As seen in the previous part, the exponential L´evy market is in general incomplete, and hence, options are not superfluous assets whose payoff can be perfectly replicated in an ideal frictionless market. The option prices are themselves subject to modeling. It is natural to adopt an EMM that preserve the L´evy structure of the log return processXt Dlog.St=S0/as in the previous section. From now on, we adopt exactly this option pricing model and assume that the time-t price of a European claim with maturityTand payoffX is given by
…t DEQ˚
er.Tt/XˇˇSu;ut
; whereQis an EMM such that
St DS0ertCXt;
withXbeing a L´evy process underQ. Throughout,.2; b; /denotes the L´evy triplet ofXunderQ.
Note that in the case of a simple claim X D ˚.ST/;there exists a function C WRCRC!RCsuch that…t WDC.t; St.!//:Indeed, by the Markov property, one can easily see that
C.t; x/Der.Tt/EQ
h
˚
xer.Tt/CXTt i
: (4.39)
The following theorem shows thatCsatisfies an integro-partial differential equation (IPDE). The IPDE equation below is well-known in the literature (see e.g. Chan 1999and Raible2000) and its proof can be found in, e.g., (Cont and Tankov,2004, Proposition 12.1).
Proposition 6. Suppose the following conditions:
1. R
jxj1e2x.dx/ <1;
2. Either > 0or lim inf"&0"ˇR
jxj"jxj2.dx/ <1:
3. j˚.x/˚.y/j cjxyj, for allx; yand somec > 0.
Then, the functionC.t; x/in (4.39) is continuous onŒ0; T Œ0;1/,C1;2on.0; T / .0;1/and verifies the integro-partial differential equation:
@C.t; x/
@t Crx@C
@x.t; x/C 1
22x2@2C
@x2.t; x/rC.t; x/
C Z
R0
C.t; xey/C.t; x/x.ey1/@C
@x.t; x/
.dy/D0;
onŒ0; T /.0;1/with terminal conditionC.T; x/D˚.x/;for allx.
Acknowledgement Research partially supported by a grant from the US National Science Foundation (DMS-0906919). The author is indebted to the editors for their helpful suggestions that improve this chapter considerably.
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Multivariate Time Series Models for Asset Prices
Christian M. Hafner and Hans Manner
Abstract The modelling of multivariate financial time series has attracted an enormous interest recently, both from a theoretical and practical perspective.
Focusing on factor type models that reduce the dimensionality and other models that are tractable in high dimensions, we review models for volatility, correlation and dependence, and show their application to quantities of interest such as value-at-risk or minimum-variance portfolio. In an application to a 69-dimensional asset price time series, we compare the performance of factor-based multivariate GARCH, stochastic volatility and dynamic copula models.