christmas talk07

Introduction Mathematical Results Application to Financial Market Models Extensions and future work

The size of the largest fluctuations in a financialmarket 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 TalkDublin City University

Dec 14th 2007

Supported by the Irish Research Council for Science, Engineering and Technology

Outline

1 Introduction

2 Mathematical Results

3 Application to Financial Market Models

4 Extensions and future work

Outline

1 Introduction

Introduction

We consider the size of the large fluctuations of a stochasticdifferential equation (S.D.E) with Markovian switching,concentrating on processes which obey the law of the iteratedlogarithm:

lim supt→∞

|X (t)|√2t log log t

= c a.s.

The results are applied to financial market models which aresubject to random regime shifts (confident to nervous) and tochanges in market sentiment (investor intuition).

Markovian switching: parameters can switch according to aMarkov jump process

We show that our security model exhibits the same long-rungrowth and deviation properties as conventional geometricBrownian motion.

Introduction

lim supt→∞

= c a.s.

Introduction

lim supt→∞

= c a.s.

Introduction

lim supt→∞

= c a.s.

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

limt→∞

log S∗(t)

t= µ− 1

2σ2, a.s. (1)

The maximum size of the large deviations from this growthtrend obey the law of the iterated logarithm

lim supt→∞

| log S∗(t)− (µ− 12σ2)t|

√2t log log t

= σ, a.s. (2)

limt→∞

log S∗(t)

t= µ− 1

2σ2, a.s. (1)

lim supt→∞

| log S∗(t)− (µ− 12σ2)t|

√2t log log t

= σ, a.s. (2)

limt→∞

log S∗(t)

t= µ− 1

2σ2, a.s. (1)

lim supt→∞

| log S∗(t)− (µ− 12σ2)t|

√2t log log t

= σ, a.s. (2)

Outline

1 Introduction

S.D.E with Markovian switching

We study the stochastic differential equation with Markovianswitching

dX (t) = f (X (t),Y (t), t) dt + g(X (t),Y (t), t) dB(t) (3)

X (0) = x0,

f , g : R× S× [0,∞) → R are continuous functions obeyinglocal Lipschitz continuity and linear growth conditions,

Y is an irreducible continuous-time Markov chain with finitestate space S and is independent of the Brownian motion B.

Under these conditions, existence and uniqueness is guaranteed.

X (0) = x0,

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 ≤ g2(x , y , t) ≤ K1

then X satisfies

lim supt→∞

≤√

K1, a.s.

Main results

Theorem 1

xf (x , y , t) ≤ ρ, 0 < K2 ≤ g2(x , y , t) ≤ K1

then X satisfies

lim supt→∞

≤√

K1, a.s.

Main results

Theorem 1

xf (x , y , t) ≤ ρ, 0 < K2 ≤ g2(x , y , t) ≤ K1

then X satisfies

lim supt→∞

≤√

K1, a.s.

Main results

Theorem 1 continued

If, moreover, there exists an L ∈ R such that

inf(x ,y ,t)∈R×S×[0,∞)

xf (x , y , t)

g2(x , y , t)=: L > −1

Then X satisfies

lim supt→∞

≥√

K2, a.s.

Outline of proof

The proofs rely on

time-change and stochastic comparison arguments,

constructing upper and lower bounds on |X | which, under anappropriate change of time and scale, are stationary processeswhose dynamics are not determined by Y .

The large deviations of these processes are determined bymeans of a classical theorem of Motoo.

Outline of proof

The proofs rely on

Outline of proof

The proofs rely on

Outline of proof

The proofs rely on

Outline

1 Introduction

Security Price Model

The large deviation results are now applied to a security pricemodel, 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

sup(x ,y ,t)∈R×S×[0,∞)

xf (x , y , t)

γ2(y)≤ ρ and

inf(x ,y ,t)∈R×S×[0,∞)

xf (x , y , t)

γ2(y)> −1

sup(x ,y ,t)∈R×S×[0,∞)

xf (x , y , t)

γ2(y)≤ ρ and

inf(x ,y ,t)∈R×S×[0,∞)

xf (x , y , t)

γ2(y)> −1

sup(x ,y ,t)∈R×S×[0,∞)

xf (x , y , t)

γ2(y)≤ ρ and

inf(x ,y ,t)∈R×S×[0,∞)

xf (x , y , t)

γ2(y)> −1

Fluctuations in the Markovian switching model

Theorem 2

Let S be the unique continuous process governed by

with S(0) = s0 > 0, where X is defined on the previous slide.Then:

limt→∞

log S(t)

t= µ− 1

2σ2∗, a.s.

lim supt→∞

| log S(t)− (µt − 12

∫ t0 γ2(Y (s))ds)|

√2t log log t

= σ∗

where σ2∗ =

∑j∈S γ2(j)πj , and π is the stationary probability

distribution of Y .

Theorem 2

limt→∞

log S(t)

t= µ− 1

2σ2∗, a.s.

lim supt→∞

| log S(t)− (µt − 12

∫ t0 γ2(Y (s))ds)|

√2t log log t

= σ∗

where σ2∗ =

distribution of Y .

Theorem 2

limt→∞

log S(t)

t= µ− 1

2σ2∗, a.s.

lim supt→∞

| log S(t)− (µt − 12

∫ t0 γ2(Y (s))ds)|

√2t log log t

= σ∗

where σ2∗ =

distribution of Y .

Comments

Despite the presence of the Markov process Y (whichintroduces regime shifts), we have shown that the new marketmodel obeys the same long-term properties of standardgeometric Brownian motion models.

However, the analysis is now more complicated because theincrements are neither stationary nor Gaussian, and it is notpossible to obtain an explicit solution.

The incorporation of investor sentiment into the model in thismanner is one of the important motivations behind thediscipline of behavioural finance.

Comments

Outline

1 Introduction

Extensions of the market model

We can also get upper and lower bounds on both positive andnegative large fluctuations under the assumption that the driftis integrable.

A result can be proven about the large fluctuations of theincremental returns when the diffusion coefficient depends onX ,Y and t.

Can be extended to the case in which the diffusion coefficientin X depends not only on the Markovian switching term butalso on a delay term, once that diffusion coefficient remainsbounded.

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 anr -dimensional standard Brownian motion.

We propose to capture the degree of non-linearity in f by the1-dimensional scalar function φ.

Use techniques from this talk to determine the largefluctuations of X in terms of φ.

Can quantify the relation between the degree ofmean-reversion and the size of the fluctuations.

Future work

christmas talk07

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