126 CHAPTER 4. NON-LINEAR DEPENDENCE AND OPTION PRICING
4.6. CONCLUDING REMARKS 127
0
20
40
60
80
0 20 40 60 80 2.45 2.5 2.55 2.6 2.65 2.7
Fig. 4.8. Prices for a European ITM call coption (K=S0) for different values ofθ+ andθ−
while at the same time the covariance is kept constant. The maximum difference is0.17.
framework of constructing multidimensional Lévy processes rather than to generalize a special distribution to the multidimensional case. But the real challenge of the idea of a Lévy copula seems to be the requirement of tractability. And it is the tractabil-ity property which is the leitmotif of this contribution. We deliberately stayed as concrete as possible in order to find a very special model which can incorporate very realistic features (leptokurtotic and skewed marginals, tail dependent jumps) and at the same time remain tractable.
However, there are still some problems with this approach concerning the model it-self as well as the proposed pricing methodology. As for the model techniques the question of how to estimate the two parameters has not been tackled, which clearly remains one of the most urgent things to do. More important than the technical side of the estimation procedure is availability of data. As we have to estimate the parameters of the risk-neutral as opposed to the statistical distribution there is a need for market prices of basket options in order to capture the risk-neutral depen-dence structure. It seems, though, that basket options are not very often traded at exchanges. An alternative approach would be to find economic reasons why it could perhaps be sufficient to take the statistical dependence structure. Of course, it would also be interesting to study the interrelation between the risk-neutral and the statistical dependence.
The pricing methodology in the KTD model immediately calls for an extension of the model to more than two dimensions because realistic basket options are mostly
128 CHAPTER 4. NON-LINEAR DEPENDENCE AND OPTION PRICING
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.02
−0.015
−0.01
−0.005
mean
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.07 0.08 0.09 0.1
standard deviation
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.5 0 0.5
skewness
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
4 6 8
kurtosis
1/θ+, 1/θ−
(a) Moments ofZT as a function of1/θ+ and1/θ−
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.0188
−0.0186
−0.0184
−0.0182
mean
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.094 0.095
standard deviation
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.5 0 0.5
skewness
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
2 4 6 8
kurtosis
1/θ+, 1/θ−
(b) Like (a), but with constant covariance
Fig. 4.9. Parameters:σ1=0.21,σ2=0.3,p1=0.45,p2=0.5,λ1=0.25,λ2=0.4,λ1+=4.1,λ2+=3.7, λ1−=3.6,λ2−=3.6. Moments ofY as dependent ofθ+andθ−, which both are driven from 1 to 20. Note that the marginals both have negative skewness.
4.6. CONCLUDING REMARKS 129 claims on whole indices rather than on just two assets. The first thing would be to extend the copula to more than two dimensions, which does not appear to be a straightforward task. Secondly, computations would become much less tractable, and it is not clear whether the feature of an analytical characteristic function would be preserved.
Finally, we could consider different (preferably infinitely often) differentiable payoff functions. If we define the strike price to be an asset, Eberlein and Papapantoleon (2004) provide an exact method (in the sense of being directly priced by Fourier inversion) to value a variety of exotic options depending on two assets. An in-teresting example of such an option is the exchange option with payoff function p(x, y) = (x −y,0)+. But as soon as this option is coupled with a strike price K > 0, i.e. p(x, y) = (x−y, K)+, we have three assets just like in the case of the two-dimensional basket option, and the method of Eberlein and Papapantoleon (2004) cannot be applied any longer. It seems that such an option can be priced by a suitable modification of our approximation method.
130 CHAPTER 4. NON-LINEAR DEPENDENCE AND OPTION PRICING
Chapter 5
Risk-minimizing hedging
5.1 Introduction
Models of financial markets with assets driven by Lévy processes are in general in-complete. This means that not every contingent claim can be hedged completely, and hence one is forced to think about hedging strategies which cover the risk as well as possible in a certain sense. The hedging strategies that we consider here are assumed to use only the underlying asset and a riskless bond. In a complete market the hedging problem of a final payoff is solved by investing in a portfolio and pursuing a self-financing trading strategy which produces the same final payoff. A possible solution would be to maintain the latter payoff requirement while foregoing the self-financing assumption. Such a strategy can be found in a trivial way1 and is not interesting at all. The idea of Föllmer and Sondermann (1986) is to replace the notion of a self-financing strategy by the weaker one of a mean self-financing strat-egy (see below). This leads to a non-trivial optimization problem, and one comes to the notion of a risk-minimizing hedging strategy, which we define in the sense of Schweizer (1991). The objective of this section is to compute a risk-minimizing strategy for options that depend on more than one asset in the framework of an exponential Lévy model.
This chapter has two main sections. The main result will not appear until Section 5.3 where we give an explicit presentation of a risk-minimizing hedging strategy and, correspondingly, of the minimal martingale measure in a multidimensional exponen-tial Lévy world where the derivative is allowed to be written on more than one asset.
This general result is of a very technical nature, and therefore the main economic ingredient might well be overlooked. Because of this, we decided to outline this pro-cedure in Section 5.2 for the case where the derivative in question depends merely on one asset in a way that makes the structure of the general result more lucid. In doing this, we perform mere formal computations without bothering about whether the calculated objects exist or not. In fact, all of the objects in this first section will exist under suitable assumptions, but this will not be seen until Section 5.3.
1See Schweizer (2001).
131
132 CHAPTER 5. RISK-MINIMIZING HEDGING There is a second reason why the introductory Section 5.2 was included. Risk-minimizing hedging can be given a twofold interpretation, depending on whether the tracking error is to be minimized under a (however obtained) risk-neutral or under the statistical probability measure. It will turn out that formally both approaches amount to the same result, but their interpretations are quite different: The first one is easier to obtain but has an economic flaw whereas the second one is mathemat-ically more challenging but more meaningful in economic terms. Section 5.2 deals with this problem, discusses the results and classifies the present literature accord-ingly. The general result in Section 5.3 deals only with the second approach. The very first thing to do is to define the notions that we are going to talk about.
Definition 5.1. Let M, N be square-integrable martingales. M and N are said to be orthogonal if hM, Ni= 0.
This is not the common way of defining the notion of orthogonal martingales.
Following the usual definition, two local martingales are then said to be orthogonal if their product is again a local martingale. However, for square-integrable martingales both definitions are equivalent.
Throughout this section we assume the asset price process to be given by a special semimartingale S with Doob-Meyer decomposition
S =S0+M +A
whereM is a square-integrable martingale withM0 = 0andAa predictable process whose components have finite variation.
Definition 5.2. LetΘdenote the space of all predictable Rn-valued processes ϑwith E
"Z T
0
ϑ0udhMiuϑu+ Z T
0 |ϑ0udAu| 2#
<∞ (5.1.1)
where hMi:= (hMi, Mji)i,j=1,...,n.
A portfolio strategy is a pair (C, ϑ) where C > 0 and ϑ ∈ Θ. Its associated value process is given by
Vtϑ=C+ Z t
0
ϑ0udSu.
We borrow from Choulli et al. (1998) the definition of a Föllmer-Schweizer de-composition:
Definition 5.3. Given a semimartingale S, we say that a square-integrable FT -measurable random variable H admits a Föllmer-Schweizer decomposition if it can be written as
H=H0+ Z T
0
ϑ0udSu+ ΓT,
where H0 is an F0-measurable random variable, ϑ∈Θ, and Γ is a square-integrable martingale starting at zero which is orthogonal to all components of M.
5.1. INTRODUCTION 133 Central to finding a risk-minimizing hedging strategy is a martingale measure with special properties, the so-called minimal martingale measure, which is con-structed for stochastic processes satisfying the structure condition from Schweizer (1994).
Definition 5.4. Let S be a special semimartingale with Doob-Meyer decomposition S=S0+M+A, where
Ai hMii, i= 1, . . . , n with predictable density υi. Moreover, let
ηti:=υitΣiit for i= 1, . . . , d (5.1.2) and
Σijt := dhMi, Mjit
dBt for i, j = 1, . . . , d.
B is understood as any fixed increasing predictable càdlàg process starting at zero with hMii B. If there exists a predictable process λˆ with
Σtλˆt =ηt, P−a.s.fort∈[0, T], (5.1.3) and for the process K, which is called theˆ mean-variance trade-off process, we have
Kˆt :=
Z . 0
ˆλ0udMu
t
= Z t
0
ˆλ0udAu <∞ P−a.s.for all t∈[0, T], then S is said to satisfy the structure condition (SC).
As we will see later, for the case where S is an n-dimensional exponential Lévy process we can choose B with Bt = t, t ∈ [0, T], such that (5.1.2) considerably simplifies to
ηti= dAit
dt for i= 1, . . . , d. (5.1.4) We give the definition of aminimal martingale measurein Föllmer and Schweizer (1991):
Definition 5.5. A martingale measure Pˆ ∼P will be called minimal if any square-integrable P-martingale Γ which is orthogonal to M under P remains a martingale under P.ˆ 2
The minimal martingale measure is in general a signed measure. In the following two sections we will give conditions under which it is indeed a probability measure.
Let a contingent claimHbe given with assumptions made in Definition 5.3. The price of this claim at timetis denoted byV(t, St) as a function of time and the price
2Föllmer and Schweizer (1991) assume in addition thatPˆ =PonF0. This condition is redundant in our case because we have assumed thatF0 is the trivialσ-algebra.
134 CHAPTER 5. RISK-MINIMIZING HEDGING of the underlying assetsS. The financial notion ‘risk-minimizing’ from the beginning of the section translates henceforward into a mathematical definition. For this we still need to introduce the cost processΓ of a strategy ϑ.
Γt :=V(t, St)− Z t
0
ϑ0udSu. The following definition is based on Schweizer (2001).
Definition 5.6. Let H be a contingent claim. A portfolio strategy (C, ϑ) with C = V(0, S0), ϑ∈ Θ and VTϑ = H P−a.s. is called risk-minimizing for H if ϑ is such that Γ is aP-martingale which is orthogonal to M.
From the definition of the cost process we see thatΓ0 =V(0, S0). This means that with a strategy(C, ϑ)that satisfies only the martingale property of Definition 5.6 the cost process oscillates around the starting point V(0, S0). Such a strategy is called mean-self-financingbecauseΓis not identically equal to V(0, S0) as in the case of a self-financing strategy but still equal toV(0, S0) in expectation. A risk-minimizing strategy takes into account that Lévy markets are in general incomplete, and a full elimination of the risk of a derivative by continuous trading in the underlying and the risk-neutral asset is not possible. Hence it contents oneself with minimizing the hedging error in a certain sense: The strategy is such that the hedging error Γ is orthogonal (in the sense of Definition 5.1) to the price process of the underlying, i.e.
it is the best possible way to cover the risk inherent in the terminal payoff H.