Lectures on Levy Processes, StochasticCalculus and Financial Applications,

Ovronnaz September 2005

3 Lecture 3: From Girsanov’s theorem to op-

tion pricing with Levy processes and Malli-

avin calculus

3.1 Levy-type Stochastic Integrals as Martingales

We continue to work with real-valued Levy-type stochastic integrals of theform

dY (t) = G(t)dt + F (t)dB(t) + H(t, x)N(dt, dx) + K(t, x)N(dt, dx). (3.1)

Our first goal is to find necessary and sufficient conditions for a Levy-typestochastic integral Y to be a martingale. At least formally, we would expectthis to be the case if the first and fourth term vanish with probability one,since the second and third terms are martingales.

First we impose some conditions on K and G. For simplicity we makethese stronger than necessary in order to avoid having to work with thegeneral notion of local martingale.

(M1) E(∫ t

0

∫|x|≥1

|K(s, x)|2ν(dx)ds)

< ∞.

(M2)∫ t

0E(|G(s)|)ds < ∞ for each t > 0.

(Note that E = B−0 = x ∈ Rd; 0 < |x| < 1 throughout this section.)

It follows from (M1) and the Cauchy-Schwarz inequality, that∫ t

0

∫|x|≥1

|K(s, x)|ν(dx)ds <

∞ (a.s.) and we can write

∫ t

0

∫

|x|≥1

K(s, x)N(dx, ds) =

∫ t

0

∫

|x|≥1

K(s, x)N(dx, ds)+

∫ t

0

∫

|x|≥1

K(s, x)ν(dx)ds,

1

for each t ≥ 0, with the compensated integral being a martingale.

Theorem 3.1 If Y is a Levy-type stochastic integral of the form (3.1) andthe assumptions (M1) and (M2) are satisfied, then Y is a martingale if andonly if

G(t) +

∫

|x|≥1

K(t, x)ν(dx) = 0 (a.s.),

for (Lebesgue) almost all t ≥ 0.

Proof. For each 0 ≤ s < t < ∞,

Y (t) = Y (s) +

∫ t

s

F (u)dB(u) +

∫ t

s

∫

|x|<1

H(u, x)N(du, dx)

+

∫ t

s

∫

|x|>1

K(u, x)N(du, dx) +

∫ t

s

(G(u) +

∫

|x|≥1

K(u, x)ν(dx)

)du.

Now assume that (Y (t), t ≥ 0) is a martingale, then we must have

Es

[∫ t

s

(G(u) +

∫

|x|≥1

K(u, x)ν(dx)

)du

]= 0.

Using the fact that by (M1) and (M2),∣∣∣∣∫ t

s

(G(u) +

∫

|x|≥1

K(u, x)ν(dx)

)du

∣∣∣∣ ≤∫ t

0

∣∣∣∣G(u) +

∫

|x|≥1

K(u, x)ν(dx)

∣∣∣∣ du

< ∞ (a.s.),

and the conditional version of dominated convergence, we deduce that

Es

∫ t

s

(G(u) +

∫

|x|≥1

K(u, x)ν(dx)

)du = 0.

(M1) and (M2) ensure that we can use the conditional Fubini theorem toobtain ∫ t

s

Es

(G(u) +

∫

|x|≥1

K(u, x)ν(dx)

)du = 0.

It follows that limh→0

1

h

∫ s+h

s

Es

(G(u) +

∫

|x|≥1

K(u, x)ν(dx)

)du = 0,

and hence by Lebesgue’s differentiation theorem we have

Es

(G(s) +

∫

|x|≥1

K(s, x)ν(dx)

)= 0,

for (Lebesgue) almost all s ≥ 0. But G(·)+∫|x|≥1

K(·, x)ν(dx) is adapted

and the result follows. The converse is immediate. ¤

2

3.2 Exponential Martingales

We continue to study Levy-type stochastic integrals satisfying (M1) to (M2).We now turn our attention to the process eY = (eY (t), t ≥ 0).

By Ito’s formula, we find for each t ≥ 0,

eY (t) = 1 +

∫ t

0

eY (s−)F (s)dB(s)

+

∫ t

0

∫

|x|<1

eY (s−)(eH(s,x) − 1)N(ds, dx) +

∫ t

0

∫

|x|≥1

eY (s−)(eK(s,x) − 1)N(ds, dx)

+

∫ t

0

eY (s−)

(G(s) +

1

2F (s)2 +

∫

|x|<1

(eH(s,x) − 1−H(s, x))ν(dx)

+

∫

|x|≥1

(eK(s,x) − 1)ν(dx)

)ds. (3.2)

Hence, by an immediate application of Theorem 3.1, we have:

Corollary 3.1 eY is a martingale if and only if

G(s)+1

2F (s)2+

∫

|x|<1

(eH(s,x)−1−H(s, x))ν(dx)+

∫

|x|≥1

(eK(s,x)−1)ν(dx) = 0,

(3.3)almost surely and for (Lebesgue) almost all s ≥ 0.

So eY is a martingale if and only if for (Lebesgue almost all) t ≥ 0,

eY (t) = 1 +

∫ t

0

eY (s−)F (s)dB(s) +

∫ t

0

∫

|x|<1

eY (s−)(eH(s,x) − 1)N(ds, dx)

+

∫ t

0

∫

|x|≥1

eY (s−)(eK(s,x) − 1)N(ds, dx). (3.4)

Furthermore, since each stochastic integral in (3.4) has zero mean, wededuce that for all t ≥ 0,

E(eY (t)) = 1.

Since eY (t) > 0 for all t ≥ 0, we can use it to define a new probabilitymeasure Qt which is absolutely continuous with respect to P , and has Radon-

Nikodym derivativedQt

dP= eY (t) . We’ll make explicit use of this below.

The process eY given by equation (3.4) is called an exponential martingale.Two important examples are:

3

1. The Brownian Case

Here Y is a Brownian integral of the form

Y (t) =

∫ t

0

F (s)dB(s) +

∫ t

0

G(s)ds,

for each t ≥ 0. The unique solution to (3.3) is G(t) = −12F (t)2 (a.e.).

We then have, for each t ≥ 0

eY (t) = exp

(∫ t

0

F (s)dB(s)− 1

2

∫ t

0

F (s)2ds

).

2. The Poisson Case

Here Y is a Poisson integral driven by a Poisson process N of intensityλ and has the form

Y (t) =

∫ t

0

K(s)dN(s) +

∫ t

0

G(s)ds,

for each t ≥ 0.

The unique solution to (3.3) is G(t) = −λ∫ t

0(eK(s) − 1)ds (a.e.). For

each t ≥ 0, we obtain

eY (t) = exp

(∫ t

0

K(s)dN(s)− λ

∫ t

0

(eK(s) − 1)ds

).

3.3 Change of Measure - Girsanov’s Theorem

If we are given two distinct probability measures P and Q on (Ω,F), wewill write EP (EQ) to denote expectation with respect to P (respectively Q).We also use the terminology P -martingale, P -Brownian motion etc, whenwe want to emphasise that P is the operative measure. We remark that Qand P are each also probability measures on (Ω,Ft), for each t ≥ 0, and wewill use the notations Qt and Pt when the measures are restricted in thisway. Suppose that Q << P , then each Qt << Pt and we sometimes writedQ

dP

∣∣∣∣t

=dQt

dPt

.

In the proof of the next lemma, we will make extensive use of the defin-ing property of the conditional expectation E(X|G) of an integrable F -measurable random variable X on to a sub-σ-algebra G, i.e. if Y is anintegrable G-measurable random variable for which

4

E(χAY ) = E(χAX) for all A ∈ G,

then Y = E(X|G) a.s.

Lemma 3.1

(dQ

dP

∣∣∣∣t

, t ≥ 0

)is a P -martingale.

Proof. For each t ≥ 0, let M(t) =dQ

dP

∣∣∣∣t

. For all 0 ≤ s ≤ t, A ∈ Fs,

EP (χAEP (M(t)|Fs)) = EP (χAM(t))

= EPt(χAM(t)) = EQt(χA)

= EQs(χA) = EPs(χAM(s))

= EP (χAM(s)).

¤Now let eY be an exponential martingale. Then since

EP (eY (t)) = EPt(eY (t)) = 1

we can define a probability measure Qt on (Ω,Ft) by

dQt

dPt

= eY (t), (3.5)

for each t ≥ 0.From now on, we will find it convenient to fix a time interval [0, T ]. We

write P = PT and Q = QT .Before we establish Girsanov’s theorem, which is the key result of this

section, we need a useful lemma.

Lemma 3.2 M = (M(t), 0 ≤ t ≤ T ) is a Q-martingale if and only if MeY =(M(t)eY (t), 0 ≤ t ≤ T ) is a P -martingale.

Proof. Let A ∈ Fs and assume that M is a Q-martingale, then for each0 ≤ s < t < ∞,

∫

A

M(t)eY (t)dP =

∫

A

M(t)eY (t)dPt =

∫

A

M(t)dQt

=

∫

A

M(t)dQ =

∫

A

M(s)dQ =

∫

A

M(s)dQs

=

∫

A

M(s)eY (s)dPs =

∫

A

M(s)eY (s)dP.

5

The converse is proved in the same way. ¤In the following we take Y to be a Brownian integral, so that for each

0 ≤ t ≤ T ,

eY (t) = exp

(∫ t

0

F (s)dB(s)− 1

2

∫ t

0

F (s)2ds

)

We define a new process W = (W (t), 0 ≤ t ≤ T ) by

W (t) = B(t)−∫ t

0

F (s)ds,

for each t ≥ 0.

Theorem 3.2 (Girsanov) W is a Q-Brownian motion.

Proof. First we use Ito’s product formula to find that for each 0 ≤ t ≤ T ,

d(W (t)eY (t)) = dW (t)eY (t) + W (t)deY (t) + dW (t)deY (t)

= eY (t)dB(t)− eY (t)F (t)dt + W (t)eY (t)F (t)dB(t) + eY (t)F (t)dt

= eY (t)[1 + W (t)F (t)]dB(t).

Hence WeY is a P -martingale and so (by Lemma 3.2), W is a Q-martingale.Moreover since W (0) = 0 (a.s.), we see that W is centered (with respect toQ).

But dW (t)2 = dt and so [W,W ](t) = t. The result now follows fromLevy’s characterisation of Brownian motion. ¤

The following partial extension of Girsanov’s theorem will be of use to usin applications to finance:-

Let M = (M(t), 0 ≤ t ≤ T ) be a martingale of the form

M(t) =

∫ t

0

∫

|x|<1

L(x, s)N(ds, dx),

where L ∈ P(t, E). Let eY be an exponential martingale. By Lemma 3.2it follows that N = (N(t), 0 ≤ t ≤ T ) is a martingale where

N(t) = M(t)−∫ t

0

∫

|x|<1

L(x, s)(eH(s,x) − 1)ν(dx)ds.

To see this, argue as in the first part of the proof of theorem 3.2 andapply Ito’s product formula to the process NeY .

6

3.4 Stochastic Calculus and Levy Processes in OptionPricing

In recent years, there has been a growing interest in the use of Levy processesand discontinuous semimartingales to model market behaviour and not onlyare these of great mathematical interest but there is growing evidence thatthey may be more realistic models than those that insist on continuous samplepaths. Our aim in this section is to give a brief introduction to some of theseideas.

Stock prices (we deal with a single stock, for simplicity) will be modelledas a random process (S(t), t ≥ 0). All options which we will consider will bemodelled as contingent claims with maturity date T > 0. This means that Zis a non-negative FT measurable random variable which represents the valueof the option to its holder. In the following k is the exercise price which isthe fixed price which the option entitles the holder to pay for one unit ofstock, irrespective of its market value. Some examples:

• European call option

Z = maxS(T )− k, 0 := (S(T )− k)+

.

• American call option

Z = sup0≤τ≤T

maxS(τ)− k, 0

.

• Asian option

Z = max

1

T

∫ T

0

(S(t)− k)dt, 0

.

We assume that the interest rate r is constant so that a principal Pinvested in a bank account at time 0 grows to Pe−rt by time t. So a sumof money Q promised at time t has a discounted value Qe−rt today. Inparticular, if (S(t), t ≥ 0) is the stock price, we define the discounted processS = (S(t), t ≥ 0) where each S(t) = e−rtS(t).

7

3.4.1 Portfolios, Arbitrage, Martingales

An investor (which may be an individual or a company) will hold their in-vestments as a combination of risky stocks and cash in the bank, say. Letα(t) and β(t) denote the amount each of these which we hold, respectively,at time t. The pair of adapted processes (α, β) where α = (α(t), t ≥ 0) andβ = (β(t), t ≥ 0) is called a portfolio or trading strategy . The total value ofall our investments at time t is denoted as V (t), so

V (t) = α(t)S(t) + β(t)ß(t).

One of the key aims of the Black-Scholes approach to option pricing isto be able to hedge the risk involved in selling options, by being able toconstruct a portfolio whose value at the expiration time T is exactly that ofthe option. To be precise, a portfolio is said to be replicating if

V (T ) = Z.

Clearly, replicating portfolios are desirable objects. Another class of in-teresting portfolios are those that are self-financing, i.e. any change in wealthV is due only to changes in the values of stocks and bank accounts and notto any injections of capital from outside. We can model this using stochasticdifferentials if we make the assumption that the stock price process S is asemimartingale. We can then write

dV (t) = α(t)dS(t) + β(t)dß(t)

= α(t)dS(t) + rβ(t)ß(t)dt.

so the infinitesimal change in V arises solely through infinitesimal changes inS and ß. Notice how we have sneakily slipped Ito calculus into the pictureby the assumption that dS(t) should be interpreted in the Ito sense. Thisis absolutely crucial. If we try to use any other integral (e.g. Lebesgue-Stieltjes), then the theory which follows will certainly no longer work.

A market is said to be complete if every contingent claim can be replicatedby a self-financing portfolio. So in a complete market, every option can behedged by a portfolio which requires no outside injections of capital betweenits starting time and the expiration time.

An arbitrage opportunity exists if the market allows risk-free profit. Con-versely the market is arbitrage free if there exists no self-financing strategy(α, β) for which V (0) = 0, V (T ) ≥ 0 and P (V (T ) > 0) > 0.

At least in discrete time, we have the following remarkable result whichillustrates how the absence of arbitrage forces the mathematical modellerinto the world of stochastic analysis.

8

Theorem 3.3 (Fundamental Theorem of Asset Pricing 1) If the mar-ket is free of arbitrage opportunities, then there exists a probability measureQ, which is equivalent to P , with respect to which the discounted process Sis a martingale.

A similar result holds in the continuous case but we need to make moretechnical assumptions - instead of absence of arbitrage we need the strongerNFLVR hypothesis (“no free lunch with vanishing risk”). The philosophy ofTheorem 3.3 will play a central role in the sequel.

Again in discrete time we have:

Theorem 3.4 (Fundamental Theorem of Asset Pricing 2) An arbitrage-free market is complete if and only if there exists a unique probability measureQ, which is equivalent to P , with respect to which the discounted process Sis a martingale.

Theorems 3.3 and 3.4 identify a key mathematical problem for us is tofind a (unique, if possible) Q, which is equivalent to P , under which S is amartingale. Such a Q is called a martingale measure or risk-neutral measure.If Q exists, but is not unique, then the market is said to be incomplete.

A martingale M = (M(t), 0 ≤ t ≤ T ) has the predictable representa-tion property if for any FT measurable X with E(|X|2) < ∞, there exists apredictable (G(s), 0 ≤ s ≤ T ) such that

X = E(X) +

∫ T

0

G(s)dM(s).

If the discounted stock price is a Q-martingale which has the predictablerepresentation property then the market is complete, i.e. we can always finda replicating self-financing strategy. Unfortunately the only Levy processeswhich have this property are Brownian motion and the compensated Poissonprocess, consequently the generic Levy market (see below) is incomplete.

Note. Let B = (B(t), t ≥ 0) be a one-dimensional Brownian motion and

g(t) = sup0 ≤ s ≤ t; B(s) = 0.

Azema’s martingale

M(t) =

√π

2sign(B(t))

√t− g(t)

9

is a martingale with respect to the filtration Ft = σM(s); 0 ≤ s ≤ t. Ithas the predictable representation property but it is not a Levy process. (Itis however a “quantum Levy process”).

In a complete market, it turns out that we have

V (t) = e−r(T−t)EQ(X|Ft),

and this is the arbitrage-free price of the claim X at time t.

3.5 Stock Prices as Levy Process

So far we have said little about the key process S which models the evolutionof stock prices. As far back as 1900, Bachelier in his PhD thesis proposed thatthis should be a Brownian motion . Indeed, this can be intuitively justifiedon the basis of the central limit theorem if one perceives the movement ofstocks as due to the “invisible hand of the market” manifested as a very largenumber of independent, identically distributed decisions. One immediateproblem with this is that it is unrealistic, as stock prices cannot becomenegative but Brownian motion can. An obvious way out of this is to takeexponentials, but let us be more specific.

Financial analysts like to study the return on their investment, which ina small time interval [t, t + δt] will be

δS(t)

S(t)=

S(t + δt)− S(t)

S(t),

and it is then natural to directly introduce the noise at this level andwrite

δS(t)

S(t)= σδX(t) + µδt,

where X = (X(t), t ≥ 0) is a semimartingale and σ, µ are parameterscalled the volatility and stock drift respectively. σ > 0 controls the strengthof the coupling to the noise while µ ∈ R represents deterministic effects,indeed if E(δX(t)) = 0 for all t ≥ 0 then µ is the mean rate of return.

We now interpret this in terms of Ito calculus by formally replacing allsmall changes which are written in terms of δ by Ito differentials. We thenfind that

dS(t) = σS(t−)dX(t) + µS(t−)dt

= S(t−)dZ(t), (3.6)

where Z(t) = σX(t) + µt.

10

We see immediately that S(t) = EZ(t) is the stochastic exponential of thesemimartingale Z as described in section 4.1. Indeed, when X is a standardBrownian motion B = (B(t), t ≥ 0), we obtain geometric Brownian motion,which has been very widely used as a model for stock prices since it was firstproposed in 1965 by the economist Paul Samuelson

S(t) = exp

(σB(t) +

(µt− 1

2σ2t

)). (3.7)

There has recently been a great deal of interest in taking X to be aLevy process. One argument in favour of this is that stock prices clearlydon’t move continuously and a more realistic approach is one that allowssmall jumps in small time intervals. Moreover, empirical studies of stockprices indicate distributions with heavy tails, which are incompatible with aGaussian model.

We will make the assumption from now on that X is indeed a Levyprocess. Note immediately that in order for stock prices to be non-negative,(SE) yields ∆X(t) > −σ−1 (a.s.) for each t > 0 and for convenience, we willdenote c = −σ−1 in the sequel. We will also impose the following conditionon the Levy measure ν,

∫(c,−1]∪[1,∞)

x2ν(dx) < ∞. This means that each

X(t) has finite first and second moments which seems to be a reasonableassumption for stock returns.

By the Levy-Ito decomposition, for each t ≥ 0, we have

X(t) = mt + κB(t) +

∫ ∞

c

xN(t, dx), (3.8)

where κ ≥ 0, and in terms of the earlier parametrisation, m = b +∫(c,−1]∪[1,∞)

xν(dx). So as to keep the notation simple, it is assumed in (3.8),

and in the discussion that follows, that 0 is omitted from the range of inte-gration.

Modelling S(t) as EZ(t), we obtain the following representation for thestock prices from (2.22):

d(log(S(t)) = κσdB(t) + (mσ + µ− 1

2κ2σ2)dt (3.9)

+

∫ ∞

c

log(1 + σx)N(dt, dx) +

∫ ∞

c

(log(1 + σx)− σx)ν(dx)dt.

Some advantages of modelling with Levy processes:

• Incomplete markets are believed to be more realistic.

11

• The Levy-Ito decomposition gives a natural description of stock move-ments as jumps from the very small (infinite activity) to the very large(finite activity).

• There are a number of explicit mathematically tractable and realisticmodels e.g. variance-gamma, normal inverse Gaussian, hyperbolic etc.

• Some empirical studies indicate the presence of “heavy tails”. ManyLevy processes have this property. Brownian motion has “light tails”.

3.6 Change of Measure

Motivated by the philosophy behind the fundamental theorem of asset pricing(Theorems 3.3 and 3.4), we seek to find measures Q, which are equivalentto P , with respect to which the discounted stock process S is a martingale.Rather than consider all possible changes of measure, we work in a restrictedcontext where we can exploit our understanding of stochastic calculus basedon Levy processes.

Let Y be a Levy-type stochastic integral which takes the form

dY (t) = G(t)dt + F (t)dB(t) +

∫

R−0H(t, x)N(dt, dx),

where in particular H ∈ P2(t,R−0) for each t ≥ 0. Note that we havedeliberately chosen a restricted form of Y to be compatible with that of theLevy process X in order to simplify the discussion below.

We consider the associated exponential process eY and we assume thatthis is a martingale (and so G is determined by F and H). Hence we can

define a new measure Q by the prescriptiondQ

dP= eY (T ). Furthermore by

Girsanov’s theorem and the extension to jump integrals, for each 0 ≤ t ≤T,A ∈ B([c,∞)),

BQ(t) = B(t)− ∫ t

0F (s)ds is a Q- Brownian motion.

NQ(t, A) = N(t, A)− νQ(t, A) is a Q-martingale, where

νQ(t, A) =∫ t

0

∫A(eH(s,x) − 1)ν(dx)ds.

Note that EQ(NQ(t, A)2) =∫ t

0

∫AEQ(eH(s,x))ν(dx)ds

We rewrite the discounted stock price S(t) = e−rtS(t) in terms of thesenew processes to find

d(log(S(t))) = κσdBQ(t) +

(mσ + µ− r − 1

2κ2σ2 + κσF (t)

12

+ σ

∫

R−0x(eH(t,x) − 1)ν(dx)

)dt

+

∫ ∞

c

log(1 + σx)NQ(dt, dx)

+

∫ ∞

c

(log(1 + σx)− σx)νQ(dt, dx). (3.10)

Now write S(t) = S1(t)S2(t), where

d(log(S1(t))) = κσdBQ(t)− 1

2κ2σ2dt

+

∫ ∞

c

log(1 + σx)NQ(dt, dx)

+

∫ ∞

c

(log(1 + σx)− σx)νQ(dt, dx),

and

d(log(S2(t))) =

(mσ + µ− r + κσF (t) + σ

∫

R−0x(eH(t,x) − 1)ν(dx)

)dt.

On applying Ito’s formula to S1, we obtain

dS1(t) = κσS1(t−)dBQ(t) +

∫

(c,∞)

σS1(t−)xNQ(dt, dx).

So S1 is a Q-martingale, and hence S is a Q-martingale if and only if

mσ + µ− r + κσF (t) + σ

∫

R−0x(eH(t,x) − 1)ν(dx) = 0 a.s. (3.11)

Now equation (3.11) clearly has an infinite number of possible solutionpairs (F,H). To see this suppose that f ∈ L1(R− 0, ν), then if (F,H) is

a solution, so too is

(F +

∫R−0 f(x)ν(dx), log

(eH − κf

x

)). Consequently

there are an infinite number of possible measures Q with respect to which Sis a martingale. So the general Levy process model gives rise to incompletemarkets. The following example is of considerable interest.

The Brownian Case

Here we have ν ≡ 0, κ 6= 0 and the unique solution to (3.11) is

F (t) =r − µ−mσ

κσa.s..

13

So in this case, the stock price is a geometric Brownian motion, and in factwe have a complete market. This leads to the famous Black-Scholes pricingformula which dates back to the pioneering work of Black and Scholes in1973.

The Poisson caseHere we take κ = 0 and ν = λδ1 for λ > m+ µ−r

σ. Writing H(t, 1) = H(t),

we find

H(t) = log

(r − µ + (λ−m)σ

λσ

)(a.s.).

3.7 Incomplete Markets

If the market is complete and if there is a suitable predictable representationtheorem available, it is clear that the Black-Scholes approach described abovecan, in principle, be applied to price contingent claims. However, if stockprices are driven by a general Levy process as in (3.8), the market will beincomplete. Provided there are no arbitrage opportunities, we know thatequivalent measures Q exist with respect to which S will be a martingale,but these will no longer be unique. In this section we briefly examine someapproaches which have been developed for incomplete markets. These involvefinding a “selection principle” to reduce the class of all possible measures Qto a subclass, within which a unique measure can be found.

1. The Follmer-Schweizer Minimal Measure

In the Black-Scholes set-up, we have a unique martingale measure Q

for whichdQ

dP

∣∣∣∣Ft

= eY (t) where d(eY (t)) = eY (t)F (t)dB(t) for 0 ≤ t ≤ T .

In the incomplete case, one approach to selecting Q would be to simplyreplace B by the martingale part of our Levy process (3.8), so that wehave

d(eY (t)) = eY (t)P (t)

(κdB(t) +

∫

(c,∞)

xN(ds, dx)

), (3.12)

for some adapted process P = (P (t), t ≥ 0). If we compare this withthe usual coefficients of exponential martingales in (3.2), we see thatwe have

κP (t) = F (t), xP (t) = eH(t,x) − 1

14

for each t ≥ 0, x > c. Substituting these conditions into (3.11) yields

P (t) =r + µ−mσ

σ(κ2 + ρ)

where ρ =∫∞

cx2ν(dx), so this procedure selects a unique martingale

measure, under the constraint that we only consider measure changes ofthe type (3.12). Terence Chan has shown that this coincides with a gen-eral procedure introduced by Follmer and Schweitzer, which works byconstructing a replicating portfolio of value V (t) = α(t)S(t) + β(t)ß(t)and discounting it to obtain V (t) = α(t)S(t) + β(t)ß(0). If we now de-fine the cumulative cost C(t) = V (t)− ∫ t

0α(s)dS(s), then Q minimises

the risk E((C(T )− C(t))2|Ft).

2. The Esscher Transform

In this section, we will make the additional assumption that

∫

|x|≥1

euxν(dx) < ∞,

for all u ∈ R. In this case we can analytically continue the Levy-Khintchine formula to obtain, for each t ≥ 0

E(e−uX(t)) = e−tψ(u),

where ψ(u) = −η(iu)

= bu− 1

2κ2u2 +

∫ ∞

c

(1− e−uy − uyχB(y))ν(dy).

Now recall the martingales Mu = (Mu(t), t ≥ 0), where each Mu(t) =expiuX(t)− tη(u). You can check directly that the martingale prop-erty is preserved under analytic continuation, and we will write Nu(t) =Miu(t) = exp−uX(t) + tψ(u). The key distinction between the mar-tingales Mu and Nu is that the former are complex-valued while thelatter are strictly positive. For each u ∈ R, we may thus define a newprobability measure by the prescription

dQu

dP

∣∣∣∣Ft

= Nu(t),

15

for each 0 ≤ t ≤ T . Qu is called the Esscher transform of P by Nu.It has a long history of application within actuarial science. ApplyingIto’s formula to Nu, we obtain

dNu(t) = Nu(t−)(−κuB(t) + (e−ux − 1)N(dt, dx)). (3.13)

On comparing this with our usual prescription (3.2) for exponentialmartingales eY , we find that

F (t) = −κu, H(t, x) = −ux

and so (3.11) yields the following condition for Qu to be a martingalemeasure:-

−κ2uσ + mσ + µ− r + σ

∫ ∞

c

x(e−ux − 1)ν(dx) = 0.

Define z(u) =∫∞

cx(e−ux − 1)ν(dx) − κ2u, for each u ∈ R. Then our

condition takes the form

z(u) =r − µ−mσ

σ.

Since z′(u) ≤ 0, we see that z is monotonic decreasing and hence in-vertible. Hence, this choice of u yields a martingale measure, under theconstraint that we only consider changes of measure of the form (3.13).

Chan has shown that this Qu minimises the relative entropy H(Q|P )where

H(Q|P ) =

∫dQ

dPlog

(dQ

dP

)dP.

3.8 Hyperbolic Levy Processes in Finance

So far we have concentrated our efforts in general discussions about Levyprocesses as models of stock prices, without looking at any particular caseother than Brownian motion. An example of a Levy process which appears tobe well-suited to modelling stock price movements is the hyperbolic processwhich we will now describe.

16

3.8.1 Hyperbolic Distributions

Let Υ ∈ B(R) and let (gθ; θ ∈ Υ) be a family of probability density functionson R such that the mapping (x, θ) → gθ(x) is jointly measurable from R×Υto R. Let ρ be another probability distribution on Υ which we call the mixingmeasure, then by Fubini’s theorem, we see that the probability mixture

h(x) =

∫

Υ

gθ(x)ρ(dθ),

yields another probability density function h on R. Indeed each h(x) ≥ 0and ∫

Rh(x)dx =

∫

Υ

(∫

Rgθ(x)dx

)ρ(dθ) =

∫

Υ

ρ(dθ) = 1.

The hyperbolic distributions which we will now introduce arise exactly inthis manner. First we need to describe the mixing measure ρ.

We begin with the following integral representation for Bessel functionsof the third kind:-

Kν(x) =1

2

∫ ∞

0

uν−1 exp

(−1

2x

(u +

1

u

))du,

where x, ν ∈ R.From this, we see immediately that fa,b

ν is a probability density functionon (0,∞), for each a, b > 0, where

fa,bν (x) =

(ab

) ν2

2Kν(√

ab)xν−1 exp

(−1

2

(ax +

b

x

)).

The distribution which this represents is called a generalised inverseGaussian and denoted GIG(ν, a, b). It clearly generalises the inverse Gaussiandistribution discussed in our first lecture . In our probability mixture, wenow take ρ to be a GIG(1, a, b), Υ = (0,∞) and gσ2 to be the probabilitydensity function of a N(µ + βσ2, σ2) where µ, β ∈ R. A straightforward but

tedious computation, where we apply the beautiful result K 12(x) =

√π

2xe−x,

yields

hα,βδ,µ (x) =

√α2 − β2

2αδK1(δ√

α2 − β2)exp −α

√δ2 + (x− µ)2 + β(x− µ), (3.14)

for all x ∈ R, where we have, in accordance with the usual convention,introduced the parameters α2 = a + β2 and δ2 = b.

17

The corresponding law is called a hyperbolic distribution as log(hα,βδ,µ ) is a

hyperbola. These distributions were first introduced by O.Barndorff-Nielsenwithin models for the distribution of particle size in wind-blown sand de-posits. They are infinitely divisible and, in fact, self-decomposable.

All the moments of a hyperbolic distribution exist and we may computethe moment generating function Mα,β

δ,µ (u) =∫R euxhα,β

δ,µ (x)dx, to obtain:

Proposition 3.1 For |u + β| < α,

Mα,βδ,µ (u) = eµu

√α2 − β2

K1(δ√

α2 − β2)

K1(δ√

α2 − (β + u)2)√α2 − (β + u)2

.

Proof. Straightforward manipulation. ¤Note that by analytic continuation, we get the characteristic function

φ(u) = M(iu) which is valid for all u ∈ R.

For simplicity, we will restrict ourselves to the symmetric case whereµ = β = 0. If we reparametrise, using ζ = δα, we obtain the two-parameterfamily of densities

hζ,δ(x) =1

2δK1(ζ)exp

−ζ

√1 +

(x

δ

)2

.

The corresponding Levy process Xζ,δ = (Xζ,δ(t), t ≥ 0) has no Gaussianpart and can be written

Xζ,δ(t) =

∫ t

0

∫

R−0xN(ds, dx),

for each t ≥ 0.

3.8.2 Option Pricing with Hyperbolic Levy Processes

The hyperbolic Levy process was first applied to option pricing by Eberleinand Keller following a suggestion by O.Barndorff-Nielsen. The analogy withsand production was that just as large rocks are broken down to smallerand smaller particles to create sand so, to quote Bingham and Kiesel “this‘energy cascade effect’ might be paralleled in the ‘information cascade effect’,whereby price-sensitive information originates in, say, a global newsflash andtrickles down through national and local level to smaller and smaller unitsof the economic and social environment.”

18

We may again model the stock price S = (S(t), t ≥ 0) as a stochasticexponential driven by a Xζ,δ so that

dS(t) = S(t−)dXζ,δ(t),

for each t ≥ 0 (we omit volatility for now and return to this point later). Adrawback of this approach is that the jumps of Xζ,δ are not bounded below.Eberlein and Keller suggested overcoming this problem by introducing astopping time τ = inft > 0; ∆Xζ,δ(t) < −1 and working with Xζ,δ insteadof Xζ,δ, where for each t ≥ 0,

Xζ,δ(t) = Xζ,δ(t)χt≤τ,

but this is clearly a somewhat contrived approach. An alternative pointof view, also put forward by Eberlein and Keller is to model stock prices byan exponential hyperbolic Levy process and utilise

S(t) = S(0)eXζ,δ(t).

As usual we discount and consider

S(t) = S(0)eXζ,δ(t)−rt,

and we require a measure Q with respect to which S = (S(t), t ≥ 0) is amartingale. As expected, the market is incomplete, and we use the Esshertransform to price the option. Hence we seek a measure of the form Qu,which satisfies

dQu

dP

∣∣∣∣Ft

= Nu(t) = exp (−uXζ,δ(t)− t log(Mζ,δ(u))).

Mζ,δ(u) herein denotes the moment generating function of Xζ,δ(1), as

given by Proposition 3.1, for |u| < α. Recalling Lemma 3.2, we see that Sis a Q-martingale if and only if SNu = (S(t)Nu(t), t ≥ 0) is a P -martingale.Now

S(t)Nu(t) = exp ((1− u)Xζ,δ(t)− t(log(Mζ,δ(u) + r)).

But we know that (exp ((1− u)Xζ,δ(t)− t(log(Mζ,δ(1− u))))), t ≥ 0) is a

martingale and comparing the last two facts, we find that S is a Q-martingaleif and only if

r = log(Mζ,δ(1− u))− log(Mζ,δ(u))

= log

[K1(

√ζ2 − δ2(1− u)2)

K1(√

ζ2 − δ2u2)

]− 1

2log

[ζ2 − δ2(1− u)2

ζ2 − δ2u2

].

19

The required value of u can now be determined from this expression bynumerical means.

We can now price a European call option with strike price k and expirationtime T . Writing S(0) = s as usual, the price is

V (0) = EQu(e−rT (S(T )− k)+)

= EQu(e−rT (seXζ,δ(t) − k)+).

If we now let f(t)ζ,δ be the pdf of Xζ,δ(t) with respect to P , we can use the

Esscher transform to show that Xζ,δ(t) also has a pdf with respect to Qu

which is given by

f(t)ζ,δ(x; u) = f

(t)ζ,δ(x)e−ux−t log(Mζ,δ(u)),

for each x ∈ R, t ≥ 0. We thus obtain the pricing formula

V (0) = s

∫ ∞

log( ks)

f(T )ζ,δ (x; 1− u)dx− e−rT k

∫ ∞

log( ks)

f(T )ζ,δ (x; u)dx.

Finally we discuss the volatility as promised. Suppose that instead of ahyperbolic process, we reverted to a Brownian motion model of logarithmicstock price growth and wrote S(t) = eZ(t) where Z(t) = σB(t) for eacht ≥ 0, then the volatility is given by σ2 = E(Z(1)2). By analogy, we definethe volatility in the hyperbolic case by σ2 = E(Xζ,δ(1)2). From the momentgenerating function and properties of Bessel functions, we obtain

σ2 =δ2K2(ζ)

ζK1(ζ).

3.9 The Generalised Black-Scholes Equation

In their original work, Black and Scholes derived a partial differential equa-tion for the price of a European option. It is worth trying to imitate thisin the general Levy market. We price our option using the Esscher trans-form, to establish that there is a measure Q such that S(t) = e−rtEX(t) is amartingale, hence

dS(t) = σS(t−)dBQ(t) +

∫

(c,∞)

σS(t−)xNQ(dt, dx).

(where we have taken κ = 1, for convenience). It follows that

dS(t) = rS(t−)dt + σS(t−)dBQ(t) +

∫

(c,∞)

σS(t−)xNQ(dt, dx).

20

We consider a contingent claim X and construct an associated functions → X(s) which describes its dependence on that of the underlying stock.The value of the option at time t is:

C(t, s) = EQ(e−r(T−t)X(S(t))|S(t) = s).

Now consider the integro-partial differential operator L defined on func-tions which are twice differentiable in the space variable and differentiable inthe time variable:

(LF )(t, x) =∂F

∂t(t, x) + rx

∂F

∂x(t, x) +

σ2x2

2

∂2F

∂x2(t, x)− rF (t, x)

+

∫

(c,∞)

[F (t, x(1 + σy))− F (t, x)− xσy

∂F

∂x(t, x)

]ν(dy)

(3.15)

Theorem 3.5 If LF = 0 with terminal boundary condition F (T, z) = X(z)then F = C.

Proof. First consider the related integro-differential operator

L0F (t, x) = LF (t, x) + rF (t, x).

It is an easy exercise in calculus to deduce that LF = 0 if and only ifL0G = 0, where G(t, x) = e−rtF (t, x). By Ito’s formula:

G(T, S(T ))−G(t, S(t)) = a Q−martingale +

∫ T

t

L0G(u, S(u−))du

= a Q−martingale,

henceG(t, s) = EQ(G(T, S(T ))|S(t) = s)

and we thus deduce that

F (t, s) = e−r(T−t)EQ(F (T, S(T ))|S(t) = s)

= e−r(T−t)EQ(X(S(T ))|S(t) = s),

as was required. ¤If we take ν = 0 so we have a stock market driven solely by Brownian

motion, then we recapture the famous Black-Scholes pde. The more generaloperator L is much more complicated, nonetheless both analytic and numer-ical methods have been devised to enable it to be applied to option pricingproblems.

21

3.10 An invitation to Malliavin calculus

Throughout this section, we will work solely with one-dimensional Brownianmotion B = (B(t), t ≥ 0). We also take the canonical version of B, i.e. werealise it as B(t)(ω) = ω(t) for ω ∈ Ω, where Ω is the space of continuouspaths from R+ to R which vanish at the origin (P is Wiener measure). Weshould however bear in mind that the development of Malliavin calculus formore general Levy processes and their financial application is an area wherethere is currently a great deal of activity, specifically through the work ofBernt Øksendal and his collaborators.

Let’s return to the predictable representation of a square-integrable claimX

X = E(X) +

∫ T

0

F (s)dB(s).

It would be very helpful to have more information about the process F .This is given by the Clark-Ocone formula which tells us that for sufficientlysmooth F

F (s) = E(Ds(X)|Fs),

where Ds is the Malliavin derivative.

Malliavin calculus (also called the stochastic calculus of variations) wasoriginally developed by Paul Malliavin at the end of the 1970s as part of hisscheme for obtaining a probabilistic technique for determining when the tran-sition densities of solutions of stochastic differential equations are smooth.The three main ingredients of this are:

• Wiener chaos

Multiple stochastic integrals are defined on symmetric multilinear square-integrable (deterministic) functions on [0, T ]n by continuous linear ex-tension of the following procedure:

if f(t1, . . . , tn) =

p∑i1,...,in=1

ai1...inχA1(t1) · · ·χAn(tn),

then the multiple Wiener integral∫[0,T ]n

f(t1, . . . , tn)dB(t1) . . . dB(tn)

is usually denoted as In(f) and is defined by

In(f) =

p∑i1,...,in=1

ai1...inB(A1) · · ·B(An),

22

where B([s, t]) = B(t)−B(s).

These are orthogonal:-

E(Im(f)In(g)) =

0 if m 6= n

n!〈f, g〉L2([0,T ]n) if m = n

Any F ∈ L2(Ω,F , P ) has a Wiener chaos expansion

F = E(F ) +∞∑

n=1

In(fn).

This should be compared with the predictable representation.

• The Malliavin Derivative

If F ∈ L2(Ω,F , P ) is such that

∞∑n=1

nn!||fn||2L2([0,T ]n) < ∞,

we can define, for each t ∈ (0, T ]

DtF =∞∑

n=1

nIn−1(fn(·, t)),

where In−1(fn(·, t)) signifies that we compute the multiple integral onlywith respect to the first n− 1 variables of t1, . . . , tn−1, t.

• The Skorohod Integral

If u = (u(t), t ≥ 0) is a process for which each u(t) ∈ L2(Ω,F , P ),then it has a Wiener chaos expansion, u(t) = E(u(t)) +

∑∞n=1 I(fn(t)).

fn will denote the extension of fn to a symmetric function of n + 1variables (t is the extra dimension). If u is such that

∞∑n=1

(n + 1)!||fn||2L2([0,T ]n+1) < ∞,

we define its Skorohod integral:

∫ T

0

u(t)δB(t) =∞∑

n=0

In+1(fn).

23

We sometimes write δ(u) =∫ T

0u(t)δB(t). δ is called the divergence.

Some properties:-

– The Malliavin derivative and Skorohod integral are mutually ad-joint, i.e. if u and F are such that these are well-defined, we havethe integration by parts formula:

E(Fδ(u)) = E(∫ T

0

(DtF )u(t)dt

).

– The Skorohod integral is a non-anticipating extension of the Itointegral, i.e. if u is predictable and square-integrable:

∫ T

0

u(t)δB(t) =

∫ T

0

u(t)dB(t).

Example

∫ T

0

B(t)δB(t) =

∫ T

0

B(t)dB(t) =1

2(B(T )2 − T ).

The same reasoning cannot be applied to∫ T

0B(T )δB(t) as B(T ) is only

F(t)-adapted for t ≥ T and hence is not Ito-integrable. However, B(T ) hasa chaos expansion with

f1 = 1 and fn = 0 for n ≥ 2.

Since (as a function of two variables) f1 = 1, we have

∫ T

0

B(T )δB(t) =

∫ T

0

∫ T

0

1dB(s)dB(t)

= 2

∫ T

0

∫ t

0

dB(s)dB(t) = 2

∫ T

0

B(t)dB(t) = B(T )2 − T.

24

References and Further Reading

These lectures have been broadly based on my recent book:

D.Applebaum Levy Processes and Stochastic Calculus, Cambridge Uni-versity Press (2004)

and from an earlier course of lectures partly derived from it, which havebeen separately published as

D.Applebaum, Levy processes in Euclidean spaces and groups in Quan-tum Independent Increment Processes I: From Classical Probability to Quan-tum Stochastic Calculus, Springer Lecture Notes in Mathematics , Vol. 1865M Schurmann, U. Franz (Eds.) 1-99,(2005)

A comprehensive account of the structure and properties of Levy processesis:

K-I.Sato, Levy Processes and Infinite Divisibility, Cambridge UniversityPress (1999)

A shorter account, from the point of view of the French school, whichconcentrates on fluctuation theory and potential theory aspects is

J.Bertoin, Levy Processes, Cambridge University Press (1996)

For an insight into the wide range of both theoretical and applied recentwork wherein Levy processes play a role, consult

O.E.Barndorff-Nielsen,T.Mikosch, S.Resnick (eds), Levy Processes: The-ory and Applications, Birkhauser, Basel (2001)

There are many books on stochastic calculus, stochastic integration etc. Avery nice comprehensive introduction, with an emphasis on Brownian motionas noise, can be found in

B.Øksendal, Stochastic Differential Equations (sixth edition), Springer-Verlag Berlin Heidelberg (2003)

For those who want the theory in full generality (i.e. based on generalsemimartingales with jumps), the standard text is

P.Protter, Stochastic Integration and Differential Equations (second edi-tion), Springer-Verlag, Berlin Heidelberg (2003)

25

There are a plethora of books and articles on stochastic aspects of optionpricing. For a very nice short but rigorous introduction, read

P.Protter, A partial introduction to financial asset pricing theory, Sto-chastic Processes and their Applications 91, 169-203 (2001)

Two books have now appeared which focus entirely on models with jumps:

R.Cont, P.Tankov, Financial Modelling with Jump Processes, Chapmanand Hall/CRC (2004)

is extremely comprehensive and also contains a lot of valuable backgroundmaterial on Levy processes.

W.Schoutens, Levy Processes in Finance: Pricing Financial Derivatives,Wiley (2003)

is shorter and aimed at a wider audience than mathematicians and sta-tisticians.

One of the best places to get an insight into the Malliavin calculus (whilekeeping technicalities to a minimum) is

B.Øksendal, An introduction to Malliavin calculus with applications toeconomics, available from http://www.nhh.no/for/dp/1996/wp0396.pdf

Note that a new book dedicated to financial applications of Malliavincalculus has just been published:

P.Malliavin, A.Thalmaier, Stochastic Calculus of Variations in Mathe-matical Finance, Springer-Verlag (2005)

For a large number of preprints on Levy processes and related topics, aswell as mini-proceedings of the 1998 and 2002 Aarhus conferences on Levyprocesses go to the MaPhySto website: http://www.maphysto.dk.

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