1.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
The size of the largest ﬂuctuations in a ﬁnancial
market model with Markovian switching
Terry Lynch1
(joint work with J. Appleby1 , X. Mao2 and H. Wu1 )
1 Dublin City University, Ireland.
2 Strathclyde University, Glasgow, U.K.
Christmas Talk
Dublin City University
Dec 14th 2007
Supported by the Irish Research Council for Science, Engineering and Technology
2.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline
1 Introduction
2 Mathematical Results
3 Application to Financial Market Models
4 Extensions and future work
3.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline
1 Introduction
2 Mathematical Results
3 Application to Financial Market Models
4 Extensions and future work
4.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Introduction
We consider the size of the large ﬂuctuations of a stochastic
diﬀerential equation (S.D.E) with Markovian switching,
concentrating on processes which obey the law of the iterated
logarithm:
|X (t)|
lim sup √ = c a.s.
t→∞ 2t log log t
The results are applied to ﬁnancial market models which are
subject to random regime shifts (conﬁdent to nervous) and to
changes in market sentiment (investor intuition).
Markovian switching: parameters can switch according to a
Markov jump process
We show that our security model exhibits the same long-run
growth and deviation properties as conventional geometric
Brownian motion.
5.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Introduction
We consider the size of the large ﬂuctuations of a stochastic
diﬀerential equation (S.D.E) with Markovian switching,
concentrating on processes which obey the law of the iterated
logarithm:
|X (t)|
lim sup √ = c a.s.
t→∞ 2t log log t
The results are applied to ﬁnancial market models which are
subject to random regime shifts (conﬁdent to nervous) and to
changes in market sentiment (investor intuition).
Markovian switching: parameters can switch according to a
Markov jump process
We show that our security model exhibits the same long-run
growth and deviation properties as conventional geometric
Brownian motion.
6.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Introduction
We consider the size of the large ﬂuctuations of a stochastic
diﬀerential equation (S.D.E) with Markovian switching,
concentrating on processes which obey the law of the iterated
logarithm:
|X (t)|
lim sup √ = c a.s.
t→∞ 2t log log t
The results are applied to ﬁnancial market models which are
subject to random regime shifts (conﬁdent to nervous) and to
changes in market sentiment (investor intuition).
Markovian switching: parameters can switch according to a
Markov jump process
We show that our security model exhibits the same long-run
growth and deviation properties as conventional geometric
Brownian motion.
7.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Introduction
We consider the size of the large ﬂuctuations of a stochastic
diﬀerential equation (S.D.E) with Markovian switching,
concentrating on processes which obey the law of the iterated
logarithm:
|X (t)|
lim sup √ = c a.s.
t→∞ 2t log log t
The results are applied to ﬁnancial market models which are
subject to random regime shifts (conﬁdent to nervous) and to
changes in market sentiment (investor intuition).
Markovian switching: parameters can switch according to a
Markov jump process
We show that our security model exhibits the same long-run
growth and deviation properties as conventional geometric
Brownian motion.
8.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Classical geometric Brownian motion (GBM)
Characterised as the unique solution of the linear S.D.E
dS∗ (t) = µS∗ (t) dt + σS∗ (t) dB(t)
where µ is the instantaneous mean rate of growth of the price,
σ its instantaneous volatility and S∗ (0) > 0.
S∗ grows exponentially according to
log S∗ (t) 1
lim = µ − σ2, a.s. (1)
t→∞ t 2
The maximum size of the large deviations from this growth
trend obey the law of the iterated logarithm
| log S∗ (t) − (µ − 1 σ 2 )t|
2
lim sup √ = σ, a.s. (2)
t→∞ 2t log log t
9.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Classical geometric Brownian motion (GBM)
Characterised as the unique solution of the linear S.D.E
dS∗ (t) = µS∗ (t) dt + σS∗ (t) dB(t)
where µ is the instantaneous mean rate of growth of the price,
σ its instantaneous volatility and S∗ (0) > 0.
S∗ grows exponentially according to
log S∗ (t) 1
lim = µ − σ2, a.s. (1)
t→∞ t 2
The maximum size of the large deviations from this growth
trend obey the law of the iterated logarithm
| log S∗ (t) − (µ − 1 σ 2 )t|
2
lim sup √ = σ, a.s. (2)
t→∞ 2t log log t
10.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Classical geometric Brownian motion (GBM)
Characterised as the unique solution of the linear S.D.E
dS∗ (t) = µS∗ (t) dt + σS∗ (t) dB(t)
where µ is the instantaneous mean rate of growth of the price,
σ its instantaneous volatility and S∗ (0) > 0.
S∗ grows exponentially according to
log S∗ (t) 1
lim = µ − σ2, a.s. (1)
t→∞ t 2
The maximum size of the large deviations from this growth
trend obey the law of the iterated logarithm
| log S∗ (t) − (µ − 1 σ 2 )t|
2
lim sup √ = σ, a.s. (2)
t→∞ 2t log log t
11.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline
1 Introduction
2 Mathematical Results
3 Application to Financial Market Models
4 Extensions and future work
12.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
S.D.E with Markovian switching
We study the stochastic diﬀerential equation with Markovian
switching
dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t) (3)
where
X (0) = x0 ,
f , g : R × S × [0, ∞) → R are continuous functions obeying
local Lipschitz continuity and linear growth conditions,
Y is an irreducible continuous-time Markov chain with ﬁnite
state space S and is independent of the Brownian motion B.
Under these conditions, existence and uniqueness is guaranteed.
13.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
S.D.E with Markovian switching
We study the stochastic diﬀerential equation with Markovian
switching
dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t) (3)
where
X (0) = x0 ,
f , g : R × S × [0, ∞) → R are continuous functions obeying
local Lipschitz continuity and linear growth conditions,
Y is an irreducible continuous-time Markov chain with ﬁnite
state space S and is independent of the Brownian motion B.
Under these conditions, existence and uniqueness is guaranteed.
14.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
S.D.E with Markovian switching
We study the stochastic diﬀerential equation with Markovian
switching
dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t) (3)
where
X (0) = x0 ,
f , g : R × S × [0, ∞) → R are continuous functions obeying
local Lipschitz continuity and linear growth conditions,
Y is an irreducible continuous-time Markov chain with ﬁnite
state space S and is independent of the Brownian motion B.
Under these conditions, existence and uniqueness is guaranteed.
15.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Main results
Theorem 1
Let X be the unique continuous solution satisfying
dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t).
If there exist ρ > 0 and real numbers K1 and K2 such that
∀ (x, y , t) ∈ R × S × [0, ∞)
xf (x, y , t) ≤ ρ, 0 < K2 ≤ g 2 (x, y , t) ≤ K1
then X satisﬁes
|X (t)|
lim sup √ ≤ K1 , a.s.
t→∞ 2t log log t
16.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Main results
Theorem 1
Let X be the unique continuous solution satisfying
dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t).
If there exist ρ > 0 and real numbers K1 and K2 such that
∀ (x, y , t) ∈ R × S × [0, ∞)
xf (x, y , t) ≤ ρ, 0 < K2 ≤ g 2 (x, y , t) ≤ K1
then X satisﬁes
|X (t)|
lim sup √ ≤ K1 , a.s.
t→∞ 2t log log t
17.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Main results
Theorem 1
Let X be the unique continuous solution satisfying
dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t).
If there exist ρ > 0 and real numbers K1 and K2 such that
∀ (x, y , t) ∈ R × S × [0, ∞)
xf (x, y , t) ≤ ρ, 0 < K2 ≤ g 2 (x, y , t) ≤ K1
then X satisﬁes
|X (t)|
lim sup √ ≤ K1 , a.s.
t→∞ 2t log log t
18.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Main results
Theorem 1 continued
If, moreover, there exists an L ∈ R such that
xf (x, y , t) 1
inf 2 (x, y , t)
=: L > −
(x,y ,t)∈R×S×[0,∞) g 2
Then X satisﬁes
|X (t)|
lim sup √ ≥ K2 , a.s.
t→∞ 2t log log t
19.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline of proof
The proofs rely on
time-change and stochastic comparison arguments,
constructing upper and lower bounds on |X | which, under an
appropriate change of time and scale, are stationary processes
whose dynamics are not determined by Y .
The large deviations of these processes are determined by
means of a classical theorem of Motoo.
20.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline of proof
The proofs rely on
time-change and stochastic comparison arguments,
constructing upper and lower bounds on |X | which, under an
appropriate change of time and scale, are stationary processes
whose dynamics are not determined by Y .
The large deviations of these processes are determined by
means of a classical theorem of Motoo.
21.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline of proof
The proofs rely on
time-change and stochastic comparison arguments,
constructing upper and lower bounds on |X | which, under an
appropriate change of time and scale, are stationary processes
whose dynamics are not determined by Y .
The large deviations of these processes are determined by
means of a classical theorem of Motoo.
22.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline of proof
The proofs rely on
time-change and stochastic comparison arguments,
constructing upper and lower bounds on |X | which, under an
appropriate change of time and scale, are stationary processes
whose dynamics are not determined by Y .
The large deviations of these processes are determined by
means of a classical theorem of Motoo.
23.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline
1 Introduction
2 Mathematical Results
3 Application to Financial Market Models
4 Extensions and future work
24.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Security Price Model
The large deviation results are now applied to a security price
model, where the security price S obeys
dS(t) = µS(t)dt + S(t)dX (t), t≥0
Here we consider the special case
dX (t) = f (X (t), Y (t), t) dt + γ(Y (t)) dB(t), t≥0
with X (0) = 0 and γ : S → R {0}.
Under the conditions
xf (x, y , t)
sup ≤ ρ and
(x,y ,t)∈R×S×[0,∞) γ 2 (y )
xf (x, y , t) 1
inf 2 (y )
>− ,
(x,y ,t)∈R×S×[0,∞) γ 2
25.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Security Price Model
The large deviation results are now applied to a security price
model, where the security price S obeys
dS(t) = µS(t)dt + S(t)dX (t), t≥0
Here we consider the special case
dX (t) = f (X (t), Y (t), t) dt + γ(Y (t)) dB(t), t≥0
with X (0) = 0 and γ : S → R {0}.
Under the conditions
xf (x, y , t)
sup ≤ ρ and
(x,y ,t)∈R×S×[0,∞) γ 2 (y )
xf (x, y , t) 1
inf 2 (y )
>− ,
(x,y ,t)∈R×S×[0,∞) γ 2
26.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Security Price Model
The large deviation results are now applied to a security price
model, where the security price S obeys
dS(t) = µS(t)dt + S(t)dX (t), t≥0
Here we consider the special case
dX (t) = f (X (t), Y (t), t) dt + γ(Y (t)) dB(t), t≥0
with X (0) = 0 and γ : S → R {0}.
Under the conditions
xf (x, y , t)
sup ≤ ρ and
(x,y ,t)∈R×S×[0,∞) γ 2 (y )
xf (x, y , t) 1
inf 2 (y )
>− ,
(x,y ,t)∈R×S×[0,∞) γ 2
27.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Fluctuations in the Markovian switching model
Theorem 2
Let S be the unique continuous process governed by
dS(t) = µS(t)dt + S(t)dX (t), t≥0
with S(0) = s0 > 0, where X is deﬁned on the previous slide.
Then:
1
log S(t) 1 2
lim = µ − σ∗ , a.s.
t→∞ t 2
2
t
| log S(t) − (µt − 1 0 γ 2 (Y (s))ds)|
2
lim sup √ = σ∗
t→∞ 2t log log t
where σ∗ = j∈S γ 2 (j)πj , and π is the stationary probability
2
distribution of Y .
28.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Fluctuations in the Markovian switching model
Theorem 2
Let S be the unique continuous process governed by
dS(t) = µS(t)dt + S(t)dX (t), t≥0
with S(0) = s0 > 0, where X is deﬁned on the previous slide.
Then:
1
log S(t) 1 2
lim = µ − σ∗ , a.s.
t→∞ t 2
2
t
| log S(t) − (µt − 1 0 γ 2 (Y (s))ds)|
2
lim sup √ = σ∗
t→∞ 2t log log t
where σ∗ = j∈S γ 2 (j)πj , and π is the stationary probability
2
distribution of Y .
29.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Fluctuations in the Markovian switching model
Theorem 2
Let S be the unique continuous process governed by
dS(t) = µS(t)dt + S(t)dX (t), t≥0
with S(0) = s0 > 0, where X is deﬁned on the previous slide.
Then:
1
log S(t) 1 2
lim = µ − σ∗ , a.s.
t→∞ t 2
2
t
| log S(t) − (µt − 1 0 γ 2 (Y (s))ds)|
2
lim sup √ = σ∗
t→∞ 2t log log t
where σ∗ = j∈S γ 2 (j)πj , and π is the stationary probability
2
distribution of Y .
30.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Comments
Despite the presence of the Markov process Y (which
introduces regime shifts), we have shown that the new market
model obeys the same long-term properties of standard
geometric Brownian motion models.
However, the analysis is now more complicated because the
increments are neither stationary nor Gaussian, and it is not
possible to obtain an explicit solution.
The incorporation of investor sentiment into the model in this
manner is one of the important motivations behind the
discipline of behavioural ﬁnance.
31.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Comments
Despite the presence of the Markov process Y (which
introduces regime shifts), we have shown that the new market
model obeys the same long-term properties of standard
geometric Brownian motion models.
However, the analysis is now more complicated because the
increments are neither stationary nor Gaussian, and it is not
possible to obtain an explicit solution.
The incorporation of investor sentiment into the model in this
manner is one of the important motivations behind the
discipline of behavioural ﬁnance.
32.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Comments
Despite the presence of the Markov process Y (which
introduces regime shifts), we have shown that the new market
model obeys the same long-term properties of standard
geometric Brownian motion models.
However, the analysis is now more complicated because the
increments are neither stationary nor Gaussian, and it is not
possible to obtain an explicit solution.
The incorporation of investor sentiment into the model in this
manner is one of the important motivations behind the
discipline of behavioural ﬁnance.
33.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline
1 Introduction
2 Mathematical Results
3 Application to Financial Market Models
4 Extensions and future work
34.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Extensions of the market model
We can also get upper and lower bounds on both positive and
negative large ﬂuctuations under the assumption that the drift
is integrable.
A result can be proven about the large ﬂuctuations of the
incremental returns when the diﬀusion coeﬃcient depends on
X , Y and t.
Can be extended to the case in which the diﬀusion coeﬃcient
in X depends not only on the Markovian switching term but
also on a delay term, once that diﬀusion coeﬃcient remains
bounded.
35.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Extensions of the market model
We can also get upper and lower bounds on both positive and
negative large ﬂuctuations under the assumption that the drift
is integrable.
A result can be proven about the large ﬂuctuations of the
incremental returns when the diﬀusion coeﬃcient depends on
X , Y and t.
Can be extended to the case in which the diﬀusion coeﬃcient
in X depends not only on the Markovian switching term but
also on a delay term, once that diﬀusion coeﬃcient remains
bounded.
36.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Extensions of the market model
We can also get upper and lower bounds on both positive and
negative large ﬂuctuations under the assumption that the drift
is integrable.
A result can be proven about the large ﬂuctuations of the
incremental returns when the diﬀusion coeﬃcient depends on
X , Y and t.
Can be extended to the case in which the diﬀusion coeﬃcient
in X depends not only on the Markovian switching term but
also on a delay term, once that diﬀusion coeﬃcient remains
bounded.
37.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Future work
We are currently working on a general d-dimensional S.D.E
dX (t) = f (X (t))dt + ΣdB(t)
where f is a d × 1 vector, Σ is a d × r matrix and B is an
r -dimensional standard Brownian motion.
We propose to capture the degree of non-linearity in f by the
1-dimensional scalar function φ.
Use techniques from this talk to determine the large
ﬂuctuations of X in terms of φ.
Can quantify the relation between the degree of
mean-reversion and the size of the ﬂuctuations.
38.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Future work
We are currently working on a general d-dimensional S.D.E
dX (t) = f (X (t))dt + ΣdB(t)
where f is a d × 1 vector, Σ is a d × r matrix and B is an
r -dimensional standard Brownian motion.
We propose to capture the degree of non-linearity in f by the
1-dimensional scalar function φ.
Use techniques from this talk to determine the large
ﬂuctuations of X in terms of φ.
Can quantify the relation between the degree of
mean-reversion and the size of the ﬂuctuations.
39.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Future work
We are currently working on a general d-dimensional S.D.E
dX (t) = f (X (t))dt + ΣdB(t)
where f is a d × 1 vector, Σ is a d × r matrix and B is an
r -dimensional standard Brownian motion.
We propose to capture the degree of non-linearity in f by the
1-dimensional scalar function φ.
Use techniques from this talk to determine the large
ﬂuctuations of X in terms of φ.
Can quantify the relation between the degree of
mean-reversion and the size of the ﬂuctuations.
40.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Future work
We are currently working on a general d-dimensional S.D.E
dX (t) = f (X (t))dt + ΣdB(t)
where f is a d × 1 vector, Σ is a d × r matrix and B is an
r -dimensional standard Brownian motion.
We propose to capture the degree of non-linearity in f by the
1-dimensional scalar function φ.
Use techniques from this talk to determine the large
ﬂuctuations of X in terms of φ.
Can quantify the relation between the degree of
mean-reversion and the size of the ﬂuctuations.
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