Lecture on advanced volatility models Erik Lindstrm FMS161/MASM18 - - PowerPoint PPT Presentation

Lecture on advanced volatility models

Erik Lindström

FMS161/MASM18 Financial Statistics

Erik Lindström Lecture on advanced volatility models

Stochastic Volatility (SV)

Let rt be a stochastic process.

◮ The log returns (observed) are given by

rt = exp(Vt/2)zt.

◮ The volatility Vt is a hidden AR process

Vt = α +βVt−1 +et.

◮ Or more general

A(·)Vt = et.

◮ More flexible than e.g. EGARCH models! ◮ Multivariate extensions.

Erik Lindström Lecture on advanced volatility models

A simulation of Taylor (1982)

50 100 150 200 250 300 350 400 450 500 −0.03 −0.02 −0.01 0.01 0.02 0.03 0.04

Erik Lindström Lecture on advanced volatility models

Long Memory Stochastic Volatility (LMSV)

The autocorr. of volatility decays slower than exp. rate

◮ The returns (observed) are given by (??)

rt = exp(Vt/2)zt.

◮ The volatility Vt is a hidden, fractionally integrated AR

process A(·)(1−q−1)bVt = et, where b ∈ (0,0.5).

◮ This gives long memory!

Erik Lindström Lecture on advanced volatility models

Long Memory Stochastic Volatility (LMSV)

◮ The long memory model can be approximated by a large

AR process, cf. (?, p 520).

◮ It can be shown that

(1−q−1)b =

∞

∑

j=0

πjq−j, where πj = Γ(j −b) Γ(j +1)Γ(−b).

Erik Lindström Lecture on advanced volatility models

Stochastic Volatility in continuous time

A popular application of stoch. volatility models is option valuation.

◮ Several parameterizations. ◮ The Heston model is the most used model, mainly due to

computational properties dSt = µStdt +

• VtStdW (S)

t

dVt = κ(θ −Vt)dt +σ

• VtdW (V)

t

dW (S)

t

dW (V)

t

= ρdt

◮ Note that the drift and squared diffusion have affine form. ◮ This reduces the task of computing prices to inversion of a

Fourier integral.

Erik Lindström Lecture on advanced volatility models

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.

Erik Lindström Lecture on advanced volatility models

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.

Erik Lindström Lecture on advanced volatility models

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 T 2

N2

• 2N → 0

◮ Chebyshev’s inequality then states that QN p

→ σ2T.

Erik Lindström Lecture on advanced volatility models

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

Erik Lindström Lecture on advanced volatility models

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 .

Erik Lindström Lecture on advanced volatility models

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.

Erik Lindström Lecture on advanced volatility models

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

Erik Lindström Lecture on advanced volatility models

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

Erik Lindström Lecture on advanced volatility models