Technical analyis and optimal allocation strategies in the presence - - PowerPoint PPT Presentation

Technical analyis and optimal allocation strategies in the presence of changes of instantaneous return rates and transaction costs

• C. Blanchet-Scalliet, R. Gibson Brandon, B. de Saporta, D. Talay,
• E. Tanré

INRIA Sophia Antipolis, France TOSCA Project-team

• H. PHAM’09 – 18/09/09

Outline

Introduction Our Model Main Result Numerical simulations

Outline

Introduction Our Model Main Result Numerical simulations

Technical Analysis

Why is technical analysis used?

◮ Technical analysis provides decision rules based on past prices

• behavior. It avoids model specification and thus model risk.

◮ With access to intra-daily financial data, short term traders in

pursuit of “quick trades” use chartist methods of price regime changes detection.

Few mathematical studies

◮ Pastukhov (2004): mathematical properties of volatility indicators. ◮ Shiryaev and Novikov (2008): exhibit an optimal one-time

rebalancing strategy in the Black-Scholes model when the drift term of the stock may change its value spontaneously at some random non-observable time,

◮ Blanchet et al. (2007): a framework to compare the

performances obtained by various strategies derived from erroneously calibrated mathematical models and from technical analysis; comparisons when the exact model is a diffusion model with one and only one change of stock returns at a random time.

Outline

Introduction Our Model Main Result Numerical simulations

Our model

The market: a deterministic short term rate r, a non risky asset with price process S0, and a stock with price process S whose instantaneous trend may only take two values µ1 and µ2 with µ1 < r < µ2. The changes of trend may occur at random times τ0 = 0, τn := ν1 + · · · + νn, where the νj are independent, the ν2n+1 (resp., ν2n) are i.i.d., exponential with parameter λ1 (resp., λ2). Thus the trend process is µ(θ) :=

• µ1 if τ2n ≤ θ < τ2n+1,

µ2 if τ2n+1 ≤ θ < τ2n+2, and the stock price process is dSθ Sθ = µ(θ)dθ + σdBθ.

Admissible strategies

Set:

◮ πθ: proportion of wealth invested at time θ in S, ◮ U: a utility function, ◮ W π: wealth process resulting from the strategy π.

An investment strategy (πθ) over [t, T] is said admissible if it is a piecewise constant càdlàg process taking values in {0; 1} which is progressively measurable w.r.t the filtration FS := (FS

θ , 0 ≤ θ ≤ T)

and satisfies E|U(W π

T )| < +∞.

The set of such admissible strategies is denoted by At.

The Optional Projection process is Fθ := P(µ(θ) = µ1 | FS

θ ).

Notice that Bθ := 1 σ

• log Sθ

S0 − θ

• µ1Fs + µ2(1 − Fs) − σ2

2

• ds
• is a FS Brownian motion, and that

dFθ = (−λ1Fθ + λ2(1 − Fθ))dθ + µ1 − µ2 σ Fθ(1 − Fθ)dBθ. We have: dSθ Sθ = (µ1Fθ + µ2(1 − Fθ))dθ + σdBθ, from which FS = FB .

We add proportional transaction costs .

◮ Given an amount W to transfer from the bank account to the

stock, the cost is g01W.

◮ If W is transfered from the stock to the bank account, then the

cost is g10W. Thus dW π

θ

W π

θ−

= (πθ(µ1Fθ + µ2(1 − Fθ) − r) + r) dθ + πθσdBθ − g01I∆πθ= − g10I∆πθ=−. The continuous part Z of V π satisfies dZ π

θ =

• πθ(µ1Fθ + µ2(1 − Fθ) − σ2

2 − r) + r

• dθ + πθσdBθ.

Initial conditions, value functions

Given t ∈ [0, T], i ∈ {0, 1} and π in At, let (F t,f, Z t,z,f,π, W t,x,f,i,π) be issued at time t from f ∈ [0, 1], z ∈ R, and from x > 0 if πt = i, and from x(1 − gij) if πt = j = 1 − i. Denote by ξt,i,π the purely discontinuous part of W t,x,f,i,π: ξt,i,π

θ

= − log(1 − g01)Iπt−i= − log(1 − g10)Iπt−i=− −

• t<s≤θ

[log(1 − g01)I∆πs= + log(1 − g10)I∆πs=−] we have W t,x,f,i,π

θ

= x exp(Z t,0,f,π

θ

− ξt,i,π

θ

). Now set ∀π ∈ At, Ji(t, x, f, π) := E[U(W t,x,f,i,π

T

)] and define the value functions as V i(t, x, f) := sup

π∈At

Ji(t, x, f, π).

An elementary inequality

Consider a utility function U which is, either the logarithmic utility function, or an element of the set U of the increasing and concave functions of class C1((0, +∞); R) which satisfy: U(0) = 0, and there exist real numbers C > 0 and 0 ≤ α ≤ 1 such that 0 < U′(x) ≤ C(1 + x−α) for all x > 0. Then there exists C > 0 such that, for all real numbers z, ˜ z and all positive real numbers x, ˜ x, and ζ,

• U
• xez−ζ

− U

• ˜

xe˜

z−ζ

• ≤ C
• 1 + x−αe−αz + ˜

x−αe−α˜

z

(|x − ˜ x| + (x + ˜ x) |z − ˜ z|)

• ez + e˜

z

, where α = 1 if U(x) = log(x).

Outline

Introduction Our Model Main Result Numerical simulations

A system of variational inequalities

       min

• −∂V 0

∂t − L0V 0; V 0(t, x, f) − V 1(t, x(1 − g01), f)

• = 0,

min

• −∂V 1

∂t − L1V 1; V 1(t, x, f) − V 0(t, x(1 − g10), f)

• = 0,

with the boundary condition V 0(T, x, f) = V 1(T, x, f) = U(x), where L0ϕ(t, x, f) := xr ∂ϕ ∂x (t, x, f) − (λ1f − λ2(1 − f)) ∂ϕ ∂f (t, x, f) + 1 2 µ1 − µ2 σ 2 f 2(1 − f)2 ∂2ϕ ∂f 2 (t, x, f), and L1ϕ(t, x, f) := x(µ1f + µ2(1 − f) − r)∂ϕ ∂x (t, x, f) − (λ1f − λ2(1 − f)) ∂ϕ ∂f (t, x, f + 1 2x2σ2 ∂2ϕ ∂x2 (t, x, f) + 1 2 µ1 − µ2 σ 2 f 2(1 − f)2 ∂2ϕ ∂f 2 (t, x, f) + x(µ1 − µ2)f(1 − f) ∂2ϕ ∂x∂f (t, x, f).

Definition of viscosity solutions

A pair of continuous functions (V 0, V 1) from [0, T] × (0, +∞) × [0, 1] to R is a viscosity upper solution to the above system if V 0(T, x, f) = V 1(T, x, f) = U(x) and if, for all i = j in {0, 1}, all bounded function φ of class C1,2([0, T] × R+ × [0, 1]) with bounded derivatives, and all local minimum (ˆ t, ˆ x,ˆ f) of V i − φ, one has min

• −∂φ

∂t (ˆ t, ˆ x,ˆ f) − Liφ(ˆ t, ˆ x,ˆ f); V i(ˆ t, ˆ x,ˆ f) − V j(ˆ t, ˆ x(1 − gij),ˆ f)

• ≥ 0.

A viscosity lower solution is defined analogously. Finally, a viscosity solution is both a upper and lower viscosity solution.

Theorem

Let Vα be the class of functions Υ which are continuous on [0, T] × [0, +∞) × [0, 1] and satisfy: for all (t, f) ∈ [0, T] × [0, 1], Υ(t, 0, f) = 0 and there exists C > 0 such that |Υ(t, x, f)| ≤ C(1 + x−α + x) for all (t, x, f) ∈ [0, T] × (0, +∞) × [0, 1]. Then the pair of value functions (V 0, V 1) is the unique viscosity solution of the above system in Vα satisfying V 0(T, x, f) = V 1(T, x, f) = U(x). If U is logarithmic, (V 0, V 1) is the unique viscosity solution in the set

• f function {log(x) + ¯

V(t, f)} where ¯ V is continuous on [0, T] × [0, 1].

A digression on the numerical resolution

The preceding theorem allows one to use numerical solutions to construct Markov allocation strategies ¯ π such that Ji(t, x, f, ¯ π) is close to V i(t, x, f). To implement such a strategy, the investor needs to estimate Ft at each time t from the observation of the prices (Sθ; θ ≤ t). For some smooth functions α1 and α2, dFθ = α1(Fθ)dθ + α2(Fθ)dSθ Sθ .

◮ One can discretize this equation by using, e.g., the Euler

scheme.

◮ Martinez, Rubenthaler and Tanré (2009) approximate F by a

method based on filtering theory which is more accurate than the Euler approximation.

Some references

We therefore had to solve a stochastic control problem which, to the best of our knowledge, had not been solved in the literature so far. Related works actually concern other dynamics:

◮ Tang and Yong (1993) study optimal switching and impulse

controls.

◮ Brekke and Øksendal (1994) consider optimal switching in an

economic activity.

◮ Pham (2007), Ly Vath, Pham (2007), Ly Vath, Pham, Villeneuve

(2008) obtained results on families of models which do not include our model.

Continuity of the Value Functions

Theorem: There exists C > 0 such that, for all i ∈ {0; 1}, 0 ≤ t ≤ ˆ t ≤ T, x and ˆ x in (0, +∞), f and ˆ f in [0, 1], one has

• V i(ˆ

t, ˆ x,ˆ f) − V i(t, x, f)

• ≤ C(1 + x−α + ˆ

x−α)

• |ˆ

x − x| + (x + ˆ x)(|ˆ f − f| + |ˆ t − t|1/2)

• .

Corollary: For all β ≥ α, 0 ≤ s ≤ t ≤ T, and i, x, f, for all admissible control π ∈ At, E

• V i(t, W s,x,f,i,π

t

, F s,f

t

) − V i(s, xe−ξs,i,π

t

, f)

• β

< C(t − s)β/2.

The Dynamic Programming Principle

Proposition: For all bounded continuous functions ϕ on R+, all stopping times τ such that t ≤ τ ≤ T, all x > 0, 0 ≤ f ≤ 1, π ∈ At, one has E[ϕ(W t,x,f,i,π

T

) | FS

τ ] = E[ϕ(W τ,W t,x,f,i,π

τ

,F t,f

τ ,πτ ,π|[τ,T]

T

)], Pτ − a.s. Theorem (cf. H. Pham’s book): Let Tt,T denote the set of all FS stopping times taking values in [t, T]. Then V i(t, x, f) = sup

π∈At

sup

τ∈Tt,T

E[V πτ (τ, W t,x,f,i,π

τ

, F t,f

τ )]

V i(t, x, f) = sup

π∈At

inf

τ∈Tt,T E[V πτ (τ, W t,x,f,i,π τ

, F t,f

τ )].

Sketch of the proof

The proof is divided in two parts: we first prove the upper bound the lower bound V i(t, x, f) ≥ supπ∈At supτ∈Tt,T E[V πτ (τ, W t,x,f,i,π

τ

, F t,f

τ )]

(which is less immediate to get than the upper bound). The upper bound (easy) Ji(t, x, f, π) = E[U(W t,x,f,i,π

T

)] = E

• E[U(W t,x,f,i,π

T

) | FS

t,τ]

• = EU(W

τ,W t,x,f,π

τ

,F t,f

τ ,πτ ,π|[τ,T]

T

) =

• k∈{0,1}

E

• U(W

τ,W t,x,f,i,π

τ

,F t,f

τ ,k,π|[τ,T]

T

)Iπτ =k

• =
• k∈{0,1}

E

• Jk(τ, W t,x,f,i,π

τ

, F t,f

τ , π|[τ,T])Iπτ =k

• ≤
• k∈{0,1}

E

• V k(τ, W t,x,f,i,π

τ

, F t,f

τ )Iπτ =k

• ≤ E[V πτ (τ, W t,x,f,π

τ

, F t,f

τ )].

It remains to take the infimum w.r.t τ and then the supremum w.r.t. π.

The lower bound (less easy) As the value functions are continuous, for all ε > 0, one can find a countable partition of [t, T] × (0, +∞) × [0, 1] with Borel subsets Bp such that, for all p, all (t, x, f) and (ˆ t, ˆ x,ˆ f) in Bp, all i, |V i(t, x, f) − V i(ˆ t, ˆ x,ˆ f)| ≤ ε. In addition, if t ≤ ˆ t, for all i, π in At such that πθ = i for θ ∈ [t,ˆ t), one has also for all p, all (t, x, f) and (ˆ t, ˆ x,ˆ f) in Bp, |Ji(t, x, f, π) − Ji(ˆ t, ˆ x,ˆ f, π)| ≤ ε. Set ρ := sup

π∈At

sup

τ∈Tt,T

E[V πτ (τ, W t,x,f,i,π

τ

, F t,f

τ )]

and choose π in At and τ in Tt,T such that ρ ≤ ε + E[V πτ (τ, W t,x,f,i,π

τ

, F t,f

τ )].

Now, for all p choose a triple (tp, xp, fp) in the closure ¯ Bp of Bp, where tp is the largest time in the trace of ¯ Bp in [t, T]. One can prove ρ ≤ 2ε + E  

∞

• p=0

V πτ (tp, xp, fp) I(τ,W t,x,f,i,π

τ

,F t,f

τ

)∈Bp

  . It then remains to construct a suitable admissible control ˆ π leading to ρ ≤ 4ε + E

• U(W t,x,f,i,ˆ

π T

)

• = 4ε + Ji(t, x, f, ˆ

π) ≤ 4ε + V i(t, x, f),

Existence of a viscosity upper solution

The pair (V 0, V 1) of value functions is a upper viscosity solution. Fix i ∈ {0; 1}. Let (ˆ t, ˆ x,ˆ f) be a local minimum of V i − ϕ, where ϕ is a function of class C1,2 defined on a neighborhood [ˆ t,ˆ t + ε] × Bε of (ˆ t, ˆ x,ˆ f), where Bε := ˆ x(1 + ε)−1, ˆ x(1 + ε)

• ×
• [ˆ

f − ε,ˆ f + ε] ∩ [0, 1]

• ,

and such that V i(ˆ t, ˆ x,ˆ f) = ϕ(ˆ t, ˆ x,ˆ f). For all controls π ∈ Aˆ

t,

W

ˆ t,(1−gij)ˆ x,ˆ f,j,π ˆ t

≤ W

ˆ t,ˆ x,ˆ f,i,π ˆ t

. Then, for all θ ≥ ˆ t, W

ˆ t,(1−gij)ˆ x,ˆ f,j,π θ

≤ W

ˆ t,ˆ x,ˆ f,i,π θ

. Therefore EU(W

ˆ t,(1−gij)ˆ x,ˆ f,j,π θ

) ≤ EU(W

ˆ t,ˆ x,ˆ f,i,π θ

), V j(ˆ t, ˆ x(1 − gij),ˆ f) ≤ V i(ˆ t, ˆ x,ˆ f).

We now aim to prove − ∂ϕ ∂t + Liϕ

• (ˆ

t, ˆ x,ˆ f) ≥ 0. Fix 0 < h < ε and choose a control π which takes the value i on the time interval [ˆ t,ˆ t + h]. The Dynamic Programming Principle with the constant stopping time ˆ t + h and Itô’s formula to ϕ(t, Wt, Ft) lead to 0 ≥ 1 hE ˆ

t+h ˆ t

∂ϕ ∂t + Liϕ

• (s, W

ˆ t,ˆ x,ˆ f,i,π s

, F

ˆ t,ˆ f s )ds

• + 1

hE

• (Vi − ϕ)(ˆ

t + h, W

ˆ t,ˆ x,ˆ f,i,π ˆ t+h

, F

ˆ t,ˆ f ˆ t+h)I(W

ˆ t, ˆ x, ˆ f,i,π ˆ t+h

,F

ˆ t, ˆ f ˆ t+h)/

∈Bε

• .

It then remains to let h tend to 0.

Existence of a viscosity lower solution

Suppose that (V 0, V 1) is not a viscosity lower solution. Then there exist i ∈ {0; 1}, a smooth function ϕ with bounded derivatives, ǫ > 0, a local maximum (ˆ t, ˆ x,ˆ f) of V i − ϕ on [ˆ t,ˆ t + ε) × Bε, and γ > 0 such that, for all t, x, f in [ˆ t,ˆ t + ε) × Bε, − ∂ϕ ∂t + Liϕ

• (t, x, f) ≥ γ,

V i(t, x, f) − V j(t, x(1 − gij), f) ≥ γ, j = i. One exhibits a contradiction by using the Dynamic Programming Principle with suitable stopping times (tricky step).

Uniqueness of the Viscosity Solution

Suppose that Υ and ψ are two viscosity solutions in Vα. Then there would exist (ˆ i,ˆ t, ˆ x,ˆ f) in {0; 1}×]0, T)×]0, +∞) × [0, 1] such that η := Υ

ˆ i(ˆ

t, ˆ x,ˆ f) − ψ

ˆ i(ˆ

t, ˆ x,ˆ f) > 0. As Υ and ψ are null and continuous at x = 0, there also would exist m > 0 such that, for all i, t, f, f and x, x′ ≤ m, |Υi(t, x, f) − ψi(t, x′, f ′)| < η 5. In addition, as Υ and ψ are in Vα, there exists C > 0 such that, for all i, t, x, f, |Υi(t, x, f)| + |ψi(t, x, f)| ≤ C(1 + x).

Define the functions Φ0 and Φ1 on [0, T] × [0, +∞)2 × [0, 1]2 by Φi(t, x, x′, f, f ′) := Υi(t, x, f) − ψi(t, x′, f ′) − νe−Dt(x2 + x′2) − 1 2ε(|x − x′|2 + |f − f ′|2) + βt − λ t , where ε > 0, D > (|µ1| + |µ2| + r + 3σ2); 0 < λ < ηˆ tT 5(T − ˆ t) ; 0 < β < η 5(T − ˆ t) ; 0 < ν < min

• CeDT, ηeDˆ

t

10ˆ x2

• .

Set H := Φ

ˆ i(ˆ

t, ˆ x, ˆ x,ˆ f,ˆ f) = η − 2νe−Dˆ

t ˆ

x2 + βˆ t − λ ˆ t .

One has |xε − x′

ε|2 + |fε − f ′ ε|2 ≤ 2ε(2C(1 + M) + βT − H + 1).

Define the function Ψ on [0, T] × [0, +∞)2 × [0, 1]2 as follows: Ψ(t, x, x′, f, f ′) = νe−Dt(x2 + x′2) + 1 2ε(|x − x′|2 + |f − f ′|2) − βt + λ t . Notice that Φi(t, x, x′, f, f ′) = Υi(t, x, f) − ψi(t, x′, f ′) − Ψ(t, x, x′, f, f ′). For all ε > 0, Ishii’s lemma implies that there exist two real numbers d and d′ and two symmetric matrices X and X ′ such that

• d,

∂Ψ ∂x , ∂Ψ ∂f

• (tε, xε, x′

ε, fε, f ′ ε), X

• ∈ P2+Υiε(tε, xε, fε)

−

• d′,

∂Ψ ∂x′ , ∂Ψ ∂f ′

• (tε, xε, x′

ε, fε, f ′ ε), X ′

• ∈ P2−ψiε(tε, x′

ε, f ′ ε),

d + d′ = ∂Ψ ∂t (tε, xε, x′

ε, fε, f ′ ε),

− 1 ε + A

• I ≤

X X ′

• ≤ A + εA2,

where A is the Hessian matrix of Φ at x, x′, f, f ′. Then, lengthy calculations. . .

Outline

Introduction Our Model Main Result Numerical simulations

Numerical approximation of the value functions

The approximation (ˆ V 0, ˆ V 1) is defined as follows:

◮ Set ˆ

V 0(T, ·) = ˆ V 1(T, ·) = 1.

◮ At each time step:

◮ Set ¯

V 0(t, ·) = ˆ V 0(t, ·) and ¯ V 1(t, ·) = ˆ V 1(t, ·).

◮ Compute ¯

V i(t − δt, f) solution of L

i ¯

V i(t, f) = 0, where L

i is the

finite difference approximation of Li.

◮ Set ˆ

V 0(t − δt, f) = max{¯ V 0(t − δt, f); (1 − g01)α ¯ V 1(t − δt, f)}, and ˆ V 1(t − δt, f) = max{¯ V 1(t − δt, f); (1 − g10)α ¯ V 0(t − δt, f)}.

One can easily show that (ˆ V 0, ˆ V 1) is the unique solution of      min

• − L

0ϕ0(t, f); ϕ0(t − δt, f) − (1 − g01)αϕ1(t − δt, f)

• = 0,

min

• − L

1ϕ1(t, f); ϕ1(t − δt, f) − (1 − g10)αϕ0(t − δt, f)

• = 0,

ˆ V 0(T, ·) = ˆ V 1(T, ·) = 1. So far, we have not proven the convergence of this scheme to the value functions (V 0, V 1). Key results in that direction are in Krylov (2000), Barles and Jacobsen (2002), Bonnans (2007). For artificial boundary conditions: see Huang and T. (2009).

Numerical results

For U(x) = √x (α = 1/2), T = 3, µ1 = −0.2, µ2 = 0.21, λ1 = λ2 = 2, σ = 0.15, g01 = g10 = 0.01, r = 0 and the discretization steps δt = 10−6 and δf = 10−3:

0.0 0.5 1.0 1.5time 1.01 1.02 1.03 1.04 2.0 0 0.20.4 2.5 0.6 f 0.8 3.0 1

Figure: Approximate value function ˆ V 0

1.004 1.008 1.012 1.016 0.1 0.2 0.3 0.4 2.5 0.5 f 2.6 0.6 2.7 0.7 2.8 0.8 time 2.9 0.9 3.0 1.0

Figure: Zoom of ˆ V 0 close to the horizon

The next figure shows ˆ V 0(t, 0.05) for 0 ≤ t ≤ 3. It illustrates that the time derivative is discontinuous.

time 3.0 2.5 2.0 1.5 1.0 0.5 0.0 1.045 1.040 1.035 1.030 1.025 1.020 1.015 1.010 1.005 1.000 0.995 Value function

Figure: Smoothness in time of ˆ V 0

f 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1.0030 1.0025 1.0020 1.0015 1.0010 1.0005 1.0000 0.9995 Value function Figure: Smoothness ˆ V 0(t, ·) for t = 2.9 (higher curve), t = 2.91, t = 2.92,

Efficient strategy

Here µ2 > µ1, hence the investor should invest in the stock when µ(t) = µ2, i.e. when f is close to 0, and sell when µ(t) = µ1, i.e. when f is close to 1. We propose the following efficient strategy suggested by the discrete Dynamic Programming Principle .

◮ Compute (ˆ

V 0, ˆ V 1) for all t and f in the grid.

◮ At time t in the grid, compute an estimate ˆ

Ft of Ft from the

• bservation of the stock (using Martinez, Runethaler and Tanré’s

method).

◮ Buy if ˆ

V 0(t, ˆ Ft) = (1 − g01)α ˆ V 1(t, ˆ Ft), sell if ˆ V 1(t, ˆ Ft) = (1 − g10)α ˆ V 0(t, ˆ Ft).

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 buying zone selling zone Estimated F

Figure: The efficient strategy

The figure shows the buying area (lower area) where ˆ V 0 = (1 − g01)α ˆ V 1 which means that F is close enough to 0 to buy the stock, and the selling area (upper area) where ˆ V 1 = (1 − g10)α ˆ V 0 where F is close enough to 1 to sell the stock. The last area is a no-transaction area : it means that the investor has to keep his/her

• position. This area is due to the transaction costs.

We also plot also the process ˆ Ft estimated from the stock. At time t = 0, ˆ F0 ≃ 0.2: the investor buys the stock. At time t = 0.64, ˆ F enters the selling zone, so he/she invests in the bond. At time t = 1.24, the process ˆ F reenters the buying zone, etc. Note that all transactions should stop at a certain time before the time horizon T. This is due to the transaction costs: there is not enough time left to regain the price of the transaction. Far from the horizon, we can see that, approximately, ˆ Ft is small enough to buy when ˆ Ft ≤ 0.3 and is large enough to sell when ˆ Ft ≥ 0.7.

Monte Carlo simulations

Monte Carlo simulations allow us to approximate the expected utility

• f the wealth issued from the efficient strategy . We compare the

result to the approximate value function .

F0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ˆ V 0 1.061 1.057 1.053 1.049 1.045 1.043 1.041 1.039 1.038 1.037 ˜ V 0 1.061 1.056 1.052 1.049 1.045 1.043 1.040 1.039 1.038 1.037

Table: Optimality of the efficient strategy

Misspecifications

Our last result illustrates the critical effect of calibration. We compare two strategies:

◮ The misspecified strategy is the efficient strategy with

miscalibrated coefficients: the stock follows the set of parameters mentionned above: µ1 = −0.2, µ2 = 0.21, λ1 = λ2 = 2, σ = 0.15, but the agent computes the approximation of the value functions and the approximation of Ft with a set of different (misspecified) coefficients.

◮ The moving average strategy of technical analysis (with

windowing size δ = 0.8). The trader estimates the moving average of the prices M(δ)

T

:= 1 δ t

t−δ

Sudu, and he/she invests in S iff St ≥ M(δ)

t

.

Benchmarks: the efficient strategy (upper curve) with the right parameters and the buy and hold strategy (lower curve). For each strategy, we compute the utility of the corresponding wealth at each time and run 105 Monte Carlo simulations. Here, g01 = g10 = 0.005 and the misspecified parameters are: µ1 = −0.2, µ2 = 0.21, σ = 0.3, λ1 = 0.5, λ2 = 1. The miscalibrated strategy is better than the moving average one.

1.035 2.5 2.0 1.5 1.0 0.5 0.0 3.0 1.025 1.020 1.015 1.010 1.000 0.995 1.030 1.005 1.040 Wealth Time Buy and Hold Optimal Moving Average Misspecified

Figure: Comparison of strategies 1

The next figure shows another set of miscalibrated parameters: µ1 = −0.3, µ2 = 0.17, σ = 0.3, λ1 = 2, λ2 = 2. Here the misscalibration mainly concerns the trends and the volatility. The moving average strategy is now better than the miscalibrated one.

2.5 2.0 1.5 1.0 0.5 0.0 1.040 1.035 1.030 1.025 1.020 1.015 1.010 1.005 1.000 0.995 3.0 Optimal Misspecified Buy and Hold Time Wealth Moving Average

Figure: Comparison of strategies 2

Further mathematical studies are necessary to understand these misspecification effects.

Acknowledgement

This research has been carried out within the NCCR FINRISK project

• n “Credit Risk and Non-Standard Sources of Risk in Finance”.

Financial support by the National Centre of Competence in Research Financial valuation and Risk Management (NCCR FINRISK) is gratefully acknowledged. NCCR-FINRISK is a research program supported by the Swiss National Science Foundation.