In each of the models discussed in Section 3, individual asset prices are driven by at least two independent sources of randomness so that the corresponding market models are in- complete. Based on the extended Girsanov-type measure transformation recalled in Sec- tion 2.5, in this section we shall discuss existence and, where applicable, uniqueness of martingale measures, thereby exhibiting also the market price of (jump-diffusion) risk.
Uniqueness of the martingale measure will be mainly related tocompletion of the market.
We want to point out that, as will be shown in more detail in the next Section 5 on hedg- ing, it is not necessarily true that, if a market iscompletedto yield a unique martingale measure, then it is also genuinely complete in the sense that every contingent claim can be duplicated by a self financing portfolio. In fact, for marked point process with an infinite mark space, i.e., with an infinite number of sources of randomness, it will be shown in
Ch. 5: Jump-Diffusion Models 187
Section 5.1.2 that uniqueness of the martingale measure implies only some form ofap- proximate completeness. In this Section 4 we shall limit ourselves to the jump-diffusion asset price and term structure models of Section 3.1. In Section 4.1 below we treat the case of jump-diffusion models for asset prices and show that the market can relatively easily be completed to yield a unique martingale measure if the jump part corresponds to a marked point process with a finite number of marks (multivariate point processes). For an infinite number of marks the situation is studied in more detail in Section 4.2 below in the context of term structure models.
4.1. The case of jump-diffusion asset price models
We start with a jump-diffusion model, where the jump part corresponds to a univariate Poisson point process withP-intensityλt, namely (see (34) forK=1)
dSt=St−[àtdt+σtdwt+γtdNt]
=St−
(àt+γtλt)dt+σtdwt+γtdMt
(45)
with (see (2)) Mt =Nt −t
0λsds the P-martingale corresponding to Nt. The Radon–
Nikodym derivative for an absolutely continuous change of measure from P toQ, that implies a translation of the Wiener byθt and a change of the Poisson intensity fromλt to ψtλt, is (see (27) forK=1)
Lt=exp t
0
(1−ψs)λs−1 2θs2
ds+
t 0
θsdws+ t
0
logψsdNs
. (46)
Defining the Wiener and Poisson martingaleswtQandMtQby (see (22) and (2)) dwQt =dwt−θtdt,
dMtQ=dNt−ψtλtdt (47)
the dynamics ofStunderQbecome dSt=St−
(àt+σtθt+γtψtλt)dt+σtdwQt +γtdMtQ
. (48)
Taking as numeraire the usual money market accountBt, where dBt=rtBtdt, we imme- diately see thatQis a martingale measure, i.e., a measure under whichSt :=Bt−1St is a martingale, ifθt andψt0 are chosen such thatàt +σtθt+γtψtλt=rt. From here we see that, for each pair(θt, ψt)withψt0 arbitrary and
θt=σt−1(rt−àt−γtψtλt), (49)
we obtain a martingale measure, i.e., we can obtain infinitely many martingale measures, one for each choice ofψt.
Concerning the market price of riskρt, from (45) and (49) we have
ρt:=àt+γtλt−rt=γtλt−σtθt−γtψtλt = −σtθt−γtλt(ψt−1) (50)
188 W.J. Runggaldier
from where we see that (−θt)can be interpreted as risk premium per unit of diffusion volatility, whereas−λt(ψt−1)can be interpreted asrisk premium per unit of jump volatil- ity. On an arbitrage-free market all assets have, at a given timet, the same diffusion- and jump-risk premia and they determine, via the Girsanov transformation, i.e., via (46), the equivalent martingale measureQ.
We obtained infinitely many martingale measures because, for a single risky asset, we had two independent sources of randomness. One may thus expect that, by adding a further asset, one cancompletethe market to obtain a unique martingale measure. Consider then, in addition toSt in (45), an asset with priceSt satisfying
dSt= St−[ ¯àtdt+ ¯σtdwt+ ¯γtdNt]. (51) Notice thatSt could correspond to the price of a derivative asset with underlyingSt. In fact, if one is given the explicit expression of this derivative price in terms of St, i.e., St=F (t, St), then (51) is straightforwardly obtained from (45) by use of the generalized Ito formula (19). Since the two risk premiaθt andλt(ψt−1)have to be the same for all assets, we may impose (49) on both assets with pricesSt andSt respectively, namely
θt=σt−1(rt−àt−γtψtλt)= ¯σt−1(rt− ¯àt− ¯γtψtλt) (52) from where one immediately gets
ψtλt=rt(σt− ¯σt)+(àtσ¯t−σtà¯t)
σtγ¯t−γtσ¯t . (53)
Inserting this expression in (49) it follows that θt=γt(à¯t−rt)− ¯γt(àt−rt)
σtγ¯t−γtσ¯t . (54)
We have thus obtained unique risk premia and, consequently, a unique martingale measure provided the coefficients in (45) and (51) are such thatσtγ¯t−γtσ¯t=0 and thatψtλt in (53) is positive.
With the unique martingale measure we may expect to have also obtained acomplete market in the sense that, by investing in a self financing way in the two assets with prices St andSt, one can duplicate any claim. In Section 5.1.1 we shall show that, for the given market model, this is indeed the case.
It is easily seen that, if the jump part in the jump-diffusion model corresponds to a multivariate Poisson process, i.e., if instead of (45) we have (see (34))
dSt =St−
àtdt+σtdwt+ K k=1
γt(k)dNt(k)
=St−
àt+
K k=1
γt(k)λt(k)
dt+ K k=1
γt(k)dMt(k)
(55)
Ch. 5: Jump-Diffusion Models 189
withMt(k)=Nt(k)−t
0λs(k)ds, then the previous results admit a straightforward exten- sion. In particular, (49) becomes
θt=σt−1
rt−àt− K k=1
γt(k)ψt(k)λt(k)
(56) and the market price of risk is
ρt= −σtθt− K k=1
γt(k)λt(k)
ψt(k)−1 . (57)
This time the generick-th termγt(k)λt(k)(ψt(k)−1)on the right can be interpreted as risk premium per unit of jump volatility of typek.
Again we obtain infinitely many martingale measures by choosing freely ψt(k)0, (k=1, . . . , K),andθt according to (56). Having nowK+1 independent sources of ran- domness, we may expect that one cancompletethe market by addingKfurther assets to obtain a unique equivalent martingale measure. This can be done along the lines of (52)–
(54) although this time the calculations are more complicated and the conditions on the coefficients more cumbersome.
Finally, we consider the more general model (33) (or, equivalently, (32)) with a possibly infinite number of marks. Using theP-martingale measureq(ã)in (8), by analogy to (45) and (55) we may rewrite (33) as
dSt=St−
àtdt+σtdwt+
E
γ (t, y)p (dt,dy)
=St−
àt+
E
γ (t, y)λt(dy)
dt+σtdwt+
E
γ (t, y)q(dt,dy)
. (58)
Using the particular form of the intensity given in (9), we also have
E
γ (t, y)λt(dy)= ¯γtλt withγ¯t=
E
γ (t, y)mt(dy) (59)
and so (58) becomes, quite analogously to (45), dSt=St−
(àt+ ¯γtλt)dt+σtdwt+
E
γ (t, y)q(dt,dy)
. (60)
Consider then, instead of (46), the more general Radon–Nikodym derivative (25) that we rewrite here in the form analogous to (46) as
Lt=exp t
0
(1−ψsh¯s)λs−1 2θs2
ds+
t
0
θsdws+ t
0
log
ψshs(Ys) dNs
,
(61)
190 W.J. Runggaldier
where h¯s =
Ehs(y)ms(dy). Define next the Wiener and jump martingales wQt and qQ(dt,dy)by (see (22) and (8), (9) as well as (47))
dwQt =dwt−θtdt,
qQ(dt,dy)=p(dt,dy)−ψtλtht(y)mt(dy)dt. (62) The dynamics ofSt underQthen become
dSt=St−
(àt+σtθt+ Γtψtλt)dt+σtdwQt +
E
γ (t, y)qQ(dt,dy)
, (63)
whereΓt =
Eγ (t, y)ht(y)mt(dy). The measureQis now a martingale measure ifθtand ψt0 as well asht(y)0 are chosen so thatàt+σtθt+ Γtψtλt=rt, which leads to the following relation corresponding to (49)
θt=σt−1(rt−àt− Γtψtλt). (64)
Again, this leads to infinitely many martingale measures but, unless the mark space is finite, tocompletethe market in order to obtain a unique equivalent martingale measure one needs infinitely many assets. We shall discuss this situation in more detail in the context of bond markets in the next subsection.
To complete the analogy with the previous cases, notice that this time the market price of risk becomes (by (60) and (64))
ρt:=àt+ ¯γtλt−rt= ¯γtλt−σtθt− Γtψtλt= −σtθt−λt(Γtψt− ¯γt)
= −σtθt−λt
E
γ (t, y)
ψtht(y)−1
mt(dy). (65)
This time one may interpret[ψtht(y)−1]mt(dy)asrisk premium per unit of jump volatility of typey.
In this latter context of a more general model of type (45) we want to point out that a methodology to obtain all equivalent martingale measures has also been worked out in Prigent (2001).
We close this subsection by mentioning that, depending on the purpose, one can single out some specific martingale measures among the various possible ones in a jump-diffusion model, where the market has not been completed. As an example, the construction of the so-calledminimal martingale measurein a univariate Poisson jump diffusion model can be found in Runggaldier and Schweizer (1995). From a more practical point of view, an obvious possibility is always that of calibrating the model to market data.
4.2. The case of jump-diffusion term structure models
Consider first a term structure model where, under a given measureP, the (continuously compounded) forward ratesf (t, T )and the (zero coupon) bond pricesp(t, T )satisfy (36) and (37) respectively, namely
df (t, T )=α(t, T )dt+σ (t, T )dwt+
E
δ(t, T;y)p(dt,dy), (66)
Ch. 5: Jump-Diffusion Models 191
dp(t, T )=p(t−, T )
m(t, T )dt+v(t, T )dwt+
E
n(t, T;y)p(dt,dy)
. (67) We shall also make the ad hoc assumptions that all objects are specified in a way to guar- antee the validity of the various operations that will have to be performed, such as differ- entiation under the integral sign and interchange of the order of integration.
For later use we recall from Bjửrk, Kabanov and Runggaldier (1997) the relationship between the coefficients in (66) and (67): if f (t, T )satisfies (66), thenp(t, T ) satisfies (67) with
m(t, T )=r(t)+A(t, T )+1
2S(t, T )2, v(t, T )=S(t, T ),
n(t, T;y)=eD(t,T;y)−1,
(68)
wherer(t)=f (t, t)is the short rate and
A(t, T )= − T
t
α(t, s)ds, S(t, T )= −
T
t
σ (t, s)ds, D(t, T;y)= −
T
t
δ(t, s;y)ds.
(69)
In the given bond market there are, at least theoretically, infinitely many assets, namely the bonds for all possible maturitiesT > t. A martingale measureQis now a measure under which all these bond prices, discounted with respect to the money market account, are (local) martingales. We are therefore not even sure whether in such a given market model there exists a martingale measure and so our first purpose is to investigate the existence of such a measure.
Following essentially Bjửrk, Kabanov and Runggaldier (1997) and considering general marked point processes, we also take the general form of the Radon–Nikodym derivative Lt, namely (see (24) where, for simplicity, we putψt≡1)
dLt=Ltθtdwt+Lt−
E
hs(y)−1 q(ds,dy), (70)
where (see (8) and (9))
q(ds,dy)=p(ds,dy)−λsms(dy), (71)
i.e., we assume that, under P, the local characteristics of the marked point process p(ds,dy)are(λt, mt(dy)). By Theorem 2.5 we know that, under the measureQthat cor- responds toLt in (70), the local characteristics become(λt, ht(y)mt(dy))so that, defining (see also (62))
dwQt =dwt−θtdt,
qQ(dt,dy)=p(dt,dy)−λtht(y)mt(dy)ds (72)
192 W.J. Runggaldier
the bond pricesp(t, T )satisfy, underQ, the dynamics dp(t, T )=p(t−, T )
m(t, T )+v(t, T )θt+λt
E
n(t, T;y)ht(y)mt(dy)
dt +v(t, T )dwQt +
E
n(t, T;y)qQ(dt,dy)
. (73)
A necessary condition for the existence of martingale measureQis then that there ex- ist a predictable processθt and a predictableE-indexed processht(y)0 such that the conditions of Theorem 2.5 hold and
m(t, T )+v(t, T )θt+λt
E
n(t, T;y)ht(y)mt(dy)=r(t). (74) Notice that this implies for the market price of risk a relation analogous to (65), namely
ρt:=m(t, T )+λt
E
n(t, T;y)mt(dy)−r(t)
= −v(t, T )θt−λt
E
n(t, T;y)
ht(y)−1
mt(dy). (75)
We shall now translate condition (74), involving the coefficients of (67), into a condition involving the coefficients of (66), namely of the forward rates. Using (68), condition (74) becomes
A(t, T )+1
2S(t, T )2+S(t, T )θt+
E
hs(y)ν(t, T;dy)=0 (76) withν(t, T;dy):=(eD(t,T;y)−1)λtmt(dy)and withA, SandDas in (69).
When building a term structure model it is often convenient to specify all objects di- rectly under a martingale measureQand this obviously imposes some restrictions on the coefficients in the models. Concentrating on forward rates, assume that we want model (66) to be valid under a martingale measureQ, i.e., we are postulating thatP =Qand so we have to chooseθt≡0, ht(y)≡1. Notice now that (76) has to hold for all maturities so that, inserting the above choices ofθt andht(y)and differentiating with respect toT, we obtain (using also (69)) the following necessary condition
α(t, T )=σ (t, T ) T
t
σ (t, s)ds−
E
δ(t, T;y)eD(t,T;y)λtht(y)mt(dy) (77) which is a clear extension of the classicalHeath–Jarrow–Morton drift conditionfor the pure diffusion case.
Having investigated the existence of a martingale measure, we may next look for condi- tions implying its uniqueness. Concentrating again on forward rates, a necessary condition for the existence of a martingale measure has been seen to be the existence of a predictable θt and a predictableE-indexedht(y)0 such that relation (76) holds. Quite obviously then, if (76) admits a unique solution inθt andht(y), the martingale measure is unique.
Ch. 5: Jump-Diffusion Models 193
To formalize this fact, consider the following linear operator [for technical details, that for simplicity we neglect here, we refer to Bjửrk, Kabanov and Runggaldier (1997)]
Kt:
θ , h(y) →S(t,ã)θ+
E
h(y)
eD(t,ã;y)−1 λtmt(dy). (78) The operatorKt is an integral operator of the first kind and we refer to it asmartingale operator. The martingale measure is then unique if and only if, dP ,dt-a.e., we have
KerKt=0. (79)
We may now wonder whether, in the present context of infinitely many sources of ran- domness, the uniqueness of the martingale measure implies completeness in the sense that every contingent claim can be replicated by a self financing portfolio. The answer is no;
in fact, as we shall mention in Section 5.1.2 below, we obtain only a form ofapproximate completeness.
We finally remark that the relationship (39) between discretely and continuously com- pounded forward rates has allowed Glasserman and Kou (1999) to carry over the just men- tioned results for continuously compounded forward rates also to the case when one has simple forwards instead. In fact, a model of the term structure of simple forwardsL(t, T ) (see (38)) is defined in Glasserman and Kou (1999) to be arbitrage-free, if it can be em- bedded in an arbitrage-free model of instantaneous forwardsf (t, T )via (39).