Quantitative Finance ISSN: 1469-7688 (Print) 1469-7696 (Online) Journal homepage: https://www.tandfonline.com/loi/rquf20 From local volatility to local Lévy models Peter Carr , Hélyette Geman , Dilip B Madan & Marc Yor To cite this article: Peter Carr , Hélyette Geman , Dilip B Madan & Marc Yor (2004) From local volatility to local Lévy models, Quantitative Finance, 4:5, 581-588 To link to this article: https://doi.org/10.1080/14697680400024921 Published online: 19 Jun 2011. Submit your article to this journal Article views: 178 Citing articles: 8 View citing articles Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=rquf20 QUANTITATIVE FINANCE VOLUME 4(OCTOBER 2004) 581–588 R ESEARCH P APER TAYLOR &FRANCIS LTD tandf.co.uk From local volatility to local Le´vy models Peter Carr1,He´lyette Geman2, Dilip B Madan3,5 and Marc Yor4 1 Courant Institute, New York University, USA 2 Universite´Paris-Dauphine and ESSEC, France 3 Robert H Smith School of Business, University of Maryland, USA 4 Laboratoire de Probabilite´s et Mode` les ale´atoires Universite´Paris VI and Universite´Paris VII, France E-mail: [email protected] Received 15 March 2004, in final form 19 November 2004 Published 24 December 2004 Online at www.tandf.co.uk/journals/titles/14697688.asp DOI: 10.1080/14697680400024921 Abstract We define the class of local Le´vy processes. These are Le´vy processes time changed by an inhomogeneous local speed function. The local speed function is a deterministic function of time and the level of the process itself. We show how to reverse engineer the local speed function from traded option prices of all strikes and maturities. The local Le´vy processes generalize the class of local volatility models. Closed forms for local speed functions for a variety of cases are also presented. Numerical methods for recovery are also described. 1. Introduction The local uncertainty of a local volatility model is Gaussian with zero skewness and kurtosis equal to 3. Local volatility models (Derman and Kani 1994, Dupire It seems desirable in this context to accommodate a local 1994), were developed as a class of one-dimensional uncertainty that allows for both skewness and excess Markov models with continuous sample paths that levels of kurtosis. Many researchers have already noted reprice all the traded European options. These models for a variety of purposes, that one should introduce generalize the Black and Scholes (1973) and Merton the possibility of jumps (Bates 1996, Bakshi et al 1997). (1973) models by making the instantaneous volatility of We have argued in prior research that the use of a jump the stock returns a deterministic function of time and the process with infinite activity, i:e: one allowing infinitely stock price. Such a function is called the local volatility many jumps in any time interval, effectively subsumes function. The risk neutral dynamics is fully specified on the need for an additional diffusion component (Carr setting the growth rate of the stock at the instantaneous et al. 2002). We therefore replace the local diffusive risk interest rate less the dividend yield. The resulting model neutral dynamics by a local exposure to a Le´vy process. is widely used for pricing contingent claims written on This class of processes is increasingly being used in the the stock price, including a variety of path dependent study of financial market prices (Eberlein et al 1998, options. For extensions of this approach to a jump Barndorff-Nielsen and Shephard 2001, Geman et al 2001, diffusion context we cite Andersen (2000) and Andersen Eberlein et al 2003). and Andreasen (1999). Le´vy processes offer a wide class of candidates for an alternative representation. We wish to formulate in this 5 Author to whom correspondence should be addressed. paper a class of local Le´vy models that also reprice all the 1469-7688 Print/1469-7696 Online/04/050581–8 ß 2004 Taylor & Francis Ltd 581 P Carr et al QUANTITATIVE FINANCE traded European options and provide a richer risk neutral where W ¼ðWðtÞ,0 t HÞ is a standard Brownian dynamics. motion and ðS, tÞ is the local volatility function. We view the local volatility model in its equivalent The relevance of the formulation (1) is quite extensive formulation of modelling log prices as a Brownian motion from the perspective of constructing Markov processes running at the speed of the square of the local volatility that match the marginals of general stochastic processes. function. Our essential idea is to replace Brownian motion Gyo¨ngy (1986) showed that one could associate with a with a Le´vy process running at what we call the local speed general Ito process a one dimensional Markov process function. Our local speed function is still a deterministic of the type (1) with a view to matching marginals. This function of the level of the stock price and time. The Le´vy question has also been studied from other perspectives process involved in this procedure is fixed through time in Madan and Yor (2002). and it is only its speed that is space time dependent. This Let C(K, T ) denote the price at time zero, of a generalizes the role of Brownian motion, a particular Le´ European call option of maturity T and strike K. Dupire vy process, in the local volatility model. (1994) and Derman and Kani (1994) showed that one In a direct analogy with the contribution of the local may recover the local speed function from the prices of volatility model, we show how to recover the local speed traded options using the formula function from quoted option prices. Our final results C þ C þðr À ÞKC are comparable to local volatility models, except that we 2ð Þ¼ T K ð Þ K, T 2 2 : 2 employ a transform of the calendar spread in place of the K CKK calendar spread to infer the speed function. We generalize (1) by allowing for jumps in the stock price. We provide some explicit examples associated with We denote the size of the jump in the log price at any particular local Le´vy models permitting closed form time by x. The Le´vy measure kðxÞ dx specifies the arrival recovery of local speed functions from option prices. rate of jumps of size x per unit time. In analogy with the The recovery function can in these cases be seen as a local volatility function, we introduce a local speed direct generalization of the comparable result for local function a(S, t) that measures the speed at which the volatility models. For other Le´vy processes we describe Le´vy process is running at time t when the stock price is at procedures for numerical solutions, that still permit an the level S. efficient recovery of the local speed function. We also In addition to the exposure to the Brownian motion, consider the ‘arithmetic’ (Bachelier) case where options our stock price process is also exposed to the compen- are written directly on a martingale, as opposed to a sated jump martingale with compensator positive, exponential martingale. These results could be of financial interest in markets for options written on the ðdx,duÞ¼aðSðuÞ, uÞkðxÞ dx du: ð3Þ profit and loss distribution of a portfolio of hedge funds The risk neutral dynamics for the stock price are now directly. given by The outline of the paper is as follows. Section 2 presents the details of the one dimensional Markov model dS ¼ðr À ÞSðt Þ dt þ Sðt ÞðSðt Þ, tÞ dWðtÞ Z describing the risk neutral dynamics for the discounted 1 x asset price and presents the general integral equation to þ Sðt ÞðÞe À 1 ðÞðmðdx,duÞðdx,duÞ 4Þ be solved for recovery of the local speed function. The 1 derivation for the local speed recovery procedure is given where mðdx,duÞ is the counting measure associated with in section 3. In section 4 we consider a specific local Le´vy the jumps in the logarithm of the stock price. process permitting closed form recovery. The arithmetic The formulation of the compensator in (3) alters local case is developed in section 5. Numerical procedures are volatility by running the Le´vy process at a speed that presented in section 6. Section 7 concludes. is a deterministic function of the stock price and time. Alternatively, one could scale the jump sizes instead. In the case of Brownian motion, scaling and time changing 2. Local Le´vy Models are equivalent operations by the scaling property of We begin by recalling briefly the local volatility model Brownian motion, but for general Le´vy processes these and the associated procedure for recovering the local are different operations. Time changing leads to tractable volatility function from traded option prices. Let S(t) results while scaling is much more complicated. denote the price of the stock at time t,0 t H. The objective of this paper is to show how one may Suppose the continuously compounded interest rate recover the local speed function a(S, t) from traded option is r and the dividend yield is , also continuously prices in the context of a known local volatility function compounded. The risk neutral dynamics for the stock ðS, tÞ: Of particular interest is the case of pure jump price in the local volatility model is given by the following processes, i:e:¼ 0: In this case the stock has no stochastic differential equation diffusion exposure. The solution for the local speed function employs in a critical way a convolution trans- dS ¼ðr À ÞSdt þ ðS, tÞSdWðtÞð1Þ form with the ‘double exponential tail’ of the Le´vy 582 QUANTITATIVE FINANCE From local volatility to local Le´vy models measure.