Lecture on Stochastic Differential Equations Erik Lindstrm - - PowerPoint PPT Presentation

Lecture on Stochastic Differential Equations

Erik Lindström

Motivation

◮ Continuous time models are more 'interpretable'

than discrete time models, at least if you have a background in science or engineering. It is often argued that continuous time models need fewer parameters compared to discrete time models, as the parameters often can be given an interpretation. Consistent with option valuation due to path wise properties. Integration between time scales (e.g. irregularly sampled data) Heteroscedasticity is easily integrated into the models.

Motivation

◮ Continuous time models are more 'interpretable'

than discrete time models, at least if you have a background in science or engineering.

◮ It is often argued that continuous time models

need fewer parameters compared to discrete time models, as the parameters often can be given an interpretation. Consistent with option valuation due to path wise properties. Integration between time scales (e.g. irregularly sampled data) Heteroscedasticity is easily integrated into the models.

Motivation

◮ Continuous time models are more 'interpretable'

than discrete time models, at least if you have a background in science or engineering.

◮ It is often argued that continuous time models

need fewer parameters compared to discrete time models, as the parameters often can be given an interpretation.

◮ Consistent with option valuation due to path

wise properties. Integration between time scales (e.g. irregularly sampled data) Heteroscedasticity is easily integrated into the models.

Motivation

◮ Continuous time models are more 'interpretable'

than discrete time models, at least if you have a background in science or engineering.

◮ It is often argued that continuous time models

need fewer parameters compared to discrete time models, as the parameters often can be given an interpretation.

◮ Consistent with option valuation due to path

wise properties.

◮ Integration between time scales (e.g. irregularly

sampled data) Heteroscedasticity is easily integrated into the models.

Motivation

◮ Continuous time models are more 'interpretable'

than discrete time models, at least if you have a background in science or engineering.

◮ It is often argued that continuous time models

need fewer parameters compared to discrete time models, as the parameters often can be given an interpretation.

◮ Consistent with option valuation due to path

wise properties.

◮ Integration between time scales (e.g. irregularly

sampled data)

◮ Heteroscedasticity is easily integrated into the

models.

ODEs in physics

Physics is often modelled as (a system of) ordinary differential equations dX dt (t) = µ(X(t)) (1) Similar models are found in finance Bond

dB dt (t) = rB(t)

Stock

dS dt t

noise t S t , cf. RCAR CAPM

dS dt t

r noise t S t

ODEs in physics

Physics is often modelled as (a system of) ordinary differential equations dX dt (t) = µ(X(t)) (1) Similar models are found in finance Bond

dB dt (t) = rB(t)

Stock

dS dt (t) = (µ + “noise′′(t))S(t), cf. RCAR

CAPM

dS dt t

r noise t S t

ODEs in physics

Physics is often modelled as (a system of) ordinary differential equations dX dt (t) = µ(X(t)) (1) Similar models are found in finance Bond

dB dt (t) = rB(t)

Stock

dS dt (t) = (µ + “noise′′(t))S(t), cf. RCAR

CAPM

dS dt (t) = (r + βσ + σ“noise′′(t))S(t)

Noise processes

The noise process should ideally be the time derivative of random walk. Examples of continuous time processes (see Chapter 7.5) Brownian motion W t Poisson process N t , or N t t Compound Poisson process S t

N t n 1 Yn

Note that N t

N t n 1 1

Lévy process L t

Noise processes

The noise process should ideally be the time derivative of random walk. Examples of continuous time processes (see Chapter 7.5)

◮ Brownian motion W(t)

Poisson process N t , or N t t Compound Poisson process S t

N t n 1 Yn

Note that N t

N t n 1 1

Lévy process L t

Noise processes

The noise process should ideally be the time derivative of random walk. Examples of continuous time processes (see Chapter 7.5)

◮ Brownian motion W(t) ◮ Poisson process N(t), or (N(t) − λt)

Compound Poisson process S t

N t n 1 Yn

Note that N t

N t n 1 1

Lévy process L t

Noise processes

The noise process should ideally be the time derivative of random walk. Examples of continuous time processes (see Chapter 7.5)

◮ Brownian motion W(t) ◮ Poisson process N(t), or (N(t) − λt) ◮ Compound Poisson process S(t) = ∑N(t) n=1 Yn

Note that N t

N t n 1 1

Lévy process L t

Noise processes

The noise process should ideally be the time derivative of random walk. Examples of continuous time processes (see Chapter 7.5)

◮ Brownian motion W(t) ◮ Poisson process N(t), or (N(t) − λt) ◮ Compound Poisson process S(t) = ∑N(t) n=1 Yn ◮ Note that N(t) = ∑N(t) n=1 1

Lévy process L t

Noise processes

The noise process should ideally be the time derivative of random walk. Examples of continuous time processes (see Chapter 7.5)

◮ Brownian motion W(t) ◮ Poisson process N(t), or (N(t) − λt) ◮ Compound Poisson process S(t) = ∑N(t) n=1 Yn ◮ Note that N(t) = ∑N(t) n=1 1 ◮ Lévy process L(t)

Wiener process aka Standard Brownian Motion

A processes satisfying the following conditions is a Standard Brownian Motion

◮ X(0) = 0 with probability 1. ◮ The increments W(u) − W(t), W(s) − W(0) with

u > t ≥ s > 0 are independent.

◮ The increment W(t) − W(s) ∼ N(0, t − s) ◮ The process has continuous trajectories.

Time derivative of the Wiener process

Study the object ξh = W(t + h) − W(t) h (2) (Think dW(t)/dt = limh→0 ξh). Compute

◮ E[ξh] ◮ Var[ξh]

The limit does not converge in mean square sense!

Time derivative of the Wiener process

Study the object ξh = W(t + h) − W(t) h (2) (Think dW(t)/dt = limh→0 ξh). Compute

◮ E[ξh] ◮ Var[ξh]

The limit does not converge in mean square sense!

Re-interpreting ODEs

In physics, dX dt (t) = µ(X(t)) (3) really means dX t X t dt (4)

• r actually

t

dX s X t X 0

t

X s ds (5) NOTE: No derivatives needed!

Re-interpreting ODEs

In physics, dX dt (t) = µ(X(t)) (3) really means dX(t) = µ(X(t))dt (4)

• r actually

t

dX s X t X 0

t

X s ds (5) NOTE: No derivatives needed!

Re-interpreting ODEs

In physics, dX dt (t) = µ(X(t)) (3) really means dX(t) = µ(X(t))dt (4)

• r actually

∫ t dX(s) = X(t) − X(0) = ∫ t µ(X(s))ds (5) NOTE: No derivatives needed!

Stochastic differential equations

Interpret dX dt = ( µ(X(t)) + “noise′′(t) ) (6) as X(t) − X(0) ≈ ∫ t ( µ(X(s)) + “noise′′(s) ) ds (7) ≈ ∫ t µ(X(s))ds + ∫ σ(X(s))dv ds ds (8) The mathematically correct approach is to define Stochastic Differential Equations as X t X 0 X s ds X s dW s (9)

Stochastic differential equations

Interpret dX dt = ( µ(X(t)) + “noise′′(t) ) (6) as X(t) − X(0) ≈ ∫ t ( µ(X(s)) + “noise′′(s) ) ds (7) ≈ ∫ t µ(X(s))ds + ∫ σ(X(s))dv ds ds (8) The mathematically correct approach is to define Stochastic Differential Equations as X(t) − X(0) = ∫ µ(X(s))ds + ∫ σ(X(s))dW(s) (9)

Integrals

The ∫ µ(X(s))ds (10) integral is an ordinary Riemann integral, whereas the X s dW s (11) integral is an Ito integral.

Integrals

The ∫ µ(X(s))ds (10) integral is an ordinary Riemann integral,whereas the ∫ σ(X(s))dW(s) (11) integral is an It¯

• integral.

e It¯

• integral

The It¯

• integral is defined (for a piece-wise constant

integrand σ(s, ω)) as

b

∫

a

σ(s, ω)dW(s) =

n−1

∑

k=0

σ(tk, ω)(W(tk+1) − W(tk)). (12) General functions are approximated by piece-wise constant functions, while letting the discretization tend to zero. The limit is computed in L2 sense.

e It¯

• integral

The It¯

• integral is defined (for a piece-wise constant

integrand σ(s, ω)) as

b

∫

a

σ(s, ω)dW(s) =

n−1

∑

k=0

σ(tk, ω)(W(tk+1) − W(tk)). (12) General functions are approximated by piece-wise constant functions, while letting the discretization tend to zero. The limit is computed in L2(P) sense.

Properties

Stochastic integrals are martingales. Definition: A stochastic process {X(t), t ≥ 0} is called a martingale with respect to a filtration {F(t)}t≥0 if

◮ X(t) is F(t)-measurable for all t ◮ E [|X(t)|] < ∞ for all t, and ◮ E [X(t)|F(s)] = X(s) for all s ≤ t.

Proof: E X t s E X s X t X t s (13) X s E u dW u s (14) X s E E

n 1 k

tk W tk

1

W tk tk s X s (15)

Properties

Stochastic integrals are martingales. Definition: A stochastic process {X(t), t ≥ 0} is called a martingale with respect to a filtration {F(t)}t≥0 if

◮ X(t) is F(t)-measurable for all t ◮ E [|X(t)|] < ∞ for all t, and ◮ E [X(t)|F(s)] = X(s) for all s ≤ t.

Proof: E[X(t)|F(s)] = E[X(s) + (X(t) − X(t)|F(s)] (13) = X(s) + E[ ∫ σ(u, ω)dW(u)|F(s)] (14) = X(s) + E[E[

n−1

∑

k=0

σ(tk, ω)(W(tk+1) − W(tk))|F(tk)]|F(s)] = X(s) (15)

Other properties (eorem 7.1)

◮ Stochastic integrals are linear operators ◮ The unconditional expectation of a stochastic

integral is zero

◮ Stochastic integrals are measurable wrt the

Filtration of the driving Brownian motion

◮ The It¯

• isometry is useful when computing the

covariance E [(∫ σ(s)dW(s) )2] = ∫ E [ σ2(s) ] ds (16)

Interpretation

It can be shown that

◮ The drift is given by

µ(t, X(t)) = lim

h→0

1 hE [X(t + h) − X(t)] (17) While the squared diffusion is given by

T t X t

lim

h

1 hVar X t h X t (18)

Interpretation

It can be shown that

◮ The drift is given by

µ(t, X(t)) = lim

h→0

1 hE [X(t + h) − X(t)] (17)

◮ While the squared diffusion is given by

σσT(t, X(t)) = lim

h→0

1 hVar [X(t + h) − X(t)] (18)

Simple Monte Carlo simulation

The system X(t) − X(0) = ∫ µ(X(s))ds + ∫ σ(X(s))dW(s) (19) can be simulated through the Euler-Maruyama scheme, see Chap 12 in the book. The scheme is given by X((n + 1)h) = X(nh) + µ(X(nh))h+ (20) + σ(X(nh)) (W((n + 1)h) − W(nh)) .

Continuous time volatility

◮ We can compute the volatility in a continuous

time model.

◮ Advantage: A continuous time model can use

data from any time scale, and does not assume that data is equidistantly sampled.

◮ Can derive a limit theory when data is sampled

at high frequency.

◮ This is based on the general theory on quadratic

variation.

Quadratic variation

◮ Let {S} be a general semimartingale. ◮ Let πN = {0 = τ0 < τ1 < . . . < τN = T} be a

partition of [0, T], and denote ∆ = τn − τn−1, where ∆ = T/N.

◮ Define

QN =

N

∑

n=1

(S(τn) − S(τn−1))2 .

◮ What are the properties of QN? ◮ QN converges to the quadratic variation.

Quadratic variation, cont

Let St = σWt.

◮ Then

QN =

N

∑

n=1

(S(τn) − S(τn−1))2 .

◮ Note that (S(τn) − S(τn−1))2 ∼ σ2∆χ2(1). ◮ Remember E[χ2(p)] = p, V[χ2(p)] = 2p.

What are the properties of QN? E QN

2

E

2 N 2

N

2T.

QN

2 2 2 N 4 T2 N2

2N Chebyshev's inequality then states that QN

p 2T.

Quadratic variation, cont

Let St = σWt.

◮ Then

QN =

N

∑

n=1

(S(τn) − S(τn−1))2 .

◮ Note that (S(τn) − S(τn−1))2 ∼ σ2∆χ2(1). ◮ Remember E[χ2(p)] = p, V[χ2(p)] = 2p. ◮ What are the properties of QN? ◮ E[QN] = σ2∆E[χ2(N)] = σ2∆N = σ2T. ◮ V[QN] =

( σ2∆ )2 V[χ2(N)] = ( σ4 T2

N2

) 2N → 0

◮ Chebyshev's inequality then states that

QN

p

→ σ2T.

Quadratic variation of daily log returns for the Black-Scholes model

50 100 150 200 250 300 350 400 450 500 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Quadratic variation, cont

◮ For a diffusion process

dXt = µ(t, Xt)dt + σ(t, Xt)dWt, the quadratic variation converge to QN → ∫ σ2(s, Xs)ds. For a jump diffusion dXt t Xt dt t Xt dWt dZt where Z is a Poisson process Nt with random jumps of size Ji the quadratic variation yields QN

2 s Xs ds Nt i

J2

i

Quadratic variation, cont

◮ For a diffusion process

dXt = µ(t, Xt)dt + σ(t, Xt)dWt, the quadratic variation converge to QN → ∫ σ2(s, Xs)ds.

◮ For a jump diffusion

dXt = µ(t, Xt)dt + σ(t, Xt)dWt + dZt, where {Z} is a Poisson process Nt with random jumps of size Ji the quadratic variation yields QN → ∫ σ2(s, Xs)ds +

Nt

∑

i=0

J2

i .

Realized variation

◮ The quadratic (realized) variation is estimated as

QVN =

N

∑

n=1

(S(τn) − S(τn−1))2 .

◮ The Bipower variation is estimated as

BPVN = π 2

N

∑

n=1

|S(τn+1) − S(τn)||S(τn) − S(τn−1)|.

◮ It can be shown that the Bipower variation

converge to BPVN → ∫ σ2(s, Xs)ds, for a jump diffusion process (and even for a general semimartingale).

◮ The difference between the realized variation

and Bipower variation is used to estimate the size of the jump component.

Example: Realised variation for daily log return of Black-Scholes

100 200 300 400 500 600 700 800 900 0.05 0.1 0.15 100 200 300 400 500 600 700 800 900 −3 −2 −1 1 2 x 10

−3

QV−BPV (jumps ?) QV BPV

Example: Realised variation for daily log return of OMXS30

1995 2000 2005 2010 0.2 0.4 0.6 0.8 1 1995 2000 2005 2010 0.01 0.02 0.03 0.04 QV−BPV (jumps ?) QV BPV

Practical considerations

◮ Theory suggests that ∆ → 0 would be a good

thing. Practice suggests otherwise, cf. stylized facts. Problem is micro structure noise. Several strategies for correcting for this.

Practical considerations

◮ Theory suggests that ∆ → 0 would be a good

thing.

◮ Practice suggests otherwise, cf. stylized facts.

Problem is micro structure noise. Several strategies for correcting for this.

Practical considerations

◮ Theory suggests that ∆ → 0 would be a good

thing.

◮ Practice suggests otherwise, cf. stylized facts. ◮ Problem is micro structure noise. ◮ Several strategies for correcting for this.

Solving SDEs

Generally rather difficult... Use the definitions if possible. The Ito formula states the if dX t X t dt X t dW t (21) Y t F t X t C1 2 (22) Then the Ito formula applies dY t Ft FX 1 2

TFXX

dt FXdW t (23) where the dependence on X t is suppressed and Ft F t FX F X ``Proof'': Essentially Taylor expansions, and using that X and hence Y is continuous.

Solving SDEs

Generally rather difficult... Use the definitions if possible. The Ito formula states the if dX t X t dt X t dW t (21) Y t F t X t C1 2 (22) Then the Ito formula applies dY t Ft FX 1 2

TFXX

dt FXdW t (23) where the dependence on X t is suppressed and Ft F t FX F X ``Proof'': Essentially Taylor expansions, and using that X and hence Y is continuous.

Solving SDEs

Generally rather difficult... Use the definitions if possible. The It¯

• formula states the if

dX(t) = µ(X(t))dt + σ(X(t))dW(t) (21) Y(t) = F(t, X(t)) ∈ C1,2 (22) Then the Ito formula applies dY t Ft FX 1 2

TFXX

dt FXdW t (23) where the dependence on X t is suppressed and Ft F t FX F X ``Proof'': Essentially Taylor expansions, and using that X and hence Y is continuous.

Solving SDEs

Generally rather difficult... Use the definitions if possible. The It¯

• formula states the if

dX(t) = µ(X(t))dt + σ(X(t))dW(t) (21) Y(t) = F(t, X(t)) ∈ C1,2 (22) Then the It¯

• formula applies

dY(t) = ( Ft + µFX + 1 2σσTFXX ) dt + σFXdW(t) (23) where the dependence on X(t) is suppressed and Ft = ∂F/∂t, FX = ∂F/∂X, . . . ``Proof'': Essentially Taylor expansions, and using that X and hence Y is continuous.

Solving SDEs

Generally rather difficult... Use the definitions if possible. The It¯

• formula states the if

dX(t) = µ(X(t))dt + σ(X(t))dW(t) (21) Y(t) = F(t, X(t)) ∈ C1,2 (22) Then the It¯

• formula applies

dY(t) = ( Ft + µFX + 1 2σσTFXX ) dt + σFXdW(t) (23) where the dependence on X(t) is suppressed and Ft = ∂F/∂t, FX = ∂F/∂X, . . . ``Proof'': Essentially Taylor expansions, and using that X and hence Y is continuous.