Exact simulation of stochastic volatility and other affine jump diffusion processes. Bendheim center for finance, princeton university, princeton. I would like to price asian and digital options under mertons jump diffusion model. The recent financial crisis has revealed major shortcomings in the existing approaches for modeling credit derivatives. A jump diffusion model for option pricing with three properties. Drift estimation for jump diffusions consider an sde with driver l t. Of particular interest in this work are jumpdiffusion processes with statedependent jump rates.
This dissertation studies two financial engineering and econometrics problems relating to two facets of the 20072008 financial crisis. Finding the maximum likelihood estimator for such processes is a. Due to the page limit, i will also skip various applications of the jump diffusion models. Weierstrass institute for applied analysis and stochastics maximum likelihood estimation for jump diffusions hilmar mai mohrenstrasse 39 10117 berlin germany tel. Estimating jump diffusion structural credit risk models. Finding the maximum likelihood estimator for such processes is a tedious task due to the multimodality of the likelihood function. As amplification, we consider a stochastic volatility model which we compare with them, including their advantages and limitations.
Bandiy graduate school of business the university of chicago and thong h. As a result, with n assets, one must solve numerically an ndimensional nonlinear equation. Predominantly vacancy in nature difficult for atoms to fit into interstitial sites because of size. Jumpdiffusion processes and available estimation methods 2. Numerical solution of jumpdiffusion libor market models. Kou 2002 and kou and wang 2004 derived option pricing results for jumpdiffusion with logdoubleexponentially distributed jump amplitudes. Jumpdiffusion processes and affine term structure models. Contingent claims, high peak, heavy tails, interest rate models, rational expectation, overreaction and underreaction jel classification. This type of model appears in an equally eclectic variety of applications 3, 17, 53. Financialsupportbynsfgrantsdms1208985anddms1512900 is gratefully acknowledged.
Bates finance department the wharton school university of pennsylvania philadelphia, pa 191046367 october, 1988 abstract this paper derives the appropriate characterization of asset market equilibrium when asset prices follow jumpdiffusion processes, and develops the general methodology for. Glasserman and kou show how marked point process intensities can be chosen to produce closedform expressions for caplets or swaptions, but for pricing general pathdependent. The hestonjumpdiffusion hjd model combines the advantages of stochastic volatility and jumpdiffusion models. Pricing options under jumpdiffusion processes david s. Jumpdiffusion models have been introduced by robert c. First passage times of a jump diffusion process advances. However, the proposed estimation is not limited to these two cases. The model shows that jump risk and endogenous default can have signi. Simulationbased estimation is a very flexible and powerful technique. In the first part, we compare the performance of various. This dissertation studies various issues related to the modeling of credit derivatives.
The jump diffusion model can separate the impact of informed trading from unanticipated public announcements. Pricing asian options under a hyperexponential jump diffusion model. The higher the rate, the more likely occurrence of a jump. Jump diffusion libor market models 3 this paper addresses the numerical solution, through discretization and simulation, of the market models with jumps developed in 8. Thus governs the probability to jump up a down jump having probabilit y 1. This paper argues that the reducedform jump diffusion model may not be appropriate for credit risk modeling. Kou department of industrial engineering and operations research, columbia university email. By generating a set of option prices assuming a jump diffusion with known parameters, we investigate two crucial challenges intrinsic to this type of model. In order to model such stock jumps we require certain properties. The proofs are deferred to the appendices or the ecompanion. Each author name for a columbia business school faculty member is linked to a faculty research page, which lists additional publications by that faculty member.
In this paper, an alternative stochasticvolatility jumpdiffusion model is proposed, which has squareroot and meanreverting stochasticvolatility process and loguniformly distributed jump amplitudes in section ii. Diffusion calibration using differential evolution. Citeseerx an application of jumpdiffusion model to. The following references might offer some insights on handling stochastic jumpdiffusion process modelling. Both the double exponential and normal jumpdiffusion models can lead to the leptokurtic feature although the kurtosis from the double exponential jumpdiffusion model. Jumpdi usion models jumpdi usion jd models are particular cases of exponential l evy models in which the frequency of jumps is nite. Nov 15, 20 this dissertation studies two financial engineering and econometrics problems relating to two facets of the 20072008 financial crisis. This section closely follows the chapter on jump diffusions in joshi, where more theoretical details are provided. A jump diffusion model for option pricing by steven kou.
Exact simulation of stochastic volatility and other ane jump. Option pricing under a mixedexponential jump diffusion model. In the first part, we construct the spatial capital asset pricing model and the spatial arbitrage pricing theory to characterize the risk premiums of futures contracts on real estate assets. In option pricing, a jumpdiffusion model is a form of mixture model, mixing a jump process and a diffusion process. The model contains a jump part that follows a poisson distribution and factors in large, unpredictable changes in the stock price, and a diffusion part that has the stock moving according to brownian motion. To that end, i will have to simulate from a jump diffusion process. This chapter considers jumpdiffusion processes to allow for price fluctuations to have two components, one consisting of the usual increments of a wiener process, the second allows for large jumps from timetotime. Jump di usion models jump di usion jd models are particular cases of exponential l evy models in which the frequency of jumps is nite.
It ignores the jump, and fits the stochastic volatility as a high and low volatility regime. This is generally faster than vacancy diffusion because there are many more interstitial sites than vacancy sites to jump to. Newest jumpdiffusion questions quantitative finance. Applied stochastic processes and control for jumpdi. Firstly, the jumps should occur in an instantaneous fashion, neglecting the possibility of a delta hedge. Steven kou columbia university in the city of new york. Chapter 2 jumpdiffusion models for asset pricing in. Due to the page limit, i will also skip various applications of the jumpdiffusion models. Jump diffusion models have been introduced by robert c. Jumpdiffusion processes the general parametric jumpdiffusion processes, as a mixture of both continuous diffusion path and discontinuous jump path, can be written as. Two jump diffusion processes wolfram demonstrations project. Because of the overshoot problems associated with general jump diffusion processes, the double exponential jump diffusion process offers a rare case in which analytical solutions for the first passage times are feasible. To formalise the jumpdiffusion process we therefore need to define the jump statistics. Quantitative modeling of credit derivatives academic commons.
Exact simulation of stochastic volatility and other ane. We focus on the lognormal jump diffusion process of merton 1976, mjd and the double exponential jump diffusion process of kou 2002, kjd due to their popularity and analytical tractability. Despite the similarities, warrants and call options. Extracting latent states from asset prices michael s. In particular, we will first introduce diffusion and jump diffusion processes part, then we will look at how to asses if a given set of asset returns has jumps part 23. This demonstration compares the two most popular diffusion processes. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Jumpdiffusion libor market models 3 this paper addresses the numerical solution, through discretization and simulation, of the market models with jumps developed in 8. Self diffusion motion of atoms within a pure host also occurs. Jump diffusion pricing models are enhancements to blackscholes that allow discontinuous price behavior jumps. Exact simulation of option greeks under stochastic volatility and jump diffusion models. The performance of the differential evolution algorithm is compared with standard optimization techniques. Is the jumpdiffusion model a good solution for credit.
Jump locations of jumpdiffusion processes with state. Probability density function of simple equation, compound poisson noise. Jumps are therefore independent of each other and the mean waiting time for the jump to occur is 1. Introduction to diffusion and jump diffusion process. Two papers of financial engineering relating to the risk of. The models share the same continuous component and initial. Ajumpdiffusionmodel foroptionpricing columbia university. Jumps are generally modelled with the poisson process with rate which defines the jump frequency. Jumpdiffusion models for asset pricing in financial engineering s. They can be considered as prototypes for a large class of more complex models such as the stochastic volatility plus jumps model of bates 1. Constant diffusion volatility no risk between crashes an exact hedge against a crash is of the log security together with its delta during a crash, gain on the variance contract which offsets exactly by the loss on the hedge. The mixedexponential jump diffusion model under the mem, the dynamics of the asset price s t.
Jump diffusion and stochastic volatility globalcapital. This paper aims to investigate the effect of the acquisition of columbia picture entertainment inc. Jorion 1989 examined jump processes in foreign exchange and the stock market. Kou 2002 and kou and wang 2004 derived option pricing results for jump diffusion with logdoubleexponentially distributed jump amplitudes. This paper is previously titled \a jump diusion model for option pricing with three. The misspecified jump diffusion model badly overestimates the jump probability and underestimates volatility of the jump and the unconditional variance of the process. Leptokurtic feature, volatility smile and analytical tractability keywords. Jump diffusion processes are the simplest generalizations of the classical blackscholes model of stock price movements that include discontinuous jumps in price. We also provide rigorous econometric analysis of the new models. The function should follow the following stochastic differential. Two papers of financial engineering relating to the risk. C\ of course, is the jump diffusion option value and is essentially a weighted average of blackscholes values. This post is the first part in a series of posts where we will be discussing jump diffusion models. To correctly value hybrid defaultable financial instruments, e.
Option pricing under a mixedexponential jump diffusion model management science 5711, pp. May 05, 2011 the recent financial crisis has revealed major shortcomings in the existing approaches for modeling credit derivatives. Exact simulation of option greeks under stochastic. The author thanks kostas kardaras, ariel neufeld, nizar touzi and jianfeng zhang for fruitful discussions, and he is grateful to the associate. In option pricing, a jump diffusion model is a form of mixture model, mixing a jump process and a diffusion process. Pricing options under jump diffusion processes david s.
In proceedings of the 2004 winter simulation conference, 160715. Mar 04, 2015 sample asset price paths from a jump diffusion model. Option pricing for a stochasticvolatility jumpdiffusion. Adjustment of option pricing models for warrant pricing both the blackscholes and jump diffusion models were developed for the pricing of call options. First passage times of a jump diffusion process advances in. Pricing doublebarrier options under a flexible jump. February 5, 2001 abstract we provide a general asymptotic theory for the fully functional estimates of the in. Simulating electricity prices with meanreversion and jump. Exact simulation of option greeks under stochastic volatility. Abstract a jump diffusion model coupled with a local volatility function has been suggested by andersen and andreasen 2000. Pdf latent jump diffusion factor estimation for commodity.
Pdf jumpdiffusion model in incomplete market han xia. Indeed, it applies to the entire class of exponential levy processes which have eexp xt jump diffusion model, introduced in 1976 by robert merton, is a model for stock price behavior that incorporates small daytoday diffusive movements together with larger, randomly occurring jumps. Gerrard, russell, and douglas wright, stochastic modelling unit 3. Estimation of a stochasticvolatility jumpdiffusion model. Sample electricity prices from january 1, 2010 to november 11, 20 are loaded and plotted below. Iii, a formal closed form solution according to heston 14 for riskneutral pricing of. In addition, it leads to several interesting probabilistic results. University of london, london wcl e 7 hx, uk abstract this paper presents a meanreverting jump diffusion model for the electricity spot price and derives he corresponding forward price in closedform. Bates finance department the wharton school university of pennsylvania philadelphia, pa 191046367 october, 1988 abstract this paper derives the appropriate characterization of asset market equilibrium when asset prices follow jump diffusion processes, and develops the general methodology for.
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