1
µ1
µ2
σ
λ
-0.3
0.1
0.25
(0,1.5)
111/111
Other numerical studies show that a single misspecified parameter is not sufficient
to allow the technical analyst to overperform the Model and Detect traders.
Astonishingly, other simulations show that the technical analyst may overperform
the misspecified optimal allocation strategy but not the misspecified model and detect
strategy.
One can also observe that, when µ2 /µ1 decreases, the performances of well specified
and misspecified model and detect strategies decrease.
112/111