But withf(x1, x2) = (ex1 −1)(ex2 −1)we have f(x1,0) =f(0, x2) = 0.
However, these conditions are not necessary: We might just as well imagine a very special situation where the two terms in (5.3.28) sum up to zero because both depen-dence structures go in different directions. In this case we would have a dependent two-dimensional process but the hedge ratios would not depend on one another.
The reason for this strange behaviour is that risk minimization is a quadratic crite-rion which just deals with the second moments, i.e. with variances and covariances.
Therefore (5.3.28), which is some kind of covariance, can stipulate an ‘independent’
hedging scheme when in factS1 and S2 are dependent due to some higher moment dependence.
5.4 Concluding remarks
There are two remarks which should be added to the above considerations: Beginning with the one-dimensional case of Section 5.2, the optimal hedge parameterϑˆof (5.2.8) is not yet explicit enough to obtain a numerical value. More precisely, the quantities φcandφd(x) in (5.2.5) involve the price functionalV(·, S)of the option in the form of its first derivative and the integral ofV(·, Sex)−V(·, S) with respect to the Lévy measure. The solution lies again in the field of Fourier inversion: AsV(·, S) can be represented as a Fourier integral15 the hedge parameter ϑˆ can also be written as a Fourier integral if the technical conditions allow for an interchange of the order of the necessary integrations. Looking a bit closer at this Fourier integral, the integrand has an analytic form in terms of the cumulant function of the used Lévy process.
This fact stresses again the importance of having nicely computable expression for the characteristic functions of risk-neutral distributions. In the case where we start with the statistical probability measure, the martingale preserving property of the minimal martingale measure is of great benefit in this respect.
The multidimensional case is more difficult. Calculating (5.3.18) by the same method as in the one-dimensional case amounts to finding a Fourier transform representation of the price processV(·, S)of the option. But here we are back at Chapter 4, Section 4.4.1, where we argued that such a representation rarely exists. Using our moment matching approach in the framework of the KTD model in this chapter is not feasible because our variableZT which is to be approximated contains the current asset prices in quite a complicated way; so differentiation is difficult. An additional reason is that our method approximates the price, but does not necessarily provide a good estimation of its derivative.
The second remark concerns extensions of risk-minimizing hedging. As this concept says how to hedge an option using only the underlying variable and the riskless bond, it is meant to improve on the usual delta hedging approach. A number of authors16 have established connections between the gamma of an option and the hedging error
15See Section 1.6.
16See Černý (2004b), paragraph 12.6.
150 CHAPTER 5. RISK-MINIMIZING HEDGING in a risk-minimizing strategy. It would surely be a very interesting approach to add options to the portfolio and to formulate a modified delta-gamma hedging approach, where delta is now given by a risk-minimizing strategy instead of the Black-Scholes delta.
Final remarks
As the main part of the option pricing literature since the 1970s this thesis is con-cerned with an extension of the method launched by Black and Scholes (1973) and Merton (1973). In doing this, it should have conveyed a threefold basic message:
First, arguing for the use of Lévy processes in finance means essentially to say that describing an evidently non-Gaussian distribution by means of its first four moments is preferable to adjusting only for the first two. Of course, this is a trivial but nevertheless important statistical statement. But in the same way as the fact that chopping down a tree is better done with a hatchet than a knife does not lead people to use the latter to this end, this statement should not be used to dismiss Lévy processes as irrelevant. On the contrary, it should be a strong case for the further use of Lévy processes in a field where seemingly small differences in results have a big impact.
Second, as we observe consistently in science, the advantages of models incorporat-ing new empirical findincorporat-ings must be weighed up with the experience that they are almost surely less tractable than older models. But tractability matters in particular in option pricing theory where the speed of the applicable pricing procedures is of overwhelming importance. Providing tractable Lévy models was exactly the main focus of this thesis where we defined implicitly tractability by the requirement of be-ing easily used for fast option pricbe-ing by Fourier techniques. Tractability is a feature of a concrete model rather than of a class of models. That is why most objects are defined in view of how well they go together to result in a tractable model: the NIG Lévy process with the flexible change of measure in Chapter 2, the tempered stable Lévy process with the linex measure change functions in Chapter 3 as well as Kou’s model with the Lévy copulaKθ+,θ− in Chapter 4.
The third message is a quite pragmatic view of the incompleteness issue. Lévy markets are generically incomplete, and for a large class of Lévy processes adding options to the market does not complete the market unless the number of options is infinite. Hence we cannot assign unique prices to options. However, an incomplete Lévy model provides a large number of degrees of freedom to be fitted to the prices of a finite number of options, and by means of a well-posed calibration procedure we are able to obtain a single risk-neutral distribution of the returns, although the market remains of course incomplete in theory. This degree of freedom argument is important to account for the phenomenon that risk-neutral distributions change over time without a corresponding change of the statistical distribution.
151
152
Appendix A
A class of tractable martingale measures
A.1 Log-likelihood function
The log-likelihood function L of n independent observations xi from a NIG dis-tributed sample is
L= nlogδα
π +n(δp
α2−β2−βµ) +β Xn
i=1
xi +
Xn
i=1
logK1(αp
δ2+ (xi−µ)2)−1
2log(δ2+ (xi−µ)2)
.
Kν denotes the modified Bessel function of the third kind of the order ν, and with the convenient abbreviationR(·)≡ KK01((··)) we get
∂L
∂α = nδα
pα2−β2 − Xn
i=1
pδ2+ (xi−µ)2R(αp
δ2+ (xi−µ)2)
∂L
∂β = Xn
i=1
xi−n µ+ δβ pα2−β2
!
∂L
∂δ = n δ +np
α2−β2− Xn
i=1
2δ
δ2+ (xi−µ)2 + αδR(αp
δ2+ (xi−µ)2) pδ2+ (xi−µ)2
!
∂L
∂µ = −nβ+ Xn
i=1
xi−µ pδ2+ (xi−µ)2
× 2
pδ2+ (xi−µ)2 +αR(αp
δ2+ (xi−µ)2)
! .
153
154 APPENDIX A. A CLASS OF TRACTABLE MARTINGALE MEASURES