Cost-efficiency in multivariate Lévy models

Abstract In this paper we determine lowest cost strategies for given payoff distributions called cost-efficient strategies in multivariate exponential Lévy models where the pricing is based on the multivariate Esscher martingale measure. This multivariate framework allows to deal with dependent price processes as arising in typical applications. Dependence of the components of the Lévy Process implies an influence even on the pricing of efficient versions of univariate payoffs.We state various relevant existence and uniqueness results for the Esscher parameter and determine cost efficient strategies in particular in the case of price processes driven by multivariate NIG- and VG-processes. From a monotonicity characterization of efficient payoffs we obtain that basket options are generally inefficient in Lévy markets when pricing is based on the Esscher measure.We determine efficient versions of the basket options in real market data and show that the proposed cost efficient strategies are also feasible from a numerical viewpoint. As a result we find that a considerable efficiency loss may arise when using the inefficient payoffs.


Introduction
In this paper we study optimal investment decisions in incomplete markets where the prices of the risky assets are driven by multivariate Lévy processes. Apart from the pricing and hedging of options on a single asset, practically all nancial applications require a multivariate model with dependence between the assets. The knowledge of the corresponding univariate marginals is not su cient since it provides no information on the dependence structure which considerably in uences the risks and returns of the value of the option. Thus, multidimensional models are capable to describe the actual nancial states in a more appropriate and accurate manner. Moreover, an abundance of payo function types such as the Basket option, Worst-o call, Worst-o put and their Best-o counterparts and many more can be treated with multivariate pricing models.
The concept of cost-e cient strategies has been introduced in [10,11] and has been extended in a series of papers in Jouini and Kallal [22], Föllmer and Schied [15] in [32,33], in [3,4], as well as in Burgert and Rüschendorf [7] and others in a fairly general setting. The aim of the method of cost e ciency is to construct to a speci ed payo distribution G a payo X T with payo distribution G (w.r.t. the underlying probability measure P) which minimizes the price w.r.t. the pricing measure Q used in the market. G could be the distribution of a given option X T . This approach thus improves concerning cost a given payo or determines to a speci ed payo distribution G a cheapest (cost e cient) payo having this payo distribution. [28] contains a discussion of various methods to specify payo distributions in applications.
The explicit form of cost-e cient strategies has been determined in the above mentioned papers mainly in the context of the Samuelson model. A detailed study of this concept for univariate exponential Lévy models was given in Hammerstein et al. [20] in the case that pricing is based on the Esscher martingale measure. In this paper also the potential gain and the hedging behaviour of cost e cient claims is investigated. As a result it turns out, that the cost-e cient payo may lead to considerably reduced cost and compares also favourably concerning hedging behaviour as checked for real market data.
In typical cases cost e cient payo s generate the payo distribution by following the trend in the market. In particular they neglect possible hedging goals of investors but only aim to optimize the cost in order to reach a distributional goal of the investment. They are thus tools for law invariant investors but don't satisfy protection or securization purposes. In recent papers in [4,5] and in [30] the method of cost e ciency has been extended to include state dependent constraints and thus to specify in which states income is requested.
The frame of the method of cost e ciency is the following. In a market model (Ω, F, (F t ) ≤t≤T , P) with nite time horizon [ , T] let S = (S t ) ≤t≤T ∈ R d be a market model for d stocks and (Z t ) ≤t≤T a pricing density for S rendering the discounted process (e −rt S t Z t ) ≤t≤T a P-martingale. The cost of a strategy with terminal payo X T then is given by (1.1) A basic and debatable assumption of the approach of cost e cient strategies is that the market participants agree on one and the same pricing measure Q. In an incomplete market this problem is not avoidable. Any noarbitrage price corresponds to a chosen market measure or equivalently to a speci c utility principle. Also the super hedging price, the empirical and risk minimizing pricing measures follows this principle and base their pricing on a worst case martingale measure, on 'minimal' martingale measures minimizing some hedging or risk functional. The assumption of a pricing measure Q allows as consequence to construct to any given payo distribution G a cheapest (cost-e cient) payo having this payo distribution. For a given payo distribution G a strategy with terminal payo (1. 2) The strategy with payo The di erence of the costs (X T ) = c(X T ) − c(X T ) is called the e ciency loss of X T . The following result characterizes cost-e cient strategies in the general context described above (see e.g. [3,4] is the cost-e cient strategy and X T = G − (F ZT (Z T )) is the most-expensive way to achieve a payo with given distribution function G. Moreover, for any payo X T ∼ G, the lower and upper cost bounds are given by Furthermore, one obtains as consequence that a random payo X T ∼ G is cost-e cient if and only if X T and Z T are countermonotonic while X T ∼ G is most-expensive if and only if X T and Z T are comonotonic. In state price models where Z T = h(S T ) (like in exponential Lévy models) path-dependent payo s are not cost-e cient and can be improved by cost-e cient payo s which are path-independent i.e. are of the form X T = g(S T ).
In this paper we apply the concept of cost-e ciency in the case of market models driven by multivariate Lévy processes in the case that pricing is based on the Esscher martingale measure. The Esscher transform has been introduced and motivated for contingent claim pricing in mathematical nance for Lévy processes in [16], [25], [12], [8], and [21] and has been extended to semimartingales and multivariate Lévy processes in [23], [13] and in [31]. The Esscher pricing principle thus is a well established pricing principle justi ed by a corresponding utility principle and by some inherent simpli cations it leads to. We show in our paper that the determination of cost e cient strategies is doable in some standard classes of multivariate Lévy models under pricing by the Esscher pricing measure. We introduce in Section 2 the multivariate Esscher transform and describe some of its basic and delicate properties on the existence and uniqueness of the riskneutral Esscher measure. In Section 3 we specify the construction of cost-e cient claims in Theorem 1.1 to the multivariate Lévy case. We nd that generally basket options are ine cient. In Section 4 we introduce some multivariate normal mean variance mixture models in particular the NIG and the VG model and use them for modelling bivariate log-returns. We estimate the Lévy parameters from daily log-returns of German stock data and compute the Esscher parameters. As application in Section 5 we calculate to a given basket option the cost-e cient option and determine the e ciency loss for the real data sets as discussed above.

The Esscher transform and risk neutral Esscher measure
The notion of Esscher transformation as a change of measure was introduced by Gerber and Shiu [16] although the concept of Esscher transformation for Lévy processes had been used in nance before on a mathematically profound basis (see e.g. Madan and Milne [25]). Since then it became an established tool in nancial and actuarial science. The Esscher measure provides the advantage that any Lévy process under the physical measure stays a Lévy process under the Esscher measure.
For t ≥ and d ∈ N, let S (i) t = S (i) e L (i) t , ≤ i ≤ d denote the price of the i-th risky asset and assume that S (i) is F -measurable. Let L (i) := (L (i) t ) t≥ and assume that L := (L ( ) , . . . , L (d) ) is a Lévy process with respect to the ltration (F t ) ≤t≤T . Both S (i) and L (i) t are real-valued. Recall that we consider strategies (Y t ) ≤t≤T on a nite trading period [ , T]. Then, apart from the cases where L = (L t ) ≤t≤T either is a Brownian motion or a Poisson process, such a Lévy market setting is incomplete. This means that the set of possible risk-neutral martingale measures is not a singleton, but typically has uncountably many elements. We therefore assume that the nancial market is incomplete, but free of arbitrage, perfectly liquid and frictionless. To introduce the Esscher martingale measure, we need several properties of the moment generating function of random vectors.
Let (Ω, F, P) be a probability space and let X be random vector with values in R d , d ∈ N. Denote by ·, · the Euclidean scalar product in R d . The moment generating function of X is given by For u , u ∈ R d and any α ∈ ( , ) holds using Hölder's unequality Thus, log(M X (u)) is a convex function. As a consequence M X (u) is convex, since we can write the moment generating function M X (u) = exp(log(M X (u)) as a composition of two convex functions. Now, consider a d-dimensional Lévy process L = (L t ) t≥ on a ltered probability space (Ω, F, (F t ) t≥ , P) satisfying the usual conditions. Due to the stationarity and independence of the increments of Lévy processes we have the relation M Lt (u) = M L (u) t for all u ∈ R d and t ≥ . (2. 2) The following basic assumption on the Lévy process, which serves here as a driver for the price process, is made for the remainder of this paper. The notation of a degenerate Lévy process can be found in Sato [29, p. 165].

Assumption (M d ) The d-dimensional random variable L is non-degenerate and possess a moment generating
The latter condition will turn out to be necessary but not always su cient for the existence of the riskneutral Esscher measure.
De nition 2.2 (Esscher transform). Let (L t ) t≥ be a d-dimensional Lévy process on some ltered probability space (Ω, F, (F t ) t≥ , P). We call Esscher transform any change of P to a locally equivalent measure Q θ with a density process Z t = dQ θ dP | Ft = Z θ t of the form where M Lt is the moment generating function of L t , and θ ∈ (a, b).
We indicate by E θ that the expectation is calculated with respect to Q θ . The process (Z θ t ) t≥ is a density process for all θ ∈ (a, b). This measure preserves the Lévy property: (L t ) t≥ remains a Lévy process under the Esscher measure Q θ . However, the discounted stock price process (e −rt S t ) t≥ will not be a martingale under all Q θ . A parameterθ is called risk neutral Esscher parameter if Qθ is a martingale measure for S. Qθ then is called the Esscher martingale measure. The Esscher parameterθ has to ful l the following condition:

(2.4)
Due to the stationary and independent increments of a Lévy process (L (i) t ) t≥ we have: Thus, the discounted price process is a martingale under Qθ if and only if the equation holds for all t ≥ and for ≤ i ≤ d. Or equivalently, where 1 i := ( , . . . , , , , . . . , ) denotes the i-th standard basis vector of R d . The above equation means thatθ ∈ (a, b) has to solve the system of equations This also explains why it is necessary to require M L to be de ned on an interval (a, b), where the length of each univariate interval (a (i) , b (i) ) is greater than one. In summary, the following characterizes Esscher measures. The Esscher parametersθ (i) of the univariate processes L (i) are identical to the components of the Esscher parameterθ of the multivariate Lévy process L if the components of L are independent. In general, as in the examples considered in this paper, with dependent components of L they may be di erent. As consequence we get:

Lemma 2.3. Let Assumption (M d ) be ful lled and suppose there is a parameterθ such that M L (ϑ) and M L
for ≤ i ≤ d, and ifθ = (θ ( ) , . . . ,θ (d) ) denotes a solution of the system of equations (2.6), then, In dependent Lévy models, however, they may be di erent.
From the latter we see that pricing in the univariate Lévy setting di er from the multivariate case when dependence in the components is present. The inclusion of further dependent components in the market model may lead to lower prices of e cient versions of options depending only on one component of the market model compared to pricing in the single component model.
For illustration we consider an option on one asset with payo f (S (i) The cost in the univariate setting, that is, where only the Lévy process L (i) is present, is given by whereas in the multivariate setting, that is, where L = (L ( ) , . . . , L (d) ) is the driving process, the cost is where for y ∈ R d the notation y  Existence and uniqueness criteria for multivariate (exponential) Lévy processes have been studied in [23] and in [31] and for d = in [27]. For models based on the stochastic exponential S = S E(X) a characterization of existence of an equivalent martingale measure is given in Tankov [31,Theorem 3]. Note that the stochastic exponential S is a local martingale if and only if X is a local martingale, assuming S i ≠ . In case X is a Lévy process this is equivalent to X being a martingale (even uniformly integrable on [ , T]). The proof of Tankov's theorem implies an existence result of an Esscher parameter if the underlying Lévy process has all exponential moments. Any exponential Lévy model S (i) = S i e L (i) can be represented as stochastic exponential model S (i) = S i E(X (i) ) with some Lévy process X (i) and conversely. For a given stochastic exponential model S i E(X (i) ) de ne Y (i) = ln E(X (i) ); then exp(Y (i) ) = E(X (i) ). For the converse direction e X (i) = E(Y (i) ) implies that Y (i) = L(e X (i) ) is the stochastic logarithm of e X (i) (see Goll  (2.8)

Q θ then is called Esscher measure for S. 2) The Esscher measure is uniquely determined if it exists.
Proposition 2.4 and Theorem 2.6 give some general conditions for existence and uniqueness of the Esscher measure. Condition (2.8) is a drift condition saying that the drift of X is zero w.r.t. Q θ . The existence and uniqueness results can easily be transferred to the case of discounted models of the form e −rt S = e Lt−rt . Only the drift parameter has to be changed. In [36] it is shown that even in cases where an Esscher measure does not provide an equivalent martingale measure a mean correcting Esscher parameter can be chosen to reproduce the price of a European call option with respect to any risk neutral measure.
For the general case however one has to rely either on an iterative construction or on more general results in Hodge theory as used to prove existence of solutions of log-Likelihood equations in [26].
By the uniqueness result in Theorem 2.6 we can now de ne in a formal way the risk-neutral Esscher measure under Assumption (M d ).
De nition 2.8 (Esscher martingale measure). The uniqueθ ∈ R d such that the process (e −rT S t ) t≥ is a martingale with respect to Qθ is called the Esscher parameter and Qθ is called the Esscher martingale measure or risk-neutral Esscher measure.

Cost bounds in multivariate Lévy models
In this section we specialize the general construction result for cost-e cient payo s in Theorem 1.1 to the case of multivariate Lévy models. The formulas for the cost bounds are given in terms of the Lévy process themselves (instead of the market models).  As consequence of the latter result we obtain

while X T is most-expensive if f is increasing and a = t ·θ for some t > , (3.3)
In the particular casesθ (i) > for all i resp.θ (i) < for all i we obtain a direct connection of cost-e ciency to monotonic behaviour in L T .

Corollary 3.4. Let (L t ) t≥ be a Lévy process with continuous distribution function F LT at maturity T > , and
assume that a solutionθ of (2.5) exists.

Ifθ (i) < for all ≤ i ≤ d, then a cost-e cient payo X T ∼ G is componentwise increasing in L T . 2. Ifθ (i) > for all ≤ i ≤ d, then a cost-e cient payo X T ∼ G is componentwise decreasing in L T .
For the most-expensive strategy, the reverse holds true.
Proof. Let all components of the risk-neutral Esscher parameterθ have a negative sign and let X T ∼ G be a cost-e cient payo . Then, due to Proposition 3.1 resp. Corollary 3.2 X T = G − ( −F θ ,LT ( θ , L T )) is decreasing in θ , L T . Moreover, sinceθ (i) < for all ≤ i ≤ d the function h(L T ) = θ , L T is componentwise decreasing in L T . Thus, the strategy X T is componentwise increasing in L T . The other cases can be shown analogously.
Corollary 3.4 allows in the cases whereθ (i) < orθ (i) > for all ≤ i ≤ d to identify ine cient payo s from its monotonic behaviour in the coordinates of L T .

Multivariate Lévy processes and application to real market data
In this section we recall some properties of multivariate normal mixture models its densities and moment generating functions as needed for the computation of the risk-neutral Esscher parameters for some class of Lévy models. For two sets of real market data we give a statistical analysis in terms of three di erent multivariate Lévy models the NIG, the VG and the normal model.

Normal mean variance mixture models
Normal mean variance mixtures are valuable models for analysing data from a variety of heavy-tailed and skew empirical distributions. They have been used a lot in the more recent literature for nancial data but also in various other areas. Detailed expositions are given in Barndor -Nielsen [1], Blaesild [6] and Barndor -Nielsen et al. [2]. Some recent developments in particular for dependence modelling are given in [24].
An R d -valued random variable X is said to have multivariate normal mean-variance mixture distribution if X where µ, β ∈ R d , A is a real-valued d × d matrix such that ∆ := AA is positive de nite, W is a standard normal distributed random vector (W ∼ N d ( , I d )) and Z ∼ F Z is a real-valued, non-negative random variable independent of W. An equivalent de nition is the following: A probability measure Q on (R d , B d ) is said to be a multivariate normal mean-variance mixture if where the mixing distribution F Z is a probability measure on (R+, B+). A practical short hand notation of equation ( Multivariate generalized hyperbolic distributions are de ned as normal mean-variance mixtures with Generalized inverse Gaussian (GIG) mixing distributions: where it is usually assumed without loss of generality that det(∆) = , which we shall do in the following. Due to the parameter restrictions of GIG distributions, the other GH parameters have to ful l the constraints λ ∈ R, α, δ ∈ R+, β, µ ∈ R d and The meaning and in uence of the parameters is similar as in the univariate case. The representation in (4.1) entails that the in nite divisibility of the mixing Generalized inverse Gaussian distributions transfers to the GH d distribution. In consequence there exists a Lévy process (L t ) t≥ with L(L ) = GH d (λ, α, β, δ, µ, ∆) (see e.g. Sato [29, Theorem 7.10 (iii)]). The following properties of GH d distributions and in particular of NIG and VG distributions are given in Hammerstein [19].
) and a positive-de nite d × d covariance matrix Σ = Cov(L (i) , L (j) ) , ≤ i, j ≤ d, that is, The moment generating function of L is equal to (4.6) Figure 1: Moment generating function for a bivariate NIG process. The parameters used to derive the moment generating function are listed in Table 1.

Application to real market data
In this subsection we illustrate an application of some multivariate Lévy processes to the analysis of real market data. We consider German stock price data for Allianz and Volkswagen and for E.ON and Thyssen Krupp from May 28, 2010 to September 28, 2012. That is, we consider the Lévy process L (A,VW) = (L Allianz t , L VW t ) ≤t≤T in order to model the daily log-returns of Allianz and Volkswagen in a bivariate Lévy model, and analogously L (E.ON,TK) for E.ON and Thyssen Krupp (see Figure 2). Table 1 contains the estimated parameters from daily logreturns of Allianz and Volkswagen for the bivariate NIG, VG, and the Samuelson model. The interest rate used to calculate the Esscher parameterθ in the last column is the continuously compounded 1-Month-Euribor rate of October 1, 2012, which is r = .
· − ; note that this is the continuously compounded daily rate which we need to do daily calculations and used as well as for daily rebalancing for hedging purposes for one-dimensional options. This explains the extremely small value. The annualized Euribor rates at that time point are in the order − instead.
For the determination of the Esscher parameter we numerically solved the determining system of equations (2.5) using the estimated parameters. The alternative way proposed by Theorem 2.6 is to establish existence of the Esscher parameter rst by determining the associated stochastic exponential model, which is also based on the estimated parameters. Then check the (M d ) condition and the nite variation condition. All of these seem to be doable. Then nally solve numerically equation (2.8) in order to obtain the Esscher parameter. This alternative seems however to be more involved. Figure 1 gives the moment generating function for a bivariate NIG process L (E.ON,TK) , which models the daily log-returns of the E.ON and Thyssen Krupp stock prices from May 28, 2010 to September 28, 2012. See Table 1 for the estimated parameters used for the computations. An application of the bivariate NIG model to data of Allianz and Volkswagen is given in Figure 3. For the statistical tting of the model we used the estimated parameters from Table 1. The histogram of the daily log-returns and the model t for Allianz and Volkswagen is presented in Figure 3.
With the estimated parameters and the formulas for the moment generating functions in Proposition 4.1 it is possible to solve numerically equation (2.5) i.e. to determine the Esscher parameters (see Table 1).  Table 1.
Although the moment generating function of NIG d and VG d has an analytical representation, an analytic expression for the Esscher parameterθ is not available. For the multivariate normal distribution an explicit expression forθ is given in Gerber and Shiu [16,Section 7].
As pointed out in [20] in the univariate setting the sign of θ describes a drift; a negative sign a positive drift and a positive sign a negative drift. The size of |θ| re ects the magnitude of the drift of the price process and thus can be regarded as a measure for the strength of the market trend.
In the multivariate setting we have the following observation. From the more pronounced (positive) trend in the Allianz and VW data than in the E.ON and Thyssen Krupp data we can expect that the potential savings in the Allianz and Volkswagen case are higher than for the E.ON and Thyssen Krupp case. Note that the dependence between the stocks implies that the Esscher parameters in the joint model as in Table 1 are di erent form the parameters in the single models as considered in [20]. For example this dependence implies that in the joint model Allianz gets a slightly positive Esscher parameter, indicating a mild relative negative drift in
. . the joint model, while it has a mild positive drift in the individual model. As consequence this implies that in the joint market it is possible to make use of the higher drift in the Volkswagen market and its correlation to the Allianz market to obtain better (i.e. cheaper) constructions and improvements of options based on the Allianz stock alone.
As example we consider the standard basket option with weights w = w = . and strike K = for the E.ON, Thyssen Krupp and the Allianz, Volkswagen data. This payo is symmetrically increasing in rising markets. For S ( ) T + S ( ) T ≤ the outcome is zero, which means that such an option rewards the writer when at least one of the assets S (i) T is high while the other asset decreases at most at the same level (compare Figure 4). On the contrary, the corresponding cost-e cient payo X ba T of the basket option shows a reverse behaviour. This is consistent with Corollary 3.4 since the risk-neutral Esscher parameterθ = . . is componentwise positive. Figure 5 displays the e cient payo X ba T of the optimal long basket option on E.ON and Thyssen Krupp stocks with strike K = and maturity T = days for the NIG model. Similar calculations are done for the Allianz, Volkswagen data. All computations are based on the estimated parameters given in Table 1. The initial stock prices S (i) are the closing prices at October 1, 2012, and the time to maturity is chosen to be T = trading days, meaning that the long basket options mature on November 1, 2012. The chosen initial stock prices equal S A = . , S VW = . , S E.ON = . and S TK = . . The weights are w = w = . . The strike for Allianz and Volkswagen is K = , whereas for E.ON and Thyssen Krupp it is K = . In Table 2 the prices for the long basket option and its cost-e cient counterpart as well as the e ciency loss for Allianz and Volkswagen and for E.ON and Thyssen Krupp in all three bivariate Lévy models as discussed in Section 4 are listed. As a result for the Allianz and Volkswagen case a substantial e ciency loss is observed for basket options while in the E.ON and Thyssen Krupp case the e ciency loss is more moderate. As shown in Hammerstein et al. [20,Proposition 2.3] for the one dimensional case a greater size of |θ| leads to a higher e ciency loss. This e ect can be seen from Tables 1 and 2 in our two dimensional examples as well. Thus, we expect that an analogous result also holds true in the multivariate setting in greater generality when dependent components are present.

Numerical issues
In order determine the risk-neutral Esscher parameter i.e. to solve the system of non-linear equations as in (2.5) we use numerical methods provided by the R program. In particular, the package nleqslv provides two algorithms for solving systems of non-linear equations with either a Broyden or a full Newton method. For further information we refer to Dennis and Schnabel [9] and the related documentaries.
For evaluation of multidimensional integrals over hypercubes we used the package cubature. The calculation of standard prices c(X ba T ) needs about 10.1 seconds. Its absolute error lies in the region of − . The computational time becomes better if suitable starting values and hypercubes are chosen. For the cost-e cient versions c(X ba T ) the calculation is more involved and needs signi cantly more time. The running time varies from 270 to 1.560 seconds depending on the particular Lévy model, accuracy, starting value, and hypercube chosen if the number of simulations (of the bivariate Lévy process) is at most . For a higher number of simulations, and thus a more accurate calculation, the time of calculation of such prices can grow substantially.

Conclusion
In this paper we adapt and develop the techniques necessary to determine cost-e cient payo s in the case of multivariate exponential Lévy asset models when pricing is based on the Esscher martingale measure. We show that all calculations are doable for certain classes of multivariate Lévy models as NIG, VG or in the normal case. As application we determine cost-e cient payo s generating the same payo distribution as the ine cient basket options when pricing with the Esscher pricing measure. We describe the in uence and e ect of dependence between the components of the Lévy model to the pricing of the cost e cient payo s in the example of basket options which implies that the relative trend of a stock may switch in the joint model and lead to greater improvements compared to the construction of e cient claims in one dimensional mod-els. As a result we obtain that the e ciency loss can be considerable indicating that the use of cost-e cient payo s may be pro table. It is expected, that as in the one-dimensional Lévy case also in the multivariate case cost e cient options behave favourably concerning the hedging behaviour but this still has to be explored. In the one-dimensional case it has been shown in [20] and [35] concerning ∆-hedging as well as the basic options. Extensions of the cost e ciency method to empirical pricing measures and a heuristic approach to the choice of the payo distribution have been given in a recent paper in [28] in the case of one-dimensional Lévy processes. An extension to the multivariate case is subject of subsequent work.