Tanaka formula and Levy process

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1 Tanaka formula and Levy process Simply speaking he Tanaka formula is an exension of he Iô formula while Lévy process is an exension of Brownian moion. Because he Tanaka formula and Lévy process are wo differen conceps we will explain hem separaely. Since he goal of his aricle is o inroduce he conceps we will presen hem in a very accessible way referring o reference books like Karazas and Shreve (998) or Revuz and Yor (994) for Tanaka formula and Sao (998) for Lévy processes. ) Tanaka formula The sandard heory of sochasic calculus is mainly based on he Iô lemma. This lemma saes ha he differenial funcion of a wice differeniable funcion of a sochasic process described by a diffusion equaion is given by he normal erms in he deerminisic case plus an exra erms equal o he second derivaives funcion imes he quadraic variaion of he process. Le ( ) R W ε be a sandard Brownian moion (or Wiener process see Wiener ) R processes) and ( X ε a diffusion process whose evoluion is described by a diffusion equaion whose coefficiens α( X ) β ( ) boh and X : X ( X ) d β( X s ) dw dx are Lipschiz funcion of = α + (.).

2 The derivaive funcion of a funcion f ( ) X C wice differeniable wih second order derivaives funcion coninuous is given by: df x x ( X ) f ( X ) d+ f ( X ) dx + f ( X ) β ( X )d = (.) However Iô formula does no apply when he funcion ( x) f is no C. And mos of he opion payoffs are no C. Even simple opions payoffs like he ones of calls or pu do no fulfil he condiion of he Iô lemma. However one can show ha he Iô formula can be exended o funcion ha are only convex. The formula firs proved by Hiroshi Tanaka and compleed by Waanabe uses he concep of local ime and disribuion heory o do he generalisaion. The second order erm of he Iô formula is replaced by he formal second order derivaives (in he sense of disribuion) imes he occupaion densiy ofen called in probabiliy heory he local ime. If ( x) a convex funcion is differenial funcion is given by: f is df + = x x ( X ) f ( X ) d+ f ( X ) dx + f ( x) L ( x) dx d (.3) x where f ( ) X is he lef limi of he firs order derivaives of f wih respec o x he occupaion ime in x : x L is he local ime of he process X in x defined as he limi of L x = lim P s 0 ε 0 ε ( X x ε for s ) (.4)

3 Firs i is worh noing ha as for he Iô formula his can be exended easily o jump diffusion: if X is a jump diffusion driven by a Poisson process N wih a jump size of J (see Poisson process and jump diffusion) ( X ) d + β( Xs ) dw JdN dx = + α (.5) The addiional erms comes from he jump par leading o: df ( X ) = f ( X ) d+ f ( X ) dx + f ( x) L ( ) + x ( f ( X + ) ( ) ) J f X + x J dn x dx d (.6) The concep of local ime has a very ineresing financial inerpreaion: i is he ime value of an opion. Indeed in he case of a diffusion of he ype Dupire applying he Tanaka formula o he payoff funcion of a call of mauriy T and srike K and aking he risk neural expecaion leads o: C + ( T K) = B( 0 T)( S0 K) + B( 0 T) L ( K) T (.7) which saes ha a call (similarly a pu) opion is equal o is inrinsic value plus he discouned local ime in he srike of he process. The discouned local ime is herefore equal o he ime value of he call (similarly pu) opion. The Tanaka formula is useful o prove he Dupire model. I has also been successfully used o show how o find a saic hedge for barrier opions when assuming a local volailiy model (see Carr or Andersen e al.) how o compue passpor opion and relae i o lookback opions (Henderson and Hobson).

4 ) Lévy process Roughly speaking Lévy processes are an exension of he Brownian moion. Lévy processes are general sochasic processes wih saionary independen incremens. Examples are Brownian moion Poisson processes compounded Poisson process sable processes (such as Cauchy processes) and subordinaors (such as Gamma-processes). They form a basic class in sochasic analysis. Processes ha are derived from a Lévy process encompass all he diffusion process derived from a Brownian moion. Lévy processes can be characerised by heir Lévy-Khinchine represenaion of infiniely divisible disribuions. The Lévy-Khinchine formula says ha a Lévy can always be represened as he convoluion of a random Gaussian variable (possibly wih a raher warped covariance marix) wih a compound Poisson random variable (possible wih infinie inensiy measure). In shor if X is a Lévy process is characerisic funcion can be wrien in he form of a Brownian par and a Poisson erm: ( ) Π( dx) iλx [ exp( iλx )] = exp iλµ λ σ e + iλx { x < } E R (.) where Π is a posiive measure on R so ha Min( + x ) Π( dx) < + o as he Lévy measure. R referred Academic research have recenly focused on Lévy processes (as opposed o Brownian moion). Indeed Lévy processes are more general processes han sandard Brownian moion and already encompasses los of well know processes (like he Variance gamma process of Madan and Senea (990)

5 and many of he simple jump diffusion models (like he Meron model). Lévy processes are a way o incorporae he non log-normaliy of opion underlying (referred o as he volailiy smile as he funcion of he Black Scholes implied volailiy shows some smiley figures) wihin he random process iself. This sands in conras o sochasic and deerminisic volailiy models (See Heson (993) Dupire (994) Derman (994) ec). The pricing of opions when assuming Levy processes can be done eiher by solving he parial inegro differenial equaion (Eberlein (995) (very similarly o he solving of a jump diffusion models) or by using Lévy exponen and he Laplace ransform of he opion price. (Benhamou (00)). Models driven by Lévy processes have been derived boh for equiy derivaives bu also for ineres raes derivaives generalising he Heah Jarrow Moron condiion o his wider class of sochasic processes. Ineresingly a process driven by a Lévy process can be shown o be a paricular represenaion of any (non-singular) jump diffusion process. However he use of he propery of Lévy processes makes boh heoreical and numerical solving easier. Las bu no leas one can derive closed forms for barrier opions for process wrien as he exponen of a Lévy process

6 Eric Benhamou Swaps Sraegy London FICC Goldman Sachs Inernaional Enry caegory: mahemaical models Scope: opions Relaed aricles: Brownian moion; Io's lemma Jump diffusion. The views and opinions expressed herein are he ones of he auhor s and do no necessarily reflec hose of Goldman Sachs

7 References Andersen L. J. Andreasen and D. Eliezer (00) Saic Replicaion of Barrier Opions: Some General Resuls Journal of Compuaional Finance. Barndorff-Nielsen O.E.(998). Processes of normal inverse Gaussian ype. Finance and Sochasics Benhamou E. (000) Opion Pricing wih Lévy processes London School of Economics Working paper Carr P (998) Saic Hedging of Exoic Opions Journal of Finance. Eberlein E. and U. Keller (995). Hyperbolic disribuions in Finance Bernoulli HendersonV. D. Hobson (000) Local ime coupling and he passpor opion Finance and Sochasics 4()69-80 Karazas I. and Shreve S.E (998) Brownian Moion and Sochasic Calculus Springer Verlag Second Ediion Madan D. B. and E. Senea (990). The variance gamma (V.G.) model for share marke reurns. Journal of Business Revuz D. and Yor M. (994) Coninuous Maringales and Brownian Moion Springer Verlag. Sao K. (999) Lévy Processes and Infiniely Divisible Disribuions Cambridge Universiy Press.

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 Introduction to Mathematical Finance Lesson 5 26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random