This should keep newton happy P

jump-diffusion model can be considered as a Brownian motion plus a poisson process. The rate of jump and the jump size can be any distributions. It has bigger tails that means large jump is happened more frequently. The stock price process can be expressed as dS(t)=S(t)(mudt+sigmadB+lambda dN)

The price of call and puts are a monontone increasing function of the jump intensity. (This is the jump intensity in the pricing measure.)Digital prices can go up or down. MJ

Are there any conditions that one can impose on the Poisson component of the process such that the total jump diff process is perfectly hedgeable?If yes, please state the conditions and give a reference. If no, please give a ref to a full proof.An explanation in terms of martingales (plus references) would be greatly appreciated.

Jump-diffusion with constant jump size can be complete market(forgot condition on rate of jump). see M. Jeblanc and Monique Pontier "Optimal portfolio for asmall investor in a market with discontinuous prices", Appliedmathematics and optimization 22 (1990) pp. 287-310.

Omar, perfect hedging of a pure point jump (no B.M.) is also discussedin Cox and Ross "Valuation of Options for Alternative Stoch. Processes",J. Finacial Economics, 3, 1976, 145.

Alan and VincentWould it be possible for you each to give brief summaries of the papers you mention for those of us without the relevant journals at our fingertips?Cheers

To keep this thread more accessable to those who are only familar with one dimension Bachelier-Wiener models, I suggest you read an excellent introduction to processes in general spaces:Poisson Processes , J.F.C. Kingman, Oxford Science Publications, 1995.Alternatively, you may want to brush up on Langevin's SDE.-newton

Answer to FAQ: go read a book! Hmmm?!P

QuoteOriginally posted by: JohnnyAlan and VincentWould it be possible for you each to give brief summaries of the papers you mention for those of us without the relevant journals at our fingertips?CheersHere's part of what they do.Cox and Ross consider a simple point jump model where, in thereal world: dS/S = mu dt (when no jump) with Prob = (1 - lambda dt) dS/S = mu dt + (k-1) (when a jump) with Prob = lambda dt.They create a riskless hedge in the usual way and by doing so they derive the option valuation equation:0 = dC/dt + mu dC/dS + (mu - r S)/(1 - k) C(S + k -1, t) + [r (k -1 + S) - mu]/(1 - k) C(s, t) ,They comment that the lambda dependence has dropped out.You can also get this equation from my equation (3.2) in my "Fear of Jumps" article in Wilmott Dec. 2002, the current issue. To get from my equation to the Cox-Ross one, first you setsigma = 0, so the second derivative term disappears. I treat a power utility equilibrium. Under that, the expected total return on the stock, call it alpha,satisfies, alpha = r + (e^x0 - 1 )(lambda - lambda_Q). The relationship toCox and Ross is that alpha = lambda (e^x0 - 1) + mu/S, so mu/S = r - (e^x0 - 1) lambda_Q, and also my e^x0 S = S + k -1, using their k.These substitutions turn my (3.2) into their equation above.While it's true what they say that the explicit lambda dependence hasvanished, nevertheless, if lambda changes then, (holding r, x0 fixed) mu will change. So, unless mu doesn't influence the solution of theirequation, which seems unlikely, then the option values will change anyway.

Sorry, this should read dS = mu dt (when no jump) with Prob = (1 - lambda dt) dS = mu dt + (k-1) (when a jump) with Prob = lambda dt.

What happened to the Karatzas/Shreve chapter on the Study of the Multi-Dimensional Case, 5.5(a)?If there were such a chapter, I think it would show that the diffusion and jump processes are in different sub-spaces and that the processes are separable. The connection is that the price processes just add.-newton

Regarding option values - The price associated with a Poisson jump process is found as follows.Any Poisson process or sum of processes (almost) can be converted into many small but equal jumps at small constant time intervals. Make the time intervals very small and you have an interest rate process, but this process can't be incorporated into the BS SDE (different sub-space).So you have a Poisson jump (a rare event process) price that is constant across strike but grows in time like compound interest that is added to the BS price that varies across strike to yield the option market price.-newton

AlanThanks for your summary of the Cox, Ross paper. As you mention, the real world drift term, mu, is still present in their valuation equation:0 = dC/dt + mu dC/dS + (mu - r S)/(1 - k) C(S + k -1, t) + [r (k -1 + S) - mu]/(1 - k) C(s, t)As you say, this implies that to value a single derivative it is necessary to specify risk preferences. However, in the case of valuing two or more European style derivatives with the same expiry date on the same underlying it looks at first sight as if a general equilibrium approach would lead to preference-free pricing. i.e. the market can be made complete again by the introduction of further instruments. I haven't worked through it. Any thoughts?

This thread is starting to get very interesting and starting to get close to my thoughts of my current research. Good! My idea is that mixing a Wiener and Poisson component does lead to two degrees of uncertainty. One we can deal with by a riskfree rate while the other remains an open question. To me there are relatively few ways out:1. As Johnny proposes adding a security and thus starting to mark to the market. A problem which exists in our case (spot electricity modelling) is that there are too few options to work on this.2. Introducing a risk preference. Still I think it is hard to estimate the risk parameter which however can have large impact on the price (e.g. Naik and Lee (1990) ). 3. Taking the Esscher transform. A kind of ad-hoc approach which seems to resemble power utility function anyhow (e.g. Gerber and Shiu (1994)). In this way we come close to the approach Alan Lewis proposed in his 'Fear of jumps' and get to his unknown gammaNow I am really wondering if there is an approach without introducing a risk preference while not marking to market. Does anybody see another way out? Option pricing under a jump diffusion model seems to have some arbitrariness to me.- Allu