Determination of the Lévy Exponent in Asset Pricing Models
aa r X i v : . [ q -f i n . M F ] F e b Determination of the L´evy Exponentin Asset Pricing Models
George Bouzianis and Lane P. Hughston
Department of Computing, Goldsmiths College, University of London,New Cross, London SE14 6NW, United Kingdom
We consider the problem of determining the L´evy exponent in a L´evy model forasset prices given the price data of derivatives. The model, formulated underthe real-world measure P , consists of a pricing kernel { π t } t ≥ together with oneor more non-dividend-paying risky assets driven by the same L´evy process. If { S t } t ≥ denotes the price process of such an asset then { π t S t } t ≥ is a P -martingale.The L´evy process { ξ t } t ≥ is assumed to have exponential moments, implying theexistence of a L´evy exponent ψ ( α ) = t − log E (e αξ t ) for α in an interval A ⊂ R containing the origin as a proper subset. We show that if the prices of power-payoffderivatives, for which the payoff is H T = ( ζ T ) q for some time T >
0, are given attime 0 for a range of values of q , where { ζ t } t ≥ is the so-called benchmark portfoliodefined by ζ t = 1 /π t , then the L´evy exponent is determined up to an irrelevantlinear term. In such a setting, derivative prices embody complete information aboutprice jumps: in particular, the spectrum of the price jumps can be worked out fromcurrent market prices of derivatives. More generally, if H T = ( S T ) q for a generalnon-dividend-paying risky asset driven by a L´evy process, and if we know that thepricing kernel is driven by the same L´evy process, up to a factor of proportionality,then from the current prices of power-payoff derivatives we can infer the structureof the L´evy exponent up to a transformation ψ ( α ) → ψ ( α + µ ) − ψ ( µ ) + cα , where c and µ are constants. Keywords: Asset pricing; L´evy models; L´evy processes; L´evy exponent; exponential moments;option pricing; option replication; power payoffs.
I. INTRODUCTION
We are concerned with determining the extent to which derivative prices, taken at timezero, can be used to infer the nature of the underlying jump processes in models where assetprices can move discontinuously. To this end we consider the case of geometric L´evy models,and address the question of to what degree the present values of derivatives can be used todetermine the L´evy processes driving the prices of the underlying financial assets. Since thework of Breeden & Litzenberger (1978) and Dupire (1994), a burgeoning literature has de-veloped based on the idea that given the prices of options and other derivatives one can inferdistributional and dynamical properties of the price processes of the underlyings. Most ofthis work deals with continuous price processes. In the present work we consider discontinu-ous processes and show that in the case of exponential L´evy models the L´evy exponent canbe completely determined modulo a two-parameter family of transformations. The paper isstructured as follows. In Section II we summarize a few facts concerning L´evy processes.Then we introduce the condition that the L´evy process should admit exponential momentsand explore some of the implications of this assumption. In Section III we introduce theclass of geometric L´evy models. These models generalize the standard geometric Brownianmotion model. They enable one to see the form that the excess rate of return takes as afunction of the level of risk, measured by a volatility parameter σ , and the level of marketrisk aversion, measured by a parameter λ . In Section IV we state the framework we usefor pricing derivatives and indicate how dividends are handled. In Section V we present amethod for determining the underlying jump process given a family of power-payoff deriva-tive prices, and the result is summarized in Proposition 1. Then in Section VI we show inProposition 2 that if the asset on which the derivatives are based is known to be the naturalnumeraire, then the underlying jump process can be determined with greater precision. Weelaborate further on the representation of the L´evy exponent in terms of the prices of power-payoff derivatives and comment in particular on the feasibility of using call option pricesfor a similar purpose. Finally, in Section VII we establish analogous results for imaginarypower-payoff derivatives, where we make use of the techniques of Fourier analysis to showthat a general European-style derivative can be expressed as a portfolio of imaginary powerpayoff derivatives, providing the payoff is smooth and has good asymptotic properties. II. EXPONENTIAL MOMENTS
We shall assume that the reader is familiar with L´evy processes and their financial applica-tions (Andersen & Lipton 2013, Appelbaum 2004, Bertoin 2004, Brody, Hughston & Mackie2012, Chan 1999, Cont & Tankov 2004, Gerber & Shiu 1994, Hubalek & Sgarra 2006, Kypri-anou 2006, Norberg 2004, Protter 2005, Sato 1999, Schoutens 2004). We mainly work withone-dimensional L´evy processes. For convenience we recall some definitions and classicalresults. A random process { ξ t } taking values in R on a probability space (Ω , F , P ) is said tobe a L´evy process if (a) ξ s + t − ξ s and { ξ u } ≤ u ≤ s are independent for s, t ≥ ξ s + t − ξ s has the same law as ξ t for s, t ≥ t → P ( | ξ s + t − ξ s | > ǫ ) = 0 for ǫ > ′ ∈ F satisfying P (Ω ′ ) = 1 such that for ω ∈ Ω ′ the path { ξ t ( ω ) } t ≥ is right-continuousfor t ≥ t > ξ = 0almost surely. It follows from this definition that for t ≥ κ ∈ R the Fourier transformof ξ t can be represented in the form1 t log E [exp(i κξ t )] = i pκ − qκ + Z ∞−∞ (e i κx − − i κx {| x | < } ) ν (d x ) . (1)Here p and q > ν (d x ) is a L´evy measure. A Borel measure ν (d x ) on R is called a L´evy measure if ν ( { } ) = 0 and Z ∞−∞ ∧ x ν (d x ) < ∞ , (2)where a ∧ b = min( a, b ). The L´evy measure associated with a L´evy process has the propertythat for any measurable set B ⊂ R the expected rate at which jumps occur for which thejump size lies in B is ν ( B ). The sample paths of a L´evy process have bounded variation onevery compact interval of time almost surely if and only if q = 0 and Z ∞−∞ ∧ | x | ν (d x ) < ∞ . (3)In that case we say that { ξ t } has bounded variation. Let us write ξ t − = lim s → t ξ s for the leftlimit of the process at time t . The discontinuity at time t is then defined by ∆ ξ t = ξ t − ξ t − ,and for the L´evy measure we have ν ( B ) = 1 t E X ≤ s ≤ t { ∆ ξ s ∈ B } (4)for any t >
0. If ν ( R ) < ∞ we say that { ξ t } has finite activity, whereas if ν ( R ) = ∞ we saythat { ξ t } has infinite activity. A necessary and sufficient condition for (3) to hold is that X ≤ s ≤ t | ∆ ξ s | < ∞ (5)almost surely for every t >
0. If sup t | ∆ ξ t | ≤ c almost surely for some constant c >
0, thenwe say that { ξ t } has bounded jumps.In order for { ξ t } to give rise to a L´evy model for asset prices, we require additionally thatfor every t > ξ t should satisfy a moment condition of the form E (e αξ t ) < ∞ (6)for α in an interval A = ( β, γ ) ⊂ R containing the origin. Here we set β = inf α : E (e αξ t ) < ∞ and γ = sup α : E (e αξ t ) < ∞ . If a L´evy process satisfies this condition, we say it possessesexponential moments. In that case, it follows (Sato 1999, theorem 25.17) that there existsa so-called L´evy exponent ψ : C A → R , for C A = { α ∈ C | Re( α ) ∈ A } , such that E (e αξ t ) = e ψ ( α ) t , (7)where ψ ( α ) admits a L´evy-Khinchin representation of the form ψ ( α ) = pα + 12 qα + Z ∞−∞ (e αx − − αx {| x | < } ) ν (d x ) . (8)A necessary and sufficient condition for a L´evy process to satisfy (6) for α ∈ A is that theassociated L´evy measure should satisfy Z ∞−∞ e αx {| x | > } ν (d x ) < ∞ (9)for α ∈ A (Sato 1999, theorem 25.3). If { ξ t } admits exponential moments, then one cancheck that for p > E ( | ξ t | p ) < ∞ . (10)The argument is as follows. Now, for any α ∈ R we havecosh( αξ t ) = ∞ X k =0 ( αξ t ) k (2 k )! . (11)Therefore for any k ∈ N we have cosh( αξ t ) > ( αξ t ) k (2 k )! . (12)If we choose α so that | α | < min( | β | , γ ), which ensures that α and − α are in A , then E (cosh( αξ t )) < ∞ . Therefore, E ( | ξ t | n ) < ∞ for even n ∈ N , which implies that E ( | ξ t | p ) < ∞ for all p ∈ R + , since for each n and any random variable X it holds that E ( | X | n ) < ∞ implies E ( | X | p ) < ∞ for 0 ≤ p ≤ n . More generally, for any α ∈ A and any p ∈ R + we have E ( e αξ t | ξ t | p ) < ∞ . (13)This can be seen as follows. Since A is open, for any α ∈ A we can choose ǫ > α (1 + ǫ ) is still in A . Then by H¨older’s inequality we have E ( e αξ t | ξ t | p ) ≤ (cid:0) E ( e α (1+ ǫ ) ξ t ) (cid:1) / (1+ ǫ ) (cid:0) E ( | ξ t | p (1+ ǫ ) /ǫ ) ) (cid:1) ǫ/ (1+ ǫ ) . (14)But we have already established that the terms on the right are finite, and that gives (13).A similar argument shows that if { ξ t } admits exponential moments then Z ∞−∞ e αx | x | p {| x | > } ν (d x ) < ∞ , (15)for α ∈ A and p >
0. Setting α = 0 and p = 1, we see in particular that Z ∞−∞ | x | {| x | > } ν (d x ) < ∞ , (16)which implies that one can extend the truncated term on the right side of (8) to an integral ofthe form R | x | ν (d x ), over the whole of R , by dropping the indicator function and redefiningthe constant p in equation (8). The finiteness of integrals (13) and (15) allows one to computethe Greeks for various derivative payouts in exponential L´evy models.Examples of L´evy processes admitting exponential moments include: (a) Brownian mo-tion, for which ψ ( α ) = α , α ∈ R ; (b) the Poisson process with rate m , for which ψ ( α ) = m (e α −
1) and ν (d z ) = mδ (d z ); (c) the compound Poisson process with rate m , for which ψ ( α ) = m ( θ ( α ) − θ ( α ) is the moment generating function for thedistribution µ (d x ) of a typical jump and ν (d z ) = mµ (d x ); (d) the gamma process withrate m , for which ψ ( α ) = − m log(1 − α ), α <
1, where ν (d z ) = { z > } z − exp( − z ) d z (Dickson & Waters 1993, Heston 1993, Brody, Macrina & Hughston 2008, Yor 2007); (e)the variance gamma (VG) process, for which ψ ( α ) = − m log(1 − α / m ), where we have − / m < α < / m (Madan & Seneta 1990, Madan & Milne 1991, Madan, Carr & Chang1998); (f) the truncated stable family of L´evy processes, which includes the gamma processand the VG process as special cases (Koponen 1995, Carr, Geman, Madan & Yor 2002, An-dersen & Lipton 2013, K¨uchler & Tappe 2014); (g) hyperbolic processes (Eberlein & Keller1995, Eberlein, Keller & Prause 1998, Bingham & Kiesel 2001); (h) generalized hyperbolicprocesses (Eberlein 2001); (i) normal inverse Gaussian processes (Barndorff-Nielsen 1998);and (j) Meixner processes (Schoutens & Teugels 1998). III. GEOMETRIC L´EVY MODEL
The geometric L´evy model for asset prices can be viewed as an extension of the well-knowngeometric Brownian motion model to the L´evy regime. For simplicity, we consider a modeldriven by a one-dimensional process { ξ t } t ≥ . The generalization to higher dimensional L´evyprocesses is straightforward. We assume that { ξ t } admits exponential moments and denotethe associated L´evy exponent by { ψ ( α ) } α ∈ A for A = ( β, γ ) with β < < γ . The process { m t } t ≥ defined by m t = e αξ t − ψ ( α ) t , (17)for some choice of α ∈ A , is the corresponding geometric L´evy martingale with volatility α .By the stationary and independent increments properties we find that E s ( m t ) = m s . Herewe write E t ( · ) = E ( · | F t ), where {F t } denotes the augmented filtration generated by { ξ t } .The associated geometric L´evy model consists of a pricing kernel, a money market account,and one or more so-called investment-grade assets. See Duffie (1992), Hunt & Kennedy(2004), Cochrane (2005) for general aspects of the theory of pricing kernels in arbitrage-freeasset pricing models. For the construction of the pricing kernel { π t } t ≥ in the context of aL´evy model we let r ∈ R and λ > − λ ∈ A . Then we set π t = e − rt − λξ t − ψ ( − λ ) t . (18)We refer to the related process { ζ t } t ≥ defined by ζ t = 1 /π t as the growth-optimal portfolioor natural numeraire asset (Flesaker & Hughston 1997). It serves as a benchmark relative towhich other non-dividend-paying assets are martingales. In some calculations it is convenientto make reference to the natural numeraire instead of the pricing kernel. The money marketaccount { B t } t ≥ is taken to have the value B t = B e rt at time t , where B denotes its initialvalue in some fixed unit of account. The idea of an investment-grade asset is that it shouldoffer a rate of return that is strictly greater than the interest rate. Ordinary stocks and bondsare in this sense investment-grade, whereas put options and short positions in ordinary stocksand bonds are not. We assume for the moment that the assets pay no dividends over thetime horizons considered (dividends will be treated shortly), and we write { S t } t ≥ for thevalue process of a typical non-dividend paying risky asset in the geometric L´evy model. Werequire that the product of the pricing kernel and the asset price should be a martingale,which we take to be geometric L´evy martingale of the form π t S t = S e βξ t − ψ ( β ) t (19)for some β ∈ A . From the formulae above we deduce that S t = S e rt + σξ t + ψ ( − λ ) t − ψ ( σ − λ ) t , (20)where σ = β + λ . We assume that σ > σ ∈ A . It follows that the price can beexpressed in the form S t = S e rt + R ( λ,σ ) t + σξ t − ψ ( σ ) t , (21)where R ( λ, σ ) = ψ ( σ ) + ψ ( − λ ) − ψ ( σ − λ ) . (22)Thus we see that σ is the volatility of the asset price relative to the given L´evy processand that R ( λ, σ ) is the excess rate of return above the interest rate. The parameter λ canbe interpreted as a measure of the level of market risk aversion. A calculation shows thatthe excess rate of return is bilinear in λ and σ if and only if { ξ t } is a Brownian motion(Brody et al λ as a “market price of risk”, whichis valid for models based on a Brownian filtration, does not carry through directly to thegeneral L´evy regime. Nevertheless, the notion of excess rate of return is well defined, andunder the assumptions that we have made the strict convexity of the L´evy exponent impliesthat the excess rate of return is strictly positive. To show that R ( λ, σ ) > R ( λ, σ ) = Z ∞−∞ (e σx − − e − λx ) ν (d x ) . (23)It follows by inspection of (23) that the excess rate of return is an increasing function of thevolatility and the level of risk aversion.In the case of a single asset driven by a single L´evy process one can without loss ofgenerality set σ = 1. This can be achieved by defining a rescaled L´evy process { ¯ ξ t } bysetting ¯ ξ t = σξ t . Then we define ¯ ψ ( α ) = ψ ( σα ) and set ¯ λ = λ/σ , and we have π t = e − rt − ¯ λ ¯ ξ t − ¯ ψ ( − ¯ λ ) t , S t = S e rt + ¯ R (¯ λ, t +¯ ξ t − ¯ ψ (1) t , (24)where ¯ R (¯ λ,
1) = ¯ ψ (1) + ψ ( − ¯ λ ) − ¯ ψ (1 − ¯ λ ). If we then drop the bars, we are led back to aversion of the model already set up, but with σ = 1. Nevertheless, it can be helpful to leavethe parameter σ intact as part of the theory, since there are situations where one would liketo compare versions of the model for different values of the parameter, e.g. for sensitivityanalysis and calculations of the Greeks. On the other hand, there are also situations whereit is desirable to make use of simplifications resulting from setting σ = 1; an example of thiscan be found in the proof of Proposition 1. It should be noted that the value of the assetgiven by (21) does not depend on the drift of the L´evy process, for if we replace ξ t with ξ t + ǫt for some ǫ ∈ R then the L´evy exponent ψ ( α ) gets replaced with ψ ( α ) + ǫα , and thecombination σξ t − ψ ( σ ) t appearing in the formula for the asset price is left unchanged. Thishas the implication that if one attempts to determine the L´evy exponent from the prices ofderivatives, one will be left with an indeterminacy of the form ψ ( α ) → ψ ( α ) + cα for someunknown constant c .It is worth recalling that one of the motivations indicated by Mandelbrot (1963) for theintroduction of L´evy models in finance is the possibility of offering an explanation for theapparent existence of “fat tails” in the distributions of returns. But it seems that what hehad in mind was not the construction of specific dynamical models for price processes, butrather the introduction of infinitely-divisible distributions with infinite moments to modelthe returns on such assets, an assumption that makes the construction of dynamical modelsdifficult. From an empirical point of view, however, the requirement a L´evy process shouldhave “thin tails” is a relatively mild one: a sufficient condition for a L´evy process to admitexponential moments (and hence to have thin tails) is that the jumps should be bounded(Protter 2005, theorem 34), even if the bounds are set at arbitrarily high values. Thus, froma modern point of view the use of L´evy processes in finance stems not so much from a desireto model the distributions of the tails of returns but rather to account for the characteristicsof the jumps that asset prices can undertake. IV. DERIVATIVE PRICING
The price H at time 0 of a European style derivative that delivers a single random payment H T at time T is given by H = E ( π T H T ) . (25)The expectation is, of course, with respect to the real-world probability measure. The pricingkernel takes care of both the discounting and the probability weighting needed to give theanswer. The use of such a formula for derivative pricing is well known, but it may be usefulto recall the argument. In the general theory of asset pricing one fixes, as we have done,a probability space (Ω , F , P ), where P is interpreted as the real-world measure, togetherwith a filtration {F t } , and one assumes the existence of a pricing kernel { π t } t ≥ satisfying π t > t ≥ { S t } t ≥ anda non-decreasing cumulative dividend process { K t } t ≥ , the associated deflated gain process { ¯ S t } t ≥ defined by ¯ S t = π t S t + Z t π s d K s (26)is a martingale. So far, we have considered limited liability assets, for which the prices arenon-negative and the cumulative dividend process is increasing. By a general asset, notnecessarily of limited liability, we mean an asset with the property that its price can beexpressed as the difference between the prices of two limited liability assets, and for whichthe cumulative dividend process can be expressed as the difference between two increasingcumulative dividend processes. It follows then that the deflated gain process of a generalasset is also a martingale.The formula given above allows for the possibility of both continuously paid and discretelypaid dividends. If the dividends are entirely discrete (and paid at possibly random times),then the deflated gain process can be expressed in the form¯ S t = π t S t + X ≤ s ≤ t π s ∆( K s ) . (27)At each time at which a dividend is paid, the cumulative dividend process jumps, and thevalue of the jump is equal to the dividend. In the case of a European style derivative witha single payoff H T made at time T , we think of the payoff as a dividend, and hence thecumulative dividend process is zero up to time T , then jumps to H T at T . The sum in(27) reduces to a single term, given by the jump ∆( K T ) = H T , and we have ¯ S T = π T H T at T . Since the value of the derivative itself drops to zero the instant that the dividend ispaid, it follows by the martingale condition that ¯ S = E ( π T H T ), which gives us (25). In theliterature, by virtue of a conventional abuse of notation, one often writes the price processof the derivative in the form { H s } ≤ s ≤ T , as if somehow the terminal value of the derivative iswhat is paid; in reality, the value of the derivative itself at maturity is 0, whereas the payoff(or dividend) at time T is H T . This is consistent with the notion that the price process ofthe derivative should be right continuous with left limits.We are also in a position to check that if a risky asset pays a proportional dividend atthe constant rate δ then its price in the geometric L´evy model will be given by S t = S e ( r − δ ) t + R ( λ,σ ) t + σξ t − ψ ( σ ) t . (28)The expression is of course intuitively plausible, perhaps even obvious, by analogy with thecorresponding result in the geometric Brownian motion model. Nevertheless, we need tocheck that the process { ¯ S t } defined by (26) is a martingale. A calculation making use ofequations (18), (26) and (28) gives¯ S t = S e − δt +( σ − λ ) ξ t − ψ ( σ − λ ) t + δS Z t e − δu +( σ − λ ) ξ u − ψ ( σ − λ ) u d u. (29)Splitting the integral at time s < t and taking a conditional expectation making use ofFubini’s theorem, we get E s ¯ S t = S e − δt +( σ − λ ) ξ s − ψ ( σ − λ ) s + δS Z s e − δu +( σ − λ ) ξ u − ψ ( σ − λ ) u d u + δS Z ts e − δu +( σ − λ ) ξ s − ψ ( σ − λ ) s d u, (30)from which it follows immediately that E s ¯ S t = ¯ S s . Thus we have shown that the price definedby (28) together with the proportional dividend rate δ is such that the resulting deflatedgain process is a martingale, which demonstrates that (28) is indeed the correct expressionin the geometric L´evy model for the price of a risky asset that pays a proportional dividendat a constant rate. V. DETERMINATION OF THE L´EVY EXPONENT
Turning to the problem of the determination of the L´evy exponent from price data, weproceed to consider a one-parameter family of so-called power-payoff derivatives, for which H T = ( S T ) q , (31)where q ∈ R . Various authors have considered the pricing of similar structures in L´evymodels relating the L´evy exponent to the price (see, e.g., Carr & Lee 2009, Fajardo 2018,and references cited therein). The value H ∈ R + ∪ ∞ of a power-payoff derivative at timezero is given by H = E ( π T S qT ) . (32)Here we allow the possibility that the value of the derivative may not be finite for somevalues of q . In a model driven by Brownian motion, the asset price takes the form S T = S e ( r + λσ ) T + σW T − σ T , (33)and for the pricing kernel we have π T = e − rT − λW T − λ T , (34)where for convenience we set π = 1. It follows that π T S qT = S q e ( q − rT + qλσT +( qσ − λ ) W T − ( qσ + λ ) T . (35)A calculation then allows one to deduce that the value of the power payoff derivative,regarded as a function q , takes the form H ( q ) = S q e ( q − rT + q ( q − σ T (36)in the Brownian case. One notes that the terms involving λ cancel when the expectationis taken, so the value of the derivative only depends on S , r , σ , and q , and that H ( q ) isfinite for all values of q .In the case of a geometric L´evy model, the asset price is given by (21) and the pricingkernel is given by (18). Thus we have π T S qT = S q e ( q − rT +( qσ − λ ) ξ T +( q − ψ ( − λ ) T − qψ ( σ − λ ) T . (37)The value of the power-payoff derivative regarded as a function of q then takes the form H ( q ) = S q e ( q − rT + ψ ( qσ − λ ) T +( q − ψ ( − λ ) T − qψ ( σ − λ ) T . (38)It is perhaps remarkable that an explicit expression is obtained, but this allows one to studyin detail the relation between the type of L´evy model under consideration and the resultingvalues of derivatives. We note that H (0) = e − rT and that H (1) = S , as one would expect,and it should be evident that in general H ( q ) is finite only for a certain range of values ofthe parameter q . In particular, suppose that σ > λ > σ ∈ A and − λ ∈ A . Thenfor A = ( β, γ ) clearly qσ − λ ∈ A if and only if1 σ ( β + λ ) < q < σ ( γ + λ ) . (39)Since β < − λ and γ >
0, these inequalities ensure that the interior of the set of values of q for which H ( q ) < ∞ is an open set B that includes the origin.Now we are in a position to ask to what extent specification of the family of derivativeprices { H ( q ) } q ∈ B allows one to infer the nature of the L´evy process driving the model. Tothis end we note that from observations of H (0) and H (1) one can infer the value of r and S . Thus without loss of generality it suffices to regard the function D ( q ) = 1 T log H ( q ) S q e ( q − rT = ψ ( qσ − λ ) + ( q − ψ ( − λ ) − qψ ( σ − λ ) , (40)which is finite for q ∈ B , as representing the data supplied by the family of derivative prices. Proposition 1
Let the prices of derivatives with payoff payoffs H T = ( S T ) q for q ∈ R begiven for a non-dividend-paying risky asset { S t } t ≥ that is known to be a geometric L´evyprocess, and suppose it is known that the pricing kernel is a geometric L´evy process drivenby the same L´evy process up to a factor of proportionality. Then the L´evy exponent can beinferred up to a transformation ψ ( α ) → ψ ( α + µ ) − ψ ( µ ) + cα , where c and µ are constants.Proof. Without loss of generality one can set σ = 1. Then we have D ( q ) = ψ ( q − λ ) + ( q − ψ ( − λ ) − qψ (1 − λ ) . (41)In the setting of the problem we take D ( q ) to be given for all q ∈ R and finite in someopen set B , and we consider λ to be unknown. The goal is to determine the L´evy exponent.0Writing ˜ ψ ( α ) = ψ ( α − λ ) − ψ ( − λ ), we have D ( q ) = ˜ ψ ( q ) − q ˜ ψ (1). This implies that˜ ψ ( q ) = D ( q ) + qb for some b ∈ R . Now, it is easy to see that ψ ( α ) = ˜ ψ ( α + λ ) − ˜ ψ ( λ ). Weconclude that for some choice of λ and b the L´evy exponent takes the form ψ ( α ) = D ( α + λ ) − D ( λ ) + bα. (42)Substituting (42) into the right-hand side of (41), one can check that the solution is valid.Finally, we note that under a transformation of the form ψ ( α ) → ˆ ψ ( α ), with ˆ ψ ( α ) = ψ ( α + µ ) − ψ ( µ ) + cα , where c, µ ∈ R , we find that ˆ ψ ( α ) = D ( α + λ + µ ) − D ( λ + µ ) + bα .The effect of the transformation is λ → ˆ λ = λ + µ . Since λ is unknown, this shows that theL´evy exponent is determined only up to a transformation of the type indicated. (cid:3) VI. INTERPRETIVE REMARKS
Following on from this result, a few comments may be in order. We recall that a L´evyprocess is completely characterized by the random variable ξ t at a single instant of time t .This reflects the fact that there is a one-to-one correspondence between L´evy processes andinfinitely divisible distributions, and a L´evy process has the property that the distribution ofits value at any particular time is infinitely divisible. Taking t = 1 for convenience, we have ψ ( α ) = log E [e αξ ] in the case of a L´evy process that admits exponential moments, and wenote that the distribution of ξ is determined by the L´evy exponent { ψ ( α ) } α ∈ A . Each suchdistribution belongs in a natural way to a certain one-parameter family of distributions,which we call an Esscher family of distributions. The distribution of ξ is the function F : R → [0 ,
1] defined by F ( x ) = E [ { ξ ≤ x } ]. For the associated L´evy exponent we have ψ ( α ) = log Z + ∞−∞ e αx d F ( x ) . (43)The corresponding Esscher family F δ : R → [0 , δ ∈ A , is defined by the measure change F δ ( x ) = E [e δξ − ψ ( δ ) { ξ ≤ x } ] . (44)We may accordingly ask for the structure of the L´evy exponent associated with F δ . This is ψ δ ( α ) = log Z + ∞−∞ e αx d F δ ( x ) = log Z + ∞−∞ e δx − ψ ( δ ) e αx d F ( x ) = ψ ( α + δ ) − ψ ( δ ) . (45)So we see that by Proposition 1, the specification of the prices of power-payoff derivativesallows one to determine the Esscher family of the L´evy exponent of the underlying L´evyprocess, modulo an irrelevant linear term. L´evy processes that are equivalent in this sensecan be said to belong to the same “noise type” (Brody, Hughston & Yang 2013).On the other hand, if more information is known a priori about the nature of the under-lying asset, then a more precise determination of the L´evy exponent is possible. Considerthe case, for instance, where it is known that the asset on which the power payoff derivativeis based is the natural numeraire. Then σ = λ , and for the asset price we have ζ t = ζ e rt + R ( λ,λ ) t + λξ t − ψ ( λ ) t , (46)where the excess rate of return is given by R ( λ, λ ) = ψ ( λ ) + ψ ( − λ ). In this case we canwithout loss of generality set λ = 1. It follows then from (41) and (42) that D (1) = 0 andhence ψ ( α ) = D ( α + 1) + bα . Thus we have shown the following:1 Proposition 2
Let the prices of derivatives with power payoffs H T = ( ζ T ) q for q ∈ R begiven for a natural numeraire { ζ t } t ≥ that is known to be a geometric L´evy process. Thenthe L´evy exponent can be inferred up to a transformation of the form ψ ( α ) → ψ ( α ) + bα ,where b is a constant. In a geometric L´evy model, the pricing kernel can be written in the form π t = e − rt Λ t where the martingale { Λ t } t ≥ defined byΛ t = e − λξ t − ψ ( − λ ) t (47)determines a change of measure. Thus, for any F t -measurable random variable Z t we have˜ P ( Z t < z ) = ˜ E [ ( Z t < z ) ] = E [ Λ t ( Z t < z ) ] . (48)We refer to ˜ P as the risk-neutral measure and write ˜ E for expectation under ˜ P . The termi-nology “risk-neutral” comes from the fact that ˜ E ( S t ) = S e rt in the geometric L´evy model.Then { ˜ ψ ( a ) } has the interpretation of being the L´evy exponent associated with ξ t under therisk-neutral measure. That is to say,˜ ψ ( a ) = 1 t log ˜ E [ e aξ t ] . (49)The essence of Proposition 1 is that the family of derivative prices can be used to calculate { ˜ ψ ( a ) } , which fixes the exponent { ψ ( a ) } under P , modulo the freedom indicated.Let us write C T ( x ) for the price at time 0 of a call option with maturity T and strike x .Can one use the data { C T ( x ) } x ≥ for fixed T to ascertain the L´evy exponent in a geometricL´evy model? The answer is yes, though the method is less straightforward than the useof power-payoff derivatives, as we shall see. Now, it is known that if the random variable S T corresponding to the terminal value of the asset at time T admits a risk-neutral densityfunction, then we can use the idea of Breeden & Litzenberger (1978) to work out this densityin terms of call option data. In particular, if we write˜ θ ( x ) = dd x ˜ P ( S T ≤ x ) (50)for the density of S T under the risk-neutral measure ˜ P , we have˜ θ ( x ) = e − rT d C T ( x )d x . (51)This follows from the fact that C T ( x ) = e − rT Z ∞ ( y − x ) + ˜ θ ( y ) d y. (52)Then { ˜ θ ( x ) } x ≥ can be used to calculate the values of power-payoff derivatives via therelation ˜ E ( S qT ) = Z ∞ x q ˜ θ ( x ) d x, (53)2and from there we can work out ˜ ψ ( α ), as indicated in the previous section. The difficultywith this approach is that in a geometric L´evy model the distribution of S T does not ingeneral admit a density function, and the system of call option prices { C T ( x ) } x ≥ is notdifferentiable for all x ∈ R + . The situation can be remedied to some extent if instead wemake use of the risk-neutral distribution function { ˜ F ( x ) } x ≥ and express the option price inthe form of a Lebesgue-Stieltjes integral, writing C T ( x ) = e − rT Z ∞ ( y − x ) + d ˜ F ( y ) , (54)with the understanding that the distribution function is right-continuous. Then the deriva-tive of the option price with respect to the strike is defined for all x ∈ R + apart from pointsof discontinuity of the distribution function, and this is sufficient to enable us to recoverthe distribution function in its entirety. Once the distribution function is known, one candetermine the L´evy exponent by calculating the system of power-payoff prices, given for q ∈ R by H ( q ) = e − rT Z ∞ x q d ˜ F ( x ) . (55) VII. IMAGINARY POWER PAYOFFS
As another example of a one-parameter family of derivatives from which information can beextracted concerning the L´evy exponent when the underlying is a geometric L´evy asset weconsider a family of imaginary power payoffs, for which the terminal cash flow is given by F T ( q ) = ( S T ) iq , (56)where q ∈ R . The value of such a contract at time zero takes the form F ( q ) = E ( π T S iqT ) = E ( π T e iq log S T ) . (57)Since the payoff is a complex function, we are in effect valuing two different derivatives, eachwith its own payoff. That is to say, F ( q ) = E [ π T cos( q log S T )] + i E [ π T sin( q log S T )] . (58)Thus { F ( q ) } , q ∈ R , can be thought of as a pair of price families { F c ( q ) } and { F s ( q ) } ,for which the corresponding payoff functions are given respectively by F cT ( q ) = cos( q log S T )and F sT ( q ) = sin( q log S T ). Note that the payoffs, and hence the prices, are bounded for allvalues of q . A calculation then shows that F ( q ) = S iq e r ( iq − T e ( − iq ψ ( σ − λ )+( iq − ψ ( − λ )+ ψ ( iqσ − λ ) ) T . (59)With these ideas at hand we can use the methods of Fourier analysis to investigate moregeneral payoffs. We begin by recalling briefly a few well known facts. Let the map f : R → R be such that f ∈ L . The Fourier transform of f is the function g : R → C defined by g ( q ) = 1 √ π Z ∞−∞ e − iqx f ( x ) d x. (60)3Under various further conditions the relation between f and g can then be inverted. Forexample, if f ∈ L and is continuous, and if g ∈ L , then f ( x ) = 1 √ π Z ∞−∞ e iqx g ( q ) d q. (61)A sufficient condition for these requirements to be satisfied is that f should be a “good”function in the sense of Lighthill (1958), that is to say, that it should be everywhere differen-tiable any number of times and such that it and all its derivatives are O (cid:0) | x | − n (cid:1) as | x | → ∞ for all n ∈ N . We recall that f ( x ) is said to be O ( h ( x )) as x → ∞ iflim sup x →∞ (cid:12)(cid:12)(cid:12)(cid:12) f ( x ) h ( x ) (cid:12)(cid:12)(cid:12)(cid:12) < ∞ . (62)If f is a good function then its Fourier transform g is also good. Now consider the situationwhere the payoff of a European-style derivative with value H at time zero takes the form H T = f (log S T ) for some f ∈ L . If f is continuous and g ∈ L , then by use of (61) we canwrite the payoff in the form H T = 1 √ π Z ∞−∞ e iq log S T g ( q ) d q. (63)This expresses H T as a portfolio of imaginary power payoffs parameterized by q , where g ( q )determines the relative portfolio weighting for the given q . Multiplying each side of equation(63) with the pricing kernel π T and taking the expectation we obtain E ( π T H T ) = 1 √ π E (cid:18)Z ∞−∞ π T e iq log S T g ( q ) d q (cid:19) , (64)from which it follows that E ( π T H T ) = 1 √ π (cid:18)Z ∞−∞ E (cid:2) π T e iq log S T (cid:3) g ( q ) d q (cid:19) . (65)Inserting (57) into (65), we then have H = 1 √ π (cid:18)Z ∞−∞ F ( q ) g ( q ) d q (cid:19) . (66)Thus the price of the derivative can be expressed as the value of a portfolio of imaginarypower payoff derivatives. To check that the interchange of the expectation and the integra-tion in equation (64) is valid (Kingman & Taylor 1966, Theorem 6.5), we note that E (cid:18)Z ∞−∞ π T e iq log S T g ( q ) d q (cid:19) = Z ω ∈ Ω Z ∞−∞ π T e iq log S T g ( q ) d q P (d ω ) , (67)and that Z ω ∈ Ω Z ∞−∞ (cid:12)(cid:12) π T e iq log S T g ( q ) (cid:12)(cid:12) d q P (d ω ) = E [ π T ] Z ∞−∞ | g ( q ) | d q < ∞ . (68)4As an example we consider a European-style derivative payoff H T = f (log S T ) at time T for which f takes the form of a normal density function f ( x ) = 1 √ πu e −
12 ( x − a ) u , (69)with mean a and variance u . In the case of a geometric Brownian motion model, the randomvariable corresponding to the terminal value of the underlying asset is normally distributedwith a risk-neutral density of the form˜ θ ( x ) = 1 √ πv e −
12 ( x − b ) v , (70)where b = ( r − σ ) T and v = σ T . The price of the derivative at time zero is given by H = e − rT ˜ E [ H T ] = e − rT Z ∞−∞ ˜ θ ( x ) f ( x ) d x. (71)With some calculation, we find that H = e − rT p π ( u + v ) e −
12 ( a − b ) u + v . (72)For instance, if we set a = 0 , u = 1 and insert the aforementioned values of b and v , weobtain H = e − rT p π (1 + σ T ) e −
12 ( r − σ ) T σ T . (73)Alternatively, we can replicate the payoff of the derivative as a portfolio of imaginary powerpayoffs using the Fourier technique. Since f ∈ L , we can set g ( q ) = 12 π √ u Z ∞−∞ e − iqx e −
12 ( x − a ) u d x, (74)and a calculation gives g ( q ) = 1 √ π e − q u e − iqa . (75)By (63) and (75), and using the fact that f is a good function, one sees that the payoff ofthe derivative can be expressed in the form H T = 12 π Z ∞−∞ S iqT e − q u e − iqa d q. (76)Then by (66) we obtain the derivative price as a portfolio of imaginary power-payoff prices: H = 12 π (cid:18)Z ∞−∞ F ( q ) e − q u e − iqa d q (cid:19) . (77)5We observe, finally, that if the prices of imaginary power payoff derivatives deliveringthe cash flows defined by (56) are given for all q ∈ R , then by adapting the framework ofProposition 1 we can work out the implied L´evy exponent modulo some specified freedom.We recall that the price of a power-payoff derivative at time zero is given for power q byequation (59). Therefore if we consider the function { D ( q ) } q ∈ R defined by D ( q ) = 1 T log F ( q ) S iq e r ( iq − T , (78)we find that D ( q ) = − iq ( ψ ( σ − λ ) − ψ ( − λ )) + ψ ( iqσ − λ ) − ψ ( − λ ) = ˜ ψ ( iqσ ) − iq ˜ ψ ( σ ) , (79)where ˜ ψ ( α ) = ψ ( α − λ ) − ψ ( − λ ). Without of loss of generality we can then set σ = 1 to get D ( q ) = ˜ ψ ( iq ) − iq ˜ ψ (1) . (80)This implies that ˜ ψ ( iq ) = D ( q ) + iqb for some b ∈ R . Now, ψ ( α ) = ˜ ψ ( α + λ ) − ˜ ψ ( λ ). Itfollows that for some λ and b the L´evy exponent takes the form ψ ( iq ) = D ( iq + λ ) − D ( λ ) + ibq. (81)Thus, we see that once we have been given a range of price data for imaginary power-payoffoptions, we can work out the L´evy exponent modulo a transformation of the form ψ ( iq ) → ψ ( iq + µ ) − ψ ( µ ) + icq, (82)where c and µ are constants. Acknowledgments
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