Optimal Hedging in Incomplete Markets
OOptimal Hedging in Incomplete Markets
George Bouzianis and Lane P. Hughston
Department of Computing, Goldsmiths College, University of LondonNew Cross, London SE14 6NW, United Kingdomemail: [email protected], [email protected]
We consider the problem of optimal hedging in an incomplete market with anestablished pricing kernel. In such a market, prices are uniquely determined, butperfect hedges are usually not available. We work in the rather general setting ofa L´evy-Ito market, where assets are driven jointly by an n -dimensional Brownianmotion and an independent Poisson random measure on an n -dimensional statespace. Given a position in need of hedging and the instruments available as hedges,we demonstrate the existence of an optimal hedge portfolio, where optimality isdefined by use of an least expected squared error criterion over a specified timeframe, and where the numeraire with respect to which the hedge is optimized istaken to be the benchmark process associated with the designated pricing kernel. Key words: Incomplete markets, pricing kernels, hedge ratios, Brownian motion,L´evy processes, L´evy measures, L´evy-Ito processes, Poisson random measure, simulations.
I. INTRODUCTION
This paper is concerned with optimal hedging in incomplete markets. Hedging is important,since it lies at the heart of risk management. Historically, hedging in complete markets hasplayed a significant role in the foundations of option-pricing theory [3, 5, 9, 14, 18]. From amodern perspective, however, hedging arguments need not be invoked in the determinationof prices. Instead, pricing is achieved by use of a pricing kernel. The connection betweenthe two approaches is that in a complete market the specification of the price processes ofa sufficiently large number of assets is enough to allow one to determine the pricing kernelassociated with that market. Nevertheless, in the absence of market frictions, the prices ofall of financial assets are determined in an incomplete market, including those of derivatives,once we designate a pricing kernel. In the incomplete market situation, however, one cannot in general form a perfect hedge of a given position. This leaves us with a more precisestatement of our problem: namely, determination of the optimal strategy for hedging afinancial position in an incomplete market, given the set of hedging assets at the hedger’sdisposal. The optimal hedge corresponds to the maximal possible elimination of risk in afinancial position making use of the instruments available for this purpose.The paper is structured as follows. In Section II we briefly summarize some of themathematical ideas that we require. We define what we mean by a L´evy-Ito process and inProposition 1 we recall the general form of Ito’s formula applicable to L´evy-Ito processes.Then in Proposition 2 we give a version of the Ito formula that holds when the large jumpsare moderated, which is useful in financial applications. In Proposition 3 we comment onthe form that the Ito isometry takes in the L´evy-Ito setting. In Section III we introduce thefamily of risky assets that we work with in the hedging problem. We argue that the mostnatural approach to hedging arises when the values of the various assets under consideration a r X i v : . [ q -f i n . M F ] S e p are expressed in units of the benchmark process associated with the pricing kernel. InSection IV we consider the hedging of a position in a risky asset in a one-dimensional L´evy-Ito market in the situation where the hedging instrument is another risky asset driven by thesame one-dimensional L´evy-Ito process. In general, a perfect hedge is not possible in such amarket, so one aims for a best possible hedge instead. We take the view that the goal is thatof optimal elimination of the risk, which we characterize in a natural way using a quadraticoptimization criterion. See [2, 4, 7, 8, 10, 12, 13, 15, 17, 20, 22] for aspects of quadratichedging. In Proposition 4 we obtain a formula for the optimal hedge in the case of a singlehedging asset. We refer to the asset being hedged as the contract asset. The terminology isinherited from the language of derivatives pricing, though in the present context the assetbeing hedged need not be a derivative; indeed, the various assets involved are essentially onan equal footing. In Section V we consider the situation where we hedge the contract assetwith a position in n risky assets. In Proposition 5 we work out an expression for the optimalhedge in such a market, and in Proposition 6 we show that if there is negligible redundancyamong the hedging assets then the optimal hedge obtained with n + 1 hedging instrumentsis better than the optimal hedge obtained with n such instruments. In Section VI we look inmore detail at the case where two hedging assets are available to hedge the contract asset,and an explicit formula for the optimal hedge is given in Proposition 7. We illustrate theresults in the simplest possible situation: this is the case of a geometric L´evy asset for whichthe L´evy process is a linear combination of a Brownian motion and a Bernoulli process. Werefer to a L´evy process of this type as a Bernoulli jump diffusion. By a Bernoulli process wemean a compound Poisson process for which each jump is characterized by an independentBernoulli random variable taking one of two possible values. We consider the situation wherethe contract asset and the hedging assets are geometric Bernoulli jump diffusions driven bythe same L´evy-Ito process. We illustrate the fundamental fact that a better hedge can beobtained by using both of the hedging assets rather than just a single hedging asset, eventhough a perfect hedge is not obtainable as long as the Brownian component of the drivingprocess is present. On the other hand, if the Brownian volatility is small for the variousassets under consideration, then a nearly perfect hedge can be obtained. Finally, we set outsome useful formulae from the L´evy-Ito calculus in an Appendix. II. MATHEMATICAL PRELIMINARIES
We begin with a brief account of the mathematical context in which we formulate the hedgingproblem. Most of the material in this section is well known, but we find it convenient to setout various details. The L´evy-Ito market provides a modelling framework of considerablegenerality. In particular, it contains all of the familiar Brownian motion driven modelsand L´evy driven models as special cases. The setup is as follows. We fix a probabilityspace (Ω , F , P ) that supports an n -dimensional Brownian motion { W t } t ≥ alongside anindependent Poisson random measure { N (d x, d t ) } with mean measure ν (d x ) d t , where ν (d x )is taken to be the L´evy measure associated with an n -dimensional pure-jump L´evy process.Thus ν (d x ) is a σ -finite measure on ( R n , B ( R n )) such that ν ( { } ) = 0 and (cid:90) R n min (cid:0) , | x | (cid:1) ν (d x ) < ∞ . (1)We write {F t } t ≥ for the augmented filtration generated by { W t } and { N (d x, d t ) } . See[1, 6, 11, 16, 19] for aspects of the theory of L´evy-Ito processes. In the one-dimensional case,by a L´evy-Ito process driven by { W t } and { N (d x, d t ) } we mean a process { X t } t ≥ satisfyinga dynamical equation of the formd X t = α t d t + β t d W t + (cid:90) | x |∈ (0 , γ t ( x ) ˜ N (d x, d t ) + (cid:90) | x |≥ δ t ( x ) N (d x, d t ) , (2)where ˜ N (d x, d t ) = N (d x, d t ) − ν (d x ) d t . (3)We require that { α t } t ≥ and { β t } t ≥ be {F t } -adapted, that { γ t ( x ) } t ≥ , | x | < and { δ t ( x ) } t ≥ , | x |≥ be {F t } -predictable, and that P (cid:20) (cid:90) t (cid:18) | α s | + β s + (cid:90) | x | < γ s ( x ) ν (d x ) (cid:19) d s < ∞ (cid:21) = 1 (4)for t ≥
0. We note that the integral with respect to ˜ N (d x, d t ) in equation (2) and similar ex-pressions of this type is defined by means of a limiting procedure as the origin is approached,as described, e.g., in reference [21] at page 120. Then we have the following generalizationof Ito’s formula (see, for example, reference [1], Theorem 4.4.7): Proposition 1.
Let F : R → R admit a continuous second derivative and let { X t } be aL´evy-Ito process for which the dynamics are as in (2) . Then for t ≥ it holds that d F ( X t ) = (cid:20) α t F (cid:48) ( X t − ) + 12 β t F (cid:48)(cid:48) ( X t − ) (cid:21) d t + β t F (cid:48) ( X t − ) d W t + (cid:90) | x | < [ F ( X t − + γ t ( x )) − F ( X t − ) − γ t ( x ) F (cid:48) ( X t − )] ν (d x )d t + (cid:90) | x |∈ (0 , [ F ( X t − + γ t ( x )) − F ( X t − )] ˜ N (d x, d t )+ (cid:90) | x |≥ [ F ( X t − + δ t ( x )) − F ( X t − )] N (d x, d t ) . (5)We can use the generalized Ito formula to work out Ito product and quotient rules for suchprocesses. These results are very useful, so for the convenience of the reader we set themdown in detail in the Appendix. In some situations will be appropriate to consider processesfor which the dynamical equation takes the formd X t = α t d t + β t d W t + (cid:90) | x |∈ (0 , γ t ( x ) ˜ N (d x, d t ) + (cid:90) | x |≥ δ t ( x ) ˜ N (d x, d t ) , (6)where the integral involving the large jumps is taken with respect to the compensated Poissonrandom measure. In order for this to be possible, { δ t ( x ) } must satisfy P (cid:20)(cid:90) | x |≥ | δ t ( x ) | ν (d x ) < ∞ (cid:21) = 1 , (7)which is sufficient to ensure that the integral with respect to the compensated Poissonrandom measure exists for large jumps. If we impose the stronger condition P (cid:20)(cid:90) | x |≥ δ t ( x ) ν (d x ) < ∞ (cid:21) = 1 , (8)we can simplify and unify the notation by using a common symbol { γ t ( x ) } t ≥ , x ∈ R for thecoefficients of the compensated Poisson random measures for small jumps and large jumps.Then we write d X t = α t d t + β t d W t + (cid:90) | x | > γ t ( x ) ˜ N (d x, d t ) , (9)and the associated condition on the coefficients takes the form P (cid:20) (cid:90) t (cid:18) | α s | + β s + (cid:90) x γ s ( x ) ν (d x ) (cid:19) d s < ∞ (cid:21) = 1 , (10)in place of (4), where the subscript x denotes integration over the whole of the real line.We shall refer to processes satisfying (9) and (10) as being “symmetric” since large andsmall jumps are treated similarly. Symmetric processes turn out to be useful in financialapplications, where the stronger condition on the integrability of the jump volatility withrespect to the L´evy measure for large jumps is not unreasonable. In the symmetric caseIto’s formula takes the following form: Proposition 2.
Let F : R → R admit a continuous second derivative and let { X t } be asymmetric L´evy-Ito process for which the dynamics are as in (9) . Then for t ≥ we have d F ( X t ) = (cid:20) α t F (cid:48) ( X t − ) + 12 β t F (cid:48)(cid:48) ( X t − ) (cid:21) d t + β t F (cid:48) ( X t − ) d W t + (cid:90) x [ F ( X t − + γ t ( x )) − F ( X t − ) − γ t ( x ) F (cid:48) ( X t − )] ν (d x )d t + (cid:90) | x | > [ F ( X t − + γ t ( x )) − F ( X t − )] ˜ N (d x, d t ) . (11)The higher-dimensional analogues of Propositions 1 and 2 are straightforward. Finally, wenote that the Ito isometry can be generalized in the present context. So far, we have notimposed any integrability conditions on the processes that we have considered. For theL´evy-Ito analogue of the Ito isometry we require that the process satisfies an L condition. Proposition 3.
Let { X t } t ≥ be a L´evy-Ito process such that X t = X + (cid:90) t β s d W s + (cid:90) t (cid:90) | x | > γ s ( x ) ˜ N (d x, d s ) , (12) where X is a constant and P (cid:20) (cid:90) t (cid:18) β s + (cid:90) x γ s ( x ) ν (d x ) (cid:19) d s < ∞ (cid:21) = 1 . (13) If E [ X t ] < ∞ for t ≥ , then { X t } t ≥ is a martingale and for t ≥ it holds that E (cid:2) ( X t − X ) (cid:3) = E (cid:20) (cid:90) t (cid:18) β s + (cid:90) x γ s ( x ) ν (d x ) (cid:19) d s (cid:21) . (14)Again, the corresponding result for an n -dimensional L´evy-Ito process is straightforward. III. RISKY ASSETS
We proceed to consider the problem of optimal hedging. It should be emphasized from theoutset that we are not concerned here with the problem of derivative pricing via hedgingarguments. We assume that prices are known and we look instead at the problem of hedginga position in one asset by use of a self-financing portfolio of other assets. In a completemarket we know that an exact hedge can be obtained in such a situation; but we workin an incomplete market, where exact hedges are generally not available, so we look foran optimal hedge instead. We fix a probability space (Ω , F , P ) where P is the real-worldmeasure. The market filtration {F t } t ≥ is taken to be the augmented filtration generatedby a one-dimensional Brownian motion { W t } and an independent one-dimensional Poissonrandom measure { N (d x, d t ) } , where the Poisson random measure is that associated with aone-dimensional pure-jump L´evy process in the sense discussed in Section II.We introduce a fiat currency, which we call the domestic currency, in units of which pricesare conventionally expressed. The market is assumed to be endowed with a pricing kernel { π t } t ≥ for which the dynamics take the formd π t = − π t − (cid:20) r t d t + λ t d W t + (cid:90) | x | > Λ t ( x ) ˜ N (d x, d t ) (cid:21) . (15)We assume that the domestic short rate { r t } t ≥ and the Brownian market price of risk { λ t } t ≥ are adapted and that the jump market price of risk { Λ t ( x ) } t ≥ , x ∈ R is predictableand such that Λ t ( x ) < t ≥ x ∈ R . The solution for the pricing kernel is then π t = exp (cid:20) − (cid:90) t r s d s − (cid:90) t λ s d W s − (cid:90) t λ s d s − (cid:90) t (cid:90) | x | > κ s ( x ) ˜ N (d x, d s ) − (cid:90) t (cid:90) x (cid:0) e − κ s ( x ) − κ s ( x ) (cid:1) ν (d x ) d s (cid:21) , (16)where { κ t ( x ) } t ≥ , x ∈ R is defined by κ t ( x ) = log (cid:20) − Λ t ( x ) (cid:21) . (17)We assume that the market includes a money market asset { B t } t ≥ satisfying d B t = r t B t d t ,along with one or more risky assets. For a typical risky asset we let { S t } t ≥ denote the priceprocess, and for simplicity we assume that the asset pays no dividend. The associateddynamics are taken to be of the formd S t S t − = (cid:20) r t + λ t σ t + (cid:90) x Λ t ( x ) Σ t ( x ) ν (d x ) (cid:21) d t + σ t d W t + (cid:90) | x | > Σ t ( x ) ˜ N (d x, d t ) , (18)where { σ t } t ≥ is adapted, { Σ t ( x ) } t ≥ , x ∈ R is predictable, and Σ t ( x ) > − t ≥ x ∈ R .We shall require that the dynamics of { S t } are non-degenerate in the following sense. Let D denote the subset of Ω × [0 , T ] over which it holds that σ t + (cid:90) x Σ t ( x ) ν (d x ) = 0 . (19)We say that { S t } has non-degenerate dynamics if D has { P × Leb [0 , T ] } measure 0. Analternative way of expressing the non-degeneracy condition is as follows. Let the support ofthe L´evy measure ν (d x ) be defined as the set S ν comprising all x ∈ R such that ν ( A ) > A containing x ([21], page 148). Then D ⊂ { Ω × [0 , T ] } can be defined tobe the set D = { ω ∈ Ω , t ∈ [0 , T ] : σ t = 0 ∩ Σ t ( x ) = 0 , x ∈ S ν } . (20)It should be evident that these definitions of the degeneracy subset are equivalent, and it isuseful to keep both in mind.We require that for any risky asset the process determined by the product of the pricingkernel and the asset price should be a P -martingale. Thus we have S t = 1 π t E t [ π u S u ] (21)for 0 ≤ t ≤ u < ∞ , where E t denotes conditional expectation with respect to F t . There isanother way of expressing this condition which turns out to be useful for our purposes. Itis well known that the process { ξ t } t ≥ defined by ξ t = 1 /π t for t ≥ { S t } that pays no dividend the process { ¯ S t } defined by ¯ S t = S t /ξ t representsthe price of the original asset expressed in units of the natural numeraire. It follows thatthe “natural” price of any such asset is a martingale. Then we have S t = ξ t E t (cid:20) S u ξ u (cid:21) (22)for 0 ≤ t ≤ u < ∞ , or equivalently ¯ S t = E t (cid:2) ¯ S u (cid:3) . (23)Equation (22) shows that the domestic value of the asset at time t can be represented as theproduct of the natural numeraire (which can be interpreted as a dividend-adjusted proxyfor the market as a whole) and a fluctuating term, given by the conditional expectation ofthe natural value of the asset at some later time u . A form of (23) is used in the theory ofderivatives, for instance, when we make use of the pricing formula H t = 1 π t E t [ π T H T ] , (24)valid for 0 ≤ t < T < ∞ , which shows that the natural value ¯ H t = π t H t of the derivative at t is given by the conditional expectation of the natural value of the payoff ¯ H T = π T H T .A calculation making use of (15), (18) and Lemma 5 (see Appendix) shows that thedynamical equation satisfied by the natural value of the risky asset takes the formd ¯ S t ¯ S t − = ¯ σ t d W t + (cid:90) | x | > ¯Σ t ( x ) ˜ N (d x, d t ) , (25)where ¯ σ t = σ t − λ t and ¯Σ t ( x ) = Σ t ( x ) (cid:0) − Λ t ( x ) (cid:1) − Λ t ( x ), or equivalently σ t = ¯ σ t + λ t , Σ t ( x ) = ¯Σ t ( x ) + Λ t ( x )1 − Λ t ( x ) . (26)The relations given in (26) demonstrate that the Brownian and jump volatilities of theasset with domestic price process { S t } can each be decomposed into terms involving onlythe intrinsic “natural” volatility of the asset and terms associated with the volatility of thedomestic pricing kernel but not associated with any particular asset.One can check that as a consequence of (15) and Proposition 2 (or Lemma 6), thedynamical equation satisfied by { ξ t } takes the formd ξ t ξ t − = (cid:20) r t + λ t + (cid:90) x Λ t ( x ) − Λ t ( x ) ν (d x ) (cid:21) d t + λ t d W t + (cid:90) | x | > Λ t ( x )1 − Λ t ( x ) ˜ N (d x, d t ) , (27)which is indeed of the type appropriate to an asset that pays no dividend, as one seesby comparing (18) with (27). The benchmark process has the property that its Brown-ian proportional volatility coincides with the Brownian market price of risk and its jumpproportional volatility is given by an invertible function of the jump market price of risk.The significance of the benchmark asset in the present investigation is as follows. Weare concerned with the problem of hedging a position in a risky asset with a position in aportfolio consisting of one or more other risky assets. Now, when such a hedge is carriedout, this involves a choice of base currency with respect to which the hedge is optimized.Clearly, the choice of base currency is largely arbitrary, and it does not make sense to insiston minimizing exclusively the magnitude of the residual value of the hedge portfolio inunits of the domestic currency. Sometimes it is argued that there may be a favoured choiceof base currency – for example the currency in which a household has to meet its dailyobligations, or in which a business has to accommodate a series of cashflows in connectionwith its activities. But such considerations bring additional elements of structure into theargument, and the fact remains that there is no a priori reason why one fiat currency shouldbe favoured over another in the absence of a more detailed specification of the problem. Ofall the choices of hedging currencies there is, however, a “preferred” numeraire involving noadditional elements of structure, and this is the benchmark. So we take the view that theoptimization problem takes the form of minimizing a function of the magnitude of the valueof the hedge portfolio when that value is expressed in units of the benchmark.Proceeding with our investigation of optimal hedging, let us write { C t } t ≥ for the domesticprice process of another risky asset, which we call the contract asset . We shall assume that { C t } is strictly positive and thatd C t C t − = (cid:20) r t + λ t σ ct + (cid:90) x Λ t ( x ) Σ ct ( x ) ν (d x ) (cid:21) d t + σ ct d W t + (cid:90) | x | > Σ ct ( x ) ˜ N (d x, d t ) , (28)where { σ ct } t ≥ is adapted, { Σ ct ( x ) } t ≥ , x ∈ R is predictable, and Σ ct ( x ) > − t ≥ x ∈ R . We can think of { C t } as representing the domestic value process of the position thatwe wish to hedge, and { S t } as being the domestic value process of the hedging asset.For applications, one usually needs to impose stronger conditions on the price processesunder consideration. For example, in the case of a derivative, with payoff H T at time T ,it is reasonable to assume not merely that the payoff should satisfy E [ ¯ H T ] < ∞ , but alsothat it should satisfy E [ ¯ H T ] < ∞ . In other words, for derivative risk management, wetypically desire that some measure of the uncertainty of the payoff can be worked out, suchas its variance. Indeed, in financial markets, one does not really wish to be working withinstruments that are so volatile or ill-behaved that it is not possible to assign a meaningfulvalue to the variance of the payoff. Since, in international markets, there is no particularreason to prefer one currency to another, it makes sense to introduce a minimalist assumptionto the effect that the variance of the natural value of the payoff should be quantifiable. Thus,we shall assume at the very least that Var ¯ H T < ∞ . One could consider other choices fora measure of the riskiness of the payoff, and one could work this out in other units, butthe choice that we have indicated is convenient from a mathematical perspective since thecategory of square-integrable random variables is well understood, and the use of naturalunits is well defined already under the assumptions that we have made. One might objectthat insisting on a finite variance is too strong an assumption; but the reply can be put innormative terms – namely, that for a financial instrument to be considered as a legitimateobject of commerce, it needs in principle to be capable of being risk-managed in a reasonablyconventional manner; and the requirement that the value of the instrument can be modelledas having a finite variance is a step in this direction, an embodiment of this idea. IV. OPTIMAL HEDGING IN A L´EVY-ITO MARKET
We consider setting up a trading strategy to hedge the natural value of a position in a givenasset. Going forward, we shall for this purpose assume that all values are given in naturalunits – that is, in units of the natural benchmark numeraire. Thus, we henceforth drop theuse of the “bar” notation, and let { S t } and { C t } denote the natural prices of the hedgingasset and the contract asset, respectively. For the associated price dynamics we writed S t S t − = σ t d W t + (cid:90) | x | > Σ t ( x ) ˜ N (d x, d t ) (29)and d C t C t − = σ ct d W t + (cid:90) | x | > Σ ct ( x ) ˜ N (d x, d t ) . (30)Writing σ t ( x ) = log (1 + Σ t ( x )) , σ ct ( x ) = log (1 + Σ ct ( x )) , (31)one can use the Proposition 2 to show that the corresponding price processes are given bythe expressions S t = S exp (cid:18)(cid:90) t σ u d W u − (cid:90) t σ u d u (cid:19) × exp (cid:18)(cid:90) t (cid:90) | x | > σ u ( x ) ˜ N (d x, d u ) − (cid:90) t (cid:90) x (e σ u ( x ) − σ u ( x ) − ν (d x )d u (cid:19) (32)and C t = C exp (cid:18)(cid:90) t σ cu d W u − (cid:90) t ( σ cu ) d u (cid:19) × exp (cid:18)(cid:90) t (cid:90) | x | > σ cu ( x ) ˜ N (d x, d u ) − (cid:90) t (cid:90) x (e σ cu ( x ) − σ cu ( x ) − ν (d x )d u (cid:19) . (33)The hedging problem can be formulated as follows. The hedger is holding a position in oneunit of the contract asset. The value process of this asset is { C t } in natural units. The valueprocess of the hedging asset is { S t } in natural units. We assume that the hedging assetcan be borrowed in any quantity at no cost, and that a short position in the hedging assetcan be maintained and adjusted on a continuous basis at no cost. The value of the hedgeportfolio at time t is V t = C t − φ t S t + θ t , (34)where the predictable process { φ t } denotes the number of units of the hedging asset beingshorted, and the predictable process { θ t } denotes the number of benchmark units held inthe hedge portfolio. Initially, we have θ = φ S . That is to say, the proceeds of the initialshort sale of the hedging asset are deposited in the benchmark account. Thereafter, theportfolio is managed on a self-financing basis: thus, the change in the value of the portfolioover a small interval of time is given byd V t = d C t − φ t d S t . (35)It follows from (34) and (35) that the position in the benchmark account at time t is θ t = φ t S t − − (cid:90) t − φ u d S u , (36)or equivalently θ t = φ t S t − (cid:90) t φ u d S u , (37)where the integrals on the right-hand sides (36) and (37) are understood as being over theintervals [0 , t ) and [0 , t ], respectively. Then for the dynamics of the hedge portfolio we haved V t = (cid:0) σ ct C t − − φ t σ t S t − (cid:1) d W t + (cid:90) | x | > (cid:0) Σ ct ( x ) C t − − φ t Σ t ( x ) S t − (cid:1) ˜ N (d x, d t ) . (38)Now, if both of the assets are driven purely by the Brownian motion, and there are nojumps, then a perfect hedge can be carried out in such a way that the value of the hedgeportfolio is constant. In that case a short calculation shows that φ t = σ ct C t σ t S t and θ t = C + (cid:18) σ ct σ t − (cid:19) C t . (39)The expression for the hedge ratio will look familiar, of course, but one should keep in mindthat the hedge here is for the natural value of the contract asset, not its value in units ofthe fiat currency. In the general situation, when jumps are allowed, it is not possible to finda perfect hedge in the sense of completely erasing the riskiness of the position. Instead, weproceed as follows. We assume that the natural values of the assets under consideration aresquare-integrable in the sense that E (cid:2) S t (cid:3) < ∞ , E [ S t C t ] < ∞ , E (cid:2) C t (cid:3) < ∞ , (40)for t ≥
0, and that the self-financing hedging strategy { φ t , θ t } t ≥ is such that the portfoliovalue at any time t ≥ E (cid:2) V t (cid:3) < ∞ . (41)0Again, these assumptions are reasonable from a financial point of view, since we wouldnot wish to consider assets that fail to satisfy such conditions as suitable for trading on acommercial basis. We fix a time interval [0 , T ]. Our goal is to choose the hedging strategyso as to minimize the expected squared deviation of the value of the hedge portfolio at time T from its value at time 0. Note that once we specify the positions held in the risky assets,the self-financing condition determines the corresponding holding required in the benchmarkasset. Thus, if for any admissible choice of the strategy Φ = { φ t } ≤ t ≤ T we write∆ T (Φ) = E (cid:2) ( V T − V ) (cid:3) (42)for the corresponding mean squared error, then we have∆ T (Φ) = E (cid:34)(cid:18)(cid:90) T (cid:0) σ cu C u − − φ u σ u S u − (cid:1) d W u + (cid:90) T (cid:90) | x | > (cid:0) Σ cu ( x ) C u − − φ u Σ u ( x ) S u − (cid:1) ˜ N (d x, d u ) (cid:19) (cid:35) . Therefore, by use of Proposition 3 we obtain∆ T (Φ) = E (cid:20)(cid:90) T (cid:0) σ cu C u − − φ u σ u S u − (cid:1) d u + (cid:90) T (cid:90) | x | > (cid:0) Σ cu ( x ) C u − − φ u Σ u ( x ) S u − (cid:1) ν (d x ) d u (cid:21) . It follows that the mean squared error takes the form∆ T (Φ) = E (cid:20)(cid:90) T (cid:0) K u C u − − φ u L u S u − C u − + φ u M u S u − (cid:1) d u (cid:21) , (43)where K t = σ c t + (cid:90) x Σ ct ( x ) ν (d x ) , L t = σ t σ ct + (cid:90) x Σ t ( x ) Σ ct ( x ) ν (d x ) , M t = σ t + (cid:90) x Σ t ( x ) ν (d x ) . Thus we are led to the following.
Proposition 4.
Let { C t } be hedged with { φ t } units of { S t } and { θ t } units of the benchmark.Then the optimal hedge { ˆ φ t , ˆ θ t } ≤ t ≤ T is given a.e. - { P × Leb [0 , T ] } by ˆ φ t = σ t σ ct + (cid:82) x Σ t ( x ) Σ ct ( x ) ν (d x ) σ t + (cid:82) x Σ t ( x ) ν (d x ) C t − S t − , ˆ θ t = ˆ φ t S t − (cid:90) t ˆ φ u d S u . (44) Proof.
A standard argument using the calculus of variations establishes (44) as a candidatefor the optimal hedge. The non-degeneracy condition imposed on the hedging asset ensuresthat the denominator is non-vanishing on { Ω × [0 , T ] }\ D . To prove that the candidate isindeed optimal, we need to show that the mean squared error in any alternative hedge isno less than the mean squared error in the candidate hedge. Let Ψ = { ψ t } ≤ t ≤ T denote analternative hedge. We say that two strategies { ψ t } and { ψ t } over [0 , T ] are distinct if P × Leb [0 , T ] (cid:104)(cid:0) ψ t − ψ t (cid:1) > (cid:105) > . (45)A calculation then gives∆ T (cid:0) Ψ (cid:1) − ∆ T (cid:0) ˆΦ (cid:1) = E (cid:20)(cid:90) T ( ψ u − ˆ φ u ) S u − M u d u (cid:21) , (46)where ˆΦ = { ˆ φ t } ≤ t ≤ T , and one sees that the right hand side is nonnegative for any alternativehedge. In fact, the optimal hedge dominates any distinct alternative.1We remark, incidentally, that one can substitute the optimal hedging strategy (44) backinto the hedge portfolio value process { V t } with dynamics (35) to check that the condition(41) is satisfied. In fact, one can use (46) as a shortcut to get this result. If we compare theerror when no hedge is put on to the error when the optimal hedge is put on, then we have∆ T (cid:0) ˆΦ (cid:1) = ∆ T (cid:0) − E (cid:20)(cid:90) T ˆ φ u S u − M u d u (cid:21) , (47)which gives a bound on ∆ T (cid:0) ˆΦ (cid:1) . We just need to check that the terms on the right handside of this relation are both finite. But∆ T (0) = E (cid:20)(cid:90) T (cid:18) σ cu + (cid:90) | x | > Σ cu ( x ) ν (d x ) (cid:19) C u − d u (cid:21) , (48)which is finite on account our assumption that C T is square integrable. Then since weassume that S T is square integrable and that S T C T is integrable, for the second term we get E (cid:20)(cid:90) T ˆ φ u S u − M u d u (cid:21) = E (cid:20)(cid:90) T ρ u (cid:18) σ cu + (cid:90) | x | > Σ cu ( x ) ν (d x ) (cid:19) C u − d u (cid:21) , (49)where ρ u = (cid:18) σ u σ cu + (cid:82) | x | > Σ u ( x ) Σ cu ( x ) ν (d x ) (cid:19) (cid:18) σ u + (cid:82) | x | > Σ u ( x ) ν (d x ) (cid:19)(cid:18) σ cu + (cid:82) | x | > Σ cu ( x ) ν (d x ) (cid:19) . (50)By virtue of the Cauchy-Schwartz inequality we have 0 ≤ ρ u ≤
1, and this allows to concludethat the second term is also finite.
V. OPTIMAL HEDGING WITH MULTIPLE HEDGING ASSETS
Let us now consider the more general problem of setting up a trading strategy to hedgethe natural value of a position in a given contract asset { C t } with a collection of n hedgingassets { S it } i =1 ,...,n each with dynamics of the form (29). Thus we haved S it S it − = σ it d W t + (cid:90) | x | > Σ it ( x ) ˜ N (d x, d t ) . (51)We shall assume the collection { S it } i =1 ,...,n is non-degenerate in the sense that no one of theassets can be replicated by holding a portfolio in the remaining n − D ⊂ { Ω × [0 , T ] } bethe collection of points defined by D = (cid:26) ω ∈ Ω , t ∈ [0 , T ] (cid:12)(cid:12)(cid:12)(cid:12) ∃ ι it : n (cid:88) i =1 ι it σ it S it − = 0 ∩ n (cid:88) i =1 ι it Σ it ( x ) S it − = 0 , x ∈ S ν (cid:27) . (52)2If follows that on the complement D c := { Ω × [0 , T ] }\ D , no self-financing trading strategy {{ ι it } i =1 ,...,n , ζ t } in the n hedging assets and the benchmark exists such that n (cid:88) i =1 ι it σ it S it − = 0 and n (cid:88) i =1 ι it Σ it ( x ) S it − = 0 , (53)for x ∈ R n in the support of the L´evy measure. Our assumption is that the degeneracysubset should have { P × Leb [0 , T ] } -measure zero. We assume further that E (cid:2) S it S jt (cid:3) < ∞ , E (cid:2) S it C t (cid:3) < ∞ , E (cid:2) C t (cid:3) < ∞ , (54)for t ∈ [0 , T ], i, j = 1 , . . . , n . When n such risky assets with negligible degeneracy areavailable for hedging, the hedge portfolio for the contract asset takes the form V t = C t − n (cid:88) i =1 φ it S it + θ t , (55)and we impose the self-financing conditiond V t = d C t − n (cid:88) i =1 φ it d S it . (56)Our goal is to choose the hedging strategy Φ = { φ it } ≤ t ≤ T in such a way that the meansquared error in the portfolio value∆ T (Φ) = E (cid:2) ( V T − V ) (cid:3) (57)is minimized. Then by (30), (51), (56) and (57) we have∆ T (Φ) = E (cid:32)(cid:90) T (cid:0) σ cu C u − − n (cid:88) i =1 φ iu σ iu S iu − (cid:1) d W u + (cid:90) T (cid:90) | x | > (cid:0) Σ cu ( x ) C u − − n (cid:88) i =1 φ iu Σ iu ( x ) S iu − (cid:1) ˜ N (d x, d u ) (cid:33) , and by use of the Ito isometry we obtain∆ T (Φ) = E (cid:34)(cid:90) T (cid:0) σ cu C u − − n (cid:88) i =1 φ iu σ iu S iu − (cid:1) d u + (cid:90) T (cid:90) x (cid:0) Σ cu ( x ) C u − − n (cid:88) i =1 φ iu Σ iu ( x ) S iu − (cid:1) ν (d x )d u (cid:35) . Expanding the squares and gathering together the various terms we get∆ T (Φ) = E (cid:34)(cid:90) T (cid:32) G u + n (cid:88) i =1 n (cid:88) j =1 φ iu φ ju M iju − n (cid:88) i =1 φ iu F iu (cid:33) d u (cid:35) , (58)where M iju = S iu − S ju − (cid:20) σ iu σ ju + (cid:90) x Σ iu ( x )Σ ju ( x ) ν (d x ) (cid:21) , (59)3 F iu = S iu − C u − (cid:20) σ iu σ cu + (cid:90) x Σ iu ( x )Σ cu ( x ) ν (d x ) (cid:21) , (60) G u = C u − (cid:20) σ c u + (cid:90) x Σ cu ( x ) ν (d x ) (cid:21) . (61)Applying a perturbation { φ it } ≤ t ≤ T → { φ it + (cid:15) it } ≤ t ≤ T (62)to the hedging strategy, we find that the difference in the corresponding expressions for themean squared errors is given by∆ T (Φ + (cid:15) ) − ∆ T (Φ) = E (cid:34)(cid:90) T (cid:32) n (cid:88) i =1 n (cid:88) j =1 (cid:15) iu ( (cid:15) ju + 2 φ ju ) M iju − n (cid:88) i =1 (cid:15) iu F iu (cid:33) d u (cid:35) . (63)A sufficient condition for the right-hand side of (63) to vanish to first order in the perturbingvariables, and hence lead to a candidate optimum, is that the { φ it } should satisfy n (cid:88) j =1 M ijt φ jt = F it , a . e . - { P × Leb [0 , T ] } , (64)and we are thus led to the following. Proposition 5.
Let { C t } be hedged with { φ it } units of { S it } for i = 1 , . . . , n and { θ t } unitsof the benchmark. Then the optimal hedge takes the form ˆ φ it = n (cid:88) j =1 N ijt F jt , ˆ θ t = n (cid:88) i =1 ˆ φ it S it − (cid:90) t n (cid:88) i =1 ˆ φ iu d S iu , a . e . - { P × Leb[0 , T ] } , (65) where { N ijt } is the inverse of { M ijt } on D c .Proof. The inverse of the matrix { M ijt } exists on D c on account of the non-degeneracycondition that we have imposed on the collection of hedging assets. In particular, it followsfrom the definition of D that { ω, t } ∈ D c if and only if the inequality (cid:32) n (cid:88) i =1 ι it σ iu S iu − (cid:33) + (cid:90) x (cid:32) n (cid:88) i =1 ι it Σ iu ( x ) S iu − (cid:33) ν (d x ) > { ι it } i =1 ,...,n . But this relation is equivalent to n (cid:88) i =1 n (cid:88) j =1 M ijt ι jt ι it > , (67)which shows that { M ijt } is positive definite on D c , and hence possesses an inverse. Thesolution of equation (64) then gives a candidate optimal hedge. As in the case of a single4hedging asset, we need to show that the error in any alternative hedge is no less than theerror in the candidate solution. Putting (65) back into (58), we get∆ T (cid:0) ˆΦ (cid:1) = E (cid:34)(cid:90) T (cid:32) G u − n (cid:88) i =1 n (cid:88) j =1 N iju F ju F iu (cid:33) d u (cid:35) . (68)Then, letting { ψ it } ≤ t ≤ T be any alternative hedge that is distinct from the candidate (65),one finds that∆ T (cid:0) Ψ (cid:1) − ∆ T (cid:0) ˆΦ (cid:1) = E (cid:34)(cid:90) T n (cid:88) i =1 n (cid:88) j =1 M iju ( ψ iu − ˆ φ iu )( ψ ju − ˆ φ ju ) d u (cid:35) . (69)The right side of (69) is strictly positive, and we deduce that { ˆ φ it } is optimal and indeedthat it dominates any strategy distinct from it.Next, we wish to show that if we add a further non-redundant hedging asset to anexisting collection of n hedging assets satisfying a non-degeneracy condition, the hedge willbe improved by using all n + 1 of the hedging assets. This is a characteristic feature ofincomplete markets. Given { C t } and { S it } i =1 ,...,n , let { ˆ φ it } i =1 ,...,n denote the optimal hedgedetermined in Proposition 5, and let { S t } be another hedging asset, which is taken to benon-redundant in the sense that it cannot be realized as a portfolio formed from the original n hedging assets together with the benchmark asset.More precisely, let us now write D n (in place of D ) for the degeneracy set associatedwith the n original hedging instruments, which we have assumed to be of { P × Leb(0 , T ) } -measure zero, and let us write D n +1 for the degeneracy set of the enhanced collection of n + 1 heading instruments, which we also assume to have { P × Leb(0 , T ) } -measure zero. Itshould be evident that D n ⊂ D n +1 , since there may be points in { Ω × [0 , T ] } at which theenhanced collection degenerates even though the original collection is non-degenerate. Thenwe have the following. Proposition 6.
For any contract asset { C t } , the optimal hedge { ˆΓ it } i =0 , ,...,n obtained byuse of the enhanced collection of n + 1 hedging assets { S it } i =0 , ,...,n is strictly better than theoptimal hedge { ˆ φ it } i =1 ,...,n obtained by use of the original n hedging assets { S it } i =1 ,...,n .Proof. The argument proceeds in two steps. First, let { ˆ U t } ≤ t ≤ T denote the value process ofthe optimal hedge position { ˆ φ it , ˆ θ t } constructed from the original n hedging assets togetherwith the benchmark asset, as determined in Proposition 5. Thus, we haveˆ U t = n (cid:88) i =1 ˆ φ it S it + ˆ θ t , (70)where { ˆ φ it , ˆ θ t } is given as in (65). It follows by the self-financing condition that { ˆ U t } itselfcan be treated as an asset. Now consider a hedging strategy of the form { γ t , δ t , (cid:15) t } where { γ t } denotes the holdings in { U t } , where { δ t } denotes the holdings in { S t } , and { (cid:15) t } denotesthe holdings in the benchmark asset. It is easy to see that an optimal hedge involving apair of non-redundant risky hedging instruments will perform better than the optimal hedgeobtained by use of just one of the two risky instruments. This is because the optimal hedge5involving a single risky instrument is an example of a sub-optimal hedge involving two riskyinstruments. It follows that as a hedge for { C t } the strategy { γ t , δ t , (cid:15) t } with value process { γ t U t + δ t S t + (cid:15) t } will perform strictly better than the strategy { , ˆ θ t } with value process { U t + ˆ θ t } . That is to say, ∆ T (cid:0) γ ˆΦ , δ (cid:1) < ∆ T (cid:0) ˆΦ , (cid:1) . (71)On the other hand, we observe that if ˆΓ := { ˆΓ it } i =0 , ,...,n denotes the optimal enhancedhedging strategy involving the n + 1 assets now available for hedging, along with a position { ζ t } in the benchmark, then the portfolio { γ t ˆ φ it , δ t , (cid:15) t } i =1 ,...,n considered above at (71) ismerely an example of a hedge involving the n + 1 hedging assets, and though it might beoptimal, in general it will be suboptimal. Therefore∆ T (cid:0) ˆΓ (cid:1) ≤ ∆ T (cid:0) γ ˆΦ , δ (cid:1) , (72)and hence ∆ T (cid:0) ˆΓ (cid:1) < ∆ T (cid:0) ˆΦ , (cid:1) . (73)It follows that the optimal hedge involving n + 1 hedging instruments will perform betterthan the optimal hedge formed from any n of them. VI. SIMULATIONS
In conclusion, we propose in this section to look in more detail at the n = 2 case and considersimulating the optimal trading strategy to hedge the natural value of a position in a givencontract asset by use of two risky hedging assets. The problem will be framed in the casewhere all three of the assets are driven by a one-dimensional Brownian motion { W t } andan independent one-dimensional Poisson random measure { N (d x, d t ) } . The hedging assetseach have dynamics of the form (29). We write { S it } i =1 , for the hedging assets, and wewrite { φ it } i =1 , for the holdings in these assets. Then the rather general construction givenin Section V leads to the following: Proposition 7.
Let the contract asset { C t } be hedged over [0 , T ] with { φ t } units of { S t } , { φ t } units of { S t } , and { θ t } units of the benchmark. Then the optimal hedge is given by ˆ φ t = P t − Q t R t C t − S t − , ˆ φ t = P t − Q t R t C t − S t − , (74) on the non degeneracy subset D c , where we write P ijt = (cid:18) σ ct σ it + (cid:90) x Σ ct ( x ) Σ it ( x ) ν (d x ) (cid:19) (cid:18) σ j t + (cid:90) x Σ jt ( x ) ν (d x ) (cid:19) ,Q ijt = (cid:18) σ it σ jt + (cid:90) x Σ it ( x ) Σ jt ( x ) ν (d x ) (cid:19) (cid:18) σ ct σ jt + (cid:90) x Σ ct ( x ) Σ jt ( x ) ν (d x ) (cid:19) ,R ijt = (cid:18) σ i t + (cid:90) x Σ it ( x ) ν (d x ) (cid:19) (cid:18) σ j t + (cid:90) x Σ jt ( x ) ν (d x ) (cid:19) − (cid:18) σ it σ jt + (cid:90) x Σ it ( x ) Σ jt ( x ) ν (d x ) (cid:19) . { X t } t ≥ denote a compound Poisson processfor which the jumps arrive randomly according to a Poisson process { N t } t ≥ with rate m . Thejump sizes { Y i } i ∈ N are independent identically-distributed random variables. We assume that { Y i } i ∈ N and { N t } are independent. Let us write Y for a typical element of the set { Y i } i ∈ N .In the example under consideration we shall assume that Y has a Bernoulli distributionBern( g, h ; p ). Thus Y takes values in a set { g, h } where g, h ∈ R with P [ Y = g ] = p and P [ Y = h ] = 1 − p . The L´evy measure for such a process { X t } takes the form ν (d x ) = m (cid:0) pδ g (d x ) + (1 − p ) δ h (d x ) (cid:1) , (75)where δ g (d x ) is the Dirac measure concentrated at g and δ h (d x ) is the Dirac measure con-centrated at h . Then the price processes of the assets under consideration have dynamics ofthe form (29)-(30), with deterministic time-independent volatilities. Since we are workingwith a geometric L´evy process, the jump volatility is of the form Σ( x ) = exp( βx ) −
1, forsome β ∈ R + . The price of a typical non-dividend paying risky asset in a Bernoulli jumpdiffusion market with this set up is thus of the form S t = S exp (cid:18) σW t − σ t + βX t − mt (cid:0) p (e β g −
1) + (1 − p ) (e β h − (cid:1)(cid:19) , (76)where σ is a constant. For our simulations we consider a contract asset { C t } and a pair ofhedging assets { S t } and { S t } , each of the form (76), with a view to forming an optimalhedge of the contract asset with positions in one or both of the hedging assets.In Figure 1, we show on the left-hand side a random sample path for the L´evy process { X t } alongside the underlying Poisson process { N t } . On the right-hand side one finds thecorresponding paths for the contract asset { C t } and the two hedging assets { S t } and { S t } .The inputs for this example are as follows: S = 100, S = 100, C = 100, σ = 0 . σ = 0 . σ c = 0 . β = 0 . β = 0 . β c = 0 . m = 15, p = 0 . g = 1, h = −
1, and T = 1. The unit of time depicted on the x -axis is divided into a thousand parts.Now, we know from general theory that if the Brownian motion is non-vanishing thenthe hedge can never be perfect; but if the Brownian component is small for all three assets,then a reasonably good hedge should be obtainable using just two assets in the case of aBernoulli jump diffusion. In Figure 2 we show the effect of using either { S t } or { S t } aloneas a hedge and we plot the residual movements in the values of the hedged portfolios.7 Figure 1:
Bernoulli jump-diffusion market.
The chart on the left above shows an outcome ofchance for the L´evy process in blue, with the underlying Poisson process in red. The chart on theright above plots the value process of the contract asset in green. The high volatility hedging asset1 is shown in red, and the low volatility hedging asset 2 is shown in blue.Figure 2:
Single-asset hedges.
The chart on the left plots at each step the change in the valueof the hedge portfolio, when asset 1 alone is used as the hedge. The lengthy downward spikescorrespond to jumps, whereas the shorter spikes are due to Brownian volatility. In the chart onthe right, asset 2 alone is used as the hedge. The lengthy upward spikes correspond to jumps.
In Figure 3 we show the effect of using both hedging assets together to hedge the contractasset, and we note in particular the significant drop in the variance of the hedged portfolio.If we reduce the volatilities of the Brownian components still further, then we get a nearperfect hedge, as illustrated in Figure 4. The Brownian volatilities for Figure 4 are given by σ = 0 . σ = 0 .
001 and σ c = 0 . Figure 3:
Two-asset hedge.
The figure above plots the change in the value of the hedge portfoliowhen both hedging asset 1 and hedging asset 2 are included in the hedging strategy for the contractasset. The Brownian volatilities in this example are σ = 0 . σ = 0 .
10 and σ c = 0 . Two-asset hedge with reduced Brownian volatilities.
This figure plots the change in thevalue of the hedge portfolio when both asset 1 and asset 2 are included in the hedging strategy forthe contract, with σ = 0 . σ = 0 .
001 and σ c = 0 . y -axis is smaller than that of the previous figure. It is sometimes said that L´evy markets are incomplete except in the Brownian case, inthe situation where the number of available assets is no less than the number of Brownianmotions. But this of course is not quite true, since a pure Poisson market is also complete.If a pair of geometric L´evy assets are driven by a common Poisson process, then either canbe hedged by use of the other. A pure Bernoulli market is also complete, in the sense thatif three geometric L´evy assets are driven by a common Bernoulli process, then any one canbe hedged by use of the other two. Similarly, a compound Poisson process market with k possible outcomes at each jump is complete if k hedging assets are available. If a Browniancomponent is introduced into any of these scenarios, then the resulting market is incomplete;but if the Brownian volatilities are small, then near perfect hedges can be achieved, as wesee in Figure 4.9 APPENDIX
Here we present some useful versions of the Ito product and quotient rules for L´evy-Itoprocesses. The Brownian versions of these rules will be familiar, but the correspondingL´evy-Ito rules do not seem previously to have been presented systematically in all theirdifferent versions, so we do so here. Let { X t } t ≥ and { X t } t ≥ be L´evy-Ito processes, eachsatisfying dynamical equations of the form (2), such thatd X t = α t d t + β t d W t + (cid:90) | x |∈ (0 , γ t ( x ) ˜ N (d x, d t ) + (cid:90) | x |≥ δ t ( x ) N (d x, d t ) (77)and d X t = α t d t + β t d W t + (cid:90) | x |∈ (0 , γ t ( x ) ˜ N (d x, d t ) + (cid:90) | x |≥ δ t ( x ) N (d x, d t ) . (78) Lemma 1.
The product rule for L´evy-Ito processes takes the following form: d( X t X t ) = [ α t X t − + α t X t − + β t β t ] d t + ( β t X t − + β t X t − )d W t + (cid:90) | x | < γ t ( x ) γ t ( x ) ν (d x ) d t + (cid:90) | x |∈ (0 , ( γ t ( x ) γ t ( x ) + γ t ( x ) X t − + γ t ( x ) X t − ) ˜ N (d x, d t )+ (cid:90) | x |≥ ( δ t ( x ) δ t ( x ) + δ t ( x ) X t − + δ t ( x ) X t − ) N (d x, d t ) . (79) Proof.
This is similar to the proof of the corresponding result for Ito processes, and isobtained by applying Ito’s formula to each side of the identity X t X t = 14 (cid:0) X t + X t (cid:1) − (cid:0) X t − X t (cid:1) . (80)A calculation then gives the result claimed.Now let { X t } and { X t } be L´evy-Ito processes such that { X t } , { X t − } are strictly positive.Then we obtain the following. Lemma 2.
The quotient rule for L´evy-Ito processes is given by d (cid:18) X t X t (cid:19) = (cid:20) α t X t − − α t X t − ( X t − ) + ( β t ) X t − − β t β t X t − ( X t − ) (cid:21) d t + β t X t − − β t X t − ( X t − ) d W t + (cid:90) | x | < ( γ t ( x )) X t − − γ t ( x ) γ t ( x ) X t − ( X t − ) ( X t − + γ t ( x )) ν (d x )d t + (cid:90) | x |∈ (0 , γ t ( x ) X t − − γ t ( x ) X t − X t − ( X t − + γ t ( x )) ˜ N (d x, d t ) + (cid:90) | x |≥ δ t ( x ) X t − − δ t ( x ) X t − X t − ( X t − + δ t ( x )) N (d x, d t ) . (81) Proof.
First one uses Proposition 1 to work out the dynamics of the process { /X t } . Thenone uses Lemma 1 to work out the dynamics of the product { X t × /X t } .0For applications in finance, one often makes use of the “proportional” versions of theL´evy-Ito product and quotient rules, which are applicable if we assume that { X t } , { X t − } , { X t } , { X t − } are strictly positive. The dynamical equations for { X t } and { X t } will beassumed in Lemmas 3 and 4 to take the proportional formd X t = X t − (cid:20) α t d t + β t d W t + (cid:90) | x |∈ (0 , γ t ( x ) ˜ N (d x, d t ) + (cid:90) | x |≥ δ t ( x ) N (d x, d t ) (cid:21) (82)and d X t = X t − (cid:20) α t d t + β t d W t + (cid:90) | x |∈ (0 , γ t ( x ) ˜ N (d x, d t ) + (cid:90) | x |≥ δ t ( x ) N (d x, d t ) (cid:21) . (83)Then we have the following formulae, which arise as consequences of Lemmas 1 and 2. Lemma 3.
In the proportional case, the product rule takes the form d( X t X t ) = X t − X t − (cid:20) (cid:18) α t + α t + β t β t + (cid:90) | x | < γ t ( x ) γ t ( x ) ν (d x ) (cid:19) d t + ( β t + β t )d W t + (cid:90) | x |∈ (0 , (cid:0) γ t ( x ) γ t ( x ) + γ t ( x ) + γ t ( x ) (cid:1) ˜ N (d x, d t )+ (cid:90) | x |≥ (cid:0) δ t ( x ) δ t ( x ) + δ t ( x ) + δ t ( x ) (cid:1) N (d x, d t ) (cid:21) . (84) Lemma 4.
In the proportional case, the quotient rule takes the form d (cid:18) X t X t (cid:19) = X t − X t − (cid:20) (cid:18) α t − α t − β t ( β t − β t ) − (cid:90) | x | < γ t ( x ) γ t ( x ) − γ t ( x )1 + γ t ( x ) ν (d x ) (cid:19) d t + ( β t − β t ) d W t + (cid:90) | x |∈ (0 , γ t ( x ) − γ t ( x )1 + γ t ( x ) ˜ N (d x, d t ) + (cid:90) | x |≥ δ t ( x ) − δ t ( x )1 + δ t ( x ) N (d x, d t ) (cid:21) . (85)The various forms of the Ito product and quotient rules simplify for symmetric propor-tional processes, and the resulting formulae are extremely useful in applications. Let { X t } , { X t − } , { X t } , { X t − } be strictly positive For symmetrical processes we can writed X t = X t − (cid:20) α t d t + β t d W t + (cid:90) | x | > γ t ( x ) ˜ N (d x, d t ) (cid:21) (86)and d X t = X t − (cid:20) α t d t + β t d W t + (cid:90) | x | > γ t ( x ) ˜ N (d x, d t ) (cid:21) , (87)and we obtain the following. Lemma 5.
In the symmetric proportional case the product rule takes the form d( X t X t ) = X t − X t − (cid:20) (cid:18) α t + α t + β t β t + (cid:90) x γ t ( x ) γ t ( x ) ν (d x ) (cid:19) d t + ( β t + β t )d W t + (cid:90) | x | > (cid:0) γ t ( x ) + γ t ( x ) + γ t ( x ) γ t ( x ) (cid:1) ˜ N (d x, d t ) (cid:21) . (88)1 Lemma 6.
In the symmetric proportional case the quotient rule takes the form d (cid:18) X t X t (cid:19) = X t − X t − (cid:20) (cid:18) α t − α t − β t ( β t − β t ) − (cid:90) x γ t ( x ) γ t ( x ) − γ t ( x )1 + γ t ( x ) ν (d x ) (cid:19) d t + ( β t − β t ) d W t + (cid:90) | x | > γ t ( x ) − γ t ( x )1 + γ t ( x ) ˜ N (d x, d t ) (cid:21) . (89)The corresponding results for n -dimensional L´evy-Ito process are straightforward. Acknowledgments
The authors wish to thank D. C. Brody, A. Ciatti, S. Jaimungal and L. S´anchez-Betancourtfor helpful discussions. We are also grateful for the helpful comments of an anonymousreferee. GB acknowledges support from Timelineapp Tech Ltd, Basildon.
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