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RaspberryPhi
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Joined: June 12th, 2015, 8:37 am

Pricing with collateral, Fujii, Shimada, Takahashi (2009)

June 12th, 2015, 2:00 pm

Hi everyoneI'm currently looking at the papers by Fujii, Shimada, Takahashi, (2009,2010,2011). They consider a framework of full collateralization and assumes a collateral account V(t) following the stochastic process[$]dV(s)=(r^{d}(s)-c^{d}(s))V(s)ds+a(s)dh(s)[$]where h(s) is the time s value of the derivative with cashflow h(T) at maturity T, and a(s) is the number of positions in the derivate at time s.In their article "A Note on Construction of Multiple Swap Curves with and without Collateral" they integrate the process to get the general solution:[$]V(T)=e^{\int_{t}^{T}(r^{d}(u)-c^{d}(u))du}V(t)+\int_{t}^{T}e^{\int_{s}^{T}(r^{d}(u)-c^{d}(u))du}a(s)dh(s)[$]I have been trying to integrate the process to arrive at the solution, but I am not able to. The process looks a little like an Ohrnstein-Uhlenbeck process. It has been annoying me for a while now, and I hope someone in here is able to help. Thanks for your time.
 
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QuantOption
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Joined: February 8th, 2003, 9:00 pm

Pricing with collateral, Fujii, Shimada, Takahashi (2009)

June 15th, 2015, 8:46 pm

Let y(s)=r(s)-c(s), use the standard approach as when you solve OU process, but use integrating factor exp(int_s^T y(u)du), instead of the usual exp(-y(s)s)
 
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RaspberryPhi
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Joined: June 12th, 2015, 8:37 am

Pricing with collateral, Fujii, Shimada, Takahashi (2009)

June 16th, 2015, 10:44 am

Thanks a lot QuantOption.
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