Let me explain.Take a standard implementation of a finite difference method (explicit or Crank-Nicolson with weight = 0.5) for pricing a European call option. Suppose you want to price this with zero strike 5 year call. In other words you want to price the trivial instrument whose final payoff is the final stock price itself.Now choose a huge risk-free interest rate (like 400%), huge vol (like 200%), and small number of time-steps (like 6).Your answer should simply be the current stock price, but you will find that the answer is way off.Or at least, this will be the case if you have a "standard", or "textbook" (e.g. Paul's book) implementation.Has this bothered anyone other than me? Note that the Binomial model does not have this problem.Regardless of the interest rates/vols/, no matter how small the number of time-steps, it gives exactly the right answer when strike=0. Now that's a nice property to have.My question is -- is there a way to modify say the Crank-Nicolson scheme so that it is "calibrated" to the stock price, i.e. for the zero-strike European call, we get exactly the right answer?

Try dx = vol * lambda * sqrt(dt)lambda = sqrt(3/2) Then the grid should work with high vols,small number of timesteps and r in a normalrange.A bigger problem is your r=400%, this induce"instability" in the grid. Try an even biggerlambda (=5-10) or/ad upwind/downwind finitedifferencing.

Thanks Keanu.Stability may be an issue, but I think the bigger problemis that the truncation errors (when replacing the partial derivatives with finite diffs)become big. I was wondering if there's some Finite Diff schemewhich exactly prices the trivial zero-strike call option mentioned earlierat any step size and anywhere in the parameter space.

QuoteOriginally posted by: PrasadI was wondering if there's some Finite Diff schemewhich exactly prices the trivial zero-strike call option mentioned earlierat any step size and anywhere in the parameter space.Yes there is.Write you favourite numerical scheme (Crank-Nicholson, fully implicit, etc.).This is a system of linear equations, with the values of your function at grid points as unknowns.Now write the same system for your stock S, and over again for your zero coupon B, and solve for the coefficients of the discrete PDE such that S and B are priced exactly. In other words you will use coefficients of the PDE (for BS, these are volatility, interest rate, drift of the underlying) which are slightly perturbated, and which depend now on the step size.Letting the scheme price B exactly will give you the correction for the instantaneous interest rates.Then, letting the scheme price S exactly will give you the correction for the drift.Then, letting the scheme price a call option exactly will give you the correction for the volatility coefficient.Now use these corrected coefficients AND THE SAME space and time step sizes to price another derivative instrument.This will considerably reduce truncation error.This technique is some kind of "variance reduction," it is briefly described at the end of this Wilmott article, with references:The discrete and the continuous

Number 12 (double of six )Thanks, that's exactly the kind of angle I was looking for,and I'm glad I'm not the only one bothered by this!I was considering an approach similar to yours,but didn't pursue it because of speed concerns:for a situation where we have time-varying vol and rates,in general at each time-step we would first have to "fit"the Black-Scholes parameters (vol, interest rate) to be consistent with the bond, stock and the vanilla option.This would be a lot of work to do at each time step,and would make the pricing extremely slow.Is there a closed form formula that specifies exactly how to perturb the BS parameters to match the stock, bond and vanilla option?(I do realize that once the "fitting" is done, we can use the fitted params to price any instrument that shares the same underlying, maturity and critical dates.)

This is a very good question (also DoubleSix's answer). It reminds me of a test I use whenever I derive some new model (outside of the Black-Scholes world). Generally speaking, whatever the 'model' turns out to be, it should have two simple exact solutions:1. exp(rt)2. S(Or there has to be a very good reason why these need not be solutions.) If these are not solutions then chances are there has been some error in the model derivation.P

Could someone please comment on Prasad's experience? Why does it happen? Does the instability swamp the solution because the prop of diffusion has vanished (the second derivative being weak or zero)? Is it the culprit? Keanu's suggestion to use upwind/downwind scheme, loosely implies that introducing artificial diffusion will "re-stabilize" the solution. That seems to indirectly suggest that the 2nd derivative -> 0 is a problem. I'm just speculating here ..... some explanation would be great...... Thanks!!

QuoteOriginally posted by: PrasadIs there a closed form formula that specifies exactly how to perturb the BS parameters to match the stock, bond and vanilla option?It is an analytical formula.Write down your discrete scheme (at the current node i and time j). Force S and exp(rt) to be the exact solution, and you will be able to write on paper the coefficients of the discrete PDE that you should apply at each node.(Leave volatility for now, for this will slow down the procedure as it requires that you compute the BS call at each node)It doesn't matter whether interest rates are time varying, you will call the "adujsted" interest rate formula once every time step, however you vill have to call the adusted drift at each node (especially if your space grid is in logS). But the formula involves maybe one multiplication and one division...Paul, since you're back, I pretty much liked the formula that a forum member has used in another thread:"Paul's ITO33 approach"I think we (me & and my double) should both encourage more this kind of cross-fertilising in the future.

<blockquote>Quote<hr><i>Originally posted by: <b>WaaghBakri</b></i>Could someone please comment on Prasad's experience? Why does it happen?<hr></blockquote>I have an explicit finite difference code for pricing American options. It's a pretty standard implementation (except the coefficiens are linearized with respect to r*dt) plus a modest effort to set the grid size flexibly dependent on the option parameters. It returned me exactly the stock price under conditions specified by Prasad.I'm not sure one needs to go into such complications to solve this one.

What's wrong with the explicit scheme?V^j_{n-1} = V^j_n + r\delta t (j(V^{j+1}_n-V^j_n) - j\delta S) + \delta S 0.5\sigma^2j^2 (V^{j+1}_n-2V^j_n+V^{j-1}_n)in the rhs both the second and third terms are zero and therefore V^j_{n-1} = V^j_n.

Diceman,I agree with you, if your grid is Stock-based, we have no problem.However, try using a log-grid, using the transformation Z = ln(S/S0).In this case you can work out that a necessary condition for V^j_{n-1} = V^j_n is that the stock price should be arithmetic average of the stock prices above and below,which is only approximately true in the log-grid.

Upon DoubleSix's suggestion (see above) I read his article,and hunted down one of the references that precisely answers my question:Andersen, L. B. G and Brotherton-Ratcliffe, R. (1998). Theequity option volatility smile: an implicit finite-differenceapproach, Journal of Computational Finance 1 (2), Winter1997/98 5-37.

DoubleSix,Do you mean that the vol correction should be different at each node? Even a flat vol will thus induce a grid with a variable local vol...Could someone please send me a copy of the paper mentioned above:Andersen, L. B. G and Brotherton-Ratcliffe, R. (1998). Theequity option volatility smile: an implicit finite-differenceapproach, Journal of Computational Finance 1 (2), Winter1997/98 5-37.

Does anybody know where to find this article for free? If you have it and don't feel like posting it, you could send it to me directlyolaolson@yahoo.comThanks in advanceOlaAndersen, L. B. G and Brotherton-Ratcliffe, R. (1998). Theequity option volatility smile: an implicit finite-differenceapproach, Journal of Computational Finance 1 (2), Winter1997/98 5-37.