What is the difference between pricing a closed form and finite difference method whilst pricing liquid CDS portfolio?

Can anyone share any articles which are not too mathematical intensive and can explain the differences?

What is the difference between pricing a closed form and finite difference method whilst pricing liquid CDS portfolio?

Can anyone share any articles which are not too mathematical intensive and can explain the differences?

Can anyone share any articles which are not too mathematical intensive and can explain the differences?

- Cuchulainn
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Where's the bottleneck, exactly? your knowledge of FDM in general? What's CDS equation?What is the difference between pricing a closed form and finite difference method whilst pricing liquid CDS portfolio?

Can anyone share any articles which are not too mathematical intensive and can explain the differences?

N.B. please no posts of same question on multiple forums!

one is a formula and the other is the numerical solution of a PDE?

I'll claim a little bit of expertise on this matter, and the question is not well posed. Please try again!

To reword the question:

We have two systems one for trading and another for pricing. We are trying to integrate pricing system into our trading system so that we have one source for pricing and risk.

Our trading system uses Finite difference method to calculate the scenarios for risk and P&L where as the pricing system uses closed form approximation. We would like to intregrate closed form in trading system. We wanted to assess if that was possible. Hence I wanted to know more information on difference between the two methods so that further analysis can be done

- Cuchulainn
**Posts:**58103**Joined:****Location:**Amsterdam-
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Is it possible to back up the words with a formula/PDE?

// Once you get beyond 3 underlings FDM is out of the question.

// Once you get beyond 3 underlings FDM is out of the question.

I have a guess as to what is going on here. When you say "finite difference method", do you mean that you change one or more input parameters by a finite amount and then revalue the position (or book) to get the P&L in a scenario? As opposed to multiplying a (possibly approximate) partial derivative with respect to the same parameter by the amount of the change? If so, there is a large literature from the 90's, probably best found by searching for terms like delta method vs full revaluation. To people here in general, and Cuculainn in particular, "finite difference method" triggers an immediate association with partial differential equations. If that is indeed the topic, then we definitely need more details.

- Cuchulainn
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Indeed, this is standardised mathematical jargon. A la carte definitions abound in AI and physics.

Maybe OP means the

https://en.wikipedia.org/wiki/Finite_difference

FDM = calculus of finite differences applied to PDE.

BTW are CDSs liquid these days? In 2006 no one wanted them (except Paulson)??

The market for CDS on corporate credits in fact hit its peak liquidity around 2006, when it was very liquid, at least if compared to the underlying bonds. These days, there is minimal liquidity in single name CDS, but you can still move a billion dollar trade through the CDX IG market without leaving much of a footprint. The overall credit derivatives market is probably down something like 90% from the peak, and the fraction of the market made up of simple index trades is way up.To people here in general, and Cuculainn in particular, "finite difference method" triggers an immediate association with partial differential equations.

Indeed, this is standardised mathematical jargon. A la carte definitions abound in AI and physics.

Maybe OP means thecalculus of finite differences?Feels like some kind of sensitivity analysis?

https://en.wikipedia.org/wiki/Finite_difference

FDM = calculus of finite differences applied to PDE.

BTW are CDSs liquid these days? In 2006 no one wanted them (except Paulson)??

Especially back in older days with limited computing power, it was not uncommon for risk systems to use closed-form approximations in the interests of reducing computation times. It is not unusual to see this still today when dealing with products requiring slow path-dependent valuation like you are dealing with. Another product example would be prepayable mortgages and related MBS.

We can use grade school arithmetic to demonstrate that price approximations can still produce accurate P&L calculations. Say we want to compute the P&L between T-1 and T. If we are using a pricing approximation, our price estimates will have some amount of error, presumably a reasonably ‘small’ error (otherwise you would not use the approximation).

That is, Price(T,estimate) = Price(T,true) + Error(T)

And similarly Price(T-1,estimate) = Price(T-1,true) + Error(T-1)

By definition P&L(true) = Price(T,true) - Price(T-1,true). Hold that thought for a moment.

Our P&L estimate is by definition P&L(estimate) = Price(T,estimate) - Price(T-1,estimate)

Substituting the definitions of Price(T,estimate) and Price(T-1,estimate)

P&L(estimate) = Price(T,true) + Error(T) – [Price(T-1,true) + Error(T-1)]

Rearranging the terms

P&L(estimate) = Price(T,true) – Price(T-1,true) + Error(T) - Error(T-1)

Now unless the error in the price estimates are systemic, Error(T) - Error(T-1) more or less cancels out (the difference will be ‘very small’) and hence

P&L(estimate) = Price(T,true) – Price(T-1,true)= P&L(True)

That pricing approximation errors cancel out when differenced is the rationale for using approximations in P&L and risk systems.

We can use grade school arithmetic to demonstrate that price approximations can still produce accurate P&L calculations. Say we want to compute the P&L between T-1 and T. If we are using a pricing approximation, our price estimates will have some amount of error, presumably a reasonably ‘small’ error (otherwise you would not use the approximation).

That is, Price(T,estimate) = Price(T,true) + Error(T)

And similarly Price(T-1,estimate) = Price(T-1,true) + Error(T-1)

By definition P&L(true) = Price(T,true) - Price(T-1,true). Hold that thought for a moment.

Our P&L estimate is by definition P&L(estimate) = Price(T,estimate) - Price(T-1,estimate)

Substituting the definitions of Price(T,estimate) and Price(T-1,estimate)

P&L(estimate) = Price(T,true) + Error(T) – [Price(T-1,true) + Error(T-1)]

Rearranging the terms

P&L(estimate) = Price(T,true) – Price(T-1,true) + Error(T) - Error(T-1)

Now unless the error in the price estimates are systemic, Error(T) - Error(T-1) more or less cancels out (the difference will be ‘very small’) and hence

P&L(estimate) = Price(T,true) – Price(T-1,true)= P&L(True)

That pricing approximation errors cancel out when differenced is the rationale for using approximations in P&L and risk systems.

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