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Levy Models for Interest Rates and Foreign Exchange

Lane P. Hughston

Department of MathematicsUniversity College London

London WC1E 6BT

WBS Interest Rate ConferenceLondon, 15 March 2013

Collaborators:D. C. Brody (Brunel University), S. Jaimungal (University of Toronto),

E. Mackie (IMPA, Rio de Janeiro) and X. Yang (Imperial College)

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Levy Models for Interest Rates and Foreign Exchange WBS Interest Rate Conference

Introduction

The aim of this work is to get an understanding of the nature of the risk

premium associated with jumps in asset prices, with a view to applications tointerest rates and foreign exchange.

This is a topic about which surprisingly little is known, despite its importance.

We want to work in a rather general setting, without being tied to any particular

model, at least at the outset.

We shall assume here that the dynamics of asset prices are driven by Levyprocesses.

This is a big class of models (it includes all the Brownian motion models), so weneed not worry that we are being too restrictive.

We would like to understand the relation between risk, risk aversion, and theexcess rate of return (above the interest rate) offered by risky assets.

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Pricing kernel models for asset pricing with jumps

To this end, let us recall first the setup in the geometric Brownian motion

(GBM) model.

This is a simple model, but it captures a number of the main features of theproblem.

In the one-dimensional case, we have a Brownian motion

{Wt

}t

0 on a

probability space (,F,P) and the associated augmented filtration {Ft}t0.The model is characterised by the specification of:

(a) a pricing kernel; and

(b) a collection of one or more investment-grade assets.For simplicity, we assume for the moment the assets to be non-dividend-payingover the time horizon considered.


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Investment-grade assets and positive excess rate of return

The idea of an investment-grade asset is that it should offer a positive excess

rate of return above the interest rate.

For the pricing kernel we write:

t = erteWt

12

2t, (1)

where r > 0 is the interest rate, and > 0 is the risk aversion.

For a typical investment-grade asset we have

St = S0e(r+)teWt

12

2t, > 0, (2)

where is the volatility.

The term is called the risk premium or excess rate of return, and it ispositive under the assumptions made.

We observe that the risk premium is an increasing function of both the volatilityand the risk aversion.


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Since the risk premium is linear in each factor, the parameter is oftencalled the market price of risk in the GBM model.

It should be evident, however, that there is no a priori reason why the excessrate of return should be linear in and .

One of the reasons why the pricing kernel is a useful concept in finance is thatmarket equilibrium and absence of arbitrage are in some sense built in to the

notion that the product of the pricing kernel with the price of any asset payingno dividend gives a martingale.

In the GBM case for example we have:

tSt = S0e()Wt12()2t. (3)

In what follows we use this property of the pricing kernel to establish the generalform of a consistent Levy-driven asset pricing model.


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On the relation between risk and return when prices jump

WE shall construct a family of geometric Levy models (GLMs) in the spirit of

the GBM model.

Let {Xt}t0 be a Levy process.

We recall that a Levy process on a probability space (,F,P) is a process {Xt}such that X0 = 0, Xt

Xs is independent of

Fs for t > s (independent

increments), and

P(Xt Xs y) = P(Xt+h Xs+h y) (4)(stationary increments).

Here

{Ft

}denotes the augmented filtration generated by

{Xt}

.

In order for {Xt} to give rise to a GLM, we require thatE[eXt] < (5)

for all t 0 over the relevant time horizon, for all A for some connectedinterval A containing the origin.


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Henceforth we consider Levy processes satisfying such a moment condition.

It follows by the stationary independent and increments properties that for suchprocesses there exists a function (), the so-called Levy exponent, such that

E[eXt] = et(), (6)

for in the range indicated.

Then for A the process defined byMt = e

Xtt() (7)

is a martingale. For any A we call {Mt} the geometric Levy martingaleassociated with the Levy process {Xt}, with volatility .

Asset prices

Our geometric Levy model will be as put together as follows.

First we construct the pricing kernel {t}t0, and then we construct a family ofassociated asset prices.


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Let > 0 and assume that A. Then sett = e

rteXtt(). (8)

Now write {St}t0 for the price process of a typical investment grade asset.

For a consistent pricing model we require that {tSt}t0 should be a martingale.

In order to ensure that this is the case we assume that

tSt = S0eXtt(). (9)for some A. It follows then that

St = S0 ert e(+)Xt+t()t(), (10)

and hence

St = S0 ert eXt+t()t(), (11)where = + .

We assume that is positive and that A.


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We thus deduce that the asset price can be expressed in the form

St = S0 ert eR(,)t eXtt(), (12)

where the excess rate of return is given by

R(, ) = () + () ( ). (13)

Proposition 1. The excess rate of return in a geometric Levy model is positive,and is increasing with respect to the risk aversion and the volatility.

Proof. We observe that

() =1

tlnE

eXt

(14)

and hence

() = 1tE

XteXt

E [eXt]. (15)

It follows that

() =1

t

E

(Xt Xt)2eXt

E [eXt] , (16)L. P. Hughston - 9 - 15 November 2012

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where

Xt =E Xte

Xt

E [e

Xt]

. (17)

It follows that () > 0. Thus, the Levy exponent is a convex function.

Now suppose that we consider four values of in A such that1 < 2 3 < 4, and 3 = 1 + h and 4 = 2 + h. for some h > 0.

Then the convexity of () implies that(1) + (4) > (2) + (3). (18)

By letting either 2 or 3 be zero, and letting h be or , it follows that

() + () > ( ), (19)

since either < 0 < or else < 0 < .Thus since

R(, ) = () + () ( ), (20)it follows immediately that R(, ) > 0, as claimed.


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Figure 1: Plot of the Levy exponent () as a function of , with four values of alpha such that 3 = 1 + h and4 = 2 + h for some h > 0, showing the inequality (1) + (4) > (2) + (3)


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Next we observe (see Figure 2) that the convexity of () implies that

R(, ) =

(

) + (

) > 0, (21)

and

R(, ) = () ( ) > 0. (22)

Thus R(, ) is increasing in both and in .

Figure 2: Plots of the Levy exponent () as a function of showing that the excess rate of return is increasing bothin the risk aversion and the volatility.


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We conclude that the function

R(, ) = () + () ( ), (23)defined for > 0 and > 0, represents the excess rate of return function in ageometric Levy model.

It is positive and strictly bi-monotonic for all GLMs.

Note in particular that R(, ) is in general a nonlinear function of the risk and the risk aversion .

This suggests that the notion of market price of risk, common in the financeliterature, is somehow tied to Brownian motion.

Excess rate of return (or risk premium) is the more general notion.

Indeed, it can be shown that the only class of GLMs for which the excess rate ofreturn is bilinear is the GBM class.


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Siegels paradox in foreign exchange, and volatility bounds

When the GLM is extended to the case of foreign exchange, additional features

arise that are of interest.

It is reasonable to require a numeraire symmetry in the sense that if (e.g.) thedollar price of one pound sterling offers a positive excess rate of return, then thesterling price of one dollar should also offer a positive excess rate of return.

This applies to all investment grade currencies.

Let us examine the GBM case first, where the situation is reasonably transparent.

Let the dollar be the domestic currency, and the pound the foreign currency.

Let St be the price of one pound in dollars, and St the price of one dollar inpounds.

We write r for the domestic (dollar) interest rate, and f for the foreign(sterling) interest rate, both assumed constant.


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Then we have:

St = S0 e(rf)t eteWt

12

2t, (, > 0). (24)

It follows that the inverse exchange rate is given by:St = S0 e

(fr)t et eWt+12

2t, (25)

and thus

St = S0 e(fr)t e(

2)t eWt12

2t. (26)

In particular, we see that the excess rate of return on St is positive if and only if

it holds that > .

In equilibrium, investors on both sides of the Atlantic want to see the exchangepromising a positive excess rate of return in expectation.

The argument above shows that in a GBM model this possibility can be realised.

The requirement is that the volatility of the exchange rate should exceed thedollar market price of risk.


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Levy models for foreign exchange

Now lets look at the case of geometric Levy models for foreign exchange.

In the GLM situation the exchange rate takes the form

St = S0 e(rf)t eR(,)t eXtt(), (27)

where

R(, ) = () + (

)

(

). (28)

Thus for the inverse exchange rate we obtain:

St = S0 e(fr)t eR(,)t eXt+t(), (29)

and hence

St = S0 e(fr)t eR(,)t eXtt(), (30)

where

R(, ) = R(, ) + () + (). (31)We observe therefore that the foreign excess rate of return is given by

R(, ) = (

) + (

)

(

). (32)


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Now suppose that > . Then clearly

< < 0 < . (33)By the convexity of the Levy exponent this implies that R(, ) > 0.

On the other hand, suppose that .

Then we find that R(, ) 0, and we have thus shown the following:

Proposition 2. If the volatility exceeds the risk aversion, then both (a) theexcess rate of return on the FX rate, and (b) the excess rate of return on theinverse FX rate, are positive in a geometric Levy model for foreign exchange.

A further calculation gives us:

Proposition 3. In a geometric Levy model for foreign exchange for which thevolatility exceeds the risk aversion, the excess rate of return on the inverse FXrate is increasing with respect to the independent variables = and = .


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Examples of Levy Models

Example 1: Geometric Brownian motion. It is instructive to look at some

specific examples. In the case of GBM we have

() =1

22. (34)

It follows that

R(, ) = () + () ( )= 1

22 + 1

22 1

2( )2

= > 0. (35)

Example 2: Poisson process.Let {Nt} be a standard Poisson process with jump rate m. Then for anynonnegative integer n the distribution ofNt is

P(Nt = n) = emt(mt)

n

n !. (36)


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It follows that E[Xt] = mt, and that the Levy exponent is

() = m(e 1). (37)The associated geometric Levy martingale with volatility in this example is

Mt = exp [Nt mt(e 1)]. (38)A calculation then shows that the excess rate of return function is manifestlypositive, and increasing with respect to its arguments:

R(, ) = m(1 e)(e 1). (39)Since the jumps in the geometric Poisson model are upward, the risk that aninvestor faces is that there may be fewer jumps than one hopes for.

This point is made evident if we combine the expressions for the geometricmartingale and the excess rate of return function to obtain the following formulafor the price process of a non-dividend-paying asset with upward Poisson jumps:

St = S0 exp[rt + Nt mt e(e 1)]. (40)Thus, the effect of investor risk aversion is to reduce the downward drift rate inthe compensator term by attaching the factor e to it.

For the associated pricing kernel one has

t = exp [

rt

Nt

mt(e

1)]. (41)


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Example 3: Gamma process. By a standard gamma process with growthrate m, we mean a process {t} that starts at the value zero, hasgamma-distributed stationary and independent increments, and satisfies

E [t] = mt, Var [t] = mt. (42)

The density of t is given by

P(t dx) = 1{x > 0}xmt1ex

[mt]dx. (43)

Here [c] is the gamma function, which for real c > 0 has the integralrepresentation

[c] =

0

xc1exdx. (44)

The identity [c + 1] = c[c] implies that the mean of t is mt. The associated

moment generating function is

E[et] = (1 )mt = emt ln(1), (45)and hence the Levy exponent, which is well defined for < 1, is

() = m ln(1 ). (46)L. P. Hughston - 20 - 15 November 2012

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Now let {t} be a standard gamma process with growth rate m, and let be aconstant such that 0 < < 1.

The associated geometric martingale isMt = (1 )mtet. (47)

The jumps are upward; the compensator is a deterministic decreasing process.

If follows from (46) that the excess rate of return function is of the form

R(, ) = m ln 1 + (1 )(1 + ). (48)The difference between the numerator and the denominator in the argument ofthe logarithm in (48) is (1 + ) (1 )(1 + ) = , which is positive.Thus the ratio is greater than one, and we verify that R(, ) > 0.

The corresponding geometric gamma model model for an asset price is

St = S0 e(rf)t

1

1 +

mte t. (49)

Variants of the geometric gamma model appear in Heston (1993), Gerber &Shiu (1994), and Chan (1999).


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Example 4: Variance gamma process. It will be convenient first to discussthe symmetric variance gamma (VG) process.

Then in the next example we discuss the more general drifted VG process.Both of these processes are of interest from a mathematical perspective and as abasis for financial modelling. There is a well-known further extension of themodel, due to Carr et al. (2002), which will not be discussed here.

The VG model relies on the use of a gamma process as a subordinator.

Thus we begin with a standard gamma process {t} with rate m, as defined inthe previous example, and give it the dimensionality of time by dividing it by m.

This works since m has the dimensions of inverse time. In this way we define ascaled gamma process {t} by setting

t =

1

m t (50)and we observe that E [t] = t.

We call {t} a standard gamma subordinator.The symmetric variance gamma process {Vt}, with parameter m, is defined byletting

{Wt

}be a standard Brownian motion and setting Vt = Wt.


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The associated moment generating function is thus

E [exp(Vt)] = E [exp(Wt)] = E exp12

2t = 12

2mmt

, (51)

which is defined for 2 < 2m. Clearly must have units of inverse square-roottime, since m has units of inverse time. The associated Levy exponent is

() = m ln

1 2

2m

. (52)

As a consequence the geometric VG martingale (in the symmetric case) is

Mt =

1

2

2m

mtexp (Wt) . (53)

The excess rate of return function, which is positive and monotonic, is

R(, ) = m ln

1(

)2

2m

12

2m1

12

2m1

. (54)

The corresponding VG asset price process is thus of the following form:

St = S0 ert

1 ( )

2

2m

mt 1

2

2m

mtexp (Wt) . (55)


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Example 5: Asymmetric VG process. A well known and perhaps moreintuitive characterisation of the VG process is as follows. Let {1t } and {2t } bea pair of independent standard gamma processes, each with rate m.

Then the process defined by the difference between these two processes has bothupward and downward jumps, and is clearly symmetrical about the origin indistribution, with mean zero. In fact, if we normalise the difference by setting

Vt =12m

1t 2t , (56)

then it is easy to check that the variance of Vt is t.

If we consider the moment generating function we find by virtue of theindependence of the two gamma processes that

E [exp(Vt)] = E exp 12m 1t

2t

=

1

2m

mt 1 +

2m

mt

=

1

2

2m

mt. (57)


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It is evident (Madan & Senata 1990) that the process {Vt} thus defined has thelaw of a standard VG process.

The representation of the VG process as the normalised difference between twoindependent gamma processes suggests two generalisations.

One is that of Madan et al. (1998), where we consider an asymmetric differencebetween two independent standard gamma processes. Thus writing

Ut = 11t

2

2t , (58)

where 1 and 2 are nonnegative constants, a calculation of the respectivemoment generating functions shows that Ut is identical in law to a drifted VGprocess of the form

Ut = t + Wt, (59)

where and are constants.The relationship between , , 1, 2, and m is given by = m(1 2) and2 = 2m12 , together with

1 =1

2m

+

2 + 2m2

and 2 =

1

2m

+

2 + 2m2

. (60)


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The Levy exponent

() = m ln 1 (1 2) 12 2

(61)can also written in the form

() = m ln

1 m

2

2m2

, (62)

where the relevant range of is 1/2 < < 1/1.It is straightforward to write down the associated excess rate of return function,and the corresponding expression for an asset price.

In this example there is a single risk aversion factor.

On the other hand, one can also envisage the situation where the two gammadrivers are regarded as separate sources of risk, each being assessed

independently by the market.This situation arises in instances where investors are for some reason moreworried about downward jumps than upward ones.

It is said that studies in behavioural finance suggest that this may actually bethe case.


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One can model such a situation rigorously by introducing an asymmetric pricingkernel of the form

t = ert

(1 1)mt

(1 + 2)mt

e1

1t

e2

2t

, (63)and an asset price process of the form

St = S0 ert

1 1

1 + 1

mt 1 +

21 2

mte1

1t e2

2t . (64)

Thus (as desired) we have separate risk aversion factors for the upward jumpsand the downward jumps.

It is interesting to observe that in the case of behavioural asymmetry asregards aversion towards too many downward jumps vs not enough upward

jumps, both the asset price and the pricing kernel are driven by extended VGprocesses. But there are two distinct such processes, one driving the pricing

kernel, and the other driving the asset price.

Indeed, the pricing kernel is driven by 11t 22t , whereas the asset price is

driven by 11t 22t .


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Option prices. It is natural to ask what information can be extracted (orimplied) about the values of model parameters when one is given option prices.

In the case of the GBM model, for example, it is known that one can infer thevalue of volatility , but that the option price is independent of risk aversion .

In the case of a GLM, this is no longer the case: in general, option prices dependon both the risk aversion and the volatility.

Indeed, a variety of different situations can arise, each with its own character.

In the Poisson model, there are two nontrivial parametersthe risk aversion,and the jump rate m (the volatility is easily determined by observation of theprice process). Option prices depend on me, but not on m or separately.

Thus if we can estimate the value of the actual jump rate m by observations ofthe asset price, then can be inferred from option prices.

In the case of the gamma model, there are three nontrivial modelparametersthe risk aversion, the volatility, and the jump rate.

A calculation shows that option prices depend on m and on /(1 + ), but noton and separately, so neither nor can be determined exactly from optionprices.


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General Levy models for interest rates and foreign exchange

From a modern perspective there are two main approaches to the modelling of

interest rates.

These are the volatility approach and the pricing kernel approach.

Both have been investigated extensively in the case of a Brownian filtration, butrather less so in the general situation when bond prices are allowed to jump, and

little by way of consensus has emerged either in industry or among academics onhow best to incorporate jumps into the modeling of interest rates.

The situation with foreign exchange is in some respects worse, since there is atendency to model FX as if it were a thing in its own right, using stochasticvolatility and the like, and not seeing it as a part of an overall interest rate

framework.

But is seems that the only consistent way of viewing foreign exchange is as asystem of ratios of pricing kernels, one pricing kernel for each currency.

Lets take a closer look.


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We begin with a generalisation of the concept of a geometric Levy martingale.

Let (,

F,P) be a probability space, let

{Xt

}a Levy process that admits

exponential moments, and let {Ft} be the associated filtration.Let {t} be an {Ft}predictable process, chosen in such a way that t A fort 0, and such that the local martingale defined by

Mt = expt

0

sdXs

t

0

(s)ds (65)is a martingale.

If a predictable process {t} satisfies these conditions then we say it isadmissible.

Then we are led to consider a rather general class of asset pricing model of thefollowing form.

Let the exogenously specified short rate {rt} be {Ft}predictable, and be suchthat the unit-initialised money market account is finite almost surely for t > 0.


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Let the {Ft}-predictable risk aversion and volatility processes {t} and {t} bepositive, and be such that {t}, {t}, and {t t} are admissible in thesense noted above.

The pricing kernel is taken to be given by an expression of the form

t = exp

t

0

rs ds t

0

s dXs t

0

(s) ds

. (66)

The associated expression for the price of a typical non-dividend-paying asset is

St = S0 exp

t0

rs ds +

t0

R(s, s) ds +

t0

s dXs t

0

(s) ds

, (67)

where

R(, ) = () + () ( ). (68)is the excess rate of return function associated with the given Levy exponent.

As we have already noted, it is a remarkable property of the function R(, ),arising as a consequence of the convexity of the Levy exponent, that if thevolatility and the risk aversion are positive, then the excess rate of return ispositive, and is monotonically increasing in its arguments.


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Levy models for interest rates

The theory of interest rates can be developed in just this spirit in a way that

generalises the Heath-Jarrow-Morton (HJM) framework to the Levy category.

In particular, we are able to give a consistent treatment of the risk premiumassociated with interest rate products in such a way that the risk premium ispositive for bonds.

Interest rate models admitting jumps have been pursued by a number of authors,including: Shirakawa (1991), Jarrow & Madan (1995), Bjork et al. (1997), Bjorket al. (1997), Eberlien & Raible (1999), Raible (2000), Eberlein et al. (2005),Eberlein & Kluge (2006a,b, 2007), and Filipovic et al. (2010), to mention a few.

The approach that we shall take is to introduce a pricing kernel at the outset.

This is much better than attempting to model the interest rate system through aset of dynamical equations, such as the HJM equations, or the HJMM equations.

A rather general class of Levy interest rate models exhibiting the positive excessrate of return property can then be constructed as follows.


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The idea is to model the pricing kernel and the associated discount bond systemdirectly.

There is no need as such to introduce a system of instantaneous forward rates.

We merely require that the discount bond family should be smooth with respectto the maturity index, and that the discount volatility should approach zero forsmall bond maturity

Let the pricing kernel be given, as we have discussed, by (66), and write PtT forthe price at t of a bond that matures at T to deliver one unit of currency.

For the Levy discount bond model the bond price PtT is given by the followingexpression

P0T expt

0rsds +

t0

R(s, sT)ds +t

0sTdXs t

0(sT)ds

(69)

We shall require: (a) that {t} and {tT} are positive; (b) that {t}, {tT},and {tT t} are admissible in the sense indicated above; and (c) that {tT}should vanish as t approaches T.


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The maturity condition limtT PtT = 1 allows one to solve for the moneymarket account in terms of{tT} and {t}, as in the Brownian case:

Bt = 1P0t

expt

0stdXs +

t0

(st s)ds t0

(s)ds . (70)This expression for the money market account can be used in three ways.

First, a calculation making use of the fact that the discount bond volatility goesto zero as maturity approaches shows that the money market account is

differentiable, and one can work out an expression for the short rate process.

Second, inserting the expression for the Bt back into the bond price, we obtain:

PtT = P0tTexp

t0 (sT s)dXs

t0 (sT s)ds

expt0 (st s)dXs

t

0 (st

s)ds, (71)

where P0tT = P0T/P0t. This gives us an explicit expression for the discountbond prices which can be used for simulation and derivative pricing.

Third, if we substitute the expression for the money market account back intothe definition of the pricing kernel, we deduce the following:


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The pricing kernel in a Levy interest rate model can be expressed in terms of theinitial term structure{P0t}, the volatility{tT}, and the risk aversion {t} as:

t = P0t exp

t0

(st s)dXs t

0

(st s)ds

. (72)

This result offers a rather general method for modelling an arbitrage-free interestrate system in the physical measure when there are jumps.

In particular, if we model the volatility structure and the risk aversionexogenously, and specify the initial term structure, then we determine the pricingkernel, the discount bond system, the money market account, and the short rate.

As in the Brownian case, the volatility structure and the risk aversion process can

be modelled parametrically, up to undetermined functional degrees of freedom,to be fixed by calibration to the prices of market instruments at time zero.

Indeed, one could treat the Levy exponent itself (or more generally the associatedLevy characteristics) as part of the functional freedom of the model.


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Or, more generally, essentially the same line of reasoning can be applied in thecase of Levy-Ito models driven by Poisson random measure.

In that case the pricing kernel takes the following general form (Hughston &Jaimungal 2013):

t = P0t exp

t0

R

Vst(x)M(ds, dx) t

0

R

(exp Vst(x) 1)ds(dx)

(73)

where M(dt, dx) is a homogeneous Poisson random measure on R+

R, and

where dt (dx) is the associated mean measure.

The volatility and risk aversion processes are dependent on the jump size in agenerally nonlinear way (the corresponding functions are linear in Levy models),and we have:

Vst(x) = st(x) s(x). (74)With there formulae in hand, one can work out the corresponding expressions forthe bond prices, the short rate, and the instantaneous forward rate.


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References:

Bjork, T., Di Masi, G. , Kabinov, Y. & Runggaldier (1997) Towards a general

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