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FaridMoussaoui
Posts: 368
Joined: June 20th, 2008, 10:05 am

### Re: Is it possible to "complexify" SDEs?

Daniel, have a look to Kunita book: "Stochastic Flows and Stochastic Differential Equations". where you can find formulations like this one:
Let X be a diffusion process, strong solution to the following SDE

$dX_t = a(\theta, X_t) dt + b(\theta, X_t) dW_t, ~~X_0 = x$

Then $Y_t = \frac{\partial X_t}{\partial \theta}$ is defined as the solution of the following SDE:

$dY_t = [a'_{\theta}(\theta, X_t) + a'_{x}(\theta, X_t) Y_t ]dt + [b'_{\theta}(\theta, X_t) + b'_{x}(\theta, X_t)Y_t ]dW_t,$
$Y_0 = \frac{\partial X_0}{\partial \theta}$

where the primes denote standard derivatives.

Have a look also to Giles work on Vibrato.

Cuchulainn
Topic Author
Posts: 58952
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: Is it possible to "complexify" SDEs?

Daniel, have a look to Kunita book: "Stochastic Flows and Stochastic Differential Equations". where you can find formulations like this one:
Let X be a diffusion process, strong solution to the following SDE

$dX_t = a(\theta, X_t) dt + b(\theta, X_t) dW_t, ~~X_0 = x$

Then $Y_t = \frac{\partial X_t}{\partial \theta}$ is defined as the solution of the following SDE:

$dY_t = [a'_{\theta}(\theta, X_t) + a'_{x}(\theta, X_t) Y_t ]dt + [b'_{\theta}(\theta, X_t) + b'_{x}(\theta, X_t)Y_t ]dW_t,$
$Y_0 = \frac{\partial X_0}{\partial \theta}$

where the primes denote standard derivatives.

Have a look also to Giles work on Vibrato.
That's brilliant, Farid. It's exactly what I was looking for. Merci beaucoup.

FaridMoussaoui
Posts: 368
Joined: June 20th, 2008, 10:05 am

### Re: Is it possible to "complexify" SDEs?

Daniel, do you plan to appy the complex derivative technique on the MC level?

Cuchulainn
Topic Author
Posts: 58952
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: Is it possible to "complexify" SDEs?

Daniel, do you plan to appy the complex derivative technique on the MC level?
Hi Farid,
Yes, absolutely! It is worth a try.Now, I have done CSM for vector functions (gradient, Jacobian,..)  and for exact BS a while back.

What I did last few days was to complexify GBM SDE (i.e. all params are complex), create Euler paths and take imaginary part to get vega:

a. The results are good (feels much better than paths for S). Even convergence for convection-dominance e.g. r = .1, v = 0.01 is monotonic.
b. The presence of complex numbers does not adversely affect performance too much. (10% slower than real-valued GBM).
c. I also implemented Kunita SDE as a pair of SDEs in $(X,Y)$ using Euler. CSM is at least twice as fast and has better accuracy to date.

Nice is CSM only needs 1 SDE and each sensitivity case can be run in its own thread. With Kunita, seems you need a SDE for each sensitivity (aka boiler-plate code) and more computational cost due to new derivatives in the SDE. BTW Kunita SDE turns out what I thought it should be all along.

//
For higher-order Greeks CSM suffers from round-off errors and then multicomplex CSM enters the fray?

Cuchulainn
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Posts: 58952
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: Is it possible to "complexify" SDEs?

Sanity (double) check does Katuni approach for $dr/d\sigma$ in CIR model produce the same SDE as that produced from 1st principles using the definition of (Malliavan?) $L_p$ differentiability?
The answer is yes and the SDE is on page 5.

It would have been worrying if 2 distinctly different SDEs had come out of the woodwork.

https://projecteuclid.org/euclid.ecp/1465315573

Numerically, you have to solve for 2X2 system and boiler-;plate code. So, it will be more resource-intensive.

Cuchulainn
Topic Author
Posts: 58952
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Is it possible to "complexify" SDEs?

My read is that Daniel is experimenting with alternative ways to compute vega. It started with the Complex Step Differentiation Method -- hence the complex-valued SDEs. Then it seemed to morph into seeking a real-valued SDE or SDE pair to solve. Beyond that, I don't know and will leave to him ...
I have
1. CSM
2. Katuni stochastic flow
3. PDE (CSE).. another thread here