I was pricing a option with big dividend in the underlying. However, I got negative transition probability in a trinomial tree. Will it cause arbitrage? Does anyone have reference paper or book chapter can share?I failed to find the following paper. The paper suppose to discuss negative probabilities.Can anyone help? Thanks a lot.Mayhew, S. ?On Estimating the Risk-Neutral Probability Distribution Implied by Option Prices.?Working Paper, Purdue University, 1995.

- Cuchulainn
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It means the numerical scheme is wrong, at the very least. So, it's not correct. IMO calling something a 'negative probability' is mathematical heresyYou have to resolve the root cause of the problem. QuoteNegative probabilities in quantum theory have no physical meaning. They are mathematical artifacts resulting from obstinately applying the classical probability formalism to quantum models, where it doesn't apply by construction. I see similar analogy in this case, where the issue (seems to be) caused by a non-monotonic explicit finite difference scheme? Quotebig dividend in the underlyingYep. This is called convection dominance and well-known in fluid dynamics. A structural solution is my exponentially fitted finite difference scheme for PDE. I wonder if it can be applied in a lattice context? See Articles April 22 2013 for this methodhttp://www.datasimfinancial.com/forum/viewtopic.php?f=24&t=289

Last edited by Cuchulainn on October 19th, 2015, 10:00 pm, edited 1 time in total.

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Looking at the trinomial FD scheme, my hunch is that fitting should work as it is just an explicit scheme. Looking at it, it seems p_u and p_m are OK positive while p_d goes negative when convection is dominant. It's just a matter of replacing sig by the fitted equivalent!BTW I have seen no published work on fitting with trinomial methods. These are just FTCS (Forward-in-Time, Centred-in-Space) schemes, aka explicit Euler. Static instability disappears I reckon but you still need dx >= sig sqrt (3 dt) (to ensure dynamic stability).QuoteWill it cause arbitrage? Overshoots?// Sorry, can't help with find paper

Last edited by Cuchulainn on October 19th, 2015, 10:00 pm, edited 1 time in total.

Thanks a lot. My gut feeling is that negative probability will cause Arrow-Debrew price. Therefore, it causes arbitrage. Not sure, how should I get rid of it in trinomial tree.

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QuoteOriginally posted by: fionahaasThanks a lot. My gut feeling is that negative probability will cause Arrow-Debrew price. Therefore, it causes arbitrage. No problem. Sound reasonable. But you still need to address the root cause. Standard schemes fail, period.QuoteNot sure, how should I get rid of it in trinomial tree.Like I said couple times, by using exponential fitting. QED.Failing that, use a real PDE/FDM model.

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Just remembered; if fitting is not to your taste you could use _upwinding_ for the convection/drift term. Be careful of the sign of r - D.

Last edited by Cuchulainn on October 19th, 2015, 10:00 pm, edited 1 time in total.

For a call option with large, discrete dividend, you might want to look at the discussion in "Back to basics ...", by Haug, Haug, & Lewis

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QuoteOriginally posted by: AlanFor a call option with large, discrete dividend, you might want to look at the discussion in "Back to basics ...", by Haug, Haug, & LewisI read the article, nice.Since cash dividends create a non-lognormal process, is it an idea to model dividends as an SDE and then get a two-factor PDE as in Paul's Vol 3. The dividend date will also be uncertain.

Thanks.The article tries to make a minimal, arbitrage-free modification of some ad-hoc approaches.It does so without enlarging the number of stochastic factors. Certainly you can make the dividends a full-blown stochastic process, but things will get much messier than the approach in the article very quickly.

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QuoteOriginally posted by: AlanThanks.The article tries to make a minimal, arbitrage-free modification of some ad-hoc approaches.It does so without enlarging the number of stochastic factors. Certainly you can make the dividends a full-blown stochastic process, but things will get much messier than the approach in the article very quickly.If we take the route of starting at equation (P1a) in your paper that results in a one-factor PDE with a delta function in time in the convection term? So, we don't necessarily use/want equation (2) but instead we approximate the jump conditions at t = t_d using the finite difference scheme. Issue: you mostly see the jump defined based on backward iteration from T but in PDE we need forward marching. Admittedly the former is more intuitive an easy to compute but I have difficulty in seeing how the latter approach fits into FDM.JumpV(S, td-) = V(S - D(S), td+) In your equation (2) I presume you are using numerical quadrature to compute it so then the term S - D(S) is just a shifted mesh point and then the issue of forward versus backward does not arise? I think another advantage of the integral form (2) is that you don't need to interpolate the mesh values (introducing error) as is needed in the differential case. Is there a way to 'directly' incorporate the jump into existing PDE solvers by some kind of integration or something?

Last edited by Cuchulainn on October 25th, 2015, 11:00 pm, edited 1 time in total.

Well, backward in t from T is forward from [$]\tau \equiv T - t = 0[$]. Then, I would just halt some existing PDE solver at [$]\tau_D[$], apply the jump condition to yield a new 'initial condition', and re-start the solver at that point in time [$]\tau[$].This scheme seems fairly direct to me, and could be applied to any black-box PDE solver without knowing the detailed solver internals. Plus, it would work for any process solver -- say a euro-style Heston model solver, for example -- and regardless of how it did its time-stepping.

Last edited by Alan on October 25th, 2015, 11:00 pm, edited 1 time in total.

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QuoteOriginally posted by: AlanWell, backward in t from T is forward from [$]\tau \equiv T - t = 0[$]. Then, I would just halt some existing PDE solver at [$]\tau_D[$], apply the jump condition to yield a new 'initial condition', and re-start the solver at that point in time [$]\tau[$].This scheme seems fairly direct to me, and could be applied to any black-box PDE solver without knowing the detailed solver internals. Plus, it would work for any process solver -- say a euro-style Heston model solver, for example -- and regardless of how it did its time-stepping.One niggling issue is that the convection term has a delta function in it (it's infinite). And the jump must be explicitly introduced by the programmer, yes?Does NDSolve, for example, have the facility for defining 'jump points' in the marching scheme?

Last edited by Cuchulainn on October 25th, 2015, 11:00 pm, edited 1 time in total.

Not needed. The delta function is handled by the jump condition and never appears to the solver under my stop/re-start scheme.

IMHO, the dividend handling is solved in a cleaner / easier way in Hans Buehler, Volatility and Dividends.He write the asset S(t) = [F(0, t) - D(t)] X(t) + D(t) where X is a pure martingale process (eg dX = sigma X dW in the BS case) and D the value at t of the constant dividends.Then, you can simply solve your PDE on X instead of solving it on S, without any jump condition.

Last edited by VivienB on October 25th, 2015, 11:00 pm, edited 1 time in total.

I looked briefly. As I understand it, D(t) in Buehler is a discounted value at t of a projected deterministic dividendstream into the indefinite future. The model then implies the stock price cannot fall below D(t) without the company defaulting (on its bonds?). I would like to see some practical estimates of D(t), say for IBM. My vague impression is that D(t) would be implausibly large, but I am happyto be corrected.

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