# The difficult or "Impossible", now doable

Simulations that take hours or days, with a lot of numerical noise, can now be performed in seconds with hardly any numerical instabilityThe Algorithmic Pricer Acceleration (**APA**) method also deals with discontinuities without a problem. The following graph shows the pricing function of swap with coupon payments:

MoCaX only needed to call the original swap pricer 100 times to generate its price representation. **Accuracy was of several order of magnitude. Pricing time is in the nano-second range.**

**Monte Carlo within Monte Carlo** are no problem at all with the Algorithmic Pricer Acceleration (**APA**) method.

These are the risk profiles of an American option that uses Monte Carlo simulation in pricing. Its profile is a MC-within-MC computation.

The computational time goes from something totally impractical to something totally reasonable with the Algorithmic Pricer Acceleration (**APA**) method in MoCaX.

From September 2016, many financial institutions need to post Initial Margin on un-cleared OTC derivatives. **The industry is following ISDA’s SIMM** method to compute that margin.

In order to understand the cost of funding of Initial Margin (MVA), we need to **simulate SIMM dynamically in a Monte Carlo simulation**.

SIMM is a sensitivity-based VaR. In order to compute its funding cost we need to simulate inside an XVA Monte Carlo simulation. In order to achieve this we need to simulate the Greeks of the book of derivatives.

Simulating pricers was already difficult, now we need to simulate Greeks too.

The Algorithmic Greeks Acceleration (**AGA**) method inside MoCaX provides that. It delivers ultra-fast values of price derivatives (i.e. Greeks). Each Delta or Vega inside the Monte Carlo can be obtained in few nano-seconds. Therefore dynamic simulation of SIMM is perfectly possible.

This graph shows the SIMM simulation for a 2-year at-the-money swaption:

It can be seen how IM clusters over time: starting IM is 2.5 % of Notional; the Monte Carlo paths that end up out-of-the-money end up with zero IM, while those that end up in-the-money increase to around 4% of notional.

**This calculation could not be done promptly and with precision without the AGA algorithm inside MoCaX**.

We have passed several QuantLib pricers through **APA & AGA** in MoCaX. The results are outstanding:

These calculations were run in an i5 Intel processor for a 1D case.

**The problem**

Let’s say that we have a Risk Factor Evolution model highly driven by regulatory, auditing or model validation requirements. We need to compute risk on a Bermudan Swaption. The corporate-approved pricer for such a product is a Monte Carlo simulation based on a 1-factor Hull-White (HW) model, which is different to our rates diffusion Risk Factor Evolution model.

In order to calculate its risk in a Monte Carlo simulation, we need to

- Diffuse the yield curves using the Risk Factor Evolution model
- In each scenario and time step

- Calibrate the pricing HW model to the yield curve
- Price the Bermudan Swaption using the corporate-approved MC simulation

For example, *this has to be done up to 1 million times *(10,000 scenarios and 100 time steps) for a CVA price.

*In practice and reality, this calculation is mostly impossible with current technology*.

**The solution**

The Algorithmic Pricer Acceleration (**APA**) method can deal with this perfectly.

We can pass both the (i) HW model calibration and the (ii) Bermudan Swaption pricer through MoCaX. The output is an accurate pricing routine that includes calibration:

The performance of the Calibration + Pricing process is massively increased compared to the old one, with no relevant loss of accuracy:

**The Outcome**

*With MoCaX we can run this “impossible” risk calculation promptly and accurately*.

(*) Test run in Matlab. Both the brute force and MoCaX computing time is faster in C++. The relative time stays approximately the same.