Stochastic Volatility and Jump Modeling in Finance · 2013-01-21 · Stochastic Volatility and Jump...

Stochastic Volatility and Jump Modeling in FinanceHPCFinance 1st kick-off meeting

Elisa Nicolato

Aarhus UniversityDepartment of Economics and Business

January 21, 2013

Elisa Nicolato (Aarhus University Department of Economics and Business )Stochastic Volatility and Jump Modeling in Finance January 21, 2013 1 / 34

Outline

The aim of this talk is to give an informal, strictly non-rigorous andbasically conversational introduction to advanced modelling techniques inFinance.

Motivate why advanced modelling is often (although not always)necessary.

Hint briefly at SV models and Levy models

Attempt to highlight common traits in these two seemingly verydifferent model classes

Touch upon modelling frameworks combining the best features of SVand Jump models.


The seminal Black-Scholes model

In the BS model, the price-process of an asset S is described by a GBM

dSt = µStdt + σStdZt ⇒

d(logSt) = dXt = (µ− 1

2σ2)dt + σdZt

Implications:

The volatility is constant and log-returns are i.i.d normally distributed

Xt+∆t − Xt ∼ (µ− 1

2σ2)∆t + σ

√∆tN(0, 1)

The risk-neutral price process follows a GMB, albeit with µ = r and theprice of a call stroke at K and expiring at T is given by

e−rTEQ [(ST − K )+] = CBS (S0,K , σ,T )

with CBS denoting the BS formula.


Model vs Empirics - laughing off constant vol -1

This plot shows SPX daily log returns from 02/01/2001 to 02/31/2011.

0 500 1000 1500 2000 2500 3000−0.1

−0.05

0

0.05

0.1

0.15

Volatility doesn’t seem to be constant.Large moves follow large moves and small moves follow small moves-”volatility clustering”


Model vs Empirics - laughing off constant vol -2

This plot shows SPX daily volatility estimates for the same period

0 500 1000 1500 2000 25000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Although naive estimates (60 days-rolling window), clear indication ofnon-constant vol.


Model vs Empirics - laughing off normality -1

Frequency distribution of (10 years of) SPX daily returns compared withthe normal distribution.

−0.1 −0.05 0 0.05 0.1 0.150

50

100

150

200

250

300

Descriptive statistics: mean=0.0136% (annualized=3.42%), STD=1.37%(annualized=21.81%), skewness=0.0846, kurtosis=11.428.


Model vs Empirics - laughing off normality -2

Empirical distribution (kernel density estimator) vs fitted normal.

−0.1 −0.05 0 0.05 0.1 0.150

20

40

60Empirical vs Normal

−0.1 −0.05 0 0.05 0.1 0.15−40

−20

0

20log−scale

Stylized features: High central peak (top panel) and fat tails (bottompanel in log-scale).


Mean-Variance Mixtures Distributions: a first look

High central peak/fat tails and random variance are in fact two sidesof the same coin.

The so-called mean-variance mixtures distributions embed the twofeatures and are constructed as follows:

X ∼ µ+ θV + σ√V · N(0, 1), V ⊥ N(0, 1) (1)

where

µ, θ and σ are real parameters with σ > 0;V is a distribution on the positive real line (representing the randomvariance);V is independent of the standard normal N(0, 1).µ is a location parameter, θ is a skewness parameter, while σ regulatesthe level of the overall random variance


Variance Gamma Distribution

A notorious choice for V is the gamma distribution Γ(µ, ν) with density

fV (x) =(µν

)µ2

ν xµ2

ν−1 exp

(−µ

ν x)

Γ(µ2

ν )x > 0 (2)

The resulting mixed distribution X ∼ µ+ θV + σ√V · N(0, 1) is known as

the Variance-Gamma distribution.

Denoting with Kη(·) is a modified Bessel, the density is

fX (x) =2 exp

(θ(x − µ)/σ2

)

σ√2πν1/νΓ(1/ν)

·(

|x − µ|√2θ2/ν + σ2

)1/ν−1/2

× (3)

K1/ν−1/2

(|x − µ|

√2σ2/ν + θ2

σ2

)−∞ < x +∞

The (much simpler) c.f. φX (u) = E [exp(iuX )] is given by

φX (u) = e iµu(1− iθνu + (σ2νu2)/2

)− 1ν (4)


VG vs Normal

Fixing θ = 0, comparison between VG and Normal distribution, with thesame mean=0.0136% and STD=1.37%

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.080

10

20

30

40Normal vs VG

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08−15

−10

−5

0

5log−scale

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.080

50

100

150Normal vs VG

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08−15

−10

−5

0

5log−scale

Top panel ν = 0.5, bottom panel ν = 3.


So what have we found out?

Constant volatility is an absurd assumption.

The normal distribution is not a good fit to observed returns due tofat tails.

Mean-variance mixture distributions, such as VG; obtained byrandomizing variance in a normal display such a property andtherefore have the potential to fit the empirical distribution of returns

Now all this is information coming from historical/time-seriespreliminary statistical analysis.

How about empirical evidence against BS assumptions coming fromliquid derivative prices? In other words, is the normality assumptionvalid/acceptable in the risk-neutral world?


The Implied Volatility

In the BS model, the price of a call option is given by

CBS (S0,K , σ,T ) = e−rTEQ [(ST − K )+] = S0N(d1)− Ke−rTN(d2) (5)

with d1/2 =log(S0e

rT /K)

σ√T

± σ√T

2 .

Given an observed option price C (S0,K ,T ), the Implied Volatility is thevolatility value σBS (S0,K ,T ) to be plugged in the BS formula to matchC (S0,K ,T ), i.e. such that

C (S0,K ,T ) = CBS (S0,K , σBS (S0,K ,T ),T )

IV has been described as ”the wrong number in the wrong formula to getthe right price” (R. Rebonato).

By its very definition, for prices coming from the BS model with volatility σ

σBS (S0,K ,T ) ≡ σ for all K ,T .


Implied volatilities are in one-to-one correspondence with optionprices.

They can be seen as a useful alternative way to quote options insteadof raw prices: option prices can be compared across different strikes,maturities and underlyings.

Given a surface of option prices C (S0,K ,T ), the volatility surface

σBS(S0,K ,T ) is the associated surface of implied vols.

The volatility surface associated with BS model with volatility σ is aflat surface of height σ.


Observed Implied Vol Surface from JG, 15/09/2005

Here’s a 3D plot pf the (smoothed) SPX volatility surface as of September15,2005.


Less smooth but more real pics

Raw SPX implied vols observed on March 15, 2010, for two different expiries.S0 = 1150

0,1

0,15

0,2

0,25

Expiry:17/04/2010

0

0,05

0,9 0,95 1 1,05 1,1 1,15 1,2

K/S

/ /

0,4

Expiry 19/06/2010

0,3

0,35

0 15

0,2

0,25

0,05

0,1

0,15

0

0,8 0,85 0,9 0,95 1 1,05 1,1 1,15 1,2

K/SK/S

Typical behavior across moneyness K/S known as smile (hokey-stick) effect.

Notice negative slope around the at-the-money point, the skew.

If BS was a good model for market prices, we should observe a flat volsurface!


How to generate a smile? A naive experiment.

By risk-neutral evaluation, we have that the price of a call is given by

C (S0,K ,T ) = EQ [(eXT − K )+] (6)

where XT is the distribution of the log-price at maturity under thesimplifying assumption r = d = 0.

In the BS model

XT = log ST = log S0 + r − 1/2σ2T +√σ2TN(0, 1).

Inspired by the return analysis, we could consider σ2 as a randomvariable and model XT as a mean-variance mixture as introduced in(1).

We could fix T = 1 and choose

σ2 ∼ σ2Γ(1, ν), σ2 ∈ R

(as in the VG model introduced by Carr& Madan & Chang (1998).


VG prices

VG Prices can then be computed as the integral

CVG (S0,K ,T ) = EQ [(eXT − K )+] =

∫

R

(ex − K )f X (x)dx (7)

where f X (x) is the VG density given in (3): Although known ”explicitly”, itis definitely a mess!

Due to the numerical complexity of the integral above, one typically plugsthe integrand through the Fourier transform machinery to obtain theequivalent expression in terms of the c.f.

C (S0,K ,T ) = S0 −√S0K

π

∫ ∞

0

du

u2 + 1/4Re[e−iukφ0(u − i/2)

](8)

where k := log(K/S0) and φ0(·) denotes the c.f. of the standardizedlog-price XT − X0 = log ST

S0at maturity.

For the VG distribution (and in many other cases) φ0(·) is way moretractable than the density at the point that the integral (8) is know as”closed-form” solution.


The VG implied vol function

And now the experiment goes as follows:

For the fixed maturity (e.g. T = 1), and for some fixed parameters σand ν, compute the call prices CVG (S0,Kj ,T ) for a set of strikes Kj ,j = 1, . . . ,m.

Numerically invert

CVG (S0,Kj ,T ) = CBS(S0,Kj , σBS (S0,Kj ,T ),T )

to find the implied volatilities

plot them against K .


A smile!

And the result is as follows (green line BS prices corresponding toσ = 26%)

0.7 0.8 0.9 1 1.1 1.2 1.3 1.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35Prices

0.7 0.8 0.9 1 1.1 1.2 1.3 1.40.21

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29Implied Vols

We do manage to obtain a smile!!! We can conclude that stochasticvolatility injects curvature in an otherwise flat implied vol function.


Summing up

From the distributional point of view we have seen that mean-variancemixture laws have the potential to fit the empirical distributions of returnsas well as to capture the implied volatility smile.

From now on we focus on derivatives and we are interested in findingdynamical models in continuous time capable to match the properties ofempirically observed IV surfaces.

In other words: how do we bend the seminal BS model so that the flatvolatility surface turns into a smile?

Roughly speaking there are two main ways to score this goal:

SV Models Bend it like Heston

Levy Models Bend it like Madan


SV models: diffusion formulation

Stochastic volatility models are models generalizing the seminalBS, allowing the volatility parameter itself to be a stochastic processwith its own source of randomness.

In a quite (but not the most) general SV model, the price S of a nondividend paying underlying asset is given by:

dSt = rStdt +√vtStdBt (9)

dvt = a(vt)dt + b(vt)dWt (10)

where B and W are two Brownian Motions, possibly correlatedd〈B ,W 〉 = ρdt, while vt is a process representing the randominstantaneous variance.

Here the model is specified under the risk-neutral measure Q.


Examples of SV models

The spot is always described by dSt = rStdt +√vtStdBt

The Good: Hull and White (1987): dvt = αvtdt + ηvtdWt

GBM implying log-normal vt , no mean reversion.

The Bad: Heston (1993): dvt = −λ(vt − v)dt + η√vtdWt

Stationary, Gamma invariant law.

The Ugly: 3/2 (???): v = 1/x with x described by square root processas in Heston:

dvt = −λvt(vt − v)dt + ηv3/2t dWt

Level dependent speed of mean reversion, stationary, InverseGamma invariant law.

Common to all these models (and many others) is the fact that theinstantaneous variance process vt is Markovian in itself.


Distribution of log-price XT

The log price process is given by

Xt = X0 + rt − 1

2

∫ t

0vsds +

∫ t

0

√vsdBs

Let Vt =∫ t

0 vsds denote the integrated variance up to time t: By theDambis-Dubins-Schwarz theorem, Xt admits the followingtime-change representation

Xt = X0 + rt − 1

2Vt + BVt

where B is a Brownian motion (in a suitable filtration)Hence, the distribution of the log-price Xt is given by a mean-variancemixture

Xt − X0 ∼ rt − 12Vt +

√VtN(0, 1) (11)

However, Vt and N(0, 1) in this case are not necessarily independent,but they have correlation ρ.


Given the discussion above, it is natural to expect that any SV modelwill be capable to generate a smile. And so it is.

However, the Heston model, as any bad guy, is possibly the mostpopular both among practitioners and academics. Why?

Quoting Gatheral ”The great difference...is the existence of a fast andeasily implemented quasi-closed form solution for European options”.

That is, the c.f. φ0(u) of XT is available in terms of elementaryfunctions and therefore we may use

C (S0,K ,T ) = S0 −√S0K

π

∫∞

0

du

u2 + 1/4Re[e−iukφ0(u − i/2)

]

introduced above to compute call option prices.


The Heston c.f.

Define

a = −1

2(z2 + iz) b = λ− ρη iz d =

√b2 − 2aη2 r± =

b ± d

η2g =

r−r+

then the c.f. φ0(z) takes the form

φ0(z) = eC(z,T )+D(z,T )v0 (12)

where

D(z , τ) = r−1− e−dτ

1− ge−dτC (z , τ) = λv

{r−τ − 2

η2log

(1− ge−dτ

1− g

)}

The exponential-affine form (12) is not a coincidence but a consequence ofthe fact that v is described by an affine diffusion


A couple of words about the ”good” and the ”ugly”

In principle it is possible to compute the c.f. φ0 also for the 3/2model, but it is in terms of the Confluent Hypergeometric function (aseries expansion)

It is so ugly (and untractable) that I will not even begin to write itdown.

Still, this model is attracting attention (again) lately, due to theproperties of its invariant law in connection with the pricing of VIXoption.

For the good old Hull and White model, no explicit or semi-explicitformula is available.

Such a pity, as the log-normal law offers the best description of theempirical law of the instantaneous variance.


Let’s bend it like Heston

Here’s the 3D plot of a Heston imp. vol. surface obtained with parameters fittedto the observed surface on 15,September 2005. The values are λ = 0.54,ρ = −0.7,η = 0.30,v = 0.044, v0 = 0.044.

−1

−0.5

0

0.5 0

1

2

30.1

0.15

0.2

0.25

0.3

0.35

time to maturitylog moneyness log(K/S)

impl

ied

vols


Games you might play: parameter sensitivities

Here’s the graphical illustration of how the parameters affect the volatility shapefor a maturity fixed at T = 0.3. Left panel: the correlation parameter ρ is takingvalues −0.7 (blue),0 (green) and 0.7 (red). Right panel: the vol-of-vol parameterη is taking values 0.1 (blue),0.6 (green) and 1.1 (red). The remaining parametersare same as from fit to 15, September, 2005.

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

log moneyness log(K/S)

Imp.

Vol

s

Sensitivity w.r.t. ρ

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.60.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

log moneyness log(K/S)

Imp.

Vol

s

Sensitivity w.r.t. η


Heston: Limitations

It seems that the Heston model is not quite capable to capture the

behavior of the Skew∂σ2

BS(κ,T )∂κ for short maturities.

For short maturities, the empirically observed skew seems to be muchsteeper than the skew generated by Heston model (or any other SVmodel, actually)

Hence the need of introducing new models beyond/different than SV.


Back to square one

Let’s get back to Black-Scholes model for which (risk-neutral)log-prices follow

dX = (r − 1/2σ2)dt + σdB ∼ (r − 1/2σ2)∆t + σN(0, dt)

What is really annoying here in terms of the smile is the normality ofthe increments of the BM B , which transfers to the distributions oflog-returns.

The idea here is: can we substitute B with another continuous timeprocess L which has independent and stationary increments butdescribed by a distribution capable to bend a smile?

The answer is yes and it is captured by the class of the exponentialLevy models.


A brief introduction to Levy processes

Definition: A stochastic process (Lt)t≥0 such that L0 = 0 is called a Levy

process if

1 Independent increments: for every increasing sequence of timest0 ≤ · · · ≤ tn, the random variables Lt0 , Lt1 − Lt0 , . . . , Ltn − Ltn−1 areindependent.

2 Stationary increments: the law of Lt+h − Lt does not depend on t, i.e.Lt+h − Lt ∼ Lh.

3 Stochastic continuity: ∀ε > 0, limh→0 Q(|Lt+h − Lt | ≥ ε) = 0.

- The third property does not imply in any way that the sample paths arecontinuous. It means that for a given t, the probability of a jump to occurexactly at t is zero.

- The most prominent examples are the Brownian motion (continuous paths,the one and only) and the Poisson process (discontinuous paths).


Levy models

Definition: A probability distribution F on R is said to be infinitely

divisible if for any integer n ≥ 2 there exists n i.i.d. random variablesY1, . . . ,Yn such that Y1 + · · ·+ Yn has distribution F .

Given an infinitely divisible distribution F , then there exists a Levy process(Lt) such that L1 ∼ F ,

We can then define a risk-neutral log price with unit F distribution as follows

dXt = rdt + drift dt + dLt (13)

where the drift is determined by risk-neutrality and is given by

drift = − logφ0(−i)

where φ0(z) is the c.f. of F .

The VG distribution is infinitely divisible, implying that via (13) we candefine a log price process with VG distributions as marginals: The VG modelof Madan et al.


The Seagull Volatility Surface

Once again prices can be computed via Fourier transform. The typical volatilitysurface in the VG model looks like this

−0.3−0.2

−0.10

0.10.2 0

0.5

10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

time to maturitylog moneyness log(K/S)

impl

ied

vols

This time, the smile is not well behaved in the long-term end, i.e. for highmaturities


Injecting SV in Levy models

Levy models can capture the IV smile for a single maturity, but do notaccommodate the term structure of the surface of in the long-termend.

On the other hand SV models are better at generating IVs acrossmaturities but fail to reproduce strong skews at short maturities.

To depict the volatility surface in the strike as well as in the maturitydimensions, Carr et al. propose the Levy-Stochastic Volatility

process.

Here, the driving source of randomness X is constructed bysubordinating a general Levy process L to an absolutely continuousrandom clock Y

(Xt) := (LYt), with Yt =

∫ t

0ys ds, (14)

where the instantaneous activity rate y is given by some positiveprocess independent of L.


Stochastic Volatility and Jump Modeling in Finance · 2013-01-21 · Stochastic Volatility and Jump...

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