Parabolic partial differential equations

[Alice Liu]

FE Academic Director Henry Schellhorn recently published a paper on second-order parabolic partial differential equations with Sixian Jin in International Journal of Stochastic Analysis. Please find more about it here.

In addition to his work (mentioned above), Schellhorn also presented his research on “Density Formula for compound Poisson processes with Malliavin calculus” in the spring at the USC mathematics seminar. Co-authors are former PhD student and MSFE instructor Sixian Jin, Ivan Nourdin, and Josep Vives.

 

[Daniel Duffy]

I had a quick look but I did not seen any conclusions on the benefits and how it would compare to other methods to solve Dupire equation.

For example, is equation (7) meant to computed?

 

[Alice Liu]

Yes. Equation (7) is meant to be computed. We used it to solve for the bond price in the CIR model with time-varying coefficients (see our paper in Stochastics), the CIR model with fractional BM (see our paper in Communications on Stochastic Analysis), the CIR model with jump processes (see our hopefully forthcoming paper in SPA). We are now working on the SABR model, for which we will present a complete solution at the 9th Western conf. of Math Finance.

We have not studied the Dupire equation though. I would be happy to hear about potential other applications of our method.

 

[Daniel Duffy]

Yes. Equation (7) is meant to be computed. We used it to solve for the bond price in the CIR model with time-varying coefficients (see our paper in Stochastics), the CIR model with fractional BM (see our paper in Communications on Stochastic Analysis), the CIR model with jump processes (see our hopefully forthcoming paper in SPA). We are now working on the SABR model, for which we will present a complete solution at the 9th Western conf. of Math Finance.

We have not studied the Dupire equation though. I would be happy to hear about potential other applications of our method.

I have no hands-on experience of Malliavin calculus, but I expect equation (7) would be a computational challenge 1) n-dimensional integral 2) truncation of series (round-off?), 3) do you need to approximate D^2?

In general (and in particular), it is good to produce tables of numerical output as well.

Just curious.

 

[Alice Liu]

I have no hands-on experience of Malliavin calculus, but I expect equation (7) would be a computational challenge 1) n-dimensional integral 2) truncation of series (round-off?), 3) do you need to approximate D^2?

In general (and in particular), it is good to produce tables of numerical output as well.

Just curious.

You are completely right. Calculating the series numerically is a challenge that resisted our efforts. The main application of this series is to get analytical results. The challenge is the non-differentiability of many option payoffs, but this can often be overcome. This is why we started with bond pricing. If you “massage” the expressions, the price of calls on stocks with stochastic volatility can be transformed into expectations of Malliavin differentiable functions.

 

[Daniel Duffy]

I suppose the essential (numerical) difficulties don't just disappear. Kahan summation and/or multiprecision might help but it will be slower. And non-smoothness of payoff functions is a universal concern.