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I've implemented the Merto Jump diffusion model with normal distribution for the jump and it happens that even if I don't have a jump, the spot price using Jumpn Diffusion versus GBM are differents due to Lambda*kappa in (dS/dS-)=(mu-Lambda*kappa)dt+sigma*dz+dp.I then have the following question.Shall we really use (mu-Lambda*kappa)*dt even if there is no jump? If yes, do that means the Jump Diffusion solution in the contest of very small values of Lambda or very small time period is exp(-Lambda*kappa *dt) times the the Black-Scholes price with no jump??Also can it hapen that the call price in a jump-diffusion framework is less than the Black-Scholes price of the same option using the same volatility?? That is what I realised using the summation formula in the Merton OPTION PRICINGWHEN UNDERLYING STOCK RETURNS ARE DISCONTINUOUS (1976)

yesnothese issues are covered in my book "concepts etc"

Hey,why not using it? If there are no jumps, lambda is zero and dP has no influence, which gives you GBM.

QuoteOriginally posted by: johnywakerI've implemented the Merto Jump diffusion model with normal distribution for the jump and it happens that even if I don't have a jump, the spot price using Jumpn Diffusion versus GBM are differents due to Lambda*kappa in (dS/dS-)=(mu-Lambda*kappa)dt+sigma*dz+dp.I then have the following question.Shall we really use (mu-Lambda*kappa)*dt even if there is no jump? If yes, do that means the Jump Diffusion solution in the contest of very small values of Lambda or very small time period is exp(-Lambda*kappa *dt) times the the Black-Scholes price with no jump??The -Lambda*kappa coefficient to ensure that your process is a martingale wrt the risk neutral mesure. In other terms, it makes you fit the first moment of your process distribution with the one of the GBM. Having a difference on trajectories with no jumps in Merton vs GBM allowes you to retrieve the wanted expectation of your process (which HAS to be the same for both models).But maybe it was not what you meant ?

I use Monte Carlo simulations to price the European put options. Please see my attached matlab code.Reference:1. Glasserman P. , Monte Carlo Methods in Financial Engineering, 2004.2. Jimmy E. Hilliard and Adam Schwartz, Pricing European and American Derivatives under a Jump-Diffusion Process: A Bivariate Tree Approach, Vol. 40, No. 3, September 2005.

I'm a novice in Matlab but looking at your code the parameter kappa will always have zero value... is that correct?I have a question for you... how have you estimated the parameters of the model? In particular lambda?

You can change some or all parameter values in the jump-diffusion model. For more detailed information, please see the Merton's (1976) paper.