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FDM Boundary Conditions in QuantLib

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Haoyun XU

FDM Boundary Conditions in QuantLib

Dear all,

I wonder what are the boundary conditions used by QuantLib FDM pricing engines (e.g., FdBlackScholesBarrierEngine). I understand this may be scheme-dependent. If that's the case, you can use ImplicitEuler as an example.

Assume we have N price grids at each time step, representing stock prices ranging from Smin to Smax (both inclusive). When we discretize the PDE, we have N-2 equations (1 for each inner grid). To make it complete, we need boundary conditions at Smin & Smax for each time step.

We could do this in a product-specific way. For example, we can specify simple Dirichlet boundary conditions for vanilla European options as below:

Vanilla Call

V(Smin, t) = 0
V(Smax, t) = S - K exp (-r * (T - t))

Vanilla Put

V(Smin, t) = K * exp (-r * (T - t))
V(Smax, t) = 0

Alternatively we may impose some "generic" boundary conditions, as suggested by Wilmott. See Wilmott on QF (2nd edition) Vol 3 Chapter 77.10.

Or maybe there are other techniques to handle it, which I are not aware of.

I tried to trace the code myself, but got lost in the method ImplicitEulerScheme::step().

Could someone help on this?

Best wishes,

Henry

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Ralph Schreyer-3

Re: FDM Boundary Conditions in QuantLib

Hi,

if the boundary conditions specified in the corresponding engine are empty, generic boundary conditions are chosen, i.e. the second derivative vanishes. And for the vanilla this is good enough, see ql/pricingengines/vanilla/fdblackscholesvanillaengine.cpp

For the example you mentioned (the barrier engine), Dirichlet boundary conditions are specified, see ql/pricingengines/barrier/fdblackscholesbarrierengine.cpp

Cheers

Ralph

2014-04-23 10:05 GMT+02:00 Haoyun XU <[hidden email]>:

Dear all,

I wonder what are the boundary conditions used by QuantLib FDM pricing engines (e.g., FdBlackScholesBarrierEngine). I understand this may be scheme-dependent. If that's the case, you can use ImplicitEuler as an example.

Assume we have N price grids at each time step, representing stock prices ranging from Smin to Smax (both inclusive). When we discretize the PDE, we have N-2 equations (1 for each inner grid). To make it complete, we need boundary conditions at Smin & Smax for each time step.

We could do this in a product-specific way. For example, we can specify simple Dirichlet boundary conditions for vanilla European options as below:
Vanilla Call
V(Smin, t) = 0

V(Smax, t) = S - K exp (-r * (T - t))
Vanilla Put
V(Smin, t) = K * exp (-r * (T - t))
V(Smax, t) = 0
Alternatively we may impose some "generic" boundary conditions, as suggested by Wilmott. See Wilmott on QF (2nd edition) Vol 3 Chapter 77.10.

Or maybe there are other techniques to handle it, which I are not aware of.

I tried to trace the code myself, but got lost in the method ImplicitEulerScheme::step().

Could someone help on this?

Best wishes,
Henry

------------------------------------------------------------------------------
Start Your Social Network Today - Download eXo Platform
Build your Enterprise Intranet with eXo Platform Software
Java Based Open Source Intranet - Social, Extensible, Cloud Ready
Get Started Now And Turn Your Intranet Into A Collaboration Platform
http://p.sf.net/sfu/ExoPlatform
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QuantLib-users mailing list
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