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### High Accuracy Greeks computation under trinomial model

Posted: July 18th, 2019, 6:27 am
I am currently working on the Tian4 trinomial model for vanilla American call/puts. The performance of price accuracy turns out to be quite good. However, I am wondering if there is any way to increase the computation accuracy of delta.
The traditional way of computing delta = (up price - down price)/(uS0 - dS0) performs quite poorly in terms of accuracy.
I believe there must be some ways of weighting up price and down price or middle price unevenly that can significantly increase the accuracy.
Also, since this way of computing delta happens at time dt, this method is inherently flawed. Is there any ways to avoid it or to offset the effects of dt.
Thanks a lot.

### Re: High Accuracy Greeks computation under trinomial model

Posted: July 18th, 2019, 10:13 am
I doubt (very much) if the Trinomial method and divided differences will bring any joy. I don't want to delineate all the issues that must be addressed. In short, I think it is not very good. Certainly not under all parameter regimes.

Do people use this method in real life?

A better approach is to let FDM loose on the PDE.

Some other ways are described here.

BTW what is 'High Accuracy' in this case? Can you quantify is as a function of dx, dt?

### Re: High Accuracy Greeks computation under trinomial model

Posted: July 19th, 2019, 12:51 am
the reason why we choose to use tree models is that it is stable, accuracy controlable, intuitive and straightforward to calculate greeks.
Since we are doing option HFT, the speed is equally important as accuracy. I have tried many other approximation methods like BAW, JZ, Bjerksund etc. But all these methods exibits some erratic fashion in terms of accracy if we test them thoroughly. For example, the error of JZ greeks formula is so large near EEB that we cannot use them in practice.

### Re: High Accuracy Greeks computation under trinomial model

Posted: July 19th, 2019, 1:21 am
For your purposes, this approach may offer a handful of orders of magnitude of improvement. But I'm not really an expert on this stuff any more... High Performance American Option Pricing

### Re: High Accuracy Greeks computation under trinomial model

Posted: July 19th, 2019, 1:20 pm
You can improve the accuracy of delta computations by implementing the "overshoot" technique, which removes the dt issue you described.  This technique is described in Hull's classic textbook.

### Re: High Accuracy Greeks computation under trinomial model

Posted: July 26th, 2019, 4:01 am
If you set up the option value pde problem using x = log S with solution [\$]f(x,t)[\$], you can differentiate it to get the pde for [\$]g(x,t) = f_x[\$]. Then perhaps simply solve that second pde for [\$]g[\$] on a trinomial lattice, and convert to delta. You probably need two passes, the first to get the optimal exercise nodes from the option value solution. Might be a sufficient improvement.

But use bearish's link if you need major accuracy.

### Re: High Accuracy Greeks computation under trinomial model

Posted: July 26th, 2019, 3:19 pm
One way to compute delta is to use the SDE that it satisfies using the Kunita formula (from Farid) as discussed here.

https://forum.wilmott.com/viewtopic.php?f=8&t=101811&start=15

Not sure how that pans out for American options.

### Re: High Accuracy Greeks computation under trinomial model

Posted: July 26th, 2019, 3:21 pm
If you set up the option value pde problem using x = log S with solution [\$]f(x,t)[\$], you can differentiate it to get the pde for [\$]g(x,t) = f_x[\$]. Then perhaps simply solve that second pde for [\$]g[\$] on a trinomial lattice, and convert to delta. You probably need two passes, the first to get the optimal exercise nodes from the option value solution. Might be a sufficient improvement.

But use bearish's link if you need major accuracy.
This approach works but why not do the full FDM approach?

### Re: High Accuracy Greeks computation under trinomial model

Posted: July 26th, 2019, 5:48 pm
If you set up the option value pde problem using x = log S with solution [\$]f(x,t)[\$], you can differentiate it to get the pde for [\$]g(x,t) = f_x[\$]. Then perhaps simply solve that second pde for [\$]g[\$] on a trinomial lattice, and convert to delta. You probably need two passes, the first to get the optimal exercise nodes from the option value solution. Might be a sufficient improvement.

But use bearish's link if you need major accuracy.
This approach works but why not do the full FDM approach?

Just trying to answer the OP's query. A lot of people are comfortable with explicit binomial/trinomial lattice solns & less so with PDE numerics.

### Re: High Accuracy Greeks computation under trinomial model

Posted: July 26th, 2019, 6:04 pm
If you set up the option value pde problem using x = log S with solution [\$]f(x,t)[\$], you can differentiate it to get the pde for [\$]g(x,t) = f_x[\$]. Then perhaps simply solve that second pde for [\$]g[\$] on a trinomial lattice, and convert to delta. You probably need two passes, the first to get the optimal exercise nodes from the option value solution. Might be a sufficient improvement.

But use bearish's link if you need major accuracy.
This approach works but why not do the full FDM approach?

Just trying to answer the OP's query. A lot of people are comfortable with explicit binomial/trinomial lattice solns & less so with PDE numerics.
Fair enough. It would be nice to see how the PD E(or was it SDE??) for delta maps to the trinomial method.

### Re: High Accuracy Greeks computation under trinomial model

Posted: July 26th, 2019, 6:06 pm
dbl

### Re: High Accuracy Greeks computation under trinomial model

Posted: July 26th, 2019, 6:08 pm
If you set up the option value pde problem using x = log S with solution [\$]f(x,t)[\$], you can differentiate it to get the pde for [\$]g(x,t) = f_x[\$]. Then perhaps simply solve that second pde for [\$]g[\$] on a trinomial lattice, and convert to delta. You probably need two passes, the first to get the optimal exercise nodes from the option value solution. Might be a sufficient improvement.

But use bearish's link if you need major accuracy.
This approach works but why not do the full FDM approach?

Just trying to answer the OP's query. A lot of people are comfortable with explicit binomial/trinomial lattice solns & less so with PDE numerics.
Fair enough. Under that constraint it would be interesting to map PDE/SDE for [\$]g[\$] to trinomial.

I haven't seen anyone doing that before.

### Re: High Accuracy Greeks computation under trinomial model

Posted: July 29th, 2019, 7:33 am
Have increased the accuracy of deltas computation by roughly 20 times, using the non-cenctered difference methods. The accuracy problem is solved. Thanks, guys.