applying stochastic time changes to levy processes

7/30/2019 Applying stochastic time changes to Levy processes

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Applying stochastic time changes

to Levy processes

Liuren Wu

Zicklin School of Business, Baruch College

Option Pricing

Liuren Wu (Baruch) Stochastic time changes Option Pricing 1 / 38
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Outline

1 Stochastic time change

2 Option pricing

3 Model Design

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Outline


2 Option pricing

3 Model Design

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What Levy processes can and cannot do

Levy processes can generate different iid return innovation distributions.

Any distribution you can think of, we can specify a Levy process, withthe increments of the process matching that distribution.Caveat: The same type of distribution applies to all time horizons you may not be able to specify the distribution simultaneously atdifferent time horizons.

Levy processes cannot generate distributions that vary over time.

Returns modeled by Levy processes generate option implied volatilitysurfaces that stay the same over time (as a function of standardizedmoneyness and time to maturity).

Levy processes cannot capture the following salient features of the data:

Stochastic volatilityStochastic risk reversal (skewness)Stochastic correlation

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Capturing stochastic volatility via time changes

Discrete-time analog again: Rt+1 = t + tt+1

t+1 is an iid return innovation, with an arbitrary distributionassumption Levy process.t is the conditional volatility, t is the conditional mean return, bothof which can be time-varying, stochastic...

In continuous time, how do we model stochastic mean/volatility tractably?If the return innovation is modeled by a Brownian motion, we can letthe instantaneous variance to be stochastic and tractable, not volatility(Heston(1993), Bates (1996)).If the return innovation is modeled by a compound Poisson process, we

can let the Poisson arrival rate to be stochastic, not the mean jumpsize, jump distribution variance (Bates(2000), Pan(2002)).

If the return innovation is modeled by a general Levy process, it is tractableto randomize the time, or something proportional to time.

Variance of a Brownian motion, intensity of a Poisson process are both

proportional to time.Liuren Wu (Baruch) Stochastic time changes Option Pricing 5 / 38
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Randomize the time

Review the Levy-Khintchine Theorem:

(u) E eiuXt = et(u),(u) = iu + 12 u22 for diffusion with drift and variance 2 ,(u) =

1 eiuJ 12 u2vJ

for Mertons compound Poisson jump.

The drift , the diffusion variance 2, and the Poisson arrival rate are all

proportional to time t.

We can directly specify (t, 2t, t) as following stochastic processes.

Or we can randomize time t Tt for the same result.

We define Tt t0 vsds as the (stochastic) time change, with vt being theinstantaneous activity rate.

Depending on the Levy specification, the activity rate has the samemeaning (up to a scale) as a randomized version of the instantaneousdrift, instantaneous variance, or instantaneous arrival rate.

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Applying separate time changes

... to different Levy components

Consider a Levy process Xt (, 2, p(x)).

If we apply random time change to Xt XTt with Tt = t

0vsds, it is

equivalent to assuming that (t,

2

t, t) are all time varying, but theyare all proportional to one common source of variation vt.

If (t, 2t, t) vary separately, we need to apply separate time changes

to the three Levy components.

Decompose Xt into three Levy processes: X1

t (, 0, 0),

X

2

t

(0, 2

, 0), and X1

t

(0, 0, p(x)), and then apply separate timechanges to the three Levy processes.



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Interpretation I

We can think of t as the calendar time, and

Tt as the business time.

Business activity accumulates with calendar time, but the speed varies,depending on the business activity.

At heavy trading hours, one hour on a clock generates two hours worth ofbusiness activity (v

t= 2).

At afterhours, one hour generates half hour of activity (vt = 1/2).

Business activity tends to intensify before earnings announcements, FOMCmeeting days...

In this sense, vt captures the intensity of business activity at a certain time t.

Ane & Geman (2000, JFE): Stock returns are not normally distributed, butbecome normally distributed when they are scaled by number of trades.

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Interpretation II

We use Levy processes to model return innovations and stochastic timechanges to generate stochastic volatility and higher moments...

We can think of each Levy process as capturing one source of economicshock.

The stochastic time change on each Levy process captures the randomintensity of the impact of the economic shock on the financial security.

Return

K

i=1

XiTit

K

i=1

(Economic Shock From Source i)Stochastic impacts.

I like this interpretations. Many economic phenomena can be modeled interms of economic shocks and (time varying) intensities.

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Classification

In 1949, Bochner introduced the notion of time change to stochastic

processes. In 1973, Clark suggested that time-changed diffusions could beused to accurately describe financial time series.

At present, there are two types of clocks used to model business time:

1 Continuous clocks have the property that business time is always

strictly increasing over calendar time.2 Clocks based on increasing jump processes have staircase like paths.

The first type of business clock can be used to describe stochastic volatility which is what we do in this section.

The second type of clock can transform a diffusion into a jump process

All Levy processes considered in the previous section can be generated aschanging the clock of a diffusion with an increasing jump process(subordinator).

All semimartingales can be written as time-changed Brownian motion.



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Outline


2 Option pricing

3 Model Design

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Option pricing

To compute the time-0 price of a European option price with maturity at t,we first compute the Fourier transform of the log return ln St/S0. Then wecompute option value via Fourier inversions.

The Fourier transform of a time-changed Levy process:

Y(u) EQ

eiuXTt

= EQ eiuXTt+x(u)Ttex(u)Tt= EM

ex(u)Tt

, u D C,

where the new measure M is defined by the exponential martingale:dMdQ

t= exp (iuXTt + Ttx(u)) .

Without time-change, eiuXt+tx(u) is an exponential martingale byLevy-Khintchine Theorem.

A continuous time change does not change the martingality.Proof: Uwe Kuchler and Michael Sorensen, 1997, Exponential Families of Stochastic Processes, Springer.

M is complex valued (no longer a probability measure).



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Complex-valued measure change

When X and Tt = t

0vsds are independent, we have

Y(u) EQ

eiuXTt

= EQEQeiuXZ

Tt = Z= EQ

ex(u)Tt

,

by law of iterated expectations.

No measure change is necessary.The operation is similar to Hull and White (1987): The option value ofan independent stochastic volatility model is written as the expectationof the BMS formula over the distribution of the integrated variance,Tt =

t

0vsds.

A continuous time change does not change the martingality.

When X and vt are correlated, the measure change from Q to M hides thecorrelation under the new measure.

If we must give a name, lets call M the correlation neutral measure.

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Fourier transform

The Fourier transform of a time-changed Levy process:

Y(u) EQ

eiuXTt

= EM

ex(u)Tt

Tractability of the transform (u) depends on the tractability of

1 The characteristic exponent of the Levy process x(u)

Tractable Levy specifications include: Brownian motion, (Compound)Poisson, DPL, NIG, ... (done in previous section)

2 The Laplace transform ofTt under M.Tractable Laplace comes from activity rate dynamics: affine, quadratic,Wishart, 3/2The measure change from Q to M is defined by an exponentialmartingale.

The two (X, Tt) can be chosen separately as building blocks, for differentpurposes.



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The Laplace transform of the stochastic time TtWe have solved the characteristic exponent of the Levy process (by theLevy-Khintchine Theorem).

Now we try to solve the Laplace transform of the stochastic time,

LT() E

eTt

= E

e

t0

vsds

(1)

Recall the pricing equation for zero-coupon bonds:

B(0, t) EQ e t0 rsds (2)The two pricing equations look analogous (even though they are not related)

Both vt and rt need to be positive.If we set rt = vt, LT() is essentially the bond price.

The similarity allows us to borrow the vast literature on bond pricing:Affine class: Zero-coupon bond prices are exponential affine in thestate variable.Quadratic: Zero-coupon bond prices are exponential quadratic in thestate variable.

...Liuren Wu (Baruch) Stochastic time changes Option Pricing 15 / 38


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Review: Bond pricing dynamic term structure models

A long list of papers propose different dynamic term structure models:

Specific examples:

Vasicek, 1977, JFE: The instantaneous interest rate follows anOrnstein-Uhlenbeck process.

Cox, Ingersoll, Ross, 1985, Econometrica: The instantaneous interestrate follows a square-root process.

Many multi-factor examples ...Classifications (back-filling)

Duffie, Kan, 1996, Mathematical Finance: Spot rates are affinefunctions of state variables.Duffie, Pan, Singleton, 2000, Econometrica: Affine with jumps.

Duffie, Filipovic, Schachermayer, 2003, Annals of Applied Probability:Super mathematical representation and generalization of affine models.Leippold, Wu, 2002, JFQA: Spot rates are quadratic functions of statevariables.Filipovic, 2002, Mathematical Finance: How far can we go?Gabaix: Bond prices are affine.


Id if i d i d l


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Identifying dynamic term structure models:The forward and backward procedures

The traditional procedure:First, we make assumptions on factor dynamics (Z), market prices (),and how interest rates are related to the factors r(Z), based on whatwe think is reasonable.

Then, we derive the fair valuation of bonds based on these dynamics

and market price specifications.The back-filling (reverse engineering) procedure:

First, state the form of solution that we want for bond prices.Then, figure out what dynamics specifications generate the pricingsolutions that we want.

The dynamics are not specified to be reasonable, but specified togenerate a form of solution that we like.

It is good to be able to go both ways.

It is important not only to understand existing models, but also toderive new models that meet your work requirements.


B k filli ffi d l
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Back-filling affine models

What we want: Zero-coupon bond prices are exponential affine functions ofstate variables.

Continuously compounded spot rates are affine in state variables.It is simple and tractable. We can use spot rates as factors.

Let Z denote the state variables, let B(Zt, ) denote the time-t fair value ofa zero-coupon bond with time to maturity = T t, we have

B(Zt, ) = EQt

exp

T

t

r(Zs)ds

= expa() b()Zt

By writing B(Zt, ) and r(Zt), and solutions a(), b(), I am implicitlyfocusing on time-homogeneous models. Calendar dates do not matter.

This assumption is for (notational) simplicity more than anything else.With calendar time dependence, the notation can be changed to,B(Zt, t,T) and r(Zt, t). The solutions would be a(t,T), b(t,T).

Questions to be answered:

What is the short rate function r(Zt)?

Whats the dynamics of Zt under measure Q?Liuren Wu (Baruch) Stochastic time changes Option Pricing 18 / 38

Diff i d i
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Diffusion dynamics

To make the derivation easier, lets focus on diffusion factor dynamics:dZt = (Z)dt + (Z)dWt under Q.

We want to know: What kind of specifications for (Z), (Z) and r(Z)generate the affine solutions?

For a generic valuation problem,

f(Zt, t,T) = E

Q

t

exp

T

t r(Zs)dsT,

where T denotes terminal payoff, the value satisfies the following partialdifferential equation:

ft + Lf = rf, Lf infinitesimal generatorwith boundary condition f(T) = T.

Apply the PDE to the bond valuation problem,

Bt + BZ (Z) +

1

2

BZZ (Z)(Z) = rB

with boundary condition B(ZT, 0) = 1.Liuren Wu (Baruch) Stochastic time changes Option Pricing 19 / 38

B k filli


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Back filling

Starting with the PDE,

Bt + BZ (Z) +

1

2

BZZ (Z)(Z) = rB, B(ZT, 0) = 1.If B(Zt, ) = exp(a() b()Zt), we have

Bt = B

a() + b()Zt

, BZ = Bb(), BZZ = Bb()b(),

y(t, ) = 1 a() + b()Zt , r(Zt) = a(0) + b(0)Zt = ar + br Zt.Plug these back to the PDE,

a() + b()Zt b()(Z) + 12

b()b() (Z)(Z) = ar + br Zt

Question: What specifications of(Z) and (Z) guarantee the above PDE

to hold at all Z?Power expand (Z) and (Z)(Z) around Z and then collectcoefficients of Zp for p = 0, 1, 2 . These coefficients have to be zeroseparately for the PDE to hold at all times.


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Back filling

a() + b()Zt

b()(Z) +1

2 b()b() (Z)(Z)

= ar + br Zt

Set (Z) = am + bmZ + cmZZ + and

[(Z)(Z)]i = i + i Z + iZZ

+ , and collect terms:constant a() b()am + 12 b()b()

i = arZ b

()

b

()

bm +

1

2 b

()b

()

i =b

rZZ b()cm + 12

b()b() i = 0

The quadratic and higher-order terms are almost surely zero.

We thus have the conditions to have exponential-affine bond prices:

(Z) = am

+ bm

Z, [(Z)(Z)]i

= i

+ i

Z, r(Z) = ar

+ br

Z.

We can solve the coefficients [a(), b()] via the following ordinarydifferential equations:

a() = ar + b()am 12

b()b() i

b() = br + bm b()

12 b()b() istarting at a(0) = 0 and b(0) = 0.Liuren Wu (Baruch) Stochastic time changes Option Pricing 21 / 38

Quadratic and others


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Quadratic and others

Add jumps to the affine dynamics: The arrival rate of jumps need to beaffine in the state vector.

Can you identify the conditions for quadratic models: Bond prices areexponential quadratic in state variables?

Can you identify the conditions for cubic models: Bond prices areexponential cubic in state variables?

Affine bond prices: Recently Xavier Gabaix derive a model where bondprices are affine (not exponential affine!) in state variables.


From affine DTSM to affine activity rates
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From affine DTSM to affine activity rates

Review of affine DTSM: B(Z0, t) E e

t0

rsds = ea(t)b(t)Z0 if

rt = ar + br Zt, (Z) = ( Z), [(Z)(Z)]ii = i + i Z

By analogy, if we want:

LT(z)

E ezTt = E ez

t

0vsds = ea(t)b(t)

Z0 , we can set

vt = av + bv Zt, (Z) = ( Z), [(Z)(Z)]ii = i + i Z

Problem: We need the affine dynamics under the complex-valued measureM. Correlations between the Levy process X and the state vector Z can

make the whole thing messy. Affine dynamics under Q are no guarantee forexponential affine solution.

We use some concrete examples to show how this works.

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Example: Heston


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Example: Heston

SDE: dSt/St = (r q)dt +vtdWt withdvt = (

vt)dt + v

vtdW

vt ,E[dWtdW

vt ] = dt.

The measure change: dMdQ

= exp(iu(WTt 12Tt) + (u)Tt).The v dynamics under M:

dvt = ( vt)dt + E[iudWTtv

vtdWv

t ] + v

vtdWv

t

= ( vt)dt + iuvvtdt + vvtdWvt= (

M

vt)dt + vvtdWv

t

with M = iuv.Note that dWTt and

vtdWt are equivalent in distribution.

Since the v dynamics are affine under M, we have the Laplace transform

exponential affine in v,s(u) EQ

eiuln St/S0

= eiu(rq)tEQ

eiu(WTt

12Tt)

= eiu(rq)tEM

e(u)Tt

= eiu(rq)ta(t)b(t)v0

with a(t) = b(t) and b(t) = (u) Mb(t) 12 b(t)22v, starting ata(0) = b(0) = 0.Liuren Wu (Baruch) Stochastic time changes Option Pricing 25 / 38

Example: HestonNot all affine works
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Example: HestonNot all affine works

The key for tractability is to maintain the v dynamics affine under M.

Suppose we extend the Heston specification:dvt = ( vt)dt + v + vtdWvt , which is still affine under Q, but it isno longer affine under M:

dvt = ( vt)dt + E[iudWTt, v + vtdW

vt ] + v

vtdW

vt

unless we redefine the time change as Tt = t

0 ( + v

s)ds

or we set = 0.The constant volatility specification does not work either:dvt = ( vt)dt + vdWvt (in addition to the fact that vt can go to zerothis time).

A general affine specification vt = av + bv Zt does not always work for the

same reason, but the following two-factor specification works:

dvt = (mt vt)dt + v

vtdWv

t , dmt = m(m mt)dt + m

mtdWm

t

with E[dWmt dWv

t ] = E[dWm

t dWt] = 0.


Example: Bates jump-diffusion stochastic volatility model
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Example: Bates jump-diffusion stochastic volatility model

SDE: dSt/St = (r q)dt +vtdWt + dJ() (emuJ+ 12 vJ 1)dt withdvt = (

vt)dt + v

vtdW

vt ,E[dWtdW

vt ] = dt. The stock price

process includes compound Poisson jump process, with arrival rate .Conditional on a jump occurring, the jump size in return has a normaldistribution (J, vJ).

We can write the security return as a time-changed Levy process,

ln St/S0 = (r q)t + [WTt 12Tt] + [Xt kX(1)t], Tt =t

0vsds.

where X denotes a pure-jump Levy process with Levy density given by

(x) = 1

2vJe

(xJ)2

2vJ

The cumulant exponent is

kx(s) =

R0

(esx 1)(x)dx =

esJ+12 s

2vJ 1.


Example: Bates (1996)
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Example: Bates (1996)

The Fourier transform of the return,

s(u) EQ

e

iuln St/S0= e

iu(rq)t

E

Q e

iu(WTt12Tt)

E

Q e

iu(XtkX(1)t)= eiu(rq)tea(t)b(t)v0 eJ(u)

where [a(t), b(t)] come directly from the Heston model and thecharacteristic exponent of the pure-jump Levy process (concavity adjusted)is:

J(u) =R0

(1eiux)(x)dx+iukX(1) = (1eiuJ 12 u2vJ)+iu

eJ+

12 vJ 1

By definition, jumps are orthogonal to diffusion. Hence, the two componentscan be processed separately.

Question: Why just time change diffusion W? Why not also time changethe jump X?

ln St/S0 = (r q)t + [WTt 1

2Tt] + [XTxt kX(1)Txt ].

Also, replace the compound Possion jump with any type of jump you like

SV4 in Huang and Wu (2004).Liuren Wu (Baruch) Stochastic time changes Option Pricing 28 / 38

Outline
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Outline


2 Option pricing

3 Model Design


Model design: General principles
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Model design: General principles

Start with the risk-neutral (Q) process Thats where tractability isneeded the most dearly.

Identify the economic sources, model each with a Levy process (Xkt ).Decide whether to apply separate time changes: Xkt XkTkt to makethe impact of this economic source stochastic over time.Concavity adjust each component to guarantee the martingalecondition: EQ[St/S0] = e

(rq)t.

ln St/S0 = (r q)t +K

k=1

bkXk

Tkt kxk(bk)Tkt

,

For tractability,

Use Levy processes that generate tractable characteristic exponents(X(u)).Use time changes that generate tractable Laplace transformsOrthogonalize the economic shocks Xk such that

s(u) = eiu(rq)t

k

Eeiu

bkXkTkt

kxk

(bk)Tkt


Market prices and statistics dynamics
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Market prices and statistics dynamics

Since we can always use Euler approximation for model estimation,tractability requirement is not as strong for the statistical dynamics.

We can specify pretty much any forms for the market prices subject to (i)technical conditions, (ii) economic sensibility, and (iii) identificationconcerns.

Simple/parsimonious specification: Constant market prices of return and vol

risks (k, kv)

Mt = ertK

k=1

expkXkTkt xk (k) T

kt kvXkvTkt xkv (kv) T

kt

,

Wt

constant drift adjustment = 2.

Pure jump Levy process P(x) = exQ(x), drift adjustment: = PJ(1) QJ (1) = QJ (1 + ) QJ () QJ (1).Time change: instantaneous risk premium (vt) proportional to therisk level vt.


Example: Return on a stock
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p

Model the return on a stock to reflect shocks from two sources:

Credit risk: In case of corporate default, the stock price falls to zero.Model the impact as a Poisson Levy jump process with log return

jumps to negative infinity upon jump arrival.

Market risk: Daily market movements (small or large). Model the

impact as a diffusion or infinite-activity (infinite variation) Levy jumpprocess or both.

Apply separate time changes to the two Levy components to capture (1) theintensity variation of corporate default, (2) the market risk (volatility)

variation.

Key: Each component has a specific economic purpose.

Carr and Wu, Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation, JFEC, 2010


Example: A CAPM model
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p

Example: A CAPM model :

ln Sjt/Sj0 = (r q)t +

jXmTmt xm (j)Tmt

+

Xj

Tj

t

xj(1)Tjt.

Estimate and market prices of return and volatility risk using indexand single name options.Cross-sectional analysis of the estimates:Are the beta estimates similar to estimates from time-series stockreturn regressions?

An international CAPM:

Henry Mo, and Liuren Wu, International Capital Asset Pricing: Evidence from Options, Journal of Empirical Finance, 2007,

14(4), 465498.


Example: Return on an exchange rate
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p g

Exchange rate reflects the interaction between two economic forces.

Use two Levy processes to model the two economic forces separately.

Consider a negatively skewed distribution (downside jumps) from eacheconomic source (crash-o-phobia from both sides). Use the difference tomodel the currency return between the two economies.

Apply separate time changes to the two Levy processes to capture thestrength variation of the two economic forces.

Stochastic time changes on the two negatively skewed Levy processesgenerate both stochastic volatility and stochastic skew.

Key: Each component has its specific economic purpose.

Peter Carr, and Liuren Wu, Stochastic Skew in Currency Options, Journal of Financial Economics, 2007, 86(1), 213247.


Example: Currencies returns and sovereign CDS
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p g

Dollar price of peso drops by a significant amount when Mexico defaults onits sovereign debt.

Currency return on peso contains both market risk and credit risk.

The intensities of both types of risks are stochastic (and probablycorrelated).

Peter Carr, and Liuren Wu, Theory and Evidence on the Dynamic Interactions Between Sovereign Credit Default Swaps

and Currency Options, Journal of Banking and Finance, 2007, 31(8), 23832403.


Exchange rates and pricing kernels
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Exchange rate reflects the interaction between two economic forces.

The economic meaning becomes clearer if we model the pricing kernel ofeach economy.

Let mUS0,t and mJP0,t denote the pricing kernels of the US and Japan.

Then the dollar price of yen St is given by

ln St/S0 = ln mJP

0,tln mUS

0,t.

If we model the negative of the logarithm of each pricing kernel( ln mj0,t) as a time-changed Levy process, XjTjt (j = US, JP) withnegative skewness. Then, ln St/S0 = ln m

JP0,t ln mUS0,t = XUSTUSt X

JPTJPt

Consistent and simultaneous modeling of all currency pairs (not limitedto 2 economies).

Bakshi, Carr, and Wu, Stochastic Risk Premium, Stochastic Skewness, and Stochastic Discount Factors in InternationalEconomies JFE, 2008, 87(1), 132-156.

Reverse engineer the pricing kernel of US, UK, and Japan using currency options on dollar-yen, dollar-pound, andpound-yen.


Limitations and extensions
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Recall the rule of thumb for model design:

ln St/S0 = (r q)t +K

k=1

bkXkTkt kxk(bk)Tkt

,

Xk are orthogonal. Generate correlation among different factors viafactor loadings bk.

Limitation: Constant factor loading (bk

), together with orthogonalityassumption, puts restrictions on co-movements across assets.

Future work: How to allow flexible correlation dynamics (with independentvariation)?

Each asset (economic shock) has its own business clock.

How to model the co-movements of business clocks of multiple assets(economic sources)?

If economics ask for it, we may also need to get out of theexponential-affine setup:

Carr&Wu: Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions, wp.Liuren Wu (Baruch) Stochastic time changes Option Pricing 37 / 38

Concluding remarks
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Modeling security returns with (time-changed) Levy processes enjoys threekey virtues:

Generality: Levy process can be made to capture any return innovationdistribution; applying time changes can make this distribution varystochastic over time.

Explicit economic mapping: Each Levy component captures shocks

from one economic source. Time changes capture the relative variationof the intensities of these impacts.

Tractability: Combining any tractable Levy process (with tractable(u)) with any tractable activity rate dynamics (with a tractableLaplace) generates a tractable Fourier transform for the time changed

Levy process. The two specifications are separate.

It is a nice place to start with for generating security return dynamics thatare parsimonious, tractable, economically sensible, and statisticallyperforming well.

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applying stochastic time changes to levy processes

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