Selfsimilar additive processes and financial modelling

Selfsimilar Additive Processes and Financial ModelingCraig A.NolderDepartment of MathematicsFlorida State UniversityTallahassee,FL32306-4510,USAnolder@ABSTRACT.We present here theoretical reasons to use selfsimilar additive processes to model asset prices and a program for calibrations and implementations.L´e vy processes are stationary additive processes and are selfsimilar only in the stable case.In contrast any selfdecomposable distribution will generate selfsimilar additive processes with any positive exponent of selfsimilarity.Selfsimilar additive processes due to nonstationarity need not adhere to a Central Limit Theorem.It is hoped that because of this these models will exhibit implied volitility smirks at high maturities.KEY WORDS:selfsimilar additive processes,option pricing1IntroductionFinancial time series often exhibit selfsimilar scaling properties[11],[2],[5].As such this is a desirable feature of a mathematical model of these phenomena.Indeed the Brownian motion process in the Black-Scholes model is2-stable and hence12-selfsimilar.Mandelbrot proposed the use of more general stableprocesses infinance[11].Except in the Gaussian case however,these processes have infinite variance. As a result a stock price process modeled as an exponential stable process may not possess an equivalent martingale measure and option prices can be infinite.However P.Carr and L.Wu[6]have shown that in the extreme case of maximum negative skewness these stable exponential L´e vy processes result infinite valued price processes.At the same time the stock process retains infinite variance.As such even though the increments are independent and stationary,the conditions for a central limit are violated.Moreover the scaled distributions are similar over all time horizons due to the selfsimilarity of the processes.The combination of these properties allow a calibratedfit to a volatility smirk of S and P index data in[6]over a two year maturity span.Recently other distributions have been proposed as the basis of exponential L´e vy stock price models.Notably are the normal inverse Gaussian[3],truncated stable[5]and the generalized hyperbolic[7].Many authors have proposed the use of the L´e vy processes that these distributions generate. However,except in the stable case,these processes do not have selfsimilar scaling properties.Moreover these processes havefinite moments and,as a result of the central limit theorem,will not produce the volatility smirk which will insteadflatten with increasing maturity.On the other hand all of the above distributions are in fact selfdecomposable.Because of this one can construct selfsimilar additive processes with any exponent H>0.As such the distributions of these additive processes agree with those of the corresponding L´e vy processes at time1,but have different time evolutions.We investigate here these additive processes as models for asset pricing.Although the increments are no longer stationary,we still have spatially homogeneous Markov processes.Because ofthis the option pricing formulas are similar to the L´e vy case.As such selfsimilar additive processes offer great promise as versatile modeling tools infinance:1.Any selfdecomposable distribution defines selfsimilar processes.2.There is an entire family of these processes for each initial distribution,one for each selfsimilarexponent H>0.3.Except in the stable(when the variance is infinite)the increments are nonstationary and a normalcentral limit theorem need not apply.4.The selfsimilar property will allow fat negative tails to persist over all time horizons.A calibratedmodel shouldfit volatility smirks of implied volatilities over a wide range of maturities.We remark that many authors justify stationarity by claiming it in fact defines the temporal index. Moreover time deformations of market activity are required to attain stationarity(e.g.,[4]).Selfsimilar processes may be another way to accommodate nonstationarity.Although the distributions of time incre-ments are not the same they are related through(1)and(5).Moreover additive processes of different types can be pasted together using(3).2Additive ProcessesDefinition2.1.A real valued stochastic process{X t|t≥0}on a probability space(Ω,F,P)is an additive process if(i)for allfinite partitions0≤t0<t1...<t k,the increments X tk −X tk−1,...,X tare independent.(ii)X0=0a.s.(iii){X t}is stochastically continuous(iv)the sample paths have c`a dl`a g modifications a.s.As such an additive process with stationary increments is a L´e vy process.Given an additive process{X t}we writeµs,t for the distribution of X t−X s,0≤s≤t<∞.It follows thatµs,t is infinitely divisible andµs,t⋆µt,u=µs,u(1) for0≤s≤t≤u<∞,µs,s=δ0,0≤δ<∞,µs,t→δ0as s↑t,µs,t→δ0as t↓s.Moreover,conversely,given a system of probability measures satisfying(1)it follows from Kolmogorov’s Existence Theorem that an additive process exists with these distributions.Since the distributionµ0,t=µt=P X t is infinitely divisible for each t≥0,its characteristic exponent has characteristics(a t,νt,γt),log E(e izX t)=−a t2z2+izγt(2)+ R e izx−1−izxI[−1,1] νt(dx).The system of triplets satisfies the conditions(i)a0=0,ν0=0,γ0=0,(ii)if0≤s≤t<∞,then a s≤a tandνs(B)→νt(B)for Borel sets B,(iii)as s→t in[0,∞),a s→a t,νs(B)→νt(B)for B⊂ x |x|>ε ,ε>0andγs→γt.See[9].Conversely,given a system of infinitely divisible distributions that satisfy the conditions(3),thenthere exists an additive process such that P Xt=µt,t≥0.In this case the infinitely divisible distributionµs,t has generating triple(a t−a s,νt−νs,γt−γs).Also if{X t}is an additive process,thenP s.t(x,B)=P(X t−X s∈B−x),0≤s≤t,is a spatially homogeneous transition function and{X t}is a Markov process with this transition function and starting at zero.3Asset Process and Option PricingWe write{S t}for an asset price process that we model using an additive process{X t}:dS t=S t(r dt+σdX t).(3) here r is a riskless interest rate.The volitility parameterσcan be used both to calculate implied volitilities from market data and as a second source of randomness in(3).Under usual assumptions of an absence of arbitrage there exists an equivalent martingale measure under which the discounted stock process is a (sigma)-martingale.Let P be the market measure and Q the equivalent martingale measure.Given a European option with payoff g(S T)at expiry T,we write the fair option price at time t, 0≤t≤T,V t=e−r(T−t)E Q(g(S T)|F t).(4) 4Selfsimilar Processes and Selfdecomposable DistributionsSeveral important distributions that have recently been used infinancial modelling are selfdecompos-able.Among these are stable,truncated stable,normal inverse Gaussian and generalized hyperbolic dis-tributions.The L´e vy processes that these distributions determine appear as asset price processes in theliterature.We examine here an infinite class of additive processes canonically determined by a selfdecom-posable distribution,one for each exponent H>0.Only in the stable case with H≥12do these processesagree in law with the canonical L´e vy processes.Definition4.1.A probability measureµon R is selfdecomposable if,for any b>1,there is a probability measureρb on R such thatµ(z)= µ z b ρb(z).(5) Definition4.2.A stochastic process{X t|t≥0}on R is H-sellfsimilar,H>0,if for any a>0{X at|t≥0}d={a H X t|t≥0}.A(non-trivial)selfdecomposable distribution determines an H-selfsimilar additive process.In factlet H>0,0<s<t,and b=(ts )H in(5).Then defineµt andµs,t by µt(z)= µ(t H z)and µs,t(z)=ρ(t s)H(t H z)so that µt(z)= µs(z) µs,t(z).Withµ0=δ0,µ0,t=µt andµt,t=δ0,the corresponding measures determined above satisfy(1)and so determine an additive process{X t}.Notice it follows that for any a>0,X at d=a H X t and so{X t}is an H-selfsimilar process.For details see p.100[9].5Work in ProgressThe following programs are underway.1.Analyze market data for evidence of selfsimilarities.Obtain estimates for the exponent(s)of self-similarity.Some techiques are in[8].2.Calculate densities for selfsimilar processes whose distributions at time1are models of observeddistributions of market data.These include the normal inverse Gaussian,Generalized hyperbolic, Truncated stable and stable distributions.We remark that“stable”processes are L´e vy processes only when the exponent of selfsimilarity is larger than or equal to1.23.Calibrate the parameters in the model(volitility,selfsimilar exponent and the parameters of theinitial distributions)with market data.Calculate theoretical market prices using the model.4.Recalculate implied volitilities using market data and calibrated models.5.Calibrate a stochastic volitility model to add volitility clustering.Thefigures1–4compare a Gaussian process with a Levy process and three additive selfsimilar processes at four different times.The designations of these processes appear in the keys in this order. The graphs are at times1,1.5,3and6.The variance of the Gaussian at time1matches that of the other processes whose distributions are indentical at time1and all means are zero.The time is thefirst parameter in the key.The initial distributions of the nonGaussian processes are a truncated stable distribution with characteristic function proportional to50ln25−(25−ix)ln(25−ix)−(25+ix)ln(25+ix)as a function of x.The second parameter in the key is the exponent H in the selfsimilar processes and the third is a label: 0Gaussian,1Levy,2additive selfsimilar.We acknowledge Mack Galloway for his help in developing the software.Results of the above investigations will be published elsewhere.References[1]Boyarchenko,S.I.and Levendorskii,S.Z.,Non-Gaussian Merton-Black-Scholes Theory,World Sci-entific,2002.[2]Barndorff-Nielsen,O.E.and Prause,K.,Apparent scaling,Finance and Stochastics,5,2001,103–113.[3]Barndorff-Nielsen,O.E.and Shephard,N.,Modelling by L´e vy processes forfinancial econometrics,L´e vy Processes Theory and Applications,Birkhauser Boston,2001.[4]Cont,R.,Empirical properties of asset returns:stylized facts and statistical issues,Quantitative Fi-nance,1,2001,223–236.[5]Carr,Peter,P.,Geman,H.,Madan,D.B.and Yor,M.,Thefine structure of asset returns:An empiricalinvestigation,Journal of Business,75,2002,305–332.[6]Carr,P.and Wu,L.,Thefinite moment log stable process and option pricing,The Journal of Finance,28(2),2003,753–777.[7]Eberlein,E.,Application of generalized hyperbolic L´e vy motions tofinance,L´e vy Processes Theoryand Applications,Birkhauser Boston,2001.[8]Embrechts,P.and Maejima,M.Selfsimilar Processes,Princeton University Press,Princeton,N.J.,2992.[9]Sato,K.,L´e vy Processes and Infinitely Divisible Distributions,Cambridge University Press,1999.[10]Jacod,J.and Shiryaev,A.N.,Limit Theorems for Stochastic Processes,Springer-Verlag,1987.[11]Mandelbrot,Benoit B.,The variation of certain speculative prices,Journal of Business,36,1963,394–419.[12]Maejima,M.and Naito,Y.,Semi-selfdecomposable distributions and a new class of limit theorems,Prob.Theory and Rel.Fields,112(1),1998,13–21.[13]Maejima,M.,Sato,K.and Watanabe,T.,Remarks on semi-selfsimilar processes,preprint.。