Stochastic Modelling and Forecasting - Personalpersonal.strath.ac.uk/x.mao/talks/smfc.pdfStochastic Modelling Well-known Models Stochastic verse Deterministic Forecasting and Monte

Stochastic ModellingWell-known Models

Stochastic verse DeterministicForecasting and Monte Carlo Simulations

Stochastic Modelling and Forecasting

Xuerong Mao

Department of Statistics and Modelling ScienceUniversity of Strathclyde

Glasgow, G1 1XH

RSE/NNSFC Workshop onManagement Science and Engineering and Public Policy

Edinburgh, 17-18 March 2008

Xuerong Mao SM and Forecasting



Outline

1 Stochastic Modelling

2 Well-known Models

3 Stochastic verse DeterministicExponential growth modelLogistic Model

4 Forecasting and Monte Carlo SimulationsEM methodEM method for financial quantities




Outline


2 Well-known Models






Outline


2 Well-known Models






Outline


2 Well-known Models






One of the important problems in many branches of scienceand industry, e.g. engineering, management, finance, socialscience, is the specification of the stochastic process governingthe behaviour of an underlying quantity. We here use the termunderlying quantity to describe any interested object whosevalue is known at present but is liable to change in the future.Typical examples are

shares in a company,

commodities such as gold, oil or electricity,

number of working people,

number of pupils in primary schools.




























Now suppose that at time t the underlying quantity is x(t). Letus consider a small subsequent time interval dt , during whichx(t) changes to x(t) + dx(t). (We use the notation d · for thesmall change in any quantity over this time interval when weintend to consider it as an infinitesimal change.) By definition,the intrinsic growth rate at t is dx(t)/x(t). How might we modelthis rate?




If, given x(t) at time t , the rate of change is deterministic, sayR = R(x(t), t), then

dx(t)x(t)

= R(x(t), t)dt .

This gives the ordinary differential equation (ODE)

dx(t)dt

= R(x(t), t)x(t).




However the rate of change is in general not deterministic as itis often subjective to many factors and uncertainties e.g.system uncertainty, environmental disturbances. To model theuncertainty, we may decompose

dx(t)x(t)

= deterministic change + random change.




The deterministic change may be modeled by

R̄dt = R̄(x(t), t)dt

where R̄ = r̄(x(t), t) is the average rate of change given x(t) attime t . So

dx(t)x(t)

= R̄(x(t), t)dt + random change.

How may we model the random change?




In general, the random change is affected by many factorsindependently. By the well-known central limit theorem thischange can be represented by a normal distribution with meanzero and and variance V 2dt , namely

random change = N(0, V 2dt) = V N(0, dt),

where V = V (x(t), t) is the standard deviation of the rate ofchange given x(t) at time t , and N(0, dt) is a normal distributionwith mean zero and and variance dt . Hence

dx(t)x(t)

= R̄(x(t), t)dt + V (x(t), t)N(0, dt).




A convenient way to model N(0, dt) as a process is to use theBrownian motion B(t) (t ≥ 0) which has the followingproperties:

B(0) = 0,

dB(t) = B(t + dt)− B(t) is independent of B(t),

dB(t) follows N(0, dt).




The stochastic model can therefore be written as

dx(t)x(t)

= R̄(x(t), t)dt + V (x(t), t)dB(t),

or

dx(t) = R̄(x(t), t)x(t)dt + V (x(t), t)x(t)dB(t)

which is a stochastic differential equation (SDE).




Linear model

If both R̄ and V are constants, say

R̄(x(t), t) = µ, V (x(t), t) = σ,

then the SDE becomes

dx(t) = µx(t)dt + σx(t)dB(t).

This is

the Black–Scholes geometric Brownian motion in finance,

the exponential growth model in engineering andpopulation.




Linear model

If both R̄ and V are constants, say

R̄(x(t), t) = µ, V (x(t), t) = σ,



This is

the Black–Scholes geometric Brownian motion in finance,

the exponential growth model in engineering andpopulation.




Logistic model

IfR̄(x(t), t) = b + ax(t), V (x(t), t) = σx(t),


dx(t) = x(t)([b + ax(t)]dt + σx(t)dB(t)

).

This is the well-known Logistic model in population.




Square root process

IfR̄(x(t), t) = µ, V (x(t), t) =

σ√x(t)

,

then the SDE becomes the well-known square root process

dx(t) = µx(t)dt + σ√

x(t)dB(t).

This is used widely in engineering and finance.




Mean-reverting model

If

R̄(x(t), t) =α(µ− x(t))

x(t), V (x(t), t) =

σ√x(t)

,


dx(t) = α(µ− x(t))dt + σ√

x(t)dB(t).

This is

the mean-reverting square root process in population,

the CIR model for interest rate in finance.




Mean-reverting model

If

R̄(x(t), t) =α(µ− x(t))

x(t), V (x(t), t) =

σ√x(t)

,


dx(t) = α(µ− x(t))dt + σ√

x(t)dB(t).

This is

the mean-reverting square root process in population,

the CIR model for interest rate in finance.




Theta process

IfR̄(x(t), t) = µ, V (x(t), t) = σ(x(t))θ−1,


dx(t) = µx(t)dt + σ(x(t))θdB(t),

which is known as the theta process.




Exponential growth modelLogistic Model

Outline


2 Well-known Models







In the classical theory of population dynamics, it is assumedthat the grow rate is constant µ. Thus

dx(t)x(t)

= µdt ,

which is often written as the familiar ordinary differentialequation (ODE)

dx(t)dt

= µx(t).

This linear ODE can be solved exactly to give exponentialgrowth (or decay) in the population, i.e.

x(t) = x(0)eµt ,

where x(0) is the initial population at time t = 0.





We observe:

If µ > 0, x(t) →∞ exponentially, i.e. the population willgrow exponentially fast.

If µ < 0, x(t) → 0 exponentially, that is the population willbecome extinct.

If µ = 0, x(t) = x(0) for all t , namely the population isstationary.





We observe:








We observe:








However, if we take the uncertainty into account as explainedbefore, we may have

dx(t)x(t)

= rdt + σdB(t).

This is often written as the linear SDE


It has the explicit solution

x(t) = x(0) exp[(µ− 0.5σ2)t + σB(t)

].





Recall the properties of the Brownian motion

lim supt→∞

B(t)√2t log log t

= 1 a.s.

and

lim inft→∞

B(t)√2t log log t

= −1 a.s.





If µ > 0.5σ2, x(t) →∞ exponentially with probability 1, i.e.the population will grow exponentially fast.

If µ < 0.5σ2, x(t) → 0 exponentially with probability 1, thatis the population will become extinct.

In particular, this includes the case of 0 < µ < 0.5σ2 wherethe population will grow exponentially in the correspondingODE model but it will become extinct in the SDE model.This reveals the important feature - noise may make apopulation extinct.

If µ = 0.5σ2, lim supt→∞ x(t) = ∞ while lim inft→∞ x(t) = 0with probability 1.





































Outline


2 Well-known Models







The logistic model for single-species population dynamics isgiven by the ODE

dx(t)dt

= x(t)[b + ax(t)]. (3.1)





If a < 0 and b > 0, the equation has the global solution

x(t) =b

−a + e−bt(b + ax0)/x0(t ≥ 0) ,

which is not only positive and bounded but also

limt→∞

x(t) =b|a|

.

If a > 0, whilst retaining b > 0, then the equation has onlythe local solution

x(t) =b

−a + e−bt(b + ax0)/x0(0 ≤ t < T ) ,

which explodes to infinity at the finite time

T = −1b

log(

ax0

b + ax0

).





If a < 0 and b > 0, the equation has the global solution

x(t) =b

−a + e−bt(b + ax0)/x0(t ≥ 0) ,

which is not only positive and bounded but also

limt→∞

x(t) =b|a|

.

If a > 0, whilst retaining b > 0, then the equation has onlythe local solution

x(t) =b

−a + e−bt(b + ax0)/x0(0 ≤ t < T ) ,


T = −1b

log(

ax0

b + ax0

).





Once again, the growth rate b here is not a constant but astochastic process. Therefore, bdt should be replaced by

bdt + N(0, v2dt) = bdt + vN(0, dt) = bdt + vdB(t),

where v2 is the variance of the noise intensity. Hence the ODEevolves to an SDE

dx(t) = x(t)([b + ax(t)]dt + vdB(t)

). (3.2)





The variance may or may not depend on the state x(t). Weconsider the latter, say

v = σx(t).

Then the SDE (3.2) becomes

dx(t) = x(t)([b + ax(t)]dt + σx(t)dB(t)

). (3.3)

How is this SDE different from its corresponding ODE?





Significant Difference between ODE (3.1) and SDE(3.3)

ODE (3.1): The solution explodes to infinity at a finite timeif a > 0 and b > 0.

SDE (3.3): With probability one, the solution will no longerexplode in a finite time, even in the case when a > 0 andb > 0, as long as σ 6= 0.





Significant Difference between ODE (3.1) and SDE(3.3)

ODE (3.1): The solution explodes to infinity at a finite timeif a > 0 and b > 0.

SDE (3.3): With probability one, the solution will no longerexplode in a finite time, even in the case when a > 0 andb > 0, as long as σ 6= 0.





Example

dx(t)dt

= x(t)[1 + x(t)], t ≥ 0, x(0) = x0 > 0

has the solution

x(t) =1

−1 + e−t(1 + x0)/x0(0 ≤ t < T ) ,


T = log(

1 + x0

x0

).

However, the SDE

dx(t) = x(t)[(1 + x(t))dt + σx(t)dw(t)]

will never explode as long as σ 6= 0.Xuerong Mao SM and Forecasting




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Note on the graphs:In graph (a) the solid curve shows a stochastic trajectorygenerated by the Euler scheme for time step ∆t = 10−7 andσ = 0.25 for a one-dimensional system (3.3) with a = b = 1.The corresponding deterministic trajectory is shown by thedot-dashed curve. In Graph (b) σ = 1.0.





Key Point :When a > 0 and ε = 0 the solution explodes at the finite timet = T ; whilst conversely, no matter how small ε > 0, thesolution will not explode in a finite time. In other words,

stochastic environmental noise suppresses deterministicexplosion.




EM methodEM method for financial quantities

Most of SDEs used in practice do not have explicit solutions.How can we use these SDEs to forecast?One of the important techniques is the method of Monte Carlosimulations. There are two main motivations for suchsimulations:

using a Monte Carlo approach to compute the expectedvalue of a function of the underlying underlying quantity, forexample to value a bond or to find the expected payoff ofan option;

generating time series in order to test parameterestimation algorithms.



















Typically, let us consider the square root process

dS(t) = rS(t)dt + σ√

S(t)dB(t), 0 ≤ t ≤ T .

A numerical method, e.g. the Euler–Maruyama (EM) methodapplied to it may break down due to negative values beingsupplied to the square root function. A natural fix is to replacethe SDE by the equivalent, but computationally safer, problem

dS(t) = rS(t)dt + σ√|S(t)|dB(t), 0 ≤ t ≤ T .





Outline


2 Well-known Models







Discrete EM approximation

Given a stepsize ∆ > 0, the EM method applied to the SDEsets s0 = S(0) and computes approximations sn ≈ S(tn), wheretn = n∆, according to

sn+1 = sn(1 + r∆) + σ√|sn|∆Bn,

where ∆Bn = B(tn+1)− B(tn).





Continuous-time EM approximation

s(t) := s0 + r∫ t

0s̄(u))du + σ

∫ t

0

√|s̄(u)|dB(u),

where the “step function” s̄(t) is defined by

s̄(t) := sn, for t ∈ [tn, tn+1).

Note that s(t) and s̄(t) coincide with the discrete solution at thegridpoints; s̄(tn) = s(tn) = sn.





The ability of the EM method to approximate the true solution isguaranteed by the ability of either s(t) or s̄(t) to approximateS(t) which is described by:

Theorem

lim∆→0

E(

sup0≤t≤T

|s(t)−S(t)|2)

= lim∆→0

E(

sup0≤t≤T

|s̄(t)−S(t)|2)

= 0.





The ability of the EM method to approximate the true solution isguaranteed by the ability of either s(t) or s̄(t) to approximateS(t) which is described by:

Theorem

lim∆→0

E(

sup0≤t≤T

|s(t)−S(t)|2)

= lim∆→0

E(

sup0≤t≤T

|s̄(t)−S(t)|2)

= 0.





Outline


2 Well-known Models







Bond

If S(t) models short-term interest rate dynamics, it is pertinentto consider the expected payoff

β := E exp

(−∫ T

0S(t)dt

)from a bond. A natural approximation based on the EM methodis

β∆ := E exp

(−∆

N−1∑n=0

|sn|

), where N = T/∆.

Theorem

lim∆→0

|β − β∆| = 0.





Bond

If S(t) models short-term interest rate dynamics, it is pertinentto consider the expected payoff

β := E exp

(−∫ T

0S(t)dt

)from a bond. A natural approximation based on the EM methodis

β∆ := E exp

(−∆

N−1∑n=0

|sn|

), where N = T/∆.

Theorem

lim∆→0

|β − β∆| = 0.





Up-and-out call option

An up-and-out call option at expiry time T pays the Europeanvalue with the exercise price K if S(t) never exceeded the fixedbarrier, c, and pays zero otherwise.

Theorem

Define

V = E[(S(T )− K )+I{0≤S(t)≤c, 0≤t≤T}

],

V∆ = E[(s̄(T )− K )+I{0≤s̄(t)≤B, 0≤t≤T}

].

Thenlim

∆→0|V − V∆| = 0.





Up-and-out call option

An up-and-out call option at expiry time T pays the Europeanvalue with the exercise price K if S(t) never exceeded the fixedbarrier, c, and pays zero otherwise.

Theorem

Define

V = E[(S(T )− K )+I{0≤S(t)≤c, 0≤t≤T}

],

V∆ = E[(s̄(T )− K )+I{0≤s̄(t)≤B, 0≤t≤T}

].

Thenlim

∆→0|V − V∆| = 0.


Documents

Stochastic Modelling and Forecasting - Personalpersonal.strath.ac.uk/x.mao/talks/smfc.pdfStochastic Modelling Well-known Models Stochastic verse Deterministic Forecasting and Monte