Using Lévy Processes to Model Return Innovationsfaculty.baruch.cuny.edu › lwu › 890 ›...

Using Levy Processes to Model ReturnInnovations

Liuren Wu

Zicklin School of Business, Baruch College

Fall, 2007

Liuren Wu Levy Processes Option Pricing, Fall, 2007 1 / 37

Outline

1 Levy processes

2 Levy characteristics

3 Examples

4 Generation

5 Evidence

6 Jump design

7 Beyond Levy processes

8 Economic implications

9 Conclusion


Levy processes

A Levy process is a continuous-time process that generates stationary,independent increments ...

Think of return innovations (ε) in discrete time: Rt+1 = µt + σtεt+1.

I Normal return innovation — diffusionI Non-normal return innovation — jumps

Classic Levy specifications in finance:

I Brownian motion (Black-Scholes, Merton)I Compound Poisson process with normal jump size (Merton)

⇒ The return innovation distribution is either normal or mixture of normals.


Levy characteristics

Levy processes greatly expand our continuous-time choices of iid returninnovation distributions via the Levy triplet (µ, σ, π(x)). (π(x)–Levydensity).

The Levy-Khintchine Theorem:

φXt (u) ≡ E[e iuXt

]= e−tψ(u),

ψ(u) = −iuµ+ 12u2σ2 +

∫R0

(1− e iux + iux1|x|<1

)π(x)dx ,

Innovation distribution↔ characteristic exponent ψ(u)↔ Levy triplet (µ, σ, π(x))

I Constraint:∫ 1

0x2π(x)dx <∞.

I “Tractable:” if the integral can be carried out explicitly.

When well-defined, it is convenient to define the cumulatn exponent:

κ(s) ≡ 1

tln E

[esXt

]= sµ+

1

2s2σ2 +

∫R0

(esx − 1− sx1|x|<1

)π(x)dx .

ψ(u) = −κ(iu), κ(s) = −ψ(−is).


Model stock returns with Levy processes

Let Xt be a Levy process, κX (s) its cumulant exponent

The log return on a security can be modeled as

lnSt/S0 = µt + Xt − tκX (1)

where µ is the instantaneous drift (mean) of the stock such thatE[St ] = S0e

µt . The last term −tκX (1) is a convexity adjustment such thatXt − tκX (1) forms an exponential martingale:

E[eXt−tκX (1)

]= 1.

I Since both µ and κX (1) are deterministic components, they can becombined together: lnSt/S0 = mt + Xt , but it is more convenient toseparate them so that the mean instantaneous return µ is kept as aseparate free parameter.

I Under Q, µ = r − q.I Under this specification, we shall always set the first component of the

Levy triplet to zero (0, σ, π(x)), because it will be canceled out withthe convexity adjustment.


Characteristic function of the security return

st ≡ lnSt/S0 = µt + Xt − tκX (1)

The characteristic function for the security return is

φst (u) ≡ E[e iu ln St/S0

]= exp (− [−iuµ+ ψX (u) + iuκX (1)] t)

The characteristic exponent is

ψst (u) = −iuµ+ ψX (u) + iuκX (1)

Under Q, µ = r − q. The focus of the model specification is onXt ∼ (0, σ, π(x)), unless r and/or q are modeled to be stochastic.


Tractable examples of Levy processes1 Brownian motion (BSM) (µt + σWt): normal shocks.

2 Compound Poisson jumps (Merton, 76): Large but rare events.

π(x) = λ1√

2πvJexp

(− (x − µJ)

2

2vJ

).

3 Dampened power law (DPL):

π(x) =

{λ exp (−β+x) x−α−1, x > 0,λ exp (−β−|x |) |x |−α−1, x < 0,

λ, β± > 0,α ∈ [−1, 2)

I Finite activity when α < 0:∫

R0 π(x)dx <∞. Compound Poisson.Large and rare events.

I Infinite activity when α ≥ 0: Both small and large jumps. Jumpfrequency increases with declining jump size, and approaches infinity asx → 0.

I Infinite variation when α ≥ 1: many small jumps.

Market movements of all magnitudes, from small movements to marketcrashes.


Analytical characteristic exponents

Diffusion: ψ(u) = −iuµ+ 12u2σ2.

Merton’s compound Poisson jumps:

ψ(u) = λ(1− e iuµJ− 1

2 u2vJ

).

Dampened power law: ( for α 6= 0, 1)

ψ(u) = −λΓ(−α)[(β+ − iu)α − βα+ + (β− + iu)α − βα−

]− iuC (h)

I When α→ 2, smooth transition to diffusion (quadratic function of u).I When α = 0 (Variance-gamma by Madan et al):

ψ(u) = λ ln (1− iu/β+)`1 + iu/β−

´= λ

`ln(β+ − iu)− ln β + ln(β− + iu)− ln β−

´.

I When α = 1 (exponentially dampened Cauchy, Wu 2006):

ψ(u) = −λ`(β+ − iu) ln (β+ − iu) /β+ + λ

`β− + iu

´ln

`β− + iu

´/β−

´− iuC(h).


The Black-Scholes model

The driver is a Brownian motion Xt = σWt .

We can write the return as

lnSt/S0 = µt + σWt −1

2σ2t.

Note that κ(s) = 12 s2σ2.

The characteristic function of the return is:

φ(u) = exp

(iuµt − 1

2u2σ2t − iu

1

2σ2

)= exp

(iuµt − 1

2σ2(u2 + iu

)t

).

Under Q, µ = r − q.

The characteristic exponent of the convexity adjusted Levy process(Xt − κX (1)t) is: ψX (u) + iuκX (1) = 1

2u2σ2 + iu 12σ

2 = 12σ

2(u2 + iu).


Merton (1976)’s jump-diffusion model

The driver of this model is a Levy process that has both a diffusioncomponent and a jump component.

The Levy triplet is (0, σ, π(x)), with π(x) = λ 1√2πvJ

exp(− (x−µJ )

2

2vJ

).

I The first component of the triplet (the drift) is always normalized tozero.

I The characteristic exponent of the Levy process is

ψX (u) = 12u2σ2 + λ

(1− e iuµJ− 1

2 u2vJ

). The cumulant exponent is

κX (s) = 12 s2σ2 + λ

(esµJ+

12 s2vJ − 1

).

We can write the return as lnSt/S0 = µt +Xt −(

12σ

2 + λ(eµJ+

12 vJ − 1

))t.

The characteristic function of the return is:

φ(u) = e iuµte−12σ

2(u2+iu)te−

“λ

“1−e iuµJ−

12

u2vJ”+iuλ

“eµJ + 1

2vJ−1

””t.



Dampened power law (DPL)

The driver of this model is a pure jump Levy process, with its characteristicexponentψX (u) = −λΓ(−α)

[(β+ − iu)α − βα+ + (β− + iu)α − βα−

]− iuC (h). The

cumulant exponent isκX (s) = λΓ(−α)

[(β+ − s)α − βα+ + (β− + s)α − βα−

]+ sC (h)

We can write the return as, lnSt/S0 = µt + Xt − κX (1)t.

The characteristic function of the returnis:φ(u) = e iuµt e

−“−λΓ(−α)

h(β+−iu)α−βα

+ +“

β−+iu”α−βα

−i+iuλΓ(−α)

h(β+−1)α−βα

+ +“

β−+1”α−βα

−i”

t.


The characteristic exponent of the convexity adjusted Levy process(Xt − κX (1)t) is: ψX (u) + iuκX (1).

References:

I Carr, Geman, Madan, Yor, 2002, The Fine Structure of Asset Returns: An Empirical Investigation, Journal of

Business, 75(2), 305–332.

I Wu, 2006, Dampened Power Law: Reconciling the Tail Behavior of Financial Security Returns, Journal of

Business, 79(3), 1445–1474.


Special cases of DPL

α-stable law: No exponential dampening, β± = 0.Peter Carr, and Liuren Wu, Finite Moment Log Stable Process and Option Pricing, Journal of Finance, 2003, 58(2),

753–777.

Without exponential dampening, return moments greater than α are nolonger well defined.Characteristic function takes different form to account singularity.

Variance gamma (VG) model: α = 0. Madan, Carr, Chang, 1998, The Variance Gamma Process

and Option Pricing, European Finance Review, 2(1), 79–105.

The characteristic exponent takes a different form as α = 0 represents asingular point (Γ(0) not well defined).

Double exponential model: α = −1. Kou, 2002, A Jump-Diffusion Model for Option Pricing,

Management Science, 48(8), 1086–1101.


Other Levy examples

Other examples:

I The normal inverse Gaussian (NIG) process of Barndorff-Nielsen (1998)I The generalized hyperbolic process (Eberlein, Keller, Prause (1998))I The Meixner process (Schoutens (2003))I ...

Bottom line:

I All tractable in terms of analytical characteristic exponents ψ(u).

I We can use FFT to generate the density function of the innovation (formodel estimation).

I We can also use FFT to compute option values.


Why normal?

Traditional asset pricing theories all invariably start with a normaldistribution (or a Brownian motion in continuous time).

W.J. Youden, Experimentation and Measurement (page 55):

The normal law of error stands out in the experience of mankindas one of the broadest generalizations of natural philosophy. Itserves as the guiding instrument in researches in the physical andsocial sciences and in medicine, agriculture and engineering. It isan indispensable tool for the analysis and the interpretation of thebasic data obtained by observation and experiment.

G. Lippman, quoted in D’Arcy Thompson’s On Growth and Form V. I, p. 121:

Everybody believes in the normal approximation, theexperimenters because they believe it is a mathematical theorem,the mathematicians because they believe it is an experimentalfact!


Why normal?

Mark Kac, Statistical Independence inProbability Analysis and Number Theory, Chapter 3, The Normal Law (p. 52):

to quote a statement of Poincare, who said (partly in jest nodoubt) that there must be something mysterious about the normallaw, since mathematicians think it is a law of nature whereasphysicists are convinced that it is a mathematical theorem.

Dr. O. Lord, Language expert and scholar:

Generally Assumed Ubiquitous Symmetric Shape IdentifyingAdditive Noise

I Following the well-known law that every named quantity inmathematics was invented by somebody else, the credit for discoveringthe Gaussian bell-shaped curve should actually go to Abraham DeMoivre, who discovered it in 1733.

I Gauss and Laplace rediscovered it in 1809 and 1812 respectively intheir work on the theory of errors in observation.


The delusion of diffusion

Starting with Bachelier (1900), diffusion processes have been the mostwidely used class of stochastic processes used to describe the evolution ofasset prices over time.

The sample paths of diffusion processes are continuous over time, but arenowhere differentiable. The mathematical model is the idealization of thetrajectory of a single particle being constantly bombarded by an infinitenumber of infinitesimally small random forces.

Like a shark, a diffusion process must always be moving, or else it dies.

As Woody Allen said to Diane Keaton in describing their relationship, “Ithink what we have on our hands here is a dead shark” (Annie Hall).


Infinite variation of diffusion sample paths

In even the most active markets, one can find small enough time periodsover which there is positive probability of no price change.

Furthermore, if we sum the absolute values of price changes over a day, weget a finite number.

Diffusion processes have neither property. Over any finite time interval, thereis zero probability that the price does not change. Furthermore, the absolutevalues of price changes over a day (or any other period) sum to infinity.

If one tried to accurately draw a diffusion sample path, your pen would runout of ink before one second had elapsed.

The failure of diffusions to describe the microscopic behavior of sample pathswould not be troubling if financial theory took a more macroscopic view.

The problem is that the foundations of standard financial theories such asBlack Scholes and the intertemporal CAPM rest on the ability of investorsto continuously rebalance their portfolios.

Nobody seriously believes that anyone can trade continuously and even ifthey could, no one seriously believes that trade sizes can be kept so smallthat price impact is infinitesimal.


Time for a change

If we confront a mathematical diffusion with real-life sample paths, themodelling question becomes one of finding ways to slow down diffusions inorder to more accurately capture dynamics.

In 1949, Bochner introduced the notion of time change to stochasticprocesses. In 1973, Clark suggested that time-changed diffusions could beused to accurately describe financial time series.

Mathematically, a clock is just a weakly increasing stochastic process startedat zero. When one time changes a stochastic process, this clock is used toindex a stochastic process such as a diffusion. Clark suggested that the priceprocess runs on business time, while business time itself increases weaklyover calendar time.

The possibility that business time may not move while calendar timeinexorably marches forward is important for our purposes.


Classifying clocks

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

1 Continuous clocks have the property that business time is alwaysstrictly increasing over calendar time.

2 Clocks based on increasing jump processes have staircase like paths.

The first type of clock can be used to describe stochastic volatility models— next chapter.

The second type of clock has the capability of slowing a diffusion processdown to market speeds.

I A subordinator is a pure jump increasing process with stationaryindependent increments.


Run Brownian motions on different subordinators asbusiness clocks

If the clock is a standard Poisson process, with unit jump sizes and iidexponentially distributed inter-jump times:⇒ The resulting process is a compound Poisson process with normal jumpsizes.

If the clock is a compound Poisson process with exponentially distributedjump size with mean one:⇒ DPL with α = −1 Compound Poisson with Laplace (two-sidedexponential) jump sizes.⇒ Asymmetry can be induced by running the clock on a diffusion with drift(µt + Wt) instead of a standard Brownian motion.

If the clock is a gamma process⇒ DPL with α = 0 (variance gamma). Asymmetry can be induced byrunning the clock on a diffusion with drift (µt + Wt) instead of a standardBrownian motion.

continuous process.


General evidence on Levy return innovations

Credit risk: (compound) Poisson process

I The whole intensity-based credit modeling literature...

Market risk: Infinite-activity jumps

I Evidence from stock returns (CGMY (2002)): The α estimates forDPL on most stock return series are greater than zero.

I Evidence from options: Models with infinite-activity return innovationsprice equity index options better (Carr & Wu (2003), Huang & Wu (2004))

I Li, Wells, & Yu (2006): Infinite-activity jumps cannot be approximatedby finite-activity jumps.

The role of diffusion (in the presence of infinite-variation jumps)

I Not big, difficult to identify (CGMY (2002), Carr & Wu (2003a,b)).I Generate correlations with diffusive activity rates (Huang & Wu (2004)).I The diffusion (σ2) is identifiable in theory even in presence of

infinite-variation jumps (Aıt-Sahalia (2004), Aıt-Sahalia&Jacod 2005).


Implied volatility smiles & skews on a stock

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 20.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75AMD: 17−Jan−2006

Moneyness=ln(K/F )

σ√

τ

Impl

ied

Vola

tility Short−term smile

Long−term skew

Maturities: 32 95 186 368 732


Implied volatility skews on a stock index (SPX)

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 20.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22SPX: 17−Jan−2006

Moneyness=ln(K/F )

σ√

τ

Impl

ied

Vola

tility

More skews than smiles

Maturities: 32 60 151 242 333 704


Average implied volatility smiles on currencies

10 20 30 40 50 60 70 80 9011

11.5

12

12.5

13

13.5

14

Put delta

Ave

rage im

plie

d v

ola

tility

JPYUSD

10 20 30 40 50 60 70 80 908.2

8.4

8.6

8.8

9

9.2

9.4

9.6

9.8

Put deltaA

vera

ge im

plie

d vo

latil

ity

GBPUSD

Maturities: 1m (solid), 3m (dashed), 1y (dash-dotted)


(I) The role of jumps at very short maturities

Implied volatility smiles (skews) ↔ non-normality (asymmetry) for therisk-neutral return distribution.

IV (d) ≈ ATMV

(1 +

Skew.

6d +

Kurt.

24d2

), d =

lnK/F

σ√τ

Two mechanisms to generate return non-normality:

I Use Levy jumps to generate non-normality for the innovationdistribution.

I Use stochastic volatility to generates non-normality through mixingover multiple periods.

Over very short maturities (1 period), only jumps contribute to returnnon-normalities.


Time decay of short-term OTM optionsCarr& Wu, What Type of Process Underlies Options? A Simple Robust Test, JF, 2003, 58(6), 2581–2610.

As option maturity ↓ zero, OTM option value ↓ zero.

The speed of decay is exponential O(e−c/T ) under pure diffusion, but linearO(T ) in the presence of jumps.

Term decay plot: ln(OTM/T ) ∼ ln(T ) at fixed K :

In the presence of jumps, the Black-Scholes implied volatility for OTMoptions IV (τ,K ) explodes as τ ↓ 0.


(II) The impacts of jumps at very long horizons

Central limit theorem (CLT): Return distribution converge to normal withaggregation under certain conditions (finite return variance,...)⇒As option maturity increases, the smile should flatten.

Evidence: The skew does not flatten, but steepens!

FMLS (Carr&Wu, 2003): Maximum negatively skewed α-stable process.

I Return variance is infinite. ⇒ CLT does not apply.I Down jumps only. ⇒ Option has finite value.

But CLT seems to hold fine statistically:

0 5 10 15 20−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

Time Aggregation, Days

Skew

ness

Skewness on S&P 500 Index Return

0 5 10 15 200

5

10

15

20

25

30

35

40

45

Time Aggregation, Days

Kurto

sis

Kurtosis on S&P 500 Index Return


Reconcile P with Q via DPL jumpsWu, Dampened Power Law: Reconciling the Tail Behavior of Financial Security Returns, Journal of Business, 2006, 79(3),

1445–1474.

Model return innovations under P by DPL:

π(x) =

{λ exp (−β+x) x−α−1, x > 0,λ exp (−β−|x |) |x |−α−1, x < 0.

All return moments are finite with β± > 0. CLT applies.

Market price of jump risk (γ): dQdP∣∣t= E(−γX )

The return innovation process remains DPL under Q:

π(x) =

{λ exp (− (β+ + γ) x) x−α−1, x > 0,λ exp (− (β− − γ) |x |) |x |−α−1, x < 0.

To break CLT under Q, set γ = β− so that βQ− = 0.

Reconciling P with Q: Investors charge maximum allowed market price ondown jumps.


(III) Default risk & long-term implied vol skew

When a company defaults, its stock value jumps to zero.

It generates a steep skew in long-term stock options.

Evidence: Stock option implied volatility skews are correlated with creditdefault swap (CDS) spreads written on the same company.

02 03 04 05 06−1

0

1

2

3

4

GM: Default risk and long−term implied volatility skew

Negative skewCDS spread

Carr & Wu, Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation, wp.


Three Levy jump components in stock returns

I. Market risk (FMLS under Q, DPL under P)

I The stock index skew is strongly negative at long maturities.

II. Idiosyncratic risk (DPL under both P and Q)

I The smile on single name stocks is not as negatively skewed as that onindex at short maturities.

III. Default risk (Compound Poisson jumps).

I Long-term skew moves together with CDS spreads.

Information and identification:

I Identify market risk from stock index options.I Identify the credit risk component from the CDS market.I Identify the idiosyncratic risk from the single-name stock options.


Levy jump components in currency returns

Model currency return as the difference of the log pricing kernels betweenthe two economies.

Pricing kernel assigns market prices to systematic risks.

Market risk dominates for exchange rates between two industrializedeconomies (e.g., dollar-euro).

I Use a one-sided DPL for each economy (downward jump only).

Default risk shows up in FX for low-rating economies (say, dollar-peso).

I Peso drops by a large amount when the country (Mexico) defaults onits foreign debt.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), 2383–2403.

When pricing options on exchange rates, it is appropriate to distinguishbetween world risk versus country-specific risk.Bakshi, Carr, & Wu, Stochastic Risk Premiums, Stochastic Skewness in Currency Options, and Stochastic Discount

Factors in International Economies, JFE, forthcoming.


Pricing kernel: Review

In the absence of arbitrage, there exists at least one strictly positive process,Mt , the state-price deflator, in each economy such that the deflated gainsprocess associated with any admissible strategy is a martingale. The ratio ofMt at two horizons, Mt,T , is called a stochastic discount factor, or moreinformally, a pricing kernel.

I For an asset that has a terminal payoff ΠT , its time-t value ispt = EP

t [Mt,TΠT ].I In a discrete-time representative agent economy with an additive CRRA

utility, the pricing kernel is equal to the ratio of the marginal utilities ofconsumption,

Mt,t+1 = βu′(ct+1)

u′(ct)= β

(ct+1

ct

)−γI In an exchange economy, ct = wt , Mt,t+1 = βe−γ ln wt+1/wt , where the

log return on aggregate wealth lnwt+1/wt can be approximated by logreturn on a stock index/market portfolio.


Multiplicative representation of the pricing kernel

The multiplicative representation of the pricing kernel

Mt,T = exp

(−∫ T

t

rsds

)E

(−∫ T

t

γ>s dXs

)

r is the instantaneous riskfree rate, E is the stochastic exponentialmartingale operator,

I X denotes the risk sources in the economy. In the exchange economy,X denotes the log return on the aggregate wealth, Xt = ln wt/w0.

I γ is the market price of the risk X . If X is return on the aggregatewealth, γ would be the relative risk aversion.

I In case of Brownian risk, constant volatility, constant risk aversion, and

constant interest rate, we have Mt,T = e−rτe(−γσ(WT−Wt)− 12γ

2σ2τ).I In case of Levy risk, constant risk aversion, and constant interest rate,

we have Mt,T = e−rτe(−γ(XT−Xt)−κX (−γ)τ).


From pricing kernels to exchange rates

Let S fht denote the time-t currency-h price of currency f , with h being home

and f denoting foreign.

Currency returns are related to the pricing kernels of the two economies by

lnS fhT /S

fht = ln M f

t,T − lnMht,T

Example, S is dollar price of pound ($1.9 per pound), f would be UK, and hwould be US.

When each economy’s risk is modeled by a Levy process with constantrelative risk aversion and constant interest rates,

M it,T = e−r iτe(−γ i (X i

T−X it )−κXi (−γ i )τ) with i = h, f , we have

lnS fh

T

S fht

= (rd− r f )τ+γh(X hT −X h

t )+κX h(−γh)τ−γf (X fT −X f

t )−κX f (−γf )τ

Under the above Levy specification, what’s the expected excess return (riskpremium) on the currency investment? (assume independence between X h

and X f for simplicity)Answer: κ

Xh (γh) + κXh (−γh).


Beyond Levy processes

Levy processes can be used to generate different iid return innovationdistributions.

Yet, return distribution is not iid. It varies stochastically over time.

We need to go beyond Levy processes to capture the stochastic nature ofthe return distribution.

Applying separate stochastic time changes to different Levy componentsgenerates

I separate stochastic responses to each economic shock.I stochastic volatility, skewness, ...


Economic implications of using jumps

In the Black-Scholes world (one-factor diffusion):

I The market is complete with a bond and a stock.I The world is risk free after delta hedging.I Utility-free option pricing. Options are redundant.

In a pure-diffusion world with stochastic volatility:

I Market is complete with one (or a few) extra option(s).I The world is risk free after delta and vega hedging.

In a world with jumps of random sizes:

I The market is inherently incomplete (with stocks alone).I Need all options (+ model) to complete the market.I Derman: “Beware of economists with Greek symbols!”I Options market is informative/useful:

F Cross sections (K , T ) ⇔ Q dynamics.F Time series (t) ⇔ P dynamics.F The difference Q/P ⇔ market prices of economic risks.


Bottom line

Different types of jumps affect option pricing at both short and longmaturities.

I Implied volatility smiles at very short maturities can only beaccommodated by a jump component.

I Implied volatility skews at very long maturities ask for a jump processthat generates infinite variance.

I Credit risk exposure may also help explain the long-term skew on singlename stock options.

The choice of jump types depends on the events:

I Infinite-activity jumps ⇔ frequent market order arrival.I Finite-activity Poisson jumps ⇔ rare events (credit).

The presence of jumps of random sizes creates value for the options markets...


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