Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Skewness Premium in Lévy Markets

José Fajardo1 Ernesto Mordecki2

1Economics DepartmentIBMEC Business School

2CMATUniversidad de La Republica del Uruguay

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Outline

1 Option Market

2 Lévy Processes

3 Duality

4 Symmetry

5 Skewness Premium

6 Time-Changed Lévy Processes

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Observed moneyness biases in American call and putoptions: S&P500 options traded on CMEX

American Foreign currency call options traded inPhiladelphia Stock ExchangeThe Biases are not in the same direction, nor are theyconstant over time.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Observed moneyness biases in American call and putoptions: S&P500 options traded on CMEXAmerican Foreign currency call options traded inPhiladelphia Stock Exchange

The Biases are not in the same direction, nor are theyconstant over time.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Observed moneyness biases in American call and putoptions: S&P500 options traded on CMEXAmerican Foreign currency call options traded inPhiladelphia Stock ExchangeThe Biases are not in the same direction, nor are theyconstant over time.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Some facts

Out-of-the-money (OTM) Calls pays only if the asset pricerises above the Call’s exercise price while OTM Puts payoff only if asset price falls below the Put’s exercise price.

Call and Put prices directly reflects characteristics of theupper and lower tails of the risk neutral distribution.Then relative prices of OTM options will reflect theskewness of the risk neutral distribution.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Some facts

Out-of-the-money (OTM) Calls pays only if the asset pricerises above the Call’s exercise price while OTM Puts payoff only if asset price falls below the Put’s exercise price.Call and Put prices directly reflects characteristics of theupper and lower tails of the risk neutral distribution.

Then relative prices of OTM options will reflect theskewness of the risk neutral distribution.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Some facts

Out-of-the-money (OTM) Calls pays only if the asset pricerises above the Call’s exercise price while OTM Puts payoff only if asset price falls below the Put’s exercise price.Call and Put prices directly reflects characteristics of theupper and lower tails of the risk neutral distribution.Then relative prices of OTM options will reflect theskewness of the risk neutral distribution.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Put-Call Parity:p + S = c + Xe−rT

Just for European Options!

Same Strike

Put-Call Duality:

c(S0,K , r , δ, τ, ψ) = p(K ,S0, δ, r , τ, ψ)

European and American Options!

Different Strike

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Put-Call Parity:p + S = c + Xe−rT

Just for European Options!

Same Strike

Put-Call Duality:

c(S0,K , r , δ, τ, ψ) = p(K ,S0, δ, r , τ, ψ)

European and American Options!

Different Strike

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Put-Call Parity:p + S = c + Xe−rT

Just for European Options!

Same Strike

Put-Call Duality:

c(S0,K , r , δ, τ, ψ) = p(K ,S0, δ, r , τ, ψ)

European and American Options!

Different Strike

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Put-Call Parity:p + S = c + Xe−rT

Just for European Options!

Same Strike

Put-Call Duality:

c(S0,K , r , δ, τ, ψ) = p(K ,S0, δ, r , τ, ψ)

European and American Options!

Different Strike

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Put-Call Parity:p + S = c + Xe−rT

Just for European Options! Same Strike

Put-Call Duality:

c(S0,K , r , δ, τ, ψ) = p(K ,S0, δ, r , τ, ψ)

European and American Options! Different Strike

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

From Duality

Call Options x% out-of-the-money are priced exactly x% higherthan the corresponding OTM put:

C(F ,T ; Kc) = (1 + x)P(F ,T ; Kp), x > 0

Where F is the future price;

Kc = F (1 + x) and Kp = F/(1 + x). i.e. KcKp = F 2.

Bates’ x% rule!

Under a symmetry condition

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

From Duality

Call Options x% out-of-the-money are priced exactly x% higherthan the corresponding OTM put:

C(F ,T ; Kc) = (1 + x)P(F ,T ; Kp), x > 0

Where F is the future price;

Kc = F (1 + x) and Kp = F/(1 + x). i.e. KcKp = F 2.

Bates’ x% rule!

Under a symmetry condition

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

From Duality

Call Options x% out-of-the-money are priced exactly x% higherthan the corresponding OTM put:

C(F ,T ; Kc) = (1 + x)P(F ,T ; Kp), x > 0

Where F is the future price;

Kc = F (1 + x) and Kp = F/(1 + x). i.e. KcKp = F 2.

Bates’ x% rule!

Under a symmetry condition

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

From Duality

Call Options x% out-of-the-money are priced exactly x% higherthan the corresponding OTM put:

C(F ,T ; Kc) = (1 + x)P(F ,T ; Kp), x > 0

Where F is the future price;

Kc = F (1 + x) and Kp = F/(1 + x). i.e. KcKp = F 2.

Bates’ x% rule!

Under a symmetry condition

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

From Duality

Call Options x% out-of-the-money are priced exactly x% higherthan the corresponding OTM put:

C(F ,T ; Kc) = (1 + x)P(F ,T ; Kp), x > 0

Where F is the future price;

Kc = F (1 + x) and Kp = F/(1 + x). i.e. KcKp = F 2.

Bates’ x% rule!

Under a symmetry condition

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Skewness Premium (SK): David S. Bates

The Crash of ’87 – Was It Expected? The Evidence fromOptions Markets, Journal of Finance 46:3, 1991,1009–1044.

Dollar Jump Fears, 1984-1992: DistributionalAbnormalities Implicit in Foreign Currency FuturesOptions, Journal of International Money and Finance 15:1,1996, 65–93.The Skewness Premium: Option Pricing UnderAsymmetric Processes, Advances in Futures and OptionsResearch 9, 1997, 51-82.For which parameters SK = C

P − 1 ≶ 0?

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Skewness Premium (SK): David S. Bates

The Crash of ’87 – Was It Expected? The Evidence fromOptions Markets, Journal of Finance 46:3, 1991,1009–1044.Dollar Jump Fears, 1984-1992: DistributionalAbnormalities Implicit in Foreign Currency FuturesOptions, Journal of International Money and Finance 15:1,1996, 65–93.

The Skewness Premium: Option Pricing UnderAsymmetric Processes, Advances in Futures and OptionsResearch 9, 1997, 51-82.For which parameters SK = C

P − 1 ≶ 0?

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Skewness Premium (SK): David S. Bates

The Crash of ’87 – Was It Expected? The Evidence fromOptions Markets, Journal of Finance 46:3, 1991,1009–1044.Dollar Jump Fears, 1984-1992: DistributionalAbnormalities Implicit in Foreign Currency FuturesOptions, Journal of International Money and Finance 15:1,1996, 65–93.The Skewness Premium: Option Pricing UnderAsymmetric Processes, Advances in Futures and OptionsResearch 9, 1997, 51-82.

For which parameters SK = CP − 1 ≶ 0?

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Skewness Premium (SK): David S. Bates

The Crash of ’87 – Was It Expected? The Evidence fromOptions Markets, Journal of Finance 46:3, 1991,1009–1044.Dollar Jump Fears, 1984-1992: DistributionalAbnormalities Implicit in Foreign Currency FuturesOptions, Journal of International Money and Finance 15:1,1996, 65–93.The Skewness Premium: Option Pricing UnderAsymmetric Processes, Advances in Futures and OptionsResearch 9, 1997, 51-82.For which parameters SK = C

P − 1 ≶ 0?

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Option Market Data: 08/31/2006

0.8 0.85 0.9 0.95 1 1.050

0.04

0.08

0.12

0.16

0.2

Strike Price/ Future Price

Opt

ion

Pric

e/ F

utur

e P

rice

CallsCall splinePuts

0.8 0.85 0.9 0.95 1 1.050

0.04

0.08

0.12

0.16

0.2

Strike Price/ Future Price

Opt

ion

Pric

e/ F

utur

e P

rice

CallsPutsPut spline

Option Prices on S&P500, T=09/15/2006.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

OTM options S&P500 - 08/31/06. T=09/15/06,F=1303.82

Kc Kp = F 2/Kc x = Kc/F − 1 xobs = cobs/pint − 1 x − xobs1305 1302.641 0.00091 0.61456 -0.613661310 1297.669 0.00474 0.53280 -0.528061315 1292.735 0.00856 0.42730 -0.418721320 1287.838 0.01241 0.10891 -0.096501325 1282.979 0.01625 -0.11658 0.132831330 1278.155 0.02008 -0.45097 0.471051335 1273.368 0.02392 -0.50378 0.527701340 1268.617 0.02775 -0.61306 0.640811345 1263.901 0.03158 -0.73872 0.770311350 1259.220 0.03542 -0.81448 0.849901355 1254.573 0.03925 -0.80297 0.842221360 1249.961 0.04309 -0.82437 0.86745

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

OTM options S&P500 - 08/31/06. T= 09/15/06,F=1303.82

Kp Kc = F 2/Kp x = F/Kp − 1 xobs = cint/pobs − 1 x − xobs1250 1359.957 0.043056 -0.88837 0.9314211255 1354.539 0.0389 -0.86897 0.9078731260 1349.164 0.034778 -0.85655 0.8913311265 1343.831 0.030688 -0.78107 0.811761270 1338.541 0.02663 -0.70531 0.7319411275 1333.291 0.022604 -0.63926 0.6618691280 1328.083 0.018609 -0.51726 0.5358651285 1322.916 0.014646 -0.31216 0.3268011290 1317.788 0.010713 -0.20329 0.2140051295 1312.7 0.006811 -0.03659 0.0433971300 1307.651 0.002938 0.090739 -0.0878

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

ITM options S&P500 - 08/31/06. T=09/15/06,F=1303.82

Kc Kp = F 2/Kc x = Kc/F − 1 xobs = cobs/pint − 1 x − xobs1230 1382.070 -0.05662 0.05068 -0.107301235 1376.475 -0.05278 0.13642 -0.189201240 1370.925 -0.04895 0.11501 -0.163951245 1365.419 -0.04511 0.19770 -0.242811250 1359.957 -0.04128 0.27794 -0.319221255 1354.539 -0.03744 0.28073 -0.318171260 1349.164 -0.03361 0.53629 -0.569901265 1343.831 -0.02977 0.57498 -0.604761270 1338.541 -0.02594 0.60672 -0.632661275 1333.291 -0.02210 0.67537 -0.697481280 1328.083 -0.01827 0.69133 -0.709591285 1322.916 -0.01443 0.96631 -0.980741290 1317.788 -0.01060 0.90484 -0.915441295 1312.700 -0.00676 0.79406 -0.800821300 1307.651 -0.00293 0.78018 -0.78311

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

ITM options S&P500 - 08/31/06. T=09/15/06,F=1303.82

Kp Kc = F 2/Kp x = F/Kp − 1 xobs = cint/pobs − 1 x − xobs1305 1302.641 -0.0009 0.13084 -0.131751310 1297.669 -0.00472 0.25254 -0.257261315 1292.735 -0.0085 0.26191 -0.270411320 1287.838 -0.01226 0.24282 -0.255071325 1282.979 -0.01598 0.34642 -0.362401330 1278.155 -0.01968 0.18321 -0.202891335 1273.368 -0.02336 0.23799 -0.261351340 1268.617 -0.02700 0.14586 -0.172861345 1263.901 -0.03062 0.15264 -0.183251350 1259.220 -0.03421 0.10121 -0.135421355 1254.573 -0.03777 -0.03964 0.001871360 1249.961 -0.04131 0.02834 -0.069651365 1245.382 -0.04482 -0.01010 -0.034721375 1236.325 -0.05177 -0.04510 -0.00667

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Skewness Premium (SK)

OTM options: Usually, xobs < x . That means cp − 1 < x .

ITM options: Usually, xobs > x . That means cp − 1 > x .

Asymmetry in the market.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Skewness Premium (SK)

OTM options: Usually, xobs < x . That means cp − 1 < x .

ITM options: Usually, xobs > x . That means cp − 1 > x .

Asymmetry in the market.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Skewness Premium (SK)

OTM options: Usually, xobs < x . That means cp − 1 < x .

ITM options: Usually, xobs > x . That means cp − 1 > x .

Asymmetry in the market.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Contribution

Theoretical proposition that quantify the relation betweenOTM Calls and Puts when the underlying follows aGeometric Lévy Process.

Simply diagnostic for judging which distributions areconsistent with observed option prices.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call relationshipSkewness PremiumContribution

Contribution

Theoretical proposition that quantify the relation betweenOTM Calls and Puts when the underlying follows aGeometric Lévy Process.Simply diagnostic for judging which distributions areconsistent with observed option prices.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Lévy Processes

Consider a stochastic process X = Xtt≥0, defined on(Ω,F ,F = (Ft)t≥0,Q). We say that X = Xtt≥0 is a LévyProcess if:

X has paths RCLLX0 = 0, and has independent increments, given0 < t1 < t2 < ... < tn, the r.v.

Xt1 ,Xt2 − Xt1 , · · · ,Xtn − Xtn−1

are independents.The distribution of the increment Xt − Xs is homogenous intime, that is, depends just on the difference t − s.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Lévy Processes

Consider a stochastic process X = Xtt≥0, defined on(Ω,F ,F = (Ft)t≥0,Q). We say that X = Xtt≥0 is a LévyProcess if:

X has paths RCLL

X0 = 0, and has independent increments, given0 < t1 < t2 < ... < tn, the r.v.

Xt1 ,Xt2 − Xt1 , · · · ,Xtn − Xtn−1

are independents.The distribution of the increment Xt − Xs is homogenous intime, that is, depends just on the difference t − s.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Lévy Processes

Consider a stochastic process X = Xtt≥0, defined on(Ω,F ,F = (Ft)t≥0,Q). We say that X = Xtt≥0 is a LévyProcess if:

X has paths RCLLX0 = 0, and has independent increments, given0 < t1 < t2 < ... < tn, the r.v.

Xt1 ,Xt2 − Xt1 , · · · ,Xtn − Xtn−1

are independents.

The distribution of the increment Xt − Xs is homogenous intime, that is, depends just on the difference t − s.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Lévy Processes

Consider a stochastic process X = Xtt≥0, defined on(Ω,F ,F = (Ft)t≥0,Q). We say that X = Xtt≥0 is a LévyProcess if:

X has paths RCLLX0 = 0, and has independent increments, given0 < t1 < t2 < ... < tn, the r.v.

Xt1 ,Xt2 − Xt1 , · · · ,Xtn − Xtn−1

are independents.The distribution of the increment Xt − Xs is homogenous intime, that is, depends just on the difference t − s.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Lévy-Khintchine Formula

A key result in the theory of Lévy Processes is theLévy-Khintchine formula:

E(ezXt ) = etψ(z)

Where ψ is called characteristic exponent, and is given by:

ψ(z) = az +12σ2z2 +

∫IR(ezy − 1− zy1|y |<1)Π(dy),

where a and σ ≥ 0 are real constants, and Π is a positivemeasure in IR − 0 such that

∫(1 ∧ y2)Π(dy) <∞, called the

Lévy measure.

The triplet (a, σ2,Π) is the characteristic triplet.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Lévy-Khintchine Formula

A key result in the theory of Lévy Processes is theLévy-Khintchine formula:

E(ezXt ) = etψ(z)

Where ψ is called characteristic exponent, and is given by:

ψ(z) = az +12σ2z2 +

∫IR(ezy − 1− zy1|y |<1)Π(dy),

where a and σ ≥ 0 are real constants, and Π is a positivemeasure in IR − 0 such that

∫(1 ∧ y2)Π(dy) <∞, called the

Lévy measure. The triplet (a, σ2,Π) is the characteristic triplet.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Lévy-Khintchine Formula

A key result in the theory of Lévy Processes is theLévy-Khintchine formula:

E(ezXt ) = etψ(z)

Where ψ is called characteristic exponent, and is given by:

ψ(z) = az +12σ2z2 +

∫IR(ezy − 1− zy1|y |<1)Π(dy),

where a and σ ≥ 0 are real constants, and Π is a positivemeasure in IR − 0 such that

∫(1 ∧ y2)Π(dy) <∞, called the

Lévy measure. The triplet (a, σ2,Π) is the characteristic triplet.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Lévy-Khintchine Formula

A key result in the theory of Lévy Processes is theLévy-Khintchine formula:

E(ezXt ) = etψ(z)

Where ψ is called characteristic exponent, and is given by:

ψ(z) = az +12σ2z2 +

∫IR(ezy − 1− zy1|y |<1)Π(dy),

where a and σ ≥ 0 are real constants, and Π is a positivemeasure in IR − 0 such that

∫(1 ∧ y2)Π(dy) <∞, called the

Lévy measure.

The triplet (a, σ2,Π) is the characteristic triplet.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Lévy-Khintchine Formula

A key result in the theory of Lévy Processes is theLévy-Khintchine formula:

E(ezXt ) = etψ(z)

Where ψ is called characteristic exponent, and is given by:

ψ(z) = az +12σ2z2 +

∫IR(ezy − 1− zy1|y |<1)Π(dy),

where a and σ ≥ 0 are real constants, and Π is a positivemeasure in IR − 0 such that

∫(1 ∧ y2)Π(dy) <∞, called the

Lévy measure. The triplet (a, σ2,Π) is the characteristic triplet.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call Duality

Model

Consider a market with two assets given by

S1t = eXt , and S2

t = S20ert

where (X ) is a one dimensional Lévy process, and forsimplicity, and without loss of generality we take S1

0 = 1.

In this model we assume that the stock pays dividends withconstant rate δ ≥ 0, and that the given probability measure Q isthe chosen equivalent martingale measure.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call Duality

Model

Consider a market with two assets given by

S1t = eXt , and S2

t = S20ert

where (X ) is a one dimensional Lévy process, and forsimplicity, and without loss of generality we take S1

0 = 1.

In this model we assume that the stock pays dividends withconstant rate δ ≥ 0, and that the given probability measure Q isthe chosen equivalent martingale measure.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call Duality

Duality

Denote by MT the class of stopping times up to a fixedconstant time T , i.e:

MT = τ : 0 ≤ τ ≤ T , τ stopping time w.r.t F

for the finite horizon case and for the perpetual case we takeT = ∞ and denote by M the resulting stopping times set.

Then,for each stopping time τ ∈MT we introduce

c(S0,K , r , δ, τ, ψ) = E e−rτ (Sτ − K )+, (1)p(S0,K , r , δ, τ, ψ) = E e−rτ (K − Sτ )+. (2)

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call Duality

Duality

Denote by MT the class of stopping times up to a fixedconstant time T , i.e:

MT = τ : 0 ≤ τ ≤ T , τ stopping time w.r.t F

for the finite horizon case and for the perpetual case we takeT = ∞ and denote by M the resulting stopping times set.Then,for each stopping time τ ∈MT we introduce

c(S0,K , r , δ, τ, ψ) = E e−rτ (Sτ − K )+, (1)p(S0,K , r , δ, τ, ψ) = E e−rτ (K − Sτ )+. (2)

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call Duality

Duality

For the American finite case, prices and optimal stopping rulesτ∗c and τ∗p are defined, respectively, by:

C(S0,K , r , δ,T , ψ) = supτ∈MT

E e−rτ (Sτ − K )+

= E e−rτ∗c (Sτ∗c − K )+ (3)P(S0,K , r , δ,T , ψ) = sup

τ∈MT

E e−rτ (K − Sτ )+

= E e−rτ∗p (K − Sτ∗p )+, (4)

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call Duality

Duality

And for the American perpetual case, prices and optimalstopping rules are determined by

C(S0,K , r , δ, ψ) = supτ∈M

E e−rτ (Sτ − K )+1τ<∞

= E e−rτ∗c (Sτ∗c − K )+1τ<∞, (5)

P(S0,K , r , δ, ψ) = supτ∈M

E e−rτ (K − Sτ )+1τ<∞

= E e−rτ∗p (K − Sτ∗p )+1τ<∞. (6)

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call Duality

Lemma (Duality)

Consider a Lévy market with driving process X with characteristicexponent ψ(z). Then, for the expectations introduced in (1) and (2)we have

c(S0,K , r , δ, τ, ψ) = p(K ,S0, δ, r , τ, ψ), (7)

where

ψ(z) = az +12σ2z2 +

∫IR

(ezy − 1− zh(y)

)Π(dy) (8)

is the characteristic exponent (of a certain Lévy process) that satisfies8><>:

a = δ − r − σ2/2 −R

IR`ey − 1 − h(y)

´Π(dy),

σ = σ,

Π(dy) = e−y Π(−dy).

(9)

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call Duality

Duality

Corollary (European Options)

For the expectations introduced in (1) and (2) we have

c(S0,K , r , δ,T , ψ) = p(K ,S0, δ, r ,T , ψ), (10)

with ψ and ψ as in the Duality Lemma.

Corollary (American Options)

For the value functions in (3) and (4) we have

C(S0,K , r , δ,T , ψ) = P(K ,S0, δ, r ,T , ψ), (11)

with ψ and ψ as in the Duality Lemma.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call Duality

Duality

Corollary (European Options)

For the expectations introduced in (1) and (2) we have

c(S0,K , r , δ,T , ψ) = p(K ,S0, δ, r ,T , ψ), (10)

with ψ and ψ as in the Duality Lemma.

Corollary (American Options)

For the value functions in (3) and (4) we have

C(S0,K , r , δ,T , ψ) = P(K ,S0, δ, r ,T , ψ), (11)

with ψ and ψ as in the Duality Lemma.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Put-Call Duality

Duality

Corollary (Perpetual Options)

For prices of Perpetual Call and Put options in (5) and (6) the optimalstopping rules have, respectively, the form

τ∗c = inft ≥ 0 : St ≥ S∗c ,

τ∗p = inft ≥ 0 : St ≤ S∗p.

where the constants S∗c and S∗

p are the critical prices. Then, we have

C(S0,K , r , δ, ψ) = P(K ,S0, δ, r , ψ), (12)

with ψ and ψ as in the Duality Lemma. Furthermore, when δ > 0, forthe optimal stopping levels, we obtain the relation S∗

c S∗p = S0K .

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Dual marketsGiven a Lévy market with driving process characterized by ψ,consider a market model with two assets, a deterministicsavings account B = Btt≥0, given by

Bt = eδt , δ ≥ 0,

and a stock S = Stt≥0, modelled by

St = KeXt , S0 = K > 0,

where Xt = −Xt is a Lévy process with characteristic exponentunder Q given by ψ in (8).The process St represents the priceof KS0 dollars measured in units of stock S.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Dual marketsGiven a Lévy market with driving process characterized by ψ,consider a market model with two assets, a deterministicsavings account B = Btt≥0, given by

Bt = eδt , δ ≥ 0,

and a stock S = Stt≥0, modelled by

St = KeXt , S0 = K > 0,

where Xt = −Xt is a Lévy process with characteristic exponentunder Q given by ψ in (8).

The process St represents the priceof KS0 dollars measured in units of stock S.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Dual marketsGiven a Lévy market with driving process characterized by ψ,consider a market model with two assets, a deterministicsavings account B = Btt≥0, given by

Bt = eδt , δ ≥ 0,

and a stock S = Stt≥0, modelled by

St = KeXt , S0 = K > 0,

where Xt = −Xt is a Lévy process with characteristic exponentunder Q given by ψ in (8).The process St represents the priceof KS0 dollars measured in units of stock S.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Symmetric markets

Lets define symmetric markets by

L(e−(r−δ)t+Xt | Q

)= L

(e−(δ−r)t−Xt | Q

), (13)

meaning equality in law.

A necessary and sufficient condition for (13) to hold is

Π(dy) = e−yΠ(−dy), (14)

This ensures Π = Π, and from this follows

a− (r − δ) = a− (δ − r)

, giving (13), as always σ = σ.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Symmetric markets

Lets define symmetric markets by

L(e−(r−δ)t+Xt | Q

)= L

(e−(δ−r)t−Xt | Q

), (13)

meaning equality in law.

A necessary and sufficient condition for (13) to hold is

Π(dy) = e−yΠ(−dy), (14)

This ensures Π = Π, and from this follows

a− (r − δ) = a− (δ − r)

, giving (13), as always σ = σ.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Bates’ x%-RuleIf the call and put options have strike prices x% out-of-themoney relative to the forward price, then the call should bepriced x% higher than the put.

If r = δ, we can take the future price F as the underlying assetin Lemma 1.

CorollaryTake r = δ and assume (14) holds, we have

c(F ,Kc , r , τ, ψ) = (1 + x) p(F ,Kp, r , τ, ψ), (15)

where Kc = (1 + x)F and Kp = F/(1 + x), with x > 0.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Bates’ x%-RuleIf the call and put options have strike prices x% out-of-themoney relative to the forward price, then the call should bepriced x% higher than the put.

If r = δ, we can take the future price F as the underlying assetin Lemma 1.

CorollaryTake r = δ and assume (14) holds, we have

c(F ,Kc , r , τ, ψ) = (1 + x) p(F ,Kp, r , τ, ψ), (15)

where Kc = (1 + x)F and Kp = F/(1 + x), with x > 0.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Bates’ x%-RuleIf the call and put options have strike prices x% out-of-themoney relative to the forward price, then the call should bepriced x% higher than the put.

If r = δ, we can take the future price F as the underlying assetin Lemma 1.

CorollaryTake r = δ and assume (14) holds, we have

c(F ,Kc , r , τ, ψ) = (1 + x) p(F ,Kp, r , τ, ψ), (15)

where Kc = (1 + x)F and Kp = F/(1 + x), with x > 0.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Empirical Evidence of Symmetry

We restrict to Lévy markets with jump measure of the form

Π(dy) = eβyΠ0(dy),

where Π0(dy) is a symmetric measure, i.e. Π0(dy) = Π0(−dy),everything with respect to the risk neutral measure Q.

As a consequence of (14), market is symmetric if and only ifβ = −1/2.

In view of this, we propose to measure the asymmetryin the market through the parameter β + 1/2. Whenβ + 1/2 = 0 we have a symmetric market.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Empirical Evidence of Symmetry

We restrict to Lévy markets with jump measure of the form

Π(dy) = eβyΠ0(dy),

where Π0(dy) is a symmetric measure, i.e. Π0(dy) = Π0(−dy),everything with respect to the risk neutral measure Q.

As a consequence of (14), market is symmetric if and only ifβ = −1/2.

In view of this, we propose to measure the asymmetryin the market through the parameter β + 1/2. Whenβ + 1/2 = 0 we have a symmetric market.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Empirical Evidence of Symmetry

We restrict to Lévy markets with jump measure of the form

Π(dy) = eβyΠ0(dy),

where Π0(dy) is a symmetric measure, i.e. Π0(dy) = Π0(−dy),everything with respect to the risk neutral measure Q.

As a consequence of (14), market is symmetric if and only ifβ = −1/2.

In view of this, we propose to measure the asymmetryin the market through the parameter β + 1/2. Whenβ + 1/2 = 0 we have a symmetric market.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Esscher Transform

We can obtain an Equivalent Martingale Measure by

dQt =eθXt

EP eθXtdPt

There is a θ such that the discounted price process is amartingale respect to Q.

As a consequence:βQ = βP + θ

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Esscher Transform

We can obtain an Equivalent Martingale Measure by

dQt =eθXt

EP eθXtdPt

There is a θ such that the discounted price process is amartingale respect to Q.

As a consequence:βQ = βP + θ

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Esscher Transform

We can obtain an Equivalent Martingale Measure by

dQt =eθXt

EP eθXtdPt

There is a θ such that the discounted price process is amartingale respect to Q.

As a consequence:βQ = βP + θ

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Example 0: Diffusions with jumps

Consider the jump - diffusion model proposed by Merton(1976). The driving Lévy process in this model has Lévymeasure given by

Π(dy) = λ1

δ√

2πe−(y−µ)2/(2δ2)dy ,

and is direct to verify that condition (14) holds if and only if2µ+ δ2 = 0. This result was obtained by Bates (1997) for futureoptions.

That result is obtained as a particular case: β = µδ2

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Example 0: Diffusions with jumps

Consider the jump - diffusion model proposed by Merton(1976). The driving Lévy process in this model has Lévymeasure given by

Π(dy) = λ1

δ√

2πe−(y−µ)2/(2δ2)dy ,

and is direct to verify that condition (14) holds if and only if2µ+ δ2 = 0. This result was obtained by Bates (1997) for futureoptions.

That result is obtained as a particular case: β = µδ2

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Example 1Consider the Generalized Hyperbolic Distributions, with Lévymeasure:

Π(dy) = eβy 1

|y|

“ Z ∞

0

exp`−

p2z + α2|y|

´π2z

`J2λ

(δ√

2z) + Y 2λ

(δ√

2z)´ dz + 1λ≥0λe−α|y|

”dy

where α, βP, λ, δ are the historical parameters that satisfy theconditions 0 ≤ |βP| < α, and δ > 0; and Jλ, Yλ are the Besselfunctions of the first and second kind.

Eberlein and Prause (1998): German StocksFajardo and Farias (2004): Ibovespa

βP = −0.0035 and βQ = 80.65.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Example 1Consider the Generalized Hyperbolic Distributions, with Lévymeasure:

Π(dy) = eβy 1

|y|

“ Z ∞

0

exp`−

p2z + α2|y|

´π2z

`J2λ

(δ√

2z) + Y 2λ

(δ√

2z)´ dz + 1λ≥0λe−α|y|

”dy

where α, βP, λ, δ are the historical parameters that satisfy theconditions 0 ≤ |βP| < α, and δ > 0; and Jλ, Yλ are the Besselfunctions of the first and second kind.

Eberlein and Prause (1998): German Stocks

Fajardo and Farias (2004): Ibovespa

βP = −0.0035 and βQ = 80.65.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Example 1Consider the Generalized Hyperbolic Distributions, with Lévymeasure:

Π(dy) = eβy 1

|y|

“ Z ∞

0

exp`−

p2z + α2|y|

´π2z

`J2λ

(δ√

2z) + Y 2λ

(δ√

2z)´ dz + 1λ≥0λe−α|y|

”dy

where α, βP, λ, δ are the historical parameters that satisfy theconditions 0 ≤ |βP| < α, and δ > 0; and Jλ, Yλ are the Besselfunctions of the first and second kind.

Eberlein and Prause (1998): German StocksFajardo and Farias (2004): Ibovespa

βP = −0.0035 and βQ = 80.65.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Example 1Consider the Generalized Hyperbolic Distributions, with Lévymeasure:

Π(dy) = eβy 1

|y|

“ Z ∞

0

exp`−

p2z + α2|y|

´π2z

`J2λ

(δ√

2z) + Y 2λ

(δ√

2z)´ dz + 1λ≥0λe−α|y|

”dy

where α, βP, λ, δ are the historical parameters that satisfy theconditions 0 ≤ |βP| < α, and δ > 0; and Jλ, Yλ are the Besselfunctions of the first and second kind.

Eberlein and Prause (1998): German StocksFajardo and Farias (2004): Ibovespa

βP = −0.0035 and βQ = 80.65.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Estimated Parameters

Sample α β δ µ λ LLHBbas4 30.7740 3.5267 0.0295 -0.0051 -0.0492 3512.73Bbdc4 47.5455 -0.0006 0 0 1 3984.49Brdt4 56.4667 3.4417 0.0026 -0.0026 1.4012 3926.68Cmig4 1.4142 0.7491 0.0515 -0.0004 -2.0600 3685.43Csna3 46.1510 0.0094 0 0 0.6910 3987.52Ebtp4 3.4315 3.4316 0.0670 -0.0071 -2.1773 1415.64Elet6 1.4142 0.0120 0.0524 0 -1.8987 3539.06Ibvsp 1.7102 -0.0035 0.0357 0.0020 -1.8280 4186.31Itau4 49.9390 1.7495 0 0 1 4084.89Petr4 7.0668 0.4848 0.0416 0.0003 -1.6241 3767.41Tcsl4 1.4142 0 0.0861 0.0011 -2.6210 1329.64Tlpp4 6.8768 0.4905 0.0359 0 -1.3333 3766.28Tnep4 2.2126 2.2127 0.0786 -0.0028 -2.2980 1323.66Tnlp4 1.4142 0.0021 0.0590 0.0005 -2.1536 1508.22Vale5 25.2540 2.6134 0.0265 -0.0015 -0.6274 3958.47

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Update 12/2006Asset Subclass λ δ µ α βBbdc4 GH -1,7449 0,0364 -0,0006 11,1286 2,3276

NIG -0,5000 0,0248 -0,0009 34,0259 2,8102HYP 1,0000 0,0079 -0,0012 56,0122 3,1663

Cmig4 GH -2,5789 0,0570 -0,0007 1,4228 1,4223NIG -0,5000 0,0366 -0,0009 35,9907 1,6624HYP 1,0000 0,0205 -0,0010 52,4466 1,7550

Csna3 GH 0,2208 0,0000 0,0000 15,2576 1,2717NIG -0,5000 0,0314 -0,0007 36,4310 2,2515HYP 1,0000 0,0100 -0,0006 51,2647 2,1889

Ibov GH -1,0181 0.0259 0,0026 29,8996 -3,2944NIG -0,5000 0,0223 0,0028 40,1759 -3,5297HYP 1,0000 0,0106 0,0030 67,5016 -4,0177

Itau4 GH 1,1611 0,0000 0,0000 57,9608 1,8698NIG -0,5000 0,0267 -0,0008 39,6274 3,0841HYP 1,0000 0,0094 -0,0006 58,6820 2,7652

Petr4 GH -1,7205 0,0368 0,0012 5,0578 -0,0807NIG -0,5000 0,0245 0,0012 29,4792 -0,1395HYP 1,0000 0,0091 0,0011 53,4252 -0,0244

Tnlp4 GH -2,0605 0,0441 -0,0006 1,3513 1,3497NIG -0,5000 0,0282 -0,0009 31,9832 1,7147HYP 1,0000 0,0120 -0,0013 52,6370 2,2161

Vale5 GH -1,8872 0,0372 -0,0004 6,1221 2,2153NIG -0,5000 0,0242 -0,0006 33,9667 2,5194HYP 1,0000 0,0087 -0,0007 57,2995 2,8143

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Example 2

Consider the Meixner distribution, with Lévy measure:

Π(dy) = ce

ba y

y sinh(πy/a)dy ,

where a,b and c are parameters of the Meixner density, suchthat a > 0, −π < b < π and c > 0. Then βP = b/a.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Index a b θ βQ + 1/2Nikkei 225 0.02982825 0.12716244 0.42190524 5.18506

DAX 0.02612297 -0.50801886 -4.46513538 -23.4123FTSE-100 0.01502403 -0.014336370 -4.34746821 -4.8017

Nasdaq Comp. 0.03346698 -0.49356259 -5.95888693 -20.2066CAC-40. 0.02539854 -0.23804755 -5.77928595 -14.6518

Schoutens (2002) estimates with data 1/1/1997 to 12/31/1999

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Example 3

This CGMY model, proposed by Carr et al. (2002) ischaracterized by σ = 0 and Lévy measure given by (1), wherethe function p(y) is given by

p(y) =C

|y |1+Y e−α|y |.

The parameters satisfy C > 0, Y < 2, and G = α+ β ≥ 0,M = α− β ≥ 0, where C,G,M,Y are the parameters of themodel.

Values of β = (G−M)/2 are obtained for different assets underthe market risk neutral measure and in the general situation,the parameter β is negative and less than −1/2.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Example 3

This CGMY model, proposed by Carr et al. (2002) ischaracterized by σ = 0 and Lévy measure given by (1), wherethe function p(y) is given by

p(y) =C

|y |1+Y e−α|y |.

The parameters satisfy C > 0, Y < 2, and G = α+ β ≥ 0,M = α− β ≥ 0, where C,G,M,Y are the parameters of themodel.

Values of β = (G−M)/2 are obtained for different assets underthe market risk neutral measure and in the general situation,the parameter β is negative and less than −1/2.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Implied volatility

Any model satisfying (14) must have identicalBlack-Scholes implicit volatilities for calls and puts withstrikes ln(Kc/F ) = ln x = − ln(Kp/F ), with x > 0 arbitrary.

That is, the volatility smile curve is symmetric in themoneyness ln(K/F ).By put-call parity, European calls and puts with same strikeand maturity must have identical implicit volatilities.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Implied volatility

Any model satisfying (14) must have identicalBlack-Scholes implicit volatilities for calls and puts withstrikes ln(Kc/F ) = ln x = − ln(Kp/F ), with x > 0 arbitrary.That is, the volatility smile curve is symmetric in themoneyness ln(K/F ).

By put-call parity, European calls and puts with same strikeand maturity must have identical implicit volatilities.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Bates’ x% RuleEmpirical EvidenceExamplesOther implication

Implied volatility

Any model satisfying (14) must have identicalBlack-Scholes implicit volatilities for calls and puts withstrikes ln(Kc/F ) = ln x = − ln(Kp/F ), with x > 0 arbitrary.That is, the volatility smile curve is symmetric in themoneyness ln(K/F ).By put-call parity, European calls and puts with same strikeand maturity must have identical implicit volatilities.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Skewness Premium (SK)

The x% Skewness Premium is defined as the percentagedeviation of x% OTM call prices from x% OTM put prices.

SK (x) =c(S,T ; Xc)

p(S,T ; Xp)− 1, for European Options, (16)

SK (x) =C(S,T ; Xc)

P(S,T ; Xp)− 1, for American Options,

where Xp = F(1+x) < F < F (1 + x) = Xc , x > 0

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Skewness Premium (SK)

The x% Skewness Premium is defined as the percentagedeviation of x% OTM call prices from x% OTM put prices.

SK (x) =c(S,T ; Xc)

p(S,T ; Xp)− 1, for European Options, (16)

SK (x) =C(S,T ; Xc)

P(S,T ; Xp)− 1, for American Options,

where Xp = F(1+x) < F < F (1 + x) = Xc , x > 0

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Skewness Premium (SK)

The x% Skewness Premium is defined as the percentagedeviation of x% OTM call prices from x% OTM put prices.

SK (x) =c(S,T ; Xc)

p(S,T ; Xp)− 1, for European Options, (16)

SK (x) =C(S,T ; Xc)

P(S,T ; Xp)− 1, for American Options,

where Xp = F(1+x) < F < F (1 + x) = Xc , x > 0

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Skewness Premium (SK)

The SK was addressed for the following stochastic processes:Constant Elasticity of Variance (CEV), include arithmeticand geometric Brownian motion:

dSt = µStdt + σS1−ρt dBt

Stochastic Volatility processes, the benchmark modelbeing those for which volatility evolves independently of theasset price:

dSt = µStdt + σtStdB1t ,

dσt = δ(θ − σt)dt + ϑdB2t

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Skewness Premium (SK)

The SK was addressed for the following stochastic processes:Constant Elasticity of Variance (CEV), include arithmeticand geometric Brownian motion:

dSt = µStdt + σS1−ρt dBt

Stochastic Volatility processes, the benchmark modelbeing those for which volatility evolves independently of theasset price:

dSt = µStdt + σtStdB1t ,

dσt = δ(θ − σt)dt + ϑdB2t

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Skewness Premium (SK)

• Jump-diffusion processes, the benchmark model is theMerton’s (1976) model:

dSt = (µ− λκ)Stdt + σStdBt + Stdqt

µ expected return on asset

q is a Poisson processes generating jumps. dq and dB areindep.Y − 1 is the size of the random jump (price change due tojumps as % of price).

The jumps are i.i.dκ mean size of jumps = E(Y − 1).ln(Y ) is Normal with variance δ2.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Skewness Premium (SK)

• Jump-diffusion processes, the benchmark model is theMerton’s (1976) model:

dSt = (µ− λκ)Stdt + σStdBt + Stdqt

µ expected return on assetq is a Poisson processes generating jumps. dq and dB areindep.

Y − 1 is the size of the random jump (price change due tojumps as % of price).

The jumps are i.i.dκ mean size of jumps = E(Y − 1).ln(Y ) is Normal with variance δ2.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Skewness Premium (SK)

• Jump-diffusion processes, the benchmark model is theMerton’s (1976) model:

dSt = (µ− λκ)Stdt + σStdBt + Stdqt

µ expected return on assetq is a Poisson processes generating jumps. dq and dB areindep.Y − 1 is the size of the random jump (price change due tojumps as % of price).

The jumps are i.i.d

κ mean size of jumps = E(Y − 1).ln(Y ) is Normal with variance δ2.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Skewness Premium (SK)

• Jump-diffusion processes, the benchmark model is theMerton’s (1976) model:

dSt = (µ− λκ)Stdt + σStdBt + Stdqt

µ expected return on assetq is a Poisson processes generating jumps. dq and dB areindep.Y − 1 is the size of the random jump (price change due tojumps as % of price). The jumps are i.i.d

κ mean size of jumps = E(Y − 1).

ln(Y ) is Normal with variance δ2.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Skewness Premium (SK)

• Jump-diffusion processes, the benchmark model is theMerton’s (1976) model:

dSt = (µ− λκ)Stdt + σStdBt + Stdqt

µ expected return on assetq is a Poisson processes generating jumps. dq and dB areindep.Y − 1 is the size of the random jump (price change due tojumps as % of price). The jumps are i.i.dκ mean size of jumps = E(Y − 1).

ln(Y ) is Normal with variance δ2.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Skewness Premium (SK)

• Jump-diffusion processes, the benchmark model is theMerton’s (1976) model:

dSt = (µ− λκ)Stdt + σStdBt + Stdqt

µ expected return on assetq is a Poisson processes generating jumps. dq and dB areindep.Y − 1 is the size of the random jump (price change due tojumps as % of price). The jumps are i.i.dκ mean size of jumps = E(Y − 1).ln(Y ) is Normal with variance δ2.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Some results

For European options in general and for American options onfutures, the SK has the following properties for the abovedistributions.

SK (x) ≶ x for CEV processes with ρ ≶ 1.

SK (x) ≶ x for jump-diffusions with log-normal jumpsdepending on whether 2µ+ δ2 ≶ 0.SK (x) ≶ x for Stochastic Volatility processes dependingon whether ρSσ ≶ 0.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Some results

For European options in general and for American options onfutures, the SK has the following properties for the abovedistributions.

SK (x) ≶ x for CEV processes with ρ ≶ 1.SK (x) ≶ x for jump-diffusions with log-normal jumpsdepending on whether 2µ+ δ2 ≶ 0.

SK (x) ≶ x for Stochastic Volatility processes dependingon whether ρSσ ≶ 0.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Some results

For European options in general and for American options onfutures, the SK has the following properties for the abovedistributions.

SK (x) ≶ x for CEV processes with ρ ≶ 1.SK (x) ≶ x for jump-diffusions with log-normal jumpsdepending on whether 2µ+ δ2 ≶ 0.SK (x) ≶ x for Stochastic Volatility processes dependingon whether ρSσ ≶ 0.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Some results

Now in equation (16) consider

Xp = F (1− x) < F < F (1 + x) = Xc , x > 0.

Then,SK (x) < 0 for CEV processes only if ρ < 0.

SK (x) ≥ 0 for CEV processes only if ρ ≥ 0.

When x is small, the two SK measures will be approx. equal.For in-the-money options (x < 0), the propositions arereversed.

Calls x% in-the-money should cost 0%− x% less than puts x%in-the-money.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Some results

Now in equation (16) consider

Xp = F (1− x) < F < F (1 + x) = Xc , x > 0.

Then,SK (x) < 0 for CEV processes only if ρ < 0.SK (x) ≥ 0 for CEV processes only if ρ ≥ 0.

When x is small, the two SK measures will be approx. equal.For in-the-money options (x < 0), the propositions arereversed.

Calls x% in-the-money should cost 0%− x% less than puts x%in-the-money.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Some results

Now in equation (16) consider

Xp = F (1− x) < F < F (1 + x) = Xc , x > 0.

Then,SK (x) < 0 for CEV processes only if ρ < 0.SK (x) ≥ 0 for CEV processes only if ρ ≥ 0.

When x is small, the two SK measures will be approx. equal.

For in-the-money options (x < 0), the propositions arereversed.

Calls x% in-the-money should cost 0%− x% less than puts x%in-the-money.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Some results

Now in equation (16) consider

Xp = F (1− x) < F < F (1 + x) = Xc , x > 0.

Then,SK (x) < 0 for CEV processes only if ρ < 0.SK (x) ≥ 0 for CEV processes only if ρ ≥ 0.

When x is small, the two SK measures will be approx. equal.For in-the-money options (x < 0), the propositions arereversed.

Calls x% in-the-money should cost 0%− x% less than puts x%in-the-money.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Some results

Now in equation (16) consider

Xp = F (1− x) < F < F (1 + x) = Xc , x > 0.

Then,SK (x) < 0 for CEV processes only if ρ < 0.SK (x) ≥ 0 for CEV processes only if ρ ≥ 0.

When x is small, the two SK measures will be approx. equal.For in-the-money options (x < 0), the propositions arereversed.

Calls x% in-the-money should cost 0%− x% less than puts x%in-the-money.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Proof

By duality we have:

C(F ,T ; Fk , σk1−ρ, ρ, b) = kP(F ,T ; F/k , σ, ρ,b), fork > 0.

The idea is to exploit monotonicity of option prices on volatility.If k > 1 and ρ < 1 then σk1−ρ > σ,

C(F ,T ; Fk , σ, ρ,b) < C(F ,T ; Fk , σk1−ρ, ρ, b)

C(F ,T ; Fk , σ, ρ,b) < kP(F ,T ; F/k , σ, ρ,b)

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Proof

By duality we have:

C(F ,T ; Fk , σk1−ρ, ρ, b) = kP(F ,T ; F/k , σ, ρ,b), fork > 0.

The idea is to exploit monotonicity of option prices on volatility.If k > 1 and ρ < 1 then σk1−ρ > σ,

C(F ,T ; Fk , σ, ρ,b) < C(F ,T ; Fk , σk1−ρ, ρ, b)

C(F ,T ; Fk , σ, ρ,b) < kP(F ,T ; F/k , σ, ρ,b)

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Proof

By duality we have:

C(F ,T ; Fk , σk1−ρ, ρ, b) = kP(F ,T ; F/k , σ, ρ,b), fork > 0.

The idea is to exploit monotonicity of option prices on volatility.If k > 1 and ρ < 1 then σk1−ρ > σ,

C(F ,T ; Fk , σ, ρ,b) < C(F ,T ; Fk , σk1−ρ, ρ, b)

C(F ,T ; Fk , σ, ρ,b) < kP(F ,T ; F/k , σ, ρ,b)

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Some Questions

What are the implications of a possible mispecification ofmodels when using a fixed EMM?

Given a model specified under the physical measure,which of two EMM gives the higher option price?

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Some Questions

What are the implications of a possible mispecification ofmodels when using a fixed EMM?Given a model specified under the physical measure,which of two EMM gives the higher option price?

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Some Answers

We say that a model is convexity preserving, if for anyconvex contract function, the corresponding price is convexas a function of the price of the underlying asset at alltimes prior to maturity.

There are many examples of models that are not convexitypreserving.General Stochastic Volatility Models.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Some Answers

We say that a model is convexity preserving, if for anyconvex contract function, the corresponding price is convexas a function of the price of the underlying asset at alltimes prior to maturity.There are many examples of models that are not convexitypreserving.

General Stochastic Volatility Models.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Some Answers

We say that a model is convexity preserving, if for anyconvex contract function, the corresponding price is convexas a function of the price of the underlying asset at alltimes prior to maturity.There are many examples of models that are not convexitypreserving.General Stochastic Volatility Models.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Mispecification

European Option price is monotonically increasing in thevolatility if and only if the model is convexity preserving.

Bergman, Y. Z., Grundy, B. D. and Wiener, Z. (1996):“General Properties of Option Prices”, J. Finance 51,1573–1610.

El Karoui, N., Jeanblanc-Picque, M. and Shreve, S. (1998):“Robustness of Black and Scholes formula”, Math. Finance8, 93–126.

• Ekström, E. and Tysk, J. (2005): “Properties of Option Pricesin Models with Jumps”. Preprint.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Mispecification

European Option price is monotonically increasing in thevolatility if and only if the model is convexity preserving.

Bergman, Y. Z., Grundy, B. D. and Wiener, Z. (1996):“General Properties of Option Prices”, J. Finance 51,1573–1610.El Karoui, N., Jeanblanc-Picque, M. and Shreve, S. (1998):“Robustness of Black and Scholes formula”, Math. Finance8, 93–126.

• Ekström, E. and Tysk, J. (2005): “Properties of Option Pricesin Models with Jumps”. Preprint.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Mispecification

European Option price is monotonically increasing in thevolatility if and only if the model is convexity preserving.

Bergman, Y. Z., Grundy, B. D. and Wiener, Z. (1996):“General Properties of Option Prices”, J. Finance 51,1573–1610.El Karoui, N., Jeanblanc-Picque, M. and Shreve, S. (1998):“Robustness of Black and Scholes formula”, Math. Finance8, 93–126.

• Ekström, E. and Tysk, J. (2005): “Properties of Option Pricesin Models with Jumps”. Preprint.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Option Price Ordering

This monotonicity property is very related to the the ordering ofprices by changing the EMM.

Bellamy, N., and Jeanblanc-Picque, M. (2000): “Incompleteness of marketsdriven by a mixed diffusion”. Finance and Stochastics 4, 209–222.

Henderson, V. and Hobson, D. (2003): “Coupling and Option Price Comparisonsin a jump-diffusion model”. Stoch. Stoch. Rep. 75, 79–101.

Jakubenas, P. (2002): “On option pricing in certain incomplete markets”. ProcSteklov Inst. Math. 237, 114–133.

Henderson, V. (2005): “Analytical comparisons of option prices in stochasticvolatility models”. Math. Finance, 15, 49–59.

Gushchin, A. A., and Mordecki, E. (2002): “Bounds on option prices forsemimartingale market models”. Proc. Steklov Inst. Math. 237, 73–113.

Bergenthum, J. and L. Rüschendorf (2006):“Comparison of option prices insemimartingale models”. Finance and Stochastics 10, 222–249.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Option Price Ordering

This monotonicity property is very related to the the ordering ofprices by changing the EMM.

Bellamy, N., and Jeanblanc-Picque, M. (2000): “Incompleteness of marketsdriven by a mixed diffusion”. Finance and Stochastics 4, 209–222.

Henderson, V. and Hobson, D. (2003): “Coupling and Option Price Comparisonsin a jump-diffusion model”. Stoch. Stoch. Rep. 75, 79–101.

Jakubenas, P. (2002): “On option pricing in certain incomplete markets”. ProcSteklov Inst. Math. 237, 114–133.

Henderson, V. (2005): “Analytical comparisons of option prices in stochasticvolatility models”. Math. Finance, 15, 49–59.

Gushchin, A. A., and Mordecki, E. (2002): “Bounds on option prices forsemimartingale market models”. Proc. Steklov Inst. Math. 237, 73–113.

Bergenthum, J. and L. Rüschendorf (2006):“Comparison of option prices insemimartingale models”. Finance and Stochastics 10, 222–249.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Option Price Ordering

This monotonicity property is very related to the the ordering ofprices by changing the EMM.

Bellamy, N., and Jeanblanc-Picque, M. (2000): “Incompleteness of marketsdriven by a mixed diffusion”. Finance and Stochastics 4, 209–222.

Henderson, V. and Hobson, D. (2003): “Coupling and Option Price Comparisonsin a jump-diffusion model”. Stoch. Stoch. Rep. 75, 79–101.

Jakubenas, P. (2002): “On option pricing in certain incomplete markets”. ProcSteklov Inst. Math. 237, 114–133.

Henderson, V. (2005): “Analytical comparisons of option prices in stochasticvolatility models”. Math. Finance, 15, 49–59.

Gushchin, A. A., and Mordecki, E. (2002): “Bounds on option prices forsemimartingale market models”. Proc. Steklov Inst. Math. 237, 73–113.

Bergenthum, J. and L. Rüschendorf (2006):“Comparison of option prices insemimartingale models”. Finance and Stochastics 10, 222–249.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Option Price Ordering

This monotonicity property is very related to the the ordering ofprices by changing the EMM.

Bellamy, N., and Jeanblanc-Picque, M. (2000): “Incompleteness of marketsdriven by a mixed diffusion”. Finance and Stochastics 4, 209–222.

Henderson, V. and Hobson, D. (2003): “Coupling and Option Price Comparisonsin a jump-diffusion model”. Stoch. Stoch. Rep. 75, 79–101.

Jakubenas, P. (2002): “On option pricing in certain incomplete markets”. ProcSteklov Inst. Math. 237, 114–133.

Henderson, V. (2005): “Analytical comparisons of option prices in stochasticvolatility models”. Math. Finance, 15, 49–59.

Gushchin, A. A., and Mordecki, E. (2002): “Bounds on option prices forsemimartingale market models”. Proc. Steklov Inst. Math. 237, 73–113.

Bergenthum, J. and L. Rüschendorf (2006):“Comparison of option prices insemimartingale models”. Finance and Stochastics 10, 222–249.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Option Price Ordering

This monotonicity property is very related to the the ordering ofprices by changing the EMM.

Bellamy, N., and Jeanblanc-Picque, M. (2000): “Incompleteness of marketsdriven by a mixed diffusion”. Finance and Stochastics 4, 209–222.

Henderson, V. and Hobson, D. (2003): “Coupling and Option Price Comparisonsin a jump-diffusion model”. Stoch. Stoch. Rep. 75, 79–101.

Jakubenas, P. (2002): “On option pricing in certain incomplete markets”. ProcSteklov Inst. Math. 237, 114–133.

Henderson, V. (2005): “Analytical comparisons of option prices in stochasticvolatility models”. Math. Finance, 15, 49–59.

Gushchin, A. A., and Mordecki, E. (2002): “Bounds on option prices forsemimartingale market models”. Proc. Steklov Inst. Math. 237, 73–113.

Bergenthum, J. and L. Rüschendorf (2006):“Comparison of option prices insemimartingale models”. Finance and Stochastics 10, 222–249.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Option Price Ordering

This monotonicity property is very related to the the ordering ofprices by changing the EMM.

Bellamy, N., and Jeanblanc-Picque, M. (2000): “Incompleteness of marketsdriven by a mixed diffusion”. Finance and Stochastics 4, 209–222.

Henderson, V. and Hobson, D. (2003): “Coupling and Option Price Comparisonsin a jump-diffusion model”. Stoch. Stoch. Rep. 75, 79–101.

Jakubenas, P. (2002): “On option pricing in certain incomplete markets”. ProcSteklov Inst. Math. 237, 114–133.

Henderson, V. (2005): “Analytical comparisons of option prices in stochasticvolatility models”. Math. Finance, 15, 49–59.

Gushchin, A. A., and Mordecki, E. (2002): “Bounds on option prices forsemimartingale market models”. Proc. Steklov Inst. Math. 237, 73–113.

Bergenthum, J. and L. Rüschendorf (2006):“Comparison of option prices insemimartingale models”. Finance and Stochastics 10, 222–249.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Main results

TheoremTake r = δ and assume Π(dy) = λF (dy), λ > 0, if F is suchthat

∫eyF (dy) ≷ 1, then

c(F0,Kc , r , τ, ψ) ≷ (1 + x) p(F0,Kp, r , τ, ψ), (17)

where Kc = (1 + x)F0 and Kp = F0/(1 + x), with x > 0.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Proof

If Π(dy) = λF (dy) ⇒ Π(dy) = e−yλF (−dy).

Let λ = λ∫

e−yF (−dy) and F (dy) = e−y F (−dy)Re−y F (−dy)

. Then,∫eyF (dy) ≷ 1 ⇔ λ ≷ λ.

By Ekström and Tysk (2005), option prices are monotonic onjump intensity:

c(F0,Kc , r , τ,a, σ, λF ) ≷ c(F0,Kc , r , τ,a, σ, λF )

= (1 + x)p(F0,Kp, r , τ,a, σ, λF )

where the last equality is obtained from duality.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Proof

If Π(dy) = λF (dy) ⇒ Π(dy) = e−yλF (−dy).

Let λ = λ∫

e−yF (−dy) and F (dy) = e−y F (−dy)Re−y F (−dy)

. Then,∫eyF (dy) ≷ 1 ⇔ λ ≷ λ.

By Ekström and Tysk (2005), option prices are monotonic onjump intensity:

c(F0,Kc , r , τ,a, σ, λF ) ≷ c(F0,Kc , r , τ,a, σ, λF )

= (1 + x)p(F0,Kp, r , τ,a, σ, λF )

where the last equality is obtained from duality.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Proof

If Π(dy) = λF (dy) ⇒ Π(dy) = e−yλF (−dy).

Let λ = λ∫

e−yF (−dy) and F (dy) = e−y F (−dy)Re−y F (−dy)

. Then,∫eyF (dy) ≷ 1 ⇔ λ ≷ λ.

By Ekström and Tysk (2005), option prices are monotonic onjump intensity:

c(F0,Kc , r , τ,a, σ, λF ) ≷ c(F0,Kc , r , τ,a, σ, λF )

= (1 + x)p(F0,Kp, r , τ,a, σ, λF )

where the last equality is obtained from duality.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Proof

If Π(dy) = λF (dy) ⇒ Π(dy) = e−yλF (−dy).

Let λ = λ∫

e−yF (−dy) and F (dy) = e−y F (−dy)Re−y F (−dy)

. Then,∫eyF (dy) ≷ 1 ⇔ λ ≷ λ.

By Ekström and Tysk (2005), option prices are monotonic onjump intensity:

c(F0,Kc , r , τ,a, σ, λF ) ≷ c(F0,Kc , r , τ,a, σ, λF )

= (1 + x)p(F0,Kp, r , τ,a, σ, λF )

where the last equality is obtained from duality.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Main results

Theorem

Take r = δ and assume Π(dy) = eβyΠ0(dy), if β ≷ −1/2, then

C(F ,Kc , r , τ, ψ) ≷ (1 + x) P(F ,Kp, r , τ, ψ), (18)

where Kc = (1 + x)F and Kp = F/(1 + x), with x > 0.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Proof

We need monotonicity of call prices on the parameter β.

Wehave

β ≷ −1/2 ⇐⇒ β ≷ β := −β − 1,

then, Π(dy) = eβyΠ0(dy) has β ≷ β of Π = e−(1+β)yΠ0(dy).

By monotonicity

c(F0,Kc , r , δ, τ, a, σ,Π) ≷ c(F0,Kc , r , τ,a, σ, Π)

= (1 + x)p(F0,Kc , r , τ,a, σ,Π),

where the last equality is obtained from duality and the fact that˜Π = Π.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Proof

We need monotonicity of call prices on the parameter β. Wehave

β ≷ −1/2 ⇐⇒ β ≷ β := −β − 1,

then, Π(dy) = eβyΠ0(dy) has β ≷ β of Π = e−(1+β)yΠ0(dy).

By monotonicity

c(F0,Kc , r , δ, τ, a, σ,Π) ≷ c(F0,Kc , r , τ,a, σ, Π)

= (1 + x)p(F0,Kc , r , τ,a, σ,Π),

where the last equality is obtained from duality and the fact that˜Π = Π.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Proof

We need monotonicity of call prices on the parameter β. Wehave

β ≷ −1/2 ⇐⇒ β ≷ β := −β − 1,

then, Π(dy) = eβyΠ0(dy) has β ≷ β of Π = e−(1+β)yΠ0(dy).

By monotonicity

c(F0,Kc , r , δ, τ, a, σ,Π) ≷ c(F0,Kc , r , τ,a, σ, Π)

= (1 + x)p(F0,Kc , r , τ,a, σ,Π),

where the last equality is obtained from duality and the fact that˜Π = Π.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Merton Model

Remember that β = µδ2 , so we obtain that sufficient conditions

in the above theorems are equivalent. That is,

∫e−yF (−dy) = eµ+δ2/2 ≷ 1 ⇐⇒ β ≷ −1/2.

In general it is not true.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

Previous ResultsOption price propertiesMain Results

Merton Model

Remember that β = µδ2 , so we obtain that sufficient conditions

in the above theorems are equivalent. That is,∫e−yF (−dy) = eµ+δ2/2 ≷ 1 ⇐⇒ β ≷ −1/2.

In general it is not true.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Carr and Wu (2004)

Monroe (1978)

Stochastic Time change on Lévy Processes→ stochasticvolatilityCorrelation between Lévy processes and random clock→leverage effect.Original clock as calendar time and new random clock asbusiness time

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Carr and Wu (2004)

Monroe (1978)Stochastic Time change on Lévy Processes→ stochasticvolatility

Correlation between Lévy processes and random clock→leverage effect.Original clock as calendar time and new random clock asbusiness time

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Carr and Wu (2004)

Monroe (1978)Stochastic Time change on Lévy Processes→ stochasticvolatilityCorrelation between Lévy processes and random clock→leverage effect.

Original clock as calendar time and new random clock asbusiness time

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Carr and Wu (2004)

Monroe (1978)Stochastic Time change on Lévy Processes→ stochasticvolatilityCorrelation between Lévy processes and random clock→leverage effect.Original clock as calendar time and new random clock asbusiness time

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Time-Change

Now let t 7→ Tt , , t ≥ 0, be an increasing RCLL process, suchthat for each fixed t , Tt is a stopping time with respect to F.

Furthermore, suppose Tt is finite P − a.s., ∀t ≥ 0 and Tt →∞as t →∞.Then Tt defines a random change on time, we can alsoimpose ETt = t .

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Time-Change

Now let t 7→ Tt , , t ≥ 0, be an increasing RCLL process, suchthat for each fixed t , Tt is a stopping time with respect to F.Furthermore, suppose Tt is finite P − a.s., ∀t ≥ 0 and Tt →∞as t →∞.

Then Tt defines a random change on time, we can alsoimpose ETt = t .

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Time-Change

Now let t 7→ Tt , , t ≥ 0, be an increasing RCLL process, suchthat for each fixed t , Tt is a stopping time with respect to F.Furthermore, suppose Tt is finite P − a.s., ∀t ≥ 0 and Tt →∞as t →∞.Then Tt defines a random change on time, we can alsoimpose ETt = t .

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Time-Changed Lévy processes

Then, consider the process Yt defined by:

Yt ≡ XTt , t ≥ 0,

using different triplet for X and different time changes Tt , wecan obtain a good candidate for the underlying asset returnprocess.

We know that if Tt is another Lévy process we have that Ywould be another Lévy process

.

A more general situation is when Tt is modeled by anon-decreasing semimartingale.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Time-Changed Lévy processes

Then, consider the process Yt defined by:

Yt ≡ XTt , t ≥ 0,

using different triplet for X and different time changes Tt , wecan obtain a good candidate for the underlying asset returnprocess.We know that if Tt is another Lévy process we have that Ywould be another Lévy process.

A more general situation is when Tt is modeled by anon-decreasing semimartingale.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Time-Changed Lévy processes

Then, consider the process Yt defined by:

Yt ≡ XTt , t ≥ 0,

using different triplet for X and different time changes Tt , wecan obtain a good candidate for the underlying asset returnprocess.We know that if Tt is another Lévy process we have that Ywould be another Lévy process.A more general situation is when Tt is modeled by anon-decreasing semimartingale.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Time-Changed Lévy ProcessesIn that case

Tt = bt +

∫ t

0

∫ ∞

0yµ(dy ,ds)

where b is a drift and µ is the counting measure of jumps of thetime change as in Carr and Wu (2004) we take µ = 0 and justtake locally deterministic time changes, so we need to specifythe local intensity ν:

Tt =

∫ t

0ν(s−)ds (19)

where ν is the instantaneous activity rate, observe thatν must be non-negative.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Remarks

When Xt is the Brownian motion , ν is proportional to theinstantaneous variance rate of the Brownian motion

When Xt is a pure jump Lévy process, ν is proportional tothe Lévy intensity of jumps.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Remarks

When Xt is the Brownian motion , ν is proportional to theinstantaneous variance rate of the Brownian motionWhen Xt is a pure jump Lévy process, ν is proportional tothe Lévy intensity of jumps.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Time-Changed Lévy Processes

Now we can obtain the characteristic function of Yt :

φYt (z) = E(ez′XTt ) = E(

E(

ez′Xu/Tt = u))

.

If Tt and Xt were independent, then:

φYt (z) = LTt (ψ(z))

where LTt is the Laplace transform of Tt .

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Time-Changed Lévy Processes

Now we can obtain the characteristic function of Yt :

φYt (z) = E(ez′XTt ) = E(

E(

ez′Xu/Tt = u))

.

If Tt and Xt were independent, then:

φYt (z) = LTt (ψ(z))

where LTt is the Laplace transform of Tt .

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Time-Changed Lévy Processes

So if the Laplace transform of T and the characteristicexponent of X have closed forms, we can obtain a closed formfor φYt . Using equation (19) we have:

LTt (λ) = E(e−λR t

0 ν(s−)ds) (20)

From here we can understand λν as an instantaneous interestrate, then we can search in the bond pricing literature to obtaina closed form for φYt .

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Time-Changed Lévy Processes

When T is a Lévy processes the characteristics of Y areknown explicitly.

When T has independent increments we can obtain adeterministic characteristics for Y .When T is just a processes that turns Y in to asemimartingale we can obtain a predictable version of thecharacteristics.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Time-Changed Lévy Processes

When T is a Lévy processes the characteristics of Y areknown explicitly.When T has independent increments we can obtain adeterministic characteristics for Y .

When T is just a processes that turns Y in to asemimartingale we can obtain a predictable version of thecharacteristics.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Time-Changed Lévy Processes

When T is a Lévy processes the characteristics of Y areknown explicitly.When T has independent increments we can obtain adeterministic characteristics for Y .When T is just a processes that turns Y in to asemimartingale we can obtain a predictable version of thecharacteristics.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Duality

Consider a market with two assets given by

S1t = eYt , and S2

t = S20ert

where (Y ) is a one dimensional Time-Changed Lévy process,and for simplicity, and without loss of generality we take S1

0 = 1.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Duality

Assume that T has independent increments. Let Ψ = (B,C, ν)be the characteristic triplet of Y .

Then, as we assume that weare under the risk neutral measure, B is completely determinedby the other characteristics:

Bt = (r − δ)t − 12

∫ t

0csds −

∫ t

0

∫IR(ex − 1− x)ν(ds,dx),

where,

Bt =

∫ t

0bsds and Ct =

∫ t

0csds,

ν is obtained from the canonical decomposition of Y .

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Duality

Assume that T has independent increments. Let Ψ = (B,C, ν)be the characteristic triplet of Y . Then, as we assume that weare under the risk neutral measure, B is completely determinedby the other characteristics:

Bt = (r − δ)t − 12

∫ t

0csds −

∫ t

0

∫IR(ex − 1− x)ν(ds,dx),

where,

Bt =

∫ t

0bsds and Ct =

∫ t

0csds,

ν is obtained from the canonical decomposition of Y .

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Duality

Assume that T has independent increments. Let Ψ = (B,C, ν)be the characteristic triplet of Y . Then, as we assume that weare under the risk neutral measure, B is completely determinedby the other characteristics:

Bt = (r − δ)t − 12

∫ t

0csds −

∫ t

0

∫IR(ex − 1− x)ν(ds,dx),

where,

Bt =

∫ t

0bsds and Ct =

∫ t

0csds,

ν is obtained from the canonical decomposition of Y .

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Duality

Let us assume that τ is a stopping time with respect to thegiven filtration F , that is τ : Ω → [0,∞] belongs to Ft for allt ≥ 0; and introduce the notation

C(S0,K , r , δ, τ,Ψ) = Ee−rτ (Sτ − K )+ (21)P(S0,K , r , δ, τ,Ψ) = Ee−rτ (K − Sτ )+ (22)

If τ = T , where T is a fixed constant time, then formulas (21)and (22) give the price of the European call and put optionsrespectively.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Duality

Let us assume that τ is a stopping time with respect to thegiven filtration F , that is τ : Ω → [0,∞] belongs to Ft for allt ≥ 0; and introduce the notation

C(S0,K , r , δ, τ,Ψ) = Ee−rτ (Sτ − K )+ (21)P(S0,K , r , δ, τ,Ψ) = Ee−rτ (K − Sτ )+ (22)

If τ = T , where T is a fixed constant time, then formulas (21)and (22) give the price of the European call and put optionsrespectively.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Put-Call DualityProposition

Consider a Time-changed Lévy market with driving process Ywith characteristic triplet Ψ = (B,C, ν). Then, for theexpectations introduced in (21) and (22) we have

C(S0,K , r , δ, τ,Ψ) = P(K ,S0, δ, r , τ, Ψ), (23)

where Ψ(z) = (B, C, ν) is the characteristic triplet (of a certainsemimartingale) that satisfies:8><>:

Bt = (δ − r)t − 12

R t0 csds −

R t0

RIR

`ex − 1− x1|x|≤1

´ν(ds, dx),

C = C,ν(dy) = e−yν(−dy)

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

RemarkWhen Yt has not independent increments, the characteristictriplet is predictable.

Since P is the risk neutral probability, the process B is alsodetermined by the other characteristics.

Bt = (r − δ)t − 12

Ct −∫ t

0

∫IR(ex − 1− h(x))ν(ds,dx),

where (B,C, ν) is the triplet characteristics of Y and h is atruncation function.

Gyrsanov’s Theorem for semimartingales:

Bt = −Bt − Ct −∫ ∫

h(x)(ex − 1)ν(ds,dx) (24)

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

RemarkWhen Yt has not independent increments, the characteristictriplet is predictable.Since P is the risk neutral probability, the process B is alsodetermined by the other characteristics.

Bt = (r − δ)t − 12

Ct −∫ t

0

∫IR(ex − 1− h(x))ν(ds,dx),

where (B,C, ν) is the triplet characteristics of Y and h is atruncation function.

Gyrsanov’s Theorem for semimartingales:

Bt = −Bt − Ct −∫ ∫

h(x)(ex − 1)ν(ds,dx) (24)

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

RemarkWhen Yt has not independent increments, the characteristictriplet is predictable.Since P is the risk neutral probability, the process B is alsodetermined by the other characteristics.

Bt = (r − δ)t − 12

Ct −∫ t

0

∫IR(ex − 1− h(x))ν(ds,dx),

where (B,C, ν) is the triplet characteristics of Y and h is atruncation function.

Gyrsanov’s Theorem for semimartingales:

Bt = −Bt − Ct −∫ ∫

h(x)(ex − 1)ν(ds,dx) (24)

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

RemarkWhen Yt has not independent increments, the characteristictriplet is predictable.Since P is the risk neutral probability, the process B is alsodetermined by the other characteristics.

Bt = (r − δ)t − 12

Ct −∫ t

0

∫IR(ex − 1− h(x))ν(ds,dx),

where (B,C, ν) is the triplet characteristics of Y and h is atruncation function.

Gyrsanov’s Theorem for semimartingales:

Bt = −Bt − Ct −∫ ∫

h(x)(ex − 1)ν(ds,dx) (24)

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Symmetric markets

Lets define symmetric markets by

L(e−(r−δ)t+Yt | P

)= L

(e−(δ−r)t−Yt | P

), (25)

In view of (1), we have

ν(dy) = e−yν(−dy). (26)

This ensures ν = ν, and from this follows

B − (r − δ) = B − (δ − r)

, giving (25), as we always have C = C.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Bates’s x%-RuleIf the call and put options have strike prices x% out-of-themoney relative to the forward price, then the call should bepriced x% higher than the put.

If r = δ, we can take the future price F as the underlying assetin Proposition 1.

CorollaryTake r = δ and assume (26) holds, we have

C(F0,Kc , r , τ,Ψ) = x P(F0,Kp, r , τ,Ψ), (27)

where Kc = xF0 and Kp = F0/x, with x > 0.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Bates’s x%-RuleIf the call and put options have strike prices x% out-of-themoney relative to the forward price, then the call should bepriced x% higher than the put.

If r = δ, we can take the future price F as the underlying assetin Proposition 1.

CorollaryTake r = δ and assume (26) holds, we have

C(F0,Kc , r , τ,Ψ) = x P(F0,Kp, r , τ,Ψ), (27)

where Kc = xF0 and Kp = F0/x, with x > 0.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Bates’s x%-RuleIf the call and put options have strike prices x% out-of-themoney relative to the forward price, then the call should bepriced x% higher than the put.

If r = δ, we can take the future price F as the underlying assetin Proposition 1.

CorollaryTake r = δ and assume (26) holds, we have

C(F0,Kc , r , τ,Ψ) = x P(F0,Kp, r , τ,Ψ), (27)

where Kc = xF0 and Kp = F0/x, with x > 0.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Conclusions

Symmetry and Bates’ x% Rule. (β 6= −0.5?)

Skewness Premium: Call option x% OTM should be priced[0, x%] more than Put options x% OTM. (SK ≷ 0?)The SK can not identify which process or which parametervalues best fit observed option data.Which of the Lévy processes and associated option pricingmodels can generate the observed moneyness biases.Time-Changed Lévy Processes

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Conclusions

Symmetry and Bates’ x% Rule. (β 6= −0.5?)Skewness Premium: Call option x% OTM should be priced[0, x%] more than Put options x% OTM. (SK ≷ 0?)

The SK can not identify which process or which parametervalues best fit observed option data.Which of the Lévy processes and associated option pricingmodels can generate the observed moneyness biases.Time-Changed Lévy Processes

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Conclusions

Symmetry and Bates’ x% Rule. (β 6= −0.5?)Skewness Premium: Call option x% OTM should be priced[0, x%] more than Put options x% OTM. (SK ≷ 0?)The SK can not identify which process or which parametervalues best fit observed option data.

Which of the Lévy processes and associated option pricingmodels can generate the observed moneyness biases.Time-Changed Lévy Processes

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Conclusions

Symmetry and Bates’ x% Rule. (β 6= −0.5?)Skewness Premium: Call option x% OTM should be priced[0, x%] more than Put options x% OTM. (SK ≷ 0?)The SK can not identify which process or which parametervalues best fit observed option data.Which of the Lévy processes and associated option pricingmodels can generate the observed moneyness biases.

Time-Changed Lévy Processes

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Conclusions

Symmetry and Bates’ x% Rule. (β 6= −0.5?)Skewness Premium: Call option x% OTM should be priced[0, x%] more than Put options x% OTM. (SK ≷ 0?)The SK can not identify which process or which parametervalues best fit observed option data.Which of the Lévy processes and associated option pricingmodels can generate the observed moneyness biases.Time-Changed Lévy Processes

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Conclusions

EMM Esscher Transform and Asymmetry.

Volatility Smile and AsymmetryConvexity preserving and AssymetryOther derivatives

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Conclusions

EMM Esscher Transform and Asymmetry.Volatility Smile and Asymmetry

Convexity preserving and AssymetryOther derivatives

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Conclusions

EMM Esscher Transform and Asymmetry.Volatility Smile and AsymmetryConvexity preserving and Assymetry

Other derivatives

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Conclusions

EMM Esscher Transform and Asymmetry.Volatility Smile and AsymmetryConvexity preserving and AssymetryOther derivatives

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

References I

Fajardo, J. and E. MordeckiPut-Call Duality and Symmetry with Lévy Processes.Quantitative Finance,6:3, 219–227. 2006.

Fajardo, J. and E. MordeckiA Note on Pricing, Duality and Symmetry for TwoDimensional Lévy MarketsFrom Stochastic Analysis to Mathematical Finance -Festschrift for A.N. Shiryaev. Eds. Y. Kabanov, R. Lipsterand J. Stoyanov, Springer Verlag, New York. 2006.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

References II

Fajardo, J. and E. MordeckiDuality and Derivative pricing with Time-Changed LévyProcesses.Working paper IBMEC, 2006.

E. Eberlein and A. PapapantoleonSymmetries and pricing of exotic options in Lévy models.Exotic Option Pricing and Advanced Lévy Models. A.Kyprianou, W. Schoutens, P. Wilmott (Eds.), Wiley. 2005.

E. Eberlein, W. Kluge, A. PapapantoleonSymmetries in Lévy term structure models.Int. J. Th. Ap. Fin., 9,967–986. 2006.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

References III

E. Eberlein, A. Papapantoleon and A. N. ShiryaevOn the Duality Principle in Option Pricing: SemimartingaleSetting.Universität Freiburg. WP 92. August 2006.

Option MarketLévy Processes

DualitySymmetry

Skewness PremiumTime-Changed Lévy Processes

DualitySymmetry

Gyrsanov for Semimartingales

Lemma

Let Y be a d-dimensional semimartingale with finite variation withtriplet (B,C, ν) under P, let u, v be vectors in IRd and v ∈ [−M,M]d .Moreover let P ∼ P, with density dP

dP = e<v,YT >

E[e<v,YT >]. Then the process

Y ∗ :=< u,Y > is a P−semimartingale with triplet:

b∗s =<u,bs>+

12

(<u, csv>+< v , csu >)+

∫IRd< u, x > (e<v ,x>− 1)λs(dx)

c∗s =< u, csu >λ∗s =Λ(κs)

where Λ is a mapping λ : IRd → IR such that x 7→ Λ(x) =< u, x > andκs is a measure defined by κs(A) =

∫A e<v ,x>λs(dx).