6.1. General aspects
With the introduction of jumps and/or stochastic volatility the market becomes incomplete.
Consequently, the principle of absence of arbitrage does not lead to a uniquely defined price. One obtains actually an entire range of prices [see Eberlein and Jacod (1997), Bel- lamy and Jeanblanc (2000)] and the preference structure of the investors has to come into play to determine thepricing measure. From the point of view of pure pricing, the prob- lem then reduces to determining a specific martingale measure or, equivalently, the market price of risk. To this effect there are various possibilities and in this section we mention some of them, the last two of which will be discussed in more detail.
(i) Historically it appears that a first approach to pricing in markets that are incomplete due to jumps in the prices and to a jumping volatility has been based on general equilibriumwith a representative agent [see, e.g., Ahn and Thompson (1988), Naik and Lee (1990), Ahn (1992)].
(ii) A somewhat related and rather recent approach is that ofpricing by utility maximiza- tion, in which the density of the martingale measure (thepricing kernel) is related to the marginal utility of terminal wealth [see, e.g., Frittelli (2000) and the references therein; for a specific jump-diffusion setting see Miyahara (1998)].
(iii) An alternative possibility is given by more econometric-type approaches based ones- timating/filtering the market price of riskon the basis of market data. Related to such an approach is the approach described in Herzel (1998) for a diffusion model with a volatility that may jump at a random time and where the price of a European call turns out to be a monotone function of a parameterλcharacterizing the martingale mea- sures. There exists then a uniqueλ∗consistent with the option price thus allowing to price all the other derivatives consistently with this option. This corresponds basically to completing the market with the given option.
202 W.J. Runggaldier
(iv) Approaches based onmarket completion. In the previous Section 4.1 we have dis- cussed various ways to complete both stock as well as bond markets of the jump- diffusion type. As we have seen, this completion leads always to a unique martingale measure, but it does not necessarily imply also completeness in the sense that every claim can be hedged with a self financing portfolio. On the other hand, the unique- ness alone of an equivalent martingale measure is already sufficient to obtain a unique arbitrage-free price of a claim as the expectation of its discounted value under this measure. In all cases where one achieves also completeness in the sense of hedging (essentially all cases except when there are an infinite number of sources of random- ness) then, always by absence of arbitrage, the (unique) initial value of the self fi- nancing and hedging portfolio has to coincide with the price computed as expectation under the unique martingale measure. The approach based on market completion has been widely used an implemented in various economic setups and here we mention just Shirakawa (1990, 1991), Jeanblanc-Piqué and Pontier (1990), Naik (1993), Mer- curio and Runggaldier (1993), Jarrow and Madan (1995, 1999). It has the advantage to lead to a unique price on the basis of the principle of absence of arbitrage alone, with- out having to make assumptions on a non-priced jump risk and without the need to introduce a general equilibrium model. On the other hand it requires that the stochas- tic evolution of more than just the underlying asset has to be specified and, without specific criteria, the completion may occasionally be rather arbitrary.
(v) In the previous Section 5, in the context of hedging it was mentioned that, if the market cannot be completed, then one has to accept some residual risk and it be- comes natural to determine the hedging strategy on the basis of a risk minimization criterion. On the other hand, in the previous point (iv) we recalled the fact that, in a complete/completed market the initial value of a self financing and hedging portfo- lio has to coincide with the arbitrage-free price of the claim. By analogy, it appears then natural to define as price of a claim in a noncomplete market theinitial value of a portfolio minimizing a given hedging criterion. Quite typically, the initial value of such a portfolio turns out to be the expectation of the discounted value of the given claim under a specific martingale measure. In other words, there is a correspondence between hedging criteria and martingale measures and the choice of a specific pric- ing measure can be based on the choice of a specific hedging criterion. An approach along these lines appears thus related to the pricing approach by utility maximization, mentioned in point (ii) above. As an example, let us point out that the criterion of risk minimization discussed in Section 5.2 leads to the so-calledminimal martingale measurethat was already mentioned at the end of Section 4.1. It has been further shown in Runggaldier and Schweizer (1995) that, if in a jump-diffusion model claims are priced according to the minimal martingale measure, then convergence of asset prices implies convergence of option prices. This stability result for prices computed according to the minimal martingale measure makes the risk minimization criterion discussed in Section 5.2 an attractive criterion for hedging. [For further extensions of this stability property see Prigent (1999), Hubalek and Schachermayer (1998).]
Ch. 5: Jump-Diffusion Models 203
6.2. Computational aspects
Assume that for a jump-diffusion model we have selected a specific martingale measure according to one of the approaches mentioned in the previous Section 6.1. We have then to compute the expectation of the (discounted value of the) claim under this martingale measure. In this section we shall mention some of the possible methods to accomplish this.
We consider first the univariate jump-diffusion model (45) under a generic martingale measureQwith intensity of the Poisson processNt given byψtλt. IfQcorresponds to the unique martingale measure obtained from a market completion as in Section 4.1, thenψtλt has to be taken according to (53). For simplicity we assume that all the prices are already discounted and so we can putrt ≡0. The dynamics ofSt underQare given by (see (48), (49), (47))
dSt=St−
−γtψtλtdt+σtdwQt +γtdNt
. (111)
We want to compute the value of a European call option, namelyEQ{(ST−K)+}. For this purpose we adapt an approach from Mercurio and Runggaldier (1993), assuming first that in (111) we haveγt≡γ, i.e., the jump coefficient is constant [for this case see also Aase (1988)]. We have
EQ
(ST −K)+
=EQ EQ
(ST −K)+|NT
. (112)
For a fixedk, i.e., whenNT =k (k=0,1, . . .), using the exponential formula (17) for the specific case when (14) is given by (111), we have
ST(k)=S0eklog(1+γ )exp
− T
0
γ ψsλs+1 2σs2
ds+
T
0
σsdwQs
(113) namely
logST(k)∼N
ã;mT, σT2 (114)
with
mT =logS0+klog(1+γ )− T
0
γ ψsλs+1 2σs2
ds, σT2=
T
0
σs2ds,
(115)
i.e., ST(k) is lognormal with mean and variance given by mT and σT respectively. Next compute (withΦ(ã)the cumulative standard Gaussian distribution function)
V0(k):=EQS
0
S(k)T −K +
= +∞
logk
ex−K dN
x;mT, σT2 dx
= 1 2π σT2
+∞
logk
exe−
1 2σ2
T
(x−mT)2
dx− K 2σT2
+∞
logk
e−
1 2σ2
T
(x−mT)2
dx
204 W.J. Runggaldier
=emT−12σT2Φ
mT +σT2−logK σT
−KΦ
mT −logK σT
:=(1+γ )kG(k, S0) (116)
with
G(k, S0)=S0exp
− T
0
γ ψsλsds
Φ(x)− K
(1+γ )kΦ(y), (117)
where
x=log(S0(1+γ )k/K)T
0 (−γ ψsλs+12σs2)ds
T 0 σs2ds
,
y=x− 0Tσs2ds.
(118)
Coming back to (112) we then have EQ
(ST −K)+
=EQ V0(NT)
=∞
k=0
(1+γ )kG(k, S0) Hk
k! e−H
(119) withH=T
0 ψsλsds. Notice that, for actual computations, the infinite sum in the right in (119) has to be truncated at a sufficiently large positive integer.
The result forγt≡γ can be easily extended to the case whenγt is a piecewise constant deterministic time function. To this effect, given a positive integermand a subdivision 0=t0m< t1m<ã ã ã< tmm=T, let
γt(m)=γ01{0}(t)+ m j=1
γj1(tjm−1,tjm](t); γj>−1. (120) Furthermore, letPj(j=1, . . . , m), be independent Poisson random variables with param- etersHj =tjm
tjm−1ψsλsds. The generalization of formula (119) is then EQ
(ST −K)+
= ∞
k1,...,km=0
exp m
j=1
kjlog(1+γj)
G(k1, . . . , km, S0) m j=1
(Hj)kj (kj)! e−Hj
(121)
with
G(k1, . . . , km, S0)=S0exp
− T
0
γs(m)ψsλsds
Φ(x)− K
!m
j=1(1+γj)kjΦ(y), (122)
Ch. 5: Jump-Diffusion Models 205
where
x=log(S0
!m
j=1(1+γj)kj/K)T
0 (−γt(m)ψsλs+12σs2)ds
T 0 σs2ds
,
y=x− 0T σs2ds.
(123)
Coming finally to the case of a more general deterministic time functionγt for the jump coefficient, we assume that there exist piecewise constant deterministic time functionsγt(m) andσt(m)such that
γt(m)↑γt, σt(m)↑σt asm→ ∞. (124)
Consider then a sequence of fictitious risky assets, whose (discounted) values St(m) are martingales with respect to the same martingale measureQas isSt in (111), namely they satisfy
dSt(m)=St(m)−
−γt(m)ψtλtdt+σt(m)dwQt +γt(m)dNt
. (125)
For each of the processesSt(m)we can compute v(m)0 =EQ
ST(m)−K +
(126) according to (121)–(123). In Mercurio and Runggaldier (1993) it is now shown that
mlim→∞v0(m)=v0=EQ
(ST −K)+
, (127)
i.e., ifγt is a generic time function, that can be approximated from below by a sequence of piecewise constant time functions, then the corresponding option value can be approxi- mated arbitrarily closely by computable expressions. In Mercurio and Runggaldier (1993) it is also shown that, for givenm,v0(m) can be interpreted as initial value of a mean self financing and risk minimizing portfolio in the sense of Section 5.2 when the asset price evolves in discrete time according to the processSt(m) of (125), evaluated at the discrete time pointstj. In line with the last part of point (v) of the previous Section 6.1, we may thus consider the approximating valuesv0(m)as option values themselves, computed according to the minimal martingale measure.
After having discussed the univariate jump-diffusion model (45), we turn now to the general jump-diffusion model with a marked point process and which can equivalently be represented either by (32) or (33). We opt here for the representation (32). i.e.,
dSt=St−
àtdt+σtdwt+γ (t, Yt)dNt
. (128)
In what follows we shall make the further Assumption 6.1.
(i) γ (t, Yt)≡γ (Yt), i.e.,γis independent of the current time;
206 W.J. Runggaldier
(ii) considering the representation of the marked point process as double sequence (Tn, Yn), assume thatTn is independent of Yn and the Yn form a sequence of inde- pendent random variables, the generic oneYnhaving lawm(dy).
The driving marked point process has thus localP-characteristics(λt, m(dy)).
Suppose that we have chosen a specific martingale measure Qand that we want to computev0=EQ{H (ST)}where, typically, we may haveH (S)=(S−K)+. For this purpose, in what follows we adapt a procedure from Chapter 7 in Lamberton and Lapeyre (1997).
Recall first from Theorem 2.5 that a general absolutely continuous measure transforma- tion fromP toQtransforms theP-local characteristics intoQ-local characteristics of the form(ψtλt, ht(y)m(dy)). Recalling furthermore (63) with (62) and (64), it is easily seen that, under the measureQcorresponding to the above local characteristics, the discounted value ofSt satisfies
dSt=St−
− Γtλ¯tdt+σtdwQt +γ (Yt)dNt
, (129)
where we have putΓt=
Eγ (y)ht(y)m(dy)andλ¯t=ψtλt. Using the exponential formula (17) to integrate (129), that is of the form of (14) with the representation (15), one imme- diately finds that, for a given initial asset priceS0, the valuev0(S0)of the claimH (ST)is given by
v0(S0)=EQ
H
S0exp
− T
0
Γtλ¯t+σt2 2
dt+
T
0
σtdwtQ NT
n=1
1+γ (Yn) . (130) Next let
V (S0):=EQ
H
S0exp
− T
0
σt2 2 dt+
T
0
σtdwtQ
(131) so that, for H (S)=(S−K)+, theV (S0)is given by the Black–Scholes formula, i.e., V (S0)=BS(S0). With the use ofV (S0)we can now write
v0(S0)=EQ
V
S0exp
− T
0
Γtλ¯tdt NT
n=1
1+γ (Yn)
=∞
k=0
EQ
V
S0exp
− T
0
Γtλ¯tdt k
n=1
1+γ (Yn) Hk k! e−H
, (132)
where, due to the local characteristics underQ, we haveH=T
0 ψsλsdsand where the expectation is with respect to the joint distribution of theYn that in Assumption 6.1 were supposed to be independent. This latter expectation can be explicitly computed in special
Ch. 5: Jump-Diffusion Models 207
cases, in more complicated cases one has to use simulations. Again, for the actual compu- tations, the infinite sum has to be truncated at a sufficiently large positive integer.
We close this section by mentioning that in Glasserman and Kou (1999), for the term structure models of simple forwards in the jump-diffusion setup described therein, the authors study the pricing of some derivative securities after having characterized arbitrage- free dynamics. The derivative prices are also used to investigate what types of patterns in implied volatilities are produced through jumps.
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Chapter 6