hiCould someone kindly suggest any meterials for numerical method to jump diffusion model?I'm thinking about finite difference method, but have no idea how to implement in JDM.Thanks a lot!

QuoteOriginally posted by: dinnerhiCould someone kindly suggest any meterials for numerical method to jump diffusion model?I'm thinking about finite difference method, but have no idea how to implement in JDM.Thanks a lot!This might be a good startnataliniContYou also can use Splitting schemes as well.

What kind of products are you trying to price ? For plain-vanilla equity options there are some nice methods.But I have to look up the reference...(For other products this method might not work..)And what kind of jump-diff. are we talking about ? one possible jump, distribution of jumps ?

QuoteOriginally posted by: eye51What kind of products are you trying to price ? For plain-vanilla equity options there are some nice methods.But I have to look up the reference...(For other products this method might not work..)And what kind of jump-diff. are we talking about ? one possible jump, distribution of jumps ?Hi Eye51,Are you referring to n-factor models with jump? The articles usually refer to Merton jump model that then translates to a PIDE. Numerically we can use FDM for the PDE part and FDM (or FEM) for the integral.

Jump diffusion and finite difference works nicely if the size and the magnitude of the jump is dependent only on the value of the underlying. (i.e. the system is Markovian and time homogenous)Suppose you have an underlying X, and the probability of a jump is p(X) dt causing a change in the derivative of y(X). The effect on the expectation value of the derivative is dy = p(X) dt y(X), or dy/dt = p(X) y(X). This ends up being just another term in the PDE. The typical example of this is if you have a poisson process that determines if a company is going to go bankrupt.

QuoteSuppose you have an underlying X, and the probability of a jump is p(X) dt causing a change in the derivative of y(X). The effect on the expectation value of the derivative is dy = p(X) dt y(X), or dy/dt = p(X) y(X). This ends up being just another term in the PDE. What's the from of the new equation?

QuoteOriginally posted by: CuchulainnQuoteSuppose you have an underlying X, and the probability of a jump is p(X) dt causing a change in the derivative of y(X). The effect on the expectation value of the derivative is dy = p(X) dt y(X), or dy/dt = p(X) y(X). This ends up being just another term in the PDE. What's the from of the new equation?It's convective with a non-linear convection term. In practice a code that without the term also tends to work with the term.Also, the different from the Levy models that you refererenced. The model I am referring to are default models where the "normal" behavior of the stock is diffusion but there is a finite chance of a "jump" due to default. The use for these is to model way out of the money put options for a company that has some chance of going belly up (i.e GM). Alternatively they could be used for Renminbi models where there is "normal" stock behavior and then an external chance of a revaluation.

QuoteOriginally posted by: twofishQuoteOriginally posted by: CuchulainnQuoteSuppose you have an underlying X, and the probability of a jump is p(X) dt causing a change in the derivative of y(X). The effect on the expectation value of the derivative is dy = p(X) dt y(X), or dy/dt = p(X) y(X). This ends up being just another term in the PDE. What's the from of the new equation?It's convective with a non-linear convection term. In practice a code that without the term also tends to work with the term.Also, the different from the Levy models that you refererenced. The model I am referring to are default models where the "normal" behavior of the stock is diffusion but there is a finite chance of a "jump" due to default. The use for these is to model way out of the money put options for a company that has some chance of going belly up (i.e GM). Alternatively they could be used for Renminbi models where there is "normal" stock behavior and then an external chance of a revaluation.Do you have more information on this? Is the convection term like what? I am not familiar with this model.

QuoteOriginally posted by: CuchulainnDo you have more information on this? Is the convection term like what? I am not familiar with this model.Sure. There are a lot of papers by Vadim Linetsky at http://users.iems.northwestern.edu/~linetsky/ which describe the basic models. But one does have to do a little work to get this to work with numerical methods. What Linetsky does is to derive the equations of an option with default and then solve the problem analytically. It wouldn't be too much work to take his equations and numerically difference them, and as far as I can tell, the extra terms in the PDE don't change the nature of the PDE.I started to code up the Linetsky's equations numerically and I was trying to do so in terms of a general set of Quantlib classes dealing with default which would unifies Linetsky's work with the convertible bond formulas of Ayache-Forsythe. However, at that point, I realized pretty quickly that default processes aren't important in Shanghai convertible bonds, and I lost interest in trying to model defaults.Whether or not they are useful in modelling Chinese RMB options is something that I've spent some time thinking about, but right now I'm just knee doing other things.

QuoteSure. There are a lot of papers by Vadim Linetsky at http://users.iems.northwestern.edu/~linetsky/ which describe the basic models. There are about 20 papers Which one is it precisely (BTW the last one gives a file I/O error).

Last year ,I worked on a problem of hedging(approach by utility function) in a jump model by technique of stochastic control.The PIDE now become a HJB-PIDE.I coded it by the scheme implicit-explicit and had a little pretty result of convergence.We looked at the price in the sense of indifference pricing and find it converge to the price in BS when the jumps intensity tend to zero.If it sounds interesting for someone,i will find my report and the code(unfortunately it is in French).

QuoteOriginally posted by: vin100Last year ,I worked on a problem of hedging(approach by utility function) in a jump model by technique of stochastic control.The PIDE now become a HJB-PIDE.I coded it by the scheme implicit-explicit and had a little pretty result of convergence.We looked at the price in the sense of indifference pricing and find it converge to the price in BS when the jumps intensity tend to zero.If it sounds interesting for someone,i will find my report and the code(unfortunately it is in French).Yes, it would be nice to see how IMEX schemes are used for this PIDE. Thanks.

QuoteOriginally posted by: CuchulainnQuoteSure. There are a lot of papers by Vadim Linetsky at http://users.iems.northwestern.edu/~linetsky/ which describe the basic models. There are about 20 papers Which one is it precisely (BTW the last one gives a file I/O error).The last onehttp://users.iems.northwestern.edu/~linetsky/default.pdfYou have to do a bit of work to get numerical results, since Linetsky never actually writes down the necessary PDE.Start with equation 2.3, and then work out the PDE that corresponds to the SDE (using Ito's lemma and delta hedging arguments). What you get is Black-Scholes equation with an extra term for the hazard rate (replace the term r_t in B-S with r_t + h(S_t)). The resulting equation is your garden variety B-S equation that you can numerically difference. What you also end up with when you work out the PDE is an equation that looks very much like the Ayache-Forsythe model for convertible bonds. I don't think that this is a coincidence.Where I was about six months ago was that I was in the process of trying to code a unified framework for handling defaults in QuantLib as well as trying to figure out if any of this is relevant for non-deliverable options of the RMB. At that point I got pulled into a totally different direction.Also my intution tells me that the only reason this ends up as a PDE rather than a PIDE is that the hazard rate is time-independent. I suspect that if you put in a time-dependent hazard rate that the equation will stop being a Markov process.

QuoteOriginally posted by: CuchulainnHi Eye51,Are you referring to n-factor models with jump? The articles usually refer to Merton jump model that then translates to a PIDE. Numerically we can use FDM for the PDE part and FDM (or FEM) for the integral.Hello CuchulainnI was not referring to n-factor models. Just Black-Scholes + jump. When you have a single jump it's possible to use a simple method. For distribution of possible jumps this method does not work. For most applications the method will be too restrictive.. But for equity options it can be usefull.Maybe we can discuss this over a beer

QuoteAlso my intution tells me that the only reason this ends up as a PDE rather than a PIDE is that the hazard rate is time-independent. I suspect that if you put in a time-dependent hazard rate that the equation will stop being a Markov process.From another (physical) angle, you need integral terms along with the PDE to model heat transfer, temperature feedback, nerve progataion and this lead to Volterra or Fredholm equations. Thus, a PIDE.The Markov property assumes memoryless and takes no account of the past. This assumption might be too restictive for these kinds of problems. I have only seen Fredholm equations where the integral is taken over S space but no Volterra where the integral is taken from 0 up to the current time. I have no QF motivation for the latter kind of problem.